diff --git a/README.md b/README.md index ca33bbf28144fe45b4acc083ca1af71cc1d5e504..96adb5e655fa7001b1c98e4d738bf7b17d9cfc35 100644 --- a/README.md +++ b/README.md @@ -5,7 +5,7 @@

[📊 Dataset] [✨ GitHub] -[📄 Paper] +[📄 Paper]

[![License: MIT](https://img.shields.io/badge/License-MIT-blue.svg)](https://opensource.org/license/mit) @@ -17,7 +17,7 @@ **HiPhO** (High School Physics Olympiad Benchmark) is the **first benchmark** specifically designed to evaluate the physical reasoning abilities of (M)LLMs on **real-world Physics Olympiads from 2024–2025**.
- hipho overview five rings + hipho overview five rings
### ✨ Key Features @@ -31,7 +31,7 @@ ## 🏆 IPhO 2025 (Theory) Results
- ipho2025 results + ipho2025 results
- **Top-1 Human Score**: 29.2 / 30.0 @@ -47,7 +47,7 @@ ## 📊 Dataset Overview
- framework and stats + framework and stats
HiPhO contains: @@ -68,7 +68,7 @@ Evaluation is conducted using: ## 🖼️ Modality Categorization
- modality examples + modality examples
- 📝 **Text-Only (TO)**: Pure text, no figures @@ -83,7 +83,7 @@ Evaluation is conducted using: ## 📈 Main Results
- main results medal table + main results medal table
- **Closed-source reasoning MLLMs** lead the benchmark, earning **6–12 gold medals** (Top 5: Gemini-2.5-Pro, Gemini-2.5-Flash, GPT-5, o3, Grok-4) diff --git a/data/APhO_2025.json b/data/APhO_2025.json new file mode 100644 index 0000000000000000000000000000000000000000..eafeb37cabd627a577b71e90ef106c5f07301ab8 --- /dev/null +++ b/data/APhO_2025.json @@ -0,0 +1,1491 @@ +[ + { + "information": "None." + }, + { + "id": "APhO_2025_1_A_1", + "context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents.", + "question": "Find the values of exponents: (1) $\\beta$, (2) $\\gamma$, and (3) $\\delta$.", + "marking": [ + [ + "Award 0.2 pt if the answer correctly expresses the dimension of $G$ as $[G] = L^3 M^{-1} T^{-2}$, where $L$ is the base dimensions length, $M$ is mass, and $T$ is time. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly sets up the exponent equation $0 = 2 - \\beta$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly sets up the exponent equation $0 = \\gamma + 1$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly sets up the exponent equation $1 = \\delta - 3$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer obtains the correct value $\\beta = 2$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer obtains the correct value $\\gamma = -1$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer obtains the correct value $\\delta = 4$. Otherwise, award 0 pt." + + ] + ], + "answer": [ + "\\boxed{$\\beta = 2$}", + "\\boxed{$\\gamma = -1$}", + "\\boxed{$\\delta = 4$}" + ], + "answer_type": [ + "Numerical Value", + "Numerical Value", + "Numerical Value" + ], + "unit": [ + null, + null, + null + ], + "points": [ + 0.3, + 0.3, + 0.2 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_1_a_1.png" + ] + }, + { + "id": "APhO_2025_1_A_2", + "context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$.", + "question": "Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly calculates $\\omega = \\frac{2\\pi}{24 h} = 7.27 \\times 10^{-5} s^{-1}$, where $\\omega$ is the angular speed of the Earth's rotation. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives $h_{\\max} = 21.9 \\mathrm{km}$ from the relation $h_{\\max} \\propto \\frac{\\omega^2 R^4}{G M_E}$, where $R$ is the Earth's radius and $M_E$ is the Earth's mass. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{21.9}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "km" + ], + "points": [ + 0.2 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_1_a_1.png" + ] + }, + { + "id": "APhO_2025_1_B_1", + "context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis.", + "question": "Find (1) the direction and (2) magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$.", + "marking": [ + [ + "Award 0.2 pt if the answer correctly expresses the magnitude of $U(z) = -G \\frac{M_S}{\\sqrt{z^2 + d_{SE}^2}}$, where $M_S$ is the Sun's mass, $z$ is the axis coordinate, $d_{SE}$ is the Earth-Sun distance, $G$ is the gravitational constant. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer includes the correct negative sign in $U(z) = -G \\frac{M_S}{\\sqrt{z^2 + d_{SE}^2}}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer expresses $g_z(z)$ as a derivative $g_z(z) = - \\frac{dU}{dz}$. Partial points: award 0.1 pt if the answer does not include the negative sign. Otherwise, award 0 pt.", + "Award 0.2 pt if the derivative is calculated correctly as $g_z(z) = -G M_S \\frac{z}{(z^2 + d_{SE}^2)^{3/2}}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly obtains the approximate form for small $z$, $g_z(z) \\approx - \\frac{G M_S}{d_{SE}^3} z$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer indicates that the negative sign means $g_z$ points towards the center of the Sun ring (correct direction in figure). Otherwise, award 0 pt." + ], + [ + "Award 0.2 pt if the answer correctly writes the gravitational field of an element of the ring as $\\mathrm{d}g = G \\frac{\\mathrm{d}M}{z^2 + d_{SE}^2}$, where $\\mathrm{d}M$ is the ring element mass, $z$ is the axis coordinate, $d_{SE}$ is the Earth–Sun distance, and $G$ is the gravitational constant. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer includes a figure with correct geometry showing $\\mathrm{d}g$, $\\mathrm{d}g_z$, angle $\\theta$, $z$, and $d_{SE}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer takes only the $z$ component for symmetry reasons, writing $\\mathrm{d}g_z = - \\mathrm{d}g \\cos \\theta$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer sums or integrates over the whole ring to obtain the total field. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly calculates $g_z$ at arbitrary $z$ as $g_z = -G M_S \\frac{z}{(z^2 + d_{SE}^2)^{3/2}}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the approximate form for small $z$, $g_z \\approx - G M_S \\frac{z}{d_{SE}^3}$, is correctly given. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer indicates that the negative sign means $g_z$ points towards the center of the Sun ring (correct direction in figure). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{The direction of the gravitational field is toward the center of the Sun ring.}", + "\\boxed{$g_z(z) \\approx -\\frac{G M_S}{d_{SE}^3} z$}" + ], + "answer_type": [ + "Open-Ended", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 0.2, + 0.8 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_1_a_1.png", + "image_question/APhO_2025_1_b_1.png", + "image_question/APhO_2025_1_b_2.png" + ] + }, + { + "id": "APhO_2025_1_B_2", + "context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis. \n\n(B.1) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$.", + "question": "Find (1) the direction and (2) magnitude of the gravitational field generated by the Sun ring at a point in the ecliptic plane whose distance from the origin is $r$. Assume $r \\ll d_{SE}$.", + "marking": [ + [ + "Award 0.5 pt if the answer presents the idea of using Gauss's law to calculate the gravitational field in the plane of the Sun ring. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer correctly identifies a cylindrical Gaussian surface with axis along $z$ near the center of the Sun ring. Otherwise, award 0 pt.", + "Award 0.6 pt if the answer writes Gauss's law correctly as $g_r 2z \\times 2\\pi r + g_z \\cdot 2r^2 \\pi = 0$, where $g_r$ is the radial component and $g_z$ the axial component. Partial points: award 0.3 pt if there is a mistake in areas; award 0 pt if the error is dimensional. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly obtains the final result that $g_r$ is proportional to $r$, i.e. $g_r(r) \\propto r$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct proportionality constant $g_r(r) = \\frac{G M_S}{2 d_{SE}^3} r$. Partial points: award 0.1 pt if the prefactor is wrong but dimensionally consistent; award 0 pt if the error is dimensional. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer indicates that the field points radially outwards (correct direction in the figure). Otherwise, award 0 pt." + ], + [ + "Award 0.2 pt if the answer correctly expresses the distance $s = \\sqrt{d_{SE}^2 + r^2 - 2 d_{SE} r \\cos \\varphi}$ from trigonometry, where $r$ is the distance of point $P$ from the center, $d_{SE}$ is the Earth-Sun distance, and $\\varphi$ is the angle. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly writes the potential generated by a small element of the ring as $\\mathrm{d}U = - G M_S \\frac{\\mathrm{d}\\varphi}{2 \\pi s}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer writes the total potential $U(r)$ as an integral $U(r) = - \\frac{G M_S}{2 \\pi} \\int_0^{2\\pi} (d_{SE}^2 + r^2 - 2 d_{SE} r \\cos \\varphi)^{-1/2} \\mathrm{d}\\varphi$. Otherwise, award 0 pt.", + "Award 0.6 pt if the answer expands the integrand to second order and obtains $(1 + \\varepsilon)^{-1/2} \\approx 1 - \\frac{r^{2}}{2 d_{SE}^{2}} + \\frac{r \\cos \\varphi}{d_{SE}} + \\frac{3 r^{2} \\cos^{2}\\varphi}{2 d_{SE}^{2}}.$, where $\\varepsilon = \\frac{r^2}{d_{SE}^2} - \\frac{2r \\cos \\varphi}{d_{SE}}$ and $(1+\\varepsilon)^{-1/2} \\approx 1 - \\varepsilon/2 + 3\\varepsilon^2/8$. Partial points: award 0.1 pt if only first order is calculated; award 0.4 pt if the $\\cos^2 \\varphi$ term is missing. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer integrates over $\\varphi$ correctly: $U(r) = -\\frac{G M_{S}}{2\\pi d_{SE}} \\int_{0}^{2\\pi} \\left( 1 - \\frac{r^{2}}{2 d_{SE}^{2}} + \\frac{3 r^{2} \\cos^{2} \\varphi}{2 d_{SE}^{2}} \\right) \\mathrm{d}\\varphi$. Partial points: award 0.1 pt if the $\\cos^2 \\varphi$ term is missing. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly expresses $g_z$ as the derivative of $U(z)$: $g_r(r) = - \\frac{dU}{dr}$. Partial points: aAward 0.1 pt if the negative sign is omitted. Otherwise, award 0 pt.", + "Award 0.1 pt if the derivative $\\frac{dU}{dr}$ is calculated correctly, yielding $g_r(r) = \\frac{G M_S}{2 d_{SE}^3} r$. Otherwise, award 0 pt.", + "Award 0.3 pt if the final result shows $g_r \\propto r$. Otherwise, award 0 pt.", + "Award 0.2 pt if the proportionality constant is correct, $g_r(r) = \\frac{G M_S}{2 d_{SE}^3} r$. Partial points: award 0.1 pt if only the prefactor is wrong but dimensionally consistent; award 0 pt if dimensional error. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer indicates that the field points radially outwards (correct direction in the figure). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{The direction of the gravitational field is outward along the radial direction.}", + "\\boxed{$g_r(r) = \\frac{G M_S}{2 d_{SE}^3} r$}" + ], + "answer_type": [ + "Open-Ended", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 0.2, + 2.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_1_a_1.png", + "image_question/APhO_2025_1_b_1.png", + "image_question/APhO_2025_1_b_2.png" + ] + }, + { + "id": "APhO_2025_1_C_1", + "context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis. \n\n(B.1) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$. \n\n(B.2) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point in the ecliptic plane whose distance from the origin is $r$. Assume $r \\ll d_{SE}$. \n\n[Part C: The torque acting on the Earth] \n\nIn this section, you are asked to determine the torque exerted on the Earth due to the gravitational field obtained in Part B. For simplicity, consider the Earth as a rigid body with homogeneous mass distribution. Let us take into account that the rotational ellipsoid can be imagined as if we removed excess parts from a sphere with the equatorial radius of Earth $R_{e}$ (see Figure C.1). \n\n[figure4] \nFigure C.1. The ellipsoidal shape of the Earth can be imagined as if the excess parts were removed from a complete sphere of radius $R_{e}$.", + "question": "Find the mass $m$ of one of the two excess regions indicated in Figure C.1. Express your answer in terms of $h_{\\text{max}}$, the mass of the Earth $M_{E}$, and its polar radius $R_{p}$.", + "marking": [ + [ + "Award 0.2 pt if the answer includes the idea of transforming the ellipsoid of revolution into a perfect sphere of radius $R_e$ by stretching uniformly along the polar diameter with factor $R_e/R_p$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly computes the volume of one of the excess regions as $V = \\frac{1}{2} \\left( \\frac{4\\pi}{3} R_e^3 - \\frac{4\\pi}{3} R_e^2 R_p \\right) = \\frac{2\\pi}{3} R_e^2 h_{max}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer writes the correct expression for the Earth's density as $\\rho = \\frac{3 M_E}{4 \\pi R_e^2 R_p}$, where $M_E$ is the Earth's mass. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer obtains the final expression for the mass of one excess region as $m = \\frac{h_{max}}{2 R_p} M_E$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$m = \\frac{h_{\\max}}{2 R_p} M_E$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.8 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_1_a_1.png", + "image_question/APhO_2025_1_b_1.png", + "image_question/APhO_2025_1_b_2.png", + "image_question/APhO_2025_1_c_1.png" + ] + }, + { + "id": "APhO_2025_1_C_2", + "context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis. \n\n(B.1) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$. \n\n(B.2) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point in the ecliptic plane whose distance from the origin is $r$. Assume $r \\ll d_{SE}$. \n\n[Part C: The torque acting on the Earth] \n\nIn this section, you are asked to determine the torque exerted on the Earth due to the gravitational field obtained in Part B. For simplicity, consider the Earth as a rigid body with homogeneous mass distribution. Let us take into account that the rotational ellipsoid can be imagined as if we removed excess parts from a sphere with the equatorial radius of Earth $R_{e}$ (see Figure C.1). \n\n[figure4] \nFigure C.1. The ellipsoidal shape of the Earth can be imagined as if the excess parts were removed from a complete sphere of radius $R_{e}$. \n\n(C.1) Find the mass $m$ of one of the two excess regions indicated in Figure C.1. Express your answer in terms of $h_{\\text{max}}$, the mass of the Earth $M_{E}$, and its polar radius $R_{p}$. \n\nIt can be shown that the torque acting on the excess regions is equivalent to the torque acting on two point masses, each with a mass equal to $2m / 5$, positioned at the endpoints $A$ and $B$ of the polar diameter (see Figure C.1).", + "question": "Given this idea, find the torque $\\tau$ exerted by the Sun ring on the Earth. Express your answer in terms of $M_{E}, M_{S}, d_{SE}, R$ (the average radius), $h_{\\max}$ and the angle $\\alpha$. You can use that $h_{\\max} \\ll R$.", + "marking": [ + [ + "Award 0.1 pt if the answer mentions that the net torque acting on a perfect sphere of radius $R_e$ is zero due to symmetry. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer states the idea that the torque on the ellipsoid-shaped Earth is given by $\\vec{\\tau} = - \\vec{\\tau}'$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer correctly includes the torque contribution from $F_z$, where $F_z = \\frac{2}{5} m |g_z| = \\frac{2}{5} m G M_S \\dfrac{R \\cos \\alpha}{d_{SE}^3}$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer correctly includes the torque contribution from $F_r$, where $F_r = \\frac{2}{5} m |g_r| = \\frac{2}{5} m G M_S \\dfrac{R \\sin \\alpha}{2 d_{SE}^3}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly adds the two torque contributions with the right sign in $|\\tau'| = 2 F_z R \\sin \\alpha + 2 F_r R \\cos \\alpha$. Otherwise, award 0 pt.", + "Award 0.3 pt if the calculation is carried through to obtain the correct net torque $|\\tau'| = \\frac{6}{5} \\dfrac{G m M_S}{d_{SE}^3} R^2 \\sin \\alpha \\cos \\alpha = \\frac{3}{5} \\dfrac{G M_E M_S}{d_{SE}^3} R h_{max} \\sin \\alpha \\cos \\alpha$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct direction of $\\vec{\\tau}$: pointing into the plane (opposite to $\\vec{\\tau}'$). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$|\\tau| = \\frac{3}{5} \\cdot \\frac{G M_E M_S}{d_{SE}^3} \\cdot R h_{\\max} \\sin \\alpha \\cos \\alpha$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.8 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_1_a_1.png", + "image_question/APhO_2025_1_b_1.png", + "image_question/APhO_2025_1_b_2.png", + "image_question/APhO_2025_1_c_1.png" + ] + }, + { + "id": "APhO_2025_1_D_1", + "context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis. \n\n(B.1) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$. \n\n(B.2) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point in the ecliptic plane whose distance from the origin is $r$. Assume $r \\ll d_{SE}$. \n\n[Part C: The torque acting on the Earth] \n\nIn this section, you are asked to determine the torque exerted on the Earth due to the gravitational field obtained in Part B. For simplicity, consider the Earth as a rigid body with homogeneous mass distribution. Let us take into account that the rotational ellipsoid can be imagined as if we removed excess parts from a sphere with the equatorial radius of Earth $R_{e}$ (see Figure C.1). \n\n[figure4] \nFigure C.1. The ellipsoidal shape of the Earth can be imagined as if the excess parts were removed from a complete sphere of radius $R_{e}$. \n\n(C.1) Find the mass $m$ of one of the two excess regions indicated in Figure C.1. Express your answer in terms of $h_{\\text{max}}$, the mass of the Earth $M_{E}$, and its polar radius $R_{p}$. \n\nIt can be shown that the torque acting on the excess regions is equivalent to the torque acting on two point masses, each with a mass equal to $2m / 5$, positioned at the endpoints $A$ and $B$ of the polar diameter (see Figure C.1). \n\n(C.2) Given this idea, find the torque $\\tau$ exerted by the Sun ring on the Earth. Express your answer in terms of $M_{E}, M_{S}, d_{SE}, R$ (the average radius), $h_{\\max}$ and the angle $\\alpha$. You can use that $h_{\\max} \\ll R$. \n\n[Part D: Angular speed of the precession of the Earth's axis] \n\nThe Earth's axis of rotation moves very slowly around the $z$ axis in a conical motion. That is, it precesses.", + "question": "Give an expression for the period $T_{1}$ of precession of the Earth's axis. Express your answer in terms of $M_{S}, d_{SE}$, the angular speed $\\omega$ of the Earth's rotation, $h_{\\text{max}}, R$ and $\\alpha$.", + "marking": [ + [ + "Award 0.2 pt if the answer applies Newton's second law for rotational motion, $\\vec{\\tau} = \\dfrac{d \\vec{L}}{dt}$, where $\\tau$ is torque and $\\vec{L}$ is angular momentum. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer expresses angular momentum as $|\\vec{L}| = I \\omega$, where $I$ is the moment of inertia and $\\omega$ is the angular velocity of Earth's rotation. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer writes the correct moment of inertia for a uniform sphere as $I = \\frac{2}{5} M_E R^2$. Partial points: award 0.1 pt if the prefactor is wrong but dimensions are correct; award 0 pt if there is a dimensional error. Otherwise, award 0 pt.", + "Award 0.8 pt if the answer correctly relates the time derivative of angular momentum to precession angular speed: $| \\frac{d \\vec{L}}{dt} | = \\Omega_1 |\\vec{L}| \\sin \\alpha$, where $\\Omega_1$ is the angular speed of precession and $\\alpha$ is the half-apex angle. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer uses the relation $\\Omega_1 = \\dfrac{2\\pi}{T_1}$ between precession angular velocity and precession period. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer finds the correct precession period $T_1 = \\dfrac{4 \\pi}{3} \\dfrac{d_{SE}^3 R \\omega}{G M_S h_{\\max} \\cos \\alpha}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$T_1 = \\frac{4 \\pi}{3} \\cdot \\frac{d_{SE}^3 R \\omega}{G M_S h_{\\max} \\cos \\alpha}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.8 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_1_a_1.png", + "image_question/APhO_2025_1_b_1.png", + "image_question/APhO_2025_1_b_2.png", + "image_question/APhO_2025_1_c_1.png" + ] + }, + { + "id": "APhO_2025_1_D_2", + "context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis. \n\n(B.1) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$. \n\n(B.2) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point in the ecliptic plane whose distance from the origin is $r$. Assume $r \\ll d_{SE}$. \n\n[Part C: The torque acting on the Earth] \n\nIn this section, you are asked to determine the torque exerted on the Earth due to the gravitational field obtained in Part B. For simplicity, consider the Earth as a rigid body with homogeneous mass distribution. Let us take into account that the rotational ellipsoid can be imagined as if we removed excess parts from a sphere with the equatorial radius of Earth $R_{e}$ (see Figure C.1). \n\n[figure4] \nFigure C.1. The ellipsoidal shape of the Earth can be imagined as if the excess parts were removed from a complete sphere of radius $R_{e}$. \n\n(C.1) Find the mass $m$ of one of the two excess regions indicated in Figure C.1. Express your answer in terms of $h_{\\text{max}}$, the mass of the Earth $M_{E}$, and its polar radius $R_{p}$. \n\nIt can be shown that the torque acting on the excess regions is equivalent to the torque acting on two point masses, each with a mass equal to $2m / 5$, positioned at the endpoints $A$ and $B$ of the polar diameter (see Figure C.1). \n\n(C.2) Given this idea, find the torque $\\tau$ exerted by the Sun ring on the Earth. Express your answer in terms of $M_{E}, M_{S}, d_{SE}, R$ (the average radius), $h_{\\max}$ and the angle $\\alpha$. You can use that $h_{\\max} \\ll R$. \n\n[Part D: Angular speed of the precession of the Earth's axis] \n\nThe Earth's axis of rotation moves very slowly around the $z$ axis in a conical motion. That is, it precesses. \n\n(D.1) Give an expression for the period $T_{1}$ of precession of the Earth's axis. Express your answer in terms of $M_{S}, d_{SE}$, the angular speed $\\omega$ of the Earth's rotation, $h_{\\text{max}}, R$ and $\\alpha$.", + "question": "Calculate the precession period $T_{1}$ in years.", + "marking": [ + [ + "Award 0.2 pt if the answer gives the correct numerical result for the precession period as $T_1 \\approx 80600$ years, obtained by correctly substituting the given data into a dimensionally correct formula. Partial points: award 0 pt if the substitution is incorrect or if the formula used has a dimensional error." + ] + ], + "answer": [ + "\\boxed{80600}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "years" + ], + "points": [ + 0.2 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_1_a_1.png", + "image_question/APhO_2025_1_b_1.png", + "image_question/APhO_2025_1_b_2.png", + "image_question/APhO_2025_1_c_1.png" + ] + }, + { + "id": "APhO_2025_1_E_1", + "context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis. \n\n(B.1) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$. \n\n(B.2) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point in the ecliptic plane whose distance from the origin is $r$. Assume $r \\ll d_{SE}$. \n\n[Part C: The torque acting on the Earth] \n\nIn this section, you are asked to determine the torque exerted on the Earth due to the gravitational field obtained in Part B. For simplicity, consider the Earth as a rigid body with homogeneous mass distribution. Let us take into account that the rotational ellipsoid can be imagined as if we removed excess parts from a sphere with the equatorial radius of Earth $R_{e}$ (see Figure C.1). \n\n[figure4] \nFigure C.1. The ellipsoidal shape of the Earth can be imagined as if the excess parts were removed from a complete sphere of radius $R_{e}$. \n\n(C.1) Find the mass $m$ of one of the two excess regions indicated in Figure C.1. Express your answer in terms of $h_{\\text{max}}$, the mass of the Earth $M_{E}$, and its polar radius $R_{p}$. \n\nIt can be shown that the torque acting on the excess regions is equivalent to the torque acting on two point masses, each with a mass equal to $2m / 5$, positioned at the endpoints $A$ and $B$ of the polar diameter (see Figure C.1). \n\n(C.2) Given this idea, find the torque $\\tau$ exerted by the Sun ring on the Earth. Express your answer in terms of $M_{E}, M_{S}, d_{SE}, R$ (the average radius), $h_{\\max}$ and the angle $\\alpha$. You can use that $h_{\\max} \\ll R$. \n\n[Part D: Angular speed of the precession of the Earth's axis] \n\nThe Earth's axis of rotation moves very slowly around the $z$ axis in a conical motion. That is, it precesses. \n\n(D.1) Give an expression for the period $T_{1}$ of precession of the Earth's axis. Express your answer in terms of $M_{S}, d_{SE}$, the angular speed $\\omega$ of the Earth's rotation, $h_{\\text{max}}, R$ and $\\alpha$. \n\n(D.2) Calculate the precession period $T_{1}$ in years. \n\n[Part E: The effect of the Moon] \n\nThe value obtained in Part D is much larger than the observed value. The reason for this is that so far we have only considered the torque exerted by the Sun, and neglected the effect of the Moon. In the following calculations, assume that the Moon's orbit is in the ecliptic plane, and that the orbit of the Moon around the Earth is a circle of radius $d_{ME}$. Let us denote the mass of the Moon by $M_{M}$ and the period of precession in this modified model by $T_{2}$.", + "question": "By what factor $T_{2} / T_{1}$ does the period of precession of the Earth's axis change if we also take into account the torque exerted by the Moon? Give your answer in terms of $d_{ME}, d_{SE}, M_{S}$ and $M_{M}$.", + "marking": [ + [ + "Award 0.3 pt if the answer explicitly states that the torques exerted by the Sun and the Moon add up. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer correctly calculates the torque exerted by the Moon or states that it is proportional to $M_M/d_{ME}^3$, where $M_M$ is the Moon's mass and $d_{ME}$ is the Earth-Moon distance. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer expresses the ratio $T_2/T_1$ correctly as $\\dfrac{T_2}{T_1} = \\dfrac{M_S/d_{SE}^3}{M_M/d_{ME}^3 + M_S/d_{SE}^3}$. Partial points: award 0 pt if the expression implies $T_1 < T_2$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$T_2 / T_1 = \\frac{M_S / d_{SE}^3}{M_M / d_{ME}^3 + M_S / d_{SE}^3}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_1_a_1.png", + "image_question/APhO_2025_1_b_1.png", + "image_question/APhO_2025_1_b_2.png", + "image_question/APhO_2025_1_c_1.png" + ] + }, + { + "id": "APhO_2025_1_E_2", + "context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis. \n\n(B.1) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$. \n\n(B.2) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point in the ecliptic plane whose distance from the origin is $r$. Assume $r \\ll d_{SE}$. \n\n[Part C: The torque acting on the Earth] \n\nIn this section, you are asked to determine the torque exerted on the Earth due to the gravitational field obtained in Part B. For simplicity, consider the Earth as a rigid body with homogeneous mass distribution. Let us take into account that the rotational ellipsoid can be imagined as if we removed excess parts from a sphere with the equatorial radius of Earth $R_{e}$ (see Figure C.1). \n\n[figure4] \nFigure C.1. The ellipsoidal shape of the Earth can be imagined as if the excess parts were removed from a complete sphere of radius $R_{e}$. \n\n(C.1) Find the mass $m$ of one of the two excess regions indicated in Figure C.1. Express your answer in terms of $h_{\\text{max}}$, the mass of the Earth $M_{E}$, and its polar radius $R_{p}$. \n\nIt can be shown that the torque acting on the excess regions is equivalent to the torque acting on two point masses, each with a mass equal to $2m / 5$, positioned at the endpoints $A$ and $B$ of the polar diameter (see Figure C.1). \n\n(C.2) Given this idea, find the torque $\\tau$ exerted by the Sun ring on the Earth. Express your answer in terms of $M_{E}, M_{S}, d_{SE}, R$ (the average radius), $h_{\\max}$ and the angle $\\alpha$. You can use that $h_{\\max} \\ll R$. \n\n[Part D: Angular speed of the precession of the Earth's axis] \n\nThe Earth's axis of rotation moves very slowly around the $z$ axis in a conical motion. That is, it precesses. \n\n(D.1) Give an expression for the period $T_{1}$ of precession of the Earth's axis. Express your answer in terms of $M_{S}, d_{SE}$, the angular speed $\\omega$ of the Earth's rotation, $h_{\\text{max}}, R$ and $\\alpha$. \n\n(D.2) Calculate the precession period $T_{1}$ in years. \n\n[Part E: The effect of the Moon] \n\nThe value obtained in Part D is much larger than the observed value. The reason for this is that so far we have only considered the torque exerted by the Sun, and neglected the effect of the Moon. In the following calculations, assume that the Moon's orbit is in the ecliptic plane, and that the orbit of the Moon around the Earth is a circle of radius $d_{ME}$. Let us denote the mass of the Moon by $M_{M}$ and the period of precession in this modified model by $T_{2}$. \n\n(E.1) By what factor $T_{2} / T_{1}$ does the period of precession of the Earth's axis change if we also take into account the torque exerted by the Moon? Give your answer in terms of $d_{ME}, d_{SE}, M_{S}$ and $M_{M}$.", + "question": "By substituting the data, calculate the period of precession $T_{2}$ in years.", + "marking": [ + [ + "Award 0.2 pt if the answer gives the correct numerical result for the precession period $T_2 \\approx 25400$ years, obtained by substituting values into a dimensionally correct formula. Partial points: award 0 pt if the result does not come from explicit substitution (e.g. if it is taken from the introduction), if the substitution is incorrect, or if the formula has a dimensional error." + ] + ], + "answer": [ + "\\boxed{25400}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "years" + ], + "points": [ + 0.2 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_1_a_1.png", + "image_question/APhO_2025_1_b_1.png", + "image_question/APhO_2025_1_b_2.png", + "image_question/APhO_2025_1_c_1.png" + ] + }, + { + "id": "APhO_2025_2_A_1", + "context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass.", + "question": "It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$.", + "marking": [ + [ + "Award 0.1 pt if the answer gives the correct result for the angular momentum of the planar loop, $\\vec{L} = M R^2 \\vec{\\omega}$, where $M$ is the mass, $R$ the radius, and $\\vec{\\omega}$ the angular velocity (Award 0.1 pt if only the magnitude is given in the answer). Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct result for the magnetic dipole moment, $\\vec{\\mu} = \\frac{Q}{2} R^2 \\vec{\\omega}$, where $Q$ is the charge (Award 0.1 pt if only the magnitude is given in the answer). Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct result for the gyromagnetic ratio, $\\gamma = \\frac{Q}{2M}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\gamma = \\frac{Q}{2M}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.3 + ], + "modality": "text-only", + "field": "Electromagnetism", + "source": "APhO_2025", + "image_question": [] + }, + { + "id": "APhO_2025_2_A_2", + "context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass. \n\n(A.1) It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$. \n\nThe ring is placed in a weak uniform magnetic field $\\vec{B} = B \\hat{z}$, making an angle $\\theta$ with $\\vec{\\omega}$, see Figure A.1. \n\n[figure1] \nFigure A.1.", + "question": "Find the angular frequency $\\omega_{L}$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$.", + "marking": [ + [ + "Award 0.1 pt if the answer writes the torque equation for a magnetic dipole correctly as $\\vec{\\tau} = \\vec{\\mu} \\times \\vec{B} = \\dfrac{d \\vec{L}}{dt}$, where $\\mu$ is the magnetic dipole moment, $\\vec{B}$ the magnetic field, and $\\vec{L}$ the angular momentum. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer notes that only the perpendicular component of angular momentum, $L \\sin \\theta$, changes during precession. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer obtains the correct magnitude of the Larmor frequency as $|\\omega_L| = \\frac{\\mu}{L} B$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct sign for the Larmor frequency, $\\omega_L = - \\frac{\\mu}{L} B = - \\gamma B$, where $\\gamma$ is the gyromagnetic ratio. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\omega_L = -\\gamma B$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.4 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_2_a_1.png" + ] + }, + { + "id": "APhO_2025_2_A_3", + "context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass. \n\n(A.1) It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$. \n\nThe ring is placed in a weak uniform magnetic field $\\vec{B} = B \\hat{z}$, making an angle $\\theta$ with $\\vec{\\omega}$, see Figure A.1. \n\n[figure1] \nFigure A.1. \n\n(A.2) Find the angular frequency $\\omega_{L}$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$. \n\nNow we turn off the external magnetic field and place an identical ring at a horizontal distance $d \\gg R$ from the original ring such that the magnetic moment of the new ring $\\vec{\\mu}_{2}$ makes an angle $\\theta$ with $\\vec{\\mu}_{1}$, see Figure A.2. \n\n[figure2]\nFigure A.2.", + "question": "The magnetic interaction energy between the two rings can be written as $U = J_{0} \\vec{L}_{1} \\cdot \\vec{L}_{2}$, where $J_{0}$ is a constant and $\\vec{L}_{i}$ is the angular momentum of the $i$-th ring. Find $J_{0}$ in terms of $\\gamma$, $d$ and fundamental constants.", + "marking": [ + [ + "Award 0.1 pt if the answer writes the interaction energy correctly as $U = - \\vec{\\mu}_2 \\cdot \\vec{B}_{1 on 2} = - \\frac{\\mu_0}{4 \\pi d^3} \\mu_1 \\mu_2 \\cos(\\pi - \\theta)$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct magnetic field magnitude, $|\\vec{B}_{1 on 2}| = \\frac{\\mu_0}{4 \\pi d^3} \\mu_1$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct magnetic field direction (with the correct minus sign in $\\vec{B}_{1 on 2}$). Otherwise, award 0 pt.", + "Award 0.1 pt if the answer writes the correct magnitude for $J_0$ as $J_0 = \\frac{\\mu_0 \\gamma^2}{4 \\pi d^3}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct sign for $J_0$ in the expression $U = J_0 \\vec{L}_1 \\cdot \\vec{L}_2$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$J_0 = \\frac{\\mu_0 \\gamma^2}{4 \\pi d^3}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.5 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_2_a_1.png", + "image_question/APhO_2025_2_a_2.png" + ] + }, + { + "id": "APhO_2025_2_B_1", + "context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass. \n\n(A.1) It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$. \n\nThe ring is placed in a weak uniform magnetic field $\\vec{B} = B \\hat{z}$, making an angle $\\theta$ with $\\vec{\\omega}$, see Figure A.1. \n\n[figure1] \nFigure A.1. \n\n(A.2) Find the angular frequency $\\omega_{L}$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$. \n\nNow we turn off the external magnetic field and place an identical ring at a horizontal distance $d \\gg R$ from the original ring such that the magnetic moment of the new ring $\\vec{\\mu}_{2}$ makes an angle $\\theta$ with $\\vec{\\mu}_{1}$, see Figure A.2. \n\n[figure2]\nFigure A.2. \n\n(A.3) The magnetic interaction energy between the two rings can be written as $U = J_{0} \\vec{L}_{1} \\cdot \\vec{L}_{2}$, where $J_{0}$ is a constant and $\\vec{L}_{i}$ is the angular momentum of the $i$-th ring. Find $J_{0}$ in terms of $\\gamma$, $d$ and fundamental constants. \n\n[Part B: Spin Waves] \n\nIn what follows we investigate the dynamics of spins. A spin is a particle with intrinsic angular momentum $\\vec{S}$, which has an associated magnetic moment $\\vec{\\mu}$ related to $\\vec{S}$ via the gyromagnetic ratio as in Part A.1, $\\vec{\\mu} = \\gamma \\vec{S}$. \n\nThe magnetic dipoles of two spins interact with each other. However, this interaction is negligible compared to another interaction arising from a quantum mechanical origin, which is not present in classical systems. Interestingly, the energy associated with this quantum interaction has the same form which we found in Part A.3, scaling with $\\vec{S}_{1} \\cdot \\vec{S}_{2}$, albeit with the opposite sign. \n\nNow we will look at a very long chain of spins. The positions of the spins are fixed along the $x$-axis, with a distance $a$ separating them, see Figure B.1. We will approximate the total energy of the system by considering the interactions between nearest neighbors only, so that the energy can be written as \n$E = -J \\sum_i \\vec{S}_i \\cdot \\vec{S}_{i+1}$ \nwhere $J > 0$ is the interaction strength, and $\\vec{S}_{i}$ is the spin angular momentum vector of the $i$-th dipole, with magnitude $S$. The spin vectors are free to rotate in three dimensions. Notice that the sign of the energy is different from the last part. This interaction is purely quantum mechanical. \n\n[figure3] \nFigure B.1.", + "question": "The energy terms containing $\\vec{S}_{i}$ in the sum above can be viewed as the interaction energy between an effective magnetic field $\\vec{B}_{i, \\text{eff}}$ and the magnetic moment of $\\vec{S}_{i}$. Find $\\vec{B}_{i, \\text{eff}}$ and express your answer in terms of $J$, the gyromagnetic ratio $\\gamma$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$)", + "marking": [ + [ + "Award 0.2 pt if the answer explicitly shows the understanding that spins $i-1$ and $i+1$ are the contributors to the interaction energy of spin $i$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct effective magnetic field as $\\vec{B}{i, \\text{eff}} = \\frac{J}{\\gamma}( \\vec{S}{i-1} + \\vec{S}_{i+1})$, where $J$ is the coupling constant and $\\gamma$ the gyromagnetic ratio. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\vec{B}_{i,\\text{eff}} = \\frac{J}{\\gamma} (\\vec{S}_{i-1} + \\vec{S}_{i+1})$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.3 + ], + "modality": "text+variable figure", + "field": "Modern Physics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_2_a_1.png", + "image_question/APhO_2025_2_a_2.png", + "image_question/APhO_2025_2_b_1.png" + ] + }, + { + "id": "APhO_2025_2_B_2", + "context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass. \n\n(A.1) It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$. \n\nThe ring is placed in a weak uniform magnetic field $\\vec{B} = B \\hat{z}$, making an angle $\\theta$ with $\\vec{\\omega}$, see Figure A.1. \n\n[figure1] \nFigure A.1. \n\n(A.2) Find the angular frequency $\\omega_{L}$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$. \n\nNow we turn off the external magnetic field and place an identical ring at a horizontal distance $d \\gg R$ from the original ring such that the magnetic moment of the new ring $\\vec{\\mu}_{2}$ makes an angle $\\theta$ with $\\vec{\\mu}_{1}$, see Figure A.2. \n\n[figure2]\nFigure A.2. \n\n(A.3) The magnetic interaction energy between the two rings can be written as $U = J_{0} \\vec{L}_{1} \\cdot \\vec{L}_{2}$, where $J_{0}$ is a constant and $\\vec{L}_{i}$ is the angular momentum of the $i$-th ring. Find $J_{0}$ in terms of $\\gamma$, $d$ and fundamental constants. \n\n[Part B: Spin Waves] \n\nIn what follows we investigate the dynamics of spins. A spin is a particle with intrinsic angular momentum $\\vec{S}$, which has an associated magnetic moment $\\vec{\\mu}$ related to $\\vec{S}$ via the gyromagnetic ratio as in Part A.1, $\\vec{\\mu} = \\gamma \\vec{S}$. \n\nThe magnetic dipoles of two spins interact with each other. However, this interaction is negligible compared to another interaction arising from a quantum mechanical origin, which is not present in classical systems. Interestingly, the energy associated with this quantum interaction has the same form which we found in Part A.3, scaling with $\\vec{S}_{1} \\cdot \\vec{S}_{2}$, albeit with the opposite sign. \n\nNow we will look at a very long chain of spins. The positions of the spins are fixed along the $x$-axis, with a distance $a$ separating them, see Figure B.1. We will approximate the total energy of the system by considering the interactions between nearest neighbors only, so that the energy can be written as \n$E = -J \\sum_i \\vec{S}_i \\cdot \\vec{S}_{i+1}$ \nwhere $J > 0$ is the interaction strength, and $\\vec{S}_{i}$ is the spin angular momentum vector of the $i$-th dipole, with magnitude $S$. The spin vectors are free to rotate in three dimensions. Notice that the sign of the energy is different from the last part. This interaction is purely quantum mechanical. \n\n[figure3] \nFigure B.1. \n\n(B.1) The energy terms containing $\\vec{S}_{i}$ in the sum above can be viewed as the interaction energy between an effective magnetic field $\\vec{B}_{i, \\text{eff}}$ and the magnetic moment of $\\vec{S}_{i}$. Find $\\vec{B}_{i, \\text{eff}}$ and express your answer in terms of $J$, the gyromagnetic ratio $\\gamma$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$)", + "question": "Using the concept of effective magnetic field, express the rate of change of the $i$-th spin vector, $d \\vec{S}_{i} / d t$, in terms of $J$, $\\vec{S}_{i}$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$).", + "marking": [ + [ + "Award 0.1 pt if the answer writes the rate of change of spin using the effective magnetic field, i.e. $\\frac{d \\vec{S}_i}{dt} = \\vec{\\mu}_i \\times \\vec{B}_{i,\\text{eff}}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct explicit equation $\\frac{d \\vec{S}_i}{dt} = J \\vec{S}_i \\times ( \\vec{S}_{i-1} + \\vec{S}_{i+1})$, where $J$ is the coupling constant. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{d \\vec{S}_i}{dt} = J \\vec{S}_i \\times (\\vec{S}_{i-1} + \\vec{S}_{i+1})$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.3 + ], + "modality": "text+variable figure", + "field": "Modern Physics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_2_a_1.png", + "image_question/APhO_2025_2_a_2.png", + "image_question/APhO_2025_2_b_1.png" + ] + }, + { + "id": "APhO_2025_2_B_3", + "context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass. \n\n(A.1) It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$. \n\nThe ring is placed in a weak uniform magnetic field $\\vec{B} = B \\hat{z}$, making an angle $\\theta$ with $\\vec{\\omega}$, see Figure A.1. \n\n[figure1] \nFigure A.1. \n\n(A.2) Find the angular frequency $\\omega_{L}$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$. \n\nNow we turn off the external magnetic field and place an identical ring at a horizontal distance $d \\gg R$ from the original ring such that the magnetic moment of the new ring $\\vec{\\mu}_{2}$ makes an angle $\\theta$ with $\\vec{\\mu}_{1}$, see Figure A.2. \n\n[figure2]\nFigure A.2. \n\n(A.3) The magnetic interaction energy between the two rings can be written as $U = J_{0} \\vec{L}_{1} \\cdot \\vec{L}_{2}$, where $J_{0}$ is a constant and $\\vec{L}_{i}$ is the angular momentum of the $i$-th ring. Find $J_{0}$ in terms of $\\gamma$, $d$ and fundamental constants. \n\n[Part B: Spin Waves] \n\nIn what follows we investigate the dynamics of spins. A spin is a particle with intrinsic angular momentum $\\vec{S}$, which has an associated magnetic moment $\\vec{\\mu}$ related to $\\vec{S}$ via the gyromagnetic ratio as in Part A.1, $\\vec{\\mu} = \\gamma \\vec{S}$. \n\nThe magnetic dipoles of two spins interact with each other. However, this interaction is negligible compared to another interaction arising from a quantum mechanical origin, which is not present in classical systems. Interestingly, the energy associated with this quantum interaction has the same form which we found in Part A.3, scaling with $\\vec{S}_{1} \\cdot \\vec{S}_{2}$, albeit with the opposite sign. \n\nNow we will look at a very long chain of spins. The positions of the spins are fixed along the $x$-axis, with a distance $a$ separating them, see Figure B.1. We will approximate the total energy of the system by considering the interactions between nearest neighbors only, so that the energy can be written as \n$E = -J \\sum_i \\vec{S}_i \\cdot \\vec{S}_{i+1}$ \nwhere $J > 0$ is the interaction strength, and $\\vec{S}_{i}$ is the spin angular momentum vector of the $i$-th dipole, with magnitude $S$. The spin vectors are free to rotate in three dimensions. Notice that the sign of the energy is different from the last part. This interaction is purely quantum mechanical. \n\n[figure3] \nFigure B.1. \n\n(B.1) The energy terms containing $\\vec{S}_{i}$ in the sum above can be viewed as the interaction energy between an effective magnetic field $\\vec{B}_{i, \\text{eff}}$ and the magnetic moment of $\\vec{S}_{i}$. Find $\\vec{B}_{i, \\text{eff}}$ and express your answer in terms of $J$, the gyromagnetic ratio $\\gamma$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$) \n\n(B.2) Using the concept of effective magnetic field, express the rate of change of the $i$-th spin vector, $d \\vec{S}_{i} / d t$, in terms of $J$, $\\vec{S}_{i}$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$). \n\nFor the rest of Part B, assume that the system is highly magnetized along the $z$ direction, so we can use the approximations $S_{i, z} \\approx S$ and $d S_{i, z} / d t \\approx 0$ for each spin, see Figure B.2. In this regime, the set of equations describing the spins time evolution is satisfied by a traveling wave solution for $S_{i, x}$ and $S_{i, y}$ characterized by a wave vector $k$ and angular frequency $\\omega$. \n\n[figure4]\nFigure B.2.", + "question": "Find the relationship between $\\omega$ and $k$ (known as the dispersion relation, $\\omega(k)$) for the spin waves in terms of $J$, $S$ and $a$. Hint: express the position of the $i$-th spin as $x = a \\cdot i$.", + "marking": [ + [ + "Award 0.25 pt if the answer writes the traveling wave as a function of $kx \\pm \\omega t$ (either sign and either trigonometric functions $\\cos$ or $\\sin$ or complex exponentials are acceptable). Otherwise, award 0 pt.", + "Award 0.25 pt if the answer explicitly shows that the amplitudes of $S_x$ and $S_y$ are equal, i.e., $\\delta S_x = \\delta S_y$, where $\\delta$ is the wave amplitude. Otherwise, award 0 pt.", + "Award 0.25 pt if the answer correctly identifies the phase relation between $S_x$ and $S_y$ as a difference of $\\pi/2$, e.g., $S_{i,x} = \\delta S \\cos(kx - \\omega t)$ and $S_{i,y} = \\delta S \\sin(kx - \\omega t)$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer writes the explicit equation of motion for either $S_x$ or $S_y$, such as $\\frac{d S_{i,x}}{dt} \\approx JS(2 S_{i,y} - S_{i-1,y} - S_{i+1,y})$ or $\\frac{d S_{i,y}}{dt} \\approx -JS(2 S_{i,x} - S_{i-1,x} - S_{i+1,x})$, where $J$ is the exchange coupling constant and $S$ is the spin magnitude. Otherwise, award 0 pt.", + "Award 0.25 pt if the answer explicitly uses the approximation $S_{i,z} \\approx S$, where $S$ is the spin magnitude. Otherwise, award 0 pt.", + "Award 0.5 pt if the final dispersion relation is correctly given as $\\omega(k) = 2JS [1 - \\cos(ka)]$ (with $\\pm$ accepted), where $a$ is the lattice spacing. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\omega(k) = 2JS[1 - \\cos(ka)]$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+variable figure", + "field": "Modern Physics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_2_a_1.png", + "image_question/APhO_2025_2_a_2.png", + "image_question/APhO_2025_2_b_1.png", + "image_question/APhO_2025_2_b_2.png" + ] + }, + { + "id": "APhO_2025_2_B_4", + "context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass. \n\n(A.1) It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$. \n\nThe ring is placed in a weak uniform magnetic field $\\vec{B} = B \\hat{z}$, making an angle $\\theta$ with $\\vec{\\omega}$, see Figure A.1. \n\n[figure1] \nFigure A.1. \n\n(A.2) Find the angular frequency $\\omega_{L}$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$. \n\nNow we turn off the external magnetic field and place an identical ring at a horizontal distance $d \\gg R$ from the original ring such that the magnetic moment of the new ring $\\vec{\\mu}_{2}$ makes an angle $\\theta$ with $\\vec{\\mu}_{1}$, see Figure A.2. \n\n[figure2]\nFigure A.2. \n\n(A.3) The magnetic interaction energy between the two rings can be written as $U = J_{0} \\vec{L}_{1} \\cdot \\vec{L}_{2}$, where $J_{0}$ is a constant and $\\vec{L}_{i}$ is the angular momentum of the $i$-th ring. Find $J_{0}$ in terms of $\\gamma$, $d$ and fundamental constants. \n\n[Part B: Spin Waves] \n\nIn what follows we investigate the dynamics of spins. A spin is a particle with intrinsic angular momentum $\\vec{S}$, which has an associated magnetic moment $\\vec{\\mu}$ related to $\\vec{S}$ via the gyromagnetic ratio as in Part A.1, $\\vec{\\mu} = \\gamma \\vec{S}$. \n\nThe magnetic dipoles of two spins interact with each other. However, this interaction is negligible compared to another interaction arising from a quantum mechanical origin, which is not present in classical systems. Interestingly, the energy associated with this quantum interaction has the same form which we found in Part A.3, scaling with $\\vec{S}_{1} \\cdot \\vec{S}_{2}$, albeit with the opposite sign. \n\nNow we will look at a very long chain of spins. The positions of the spins are fixed along the $x$-axis, with a distance $a$ separating them, see Figure B.1. We will approximate the total energy of the system by considering the interactions between nearest neighbors only, so that the energy can be written as \n$E = -J \\sum_i \\vec{S}_i \\cdot \\vec{S}_{i+1}$ \nwhere $J > 0$ is the interaction strength, and $\\vec{S}_{i}$ is the spin angular momentum vector of the $i$-th dipole, with magnitude $S$. The spin vectors are free to rotate in three dimensions. Notice that the sign of the energy is different from the last part. This interaction is purely quantum mechanical. \n\n[figure3] \nFigure B.1. \n\n(B.1) The energy terms containing $\\vec{S}_{i}$ in the sum above can be viewed as the interaction energy between an effective magnetic field $\\vec{B}_{i, \\text{eff}}$ and the magnetic moment of $\\vec{S}_{i}$. Find $\\vec{B}_{i, \\text{eff}}$ and express your answer in terms of $J$, the gyromagnetic ratio $\\gamma$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$) \n\n(B.2) Using the concept of effective magnetic field, express the rate of change of the $i$-th spin vector, $d \\vec{S}_{i} / d t$, in terms of $J$, $\\vec{S}_{i}$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$). \n\nFor the rest of Part B, assume that the system is highly magnetized along the $z$ direction, so we can use the approximations $S_{i, z} \\approx S$ and $d S_{i, z} / d t \\approx 0$ for each spin, see Figure B.2. In this regime, the set of equations describing the spins time evolution is satisfied by a traveling wave solution for $S_{i, x}$ and $S_{i, y}$ characterized by a wave vector $k$ and angular frequency $\\omega$. \n\n[figure4]\nFigure B.2. \n\n(B.3) Find the relationship between $\\omega$ and $k$ (known as the dispersion relation, $\\omega(k)$) for the spin waves in terms of $J$, $S$ and $a$. Hint: express the position of the $i$-th spin as $x = a \\cdot i$. \n\nThe spin wave described above carries energy and momentum. At low energies, the relation between its energy and momentum resembles that of a massive classical particle with an effective mass $m_{\\text{eff}}$, a concept known as a quasi-particle.", + "question": "For small $k$ ($k \\ll 1 / a$), find the effective mass $m_{\\text{eff}}$ of the spin wave. Express your answer in terms of $J, S, a$ and fundamental constants.", + "marking": [ + [ + "Award 0.2 pt if the answer gives the correct Taylor expansion for small $k$, namely $\\omega(k) \\approx 2JS \\left[1 - 1 + \\frac{1}{2}(ka)^2 \\right] = JSa^2 k^2$, where $J$ is the exchange coupling constant, $S$ is the spin magnitude, and $a$ is the lattice spacing. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer uses the correct relation between momentum and wave vector, $p = \\hbar k$, where $p$ is the momentum and $\\hbar$ is the reduced Planck constant. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer uses the correct relation between energy and angular frequency, $E = \\hbar \\omega$, where $E$ is the energy and $\\omega$ is the angular frequency. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly identifies the effective mass as $m_{\\text{eff}} = \\frac{\\hbar}{2JSa^2}$, where $m_{\\text{eff}}$ is the effective mass. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$m_{\\text{eff}} = \\frac{\\hbar}{2 J S a^2}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.6 + ], + "modality": "text+variable figure", + "field": "Modern Physics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_2_a_1.png", + "image_question/APhO_2025_2_a_2.png", + "image_question/APhO_2025_2_b_1.png", + "image_question/APhO_2025_2_b_2.png" + ] + }, + { + "id": "APhO_2025_2_B_5", + "context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass. \n\n(A.1) It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$. \n\nThe ring is placed in a weak uniform magnetic field $\\vec{B} = B \\hat{z}$, making an angle $\\theta$ with $\\vec{\\omega}$, see Figure A.1. \n\n[figure1] \nFigure A.1. \n\n(A.2) Find the angular frequency $\\omega_{L}$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$. \n\nNow we turn off the external magnetic field and place an identical ring at a horizontal distance $d \\gg R$ from the original ring such that the magnetic moment of the new ring $\\vec{\\mu}_{2}$ makes an angle $\\theta$ with $\\vec{\\mu}_{1}$, see Figure A.2. \n\n[figure2]\nFigure A.2. \n\n(A.3) The magnetic interaction energy between the two rings can be written as $U = J_{0} \\vec{L}_{1} \\cdot \\vec{L}_{2}$, where $J_{0}$ is a constant and $\\vec{L}_{i}$ is the angular momentum of the $i$-th ring. Find $J_{0}$ in terms of $\\gamma$, $d$ and fundamental constants. \n\n[Part B: Spin Waves] \n\nIn what follows we investigate the dynamics of spins. A spin is a particle with intrinsic angular momentum $\\vec{S}$, which has an associated magnetic moment $\\vec{\\mu}$ related to $\\vec{S}$ via the gyromagnetic ratio as in Part A.1, $\\vec{\\mu} = \\gamma \\vec{S}$. \n\nThe magnetic dipoles of two spins interact with each other. However, this interaction is negligible compared to another interaction arising from a quantum mechanical origin, which is not present in classical systems. Interestingly, the energy associated with this quantum interaction has the same form which we found in Part A.3, scaling with $\\vec{S}_{1} \\cdot \\vec{S}_{2}$, albeit with the opposite sign. \n\nNow we will look at a very long chain of spins. The positions of the spins are fixed along the $x$-axis, with a distance $a$ separating them, see Figure B.1. We will approximate the total energy of the system by considering the interactions between nearest neighbors only, so that the energy can be written as \n$E = -J \\sum_i \\vec{S}_i \\cdot \\vec{S}_{i+1}$ \nwhere $J > 0$ is the interaction strength, and $\\vec{S}_{i}$ is the spin angular momentum vector of the $i$-th dipole, with magnitude $S$. The spin vectors are free to rotate in three dimensions. Notice that the sign of the energy is different from the last part. This interaction is purely quantum mechanical. \n\n[figure3] \nFigure B.1. \n\n(B.1) The energy terms containing $\\vec{S}_{i}$ in the sum above can be viewed as the interaction energy between an effective magnetic field $\\vec{B}_{i, \\text{eff}}$ and the magnetic moment of $\\vec{S}_{i}$. Find $\\vec{B}_{i, \\text{eff}}$ and express your answer in terms of $J$, the gyromagnetic ratio $\\gamma$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$) \n\n(B.2) Using the concept of effective magnetic field, express the rate of change of the $i$-th spin vector, $d \\vec{S}_{i} / d t$, in terms of $J$, $\\vec{S}_{i}$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$). \n\nFor the rest of Part B, assume that the system is highly magnetized along the $z$ direction, so we can use the approximations $S_{i, z} \\approx S$ and $d S_{i, z} / d t \\approx 0$ for each spin, see Figure B.2. In this regime, the set of equations describing the spins time evolution is satisfied by a traveling wave solution for $S_{i, x}$ and $S_{i, y}$ characterized by a wave vector $k$ and angular frequency $\\omega$. \n\n[figure4]\nFigure B.2. \n\n(B.3) Find the relationship between $\\omega$ and $k$ (known as the dispersion relation, $\\omega(k)$) for the spin waves in terms of $J$, $S$ and $a$. Hint: express the position of the $i$-th spin as $x = a \\cdot i$. \n\nThe spin wave described above carries energy and momentum. At low energies, the relation between its energy and momentum resembles that of a massive classical particle with an effective mass $m_{\\text{eff}}$, a concept known as a quasi-particle. \n\n(B.4) For small $k$ ($k \\ll 1 / a$), find the effective mass $m_{\\text{eff}}$ of the spin wave. Express your answer in terms of $J, S, a$ and fundamental constants. \n\nSpin waves can be experimentally probed using inelastic neutron scattering. Although neutrons have zero net charge, they have a finite spin, allowing them to interact with other spins. \n\n[figure5] \nFigure B.3.", + "question": "Suppose that initially, all the spins in the chain are pointing along the $z$ direction. A neutron with low energy travels on the $x-y$ plane making an incident angle $\\theta_{\\text{in}}$ with the chain and scatters with an angle $\\theta_{\\text{out}}$ as shown in Figure B.3. Assuming the neutron excites a single low wave vector spin wave, find the effective mass $m_{\\text{eff}}$ of the spin wave, in terms of $\\theta_{\\text{in}}, \\theta_{\\text{out}}$ and the neutron mass $m_{n}$. Assume that the chain stays at rest.", + "marking": [ + [ + "Award 0.4 pt if the answer applies conservation of momentum along the $y$-axis, i.e., $p_{\\text{in}} \\cos \\theta_{\\text{in}} = p_{\\text{out}} \\cos \\theta_{\\text{out}}$, where $p_{\\text{in}}$ and $p_{\\text{out}}$ are the incident and outgoing neutron momenta, and $\\theta_{\\text{in}}$, $\\theta_{\\text{out}}$ are their respective scattering angles. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer applies conservation of momentum along the $x$-axis, i.e., $p_s = p_{\\text{in}} \\sin \\theta_{\\text{in}} - p_{\\text{out}} \\sin \\theta_{\\text{out}}$, where $p_s$ is the spin-wave momentum. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer uses conservation of energy, i.e., $E_s = E_{\\text{in}} - E_{\\text{out}}$, where $E_s$ is the spin-wave energy, $E_{\\text{in}}$ and $E_{\\text{out}}$ are the neutron energies before and after scattering. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct relation between $E_{\\text{out}}$ and $E_{\\text{in}}$, namely $E_{\\text{out}} = \\left( \\frac{\\cos \\theta_{\\text{in}}}{\\cos \\theta_{\\text{out}}} \\right)^2 E_{\\text{in}}$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer derives the correct final expression for the effective mass, either as $m_{\\text{eff}} = \\frac{\\sin^2(\\theta_{\\text{in}} - \\theta_{\\text{out}})}{\\cos^2 \\theta_{\\text{out}} - \\cos^2 \\theta_{\\text{in}}} m_n$ or equivalently $m_{\\text{eff}} = \\frac{\\sin(\\theta_{\\text{in}} - \\theta_{\\text{out}})}{\\sin(\\theta_{\\text{in}} + \\theta_{\\text{out}})} m_n$, where $m_n$ is the neutron mass. Otherwise, award 0 pt (0.2 pt if partially correct)." + ] + ], + "answer": [ + "\\boxed{$m_{\\text{eff}} = \\frac{\\sin(\\theta_{\\text{in}} - \\theta_{\\text{out}})}{\\sin(\\theta_{\\text{in}} + \\theta_{\\text{out}})} m_n$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.3 + ], + "modality": "text+variable figure", + "field": "Modern Physics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_2_a_1.png", + "image_question/APhO_2025_2_a_2.png", + "image_question/APhO_2025_2_b_1.png", + "image_question/APhO_2025_2_b_2.png", + "image_question/APhO_2025_2_b_3.png" + ] + }, + { + "id": "APhO_2025_2_C_1", + "context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part C: Phase transitions in spin chains] \n\nNext we consider the same chain made of $N$ spins from Part B, except the spin vectors are now restricted to point either up or down along the $z$-axis, so that the spin component along $z$ can be written as $S_{i, z} = s_{i} S$, where $s_{i} = \\pm 1$, see Figure C.1. In addition to the nearest neighbor interactions, we could have an external magnetic field pointing along the $z$-axis so that the total energy of the system is given by \n$E = -\\tilde{J} \\sum_i s_i s_{i+1} - h \\sum_i s_i$ \nWe assume $\\tilde{J} \\geq 0$, and $h$ is a constant dependent on the magnetic field. The spin system is at equilibrium with a heat bath at temperature $T$. Ignore the edges of the chain. \n\n[figure1] \nFigure C.1.", + "question": "Assume first that $\\tilde{J} = 0$, what is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_{\\uparrow}$ to being anti-aligned to the magnetic field $p_{\\downarrow}$? Express $p_{\\uparrow} / p_{\\downarrow}$ in terms of $h, T$ and fundamental constants.", + "marking": [ + [ + "Award 0.2 pt if the answer uses the Boltzmann factor $p_i \\propto \\exp(-\\varepsilon_i / k_B T)$, where $\\varepsilon_i$ is the energy of state $i$, $k_B$ is the Boltzmann constant, and $T$ is the temperature. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct Boltzmann factor for the spin-up state, $p_{\\uparrow} \\sim e^{h / k_B T}$, where $h$ is the Zeeman energy. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct Boltzmann factor for the spin-down state, $p_{\\downarrow} \\sim e^{-h / k_B T}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct ratio $\\frac{p_{\\uparrow}}{p_{\\downarrow}} = e^{2h / k_B T}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$p_{\\uparrow} / p_{\\downarrow} = e^{2h/k_B T}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.5 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_2_c_1.png" + ] + }, + { + "id": "APhO_2025_2_C_2", + "context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part C: Phase transitions in spin chains] \n\nNext we consider the same chain made of $N$ spins from Part B, except the spin vectors are now restricted to point either up or down along the $z$-axis, so that the spin component along $z$ can be written as $S_{i, z} = s_{i} S$, where $s_{i} = \\pm 1$, see Figure C.1. In addition to the nearest neighbor interactions, we could have an external magnetic field pointing along the $z$-axis so that the total energy of the system is given by \n$E = -\\tilde{J} \\sum_i s_i s_{i+1} - h \\sum_i s_i$ \nWe assume $\\tilde{J} \\geq 0$, and $h$ is a constant dependent on the magnetic field. The spin system is at equilibrium with a heat bath at temperature $T$. Ignore the edges of the chain. \n\n[figure1] \nFigure C.1. \n\n(C.1) Assume first that $\\tilde{J} = 0$, what is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_{\\uparrow}$ to being anti-aligned to the magnetic field $p_{\\downarrow}$? Express $p_{\\uparrow} / p_{\\downarrow}$ in terms of $h, T$ and fundamental constants.", + "question": "Find the average polarization of the system $\\bar{s} \\equiv \\frac{1}{N} \\sum_{i} s_{i}$ for $N \\gg 1$ in terms of $h, T$ and fundamental constants. If the magnetic field $h$ can range from $-h_{0}$ to $h_{0}$, make a sketch of $\\bar{s}$ as a function of $h$ for the cases $h_{o} \\gg k_{B} T$, $h_{o} \\approx k_{B} T$ and $h_{o} \\ll k_{B} T$.", + "marking": [ + [ + "Award 0.2 pt if the answer deduces the expression for the average polarization as $\\bar{s} = p_{\\uparrow} - p_{\\downarrow}$, where $p_{\\uparrow}$ and $p_{\\downarrow}$ are the probabilities of spin-up and spin-down states, respectively. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer uses the normalization condition $p_{\\uparrow} + p_{\\downarrow} = 1$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct final result $\\bar{s} = \\tanh\\left( \\frac{h}{k_B T} \\right)$, where $h$ is the Zeeman energy, $k_B$ is the Boltzmann constant, and $T$ is the temperature. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer provides a correct sketch of $\\bar{s}$ versus $h/h_0$ in the case $h_0 \\gg k_B T$ (sharp step-like curve approaching $\\pm 1$). Otherwise, award 0 pt.", + "Award 0.2 pt if the answer provides a correct sketch of $\\bar{s}$ versus $h/h_0$ in the case $h_0 \\approx k_B T$ (smooth nonlinear S-shaped curve). Otherwise, award 0 pt.", + "Award 0.2 pt if the answer provides a correct sketch of $\\bar{s}$ versus $h/h_0$ in the case $h_0 \\ll k_B T$ (almost flat line near zero). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\bar{s} = \\tanh(\\frac{h}{k_B T})$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_2_c_1.png" + ] + }, + { + "id": "APhO_2025_2_C_3", + "context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part C: Phase transitions in spin chains] \n\nNext we consider the same chain made of $N$ spins from Part B, except the spin vectors are now restricted to point either up or down along the $z$-axis, so that the spin component along $z$ can be written as $S_{i, z} = s_{i} S$, where $s_{i} = \\pm 1$, see Figure C.1. In addition to the nearest neighbor interactions, we could have an external magnetic field pointing along the $z$-axis so that the total energy of the system is given by \n$E = -\\tilde{J} \\sum_i s_i s_{i+1} - h \\sum_i s_i$ \nWe assume $\\tilde{J} \\geq 0$, and $h$ is a constant dependent on the magnetic field. The spin system is at equilibrium with a heat bath at temperature $T$. Ignore the edges of the chain. \n\n[figure1] \nFigure C.1. \n\n(C.1) Assume first that $\\tilde{J} = 0$, what is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_{\\uparrow}$ to being anti-aligned to the magnetic field $p_{\\downarrow}$? Express $p_{\\uparrow} / p_{\\downarrow}$ in terms of $h, T$ and fundamental constants. \n\n(C.2) Find the average polarization of the system $\\bar{s} \\equiv \\frac{1}{N} \\sum_{i} s_{i}$ for $N \\gg 1$ in terms of $h, T$ and fundamental constants. If the magnetic field $h$ can range from $-h_{0}$ to $h_{0}$, make a sketch of $\\bar{s}$ as a function of $h$ for the cases $h_{o} \\gg k_{B} T$, $h_{o} \\approx k_{B} T$ and $h_{o} \\ll k_{B} T$. \n\nIn the remaining questions, we turn off the magnetic field, so $h = 0$, and set $\\tilde{J} > 0$.", + "question": "What is the energy $E_{g}$ of the ground state (the lowest energy state)? Express your answer in terms of $\\tilde{J}$ and $N$.", + "marking": [ + [ + "Award 0.1 pt if the answer recognizes that the energy of the system is minimized when all spins align in the same direction. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct ground state energy as $E_g = -\\tilde{J}(N-1) \\approx -\\tilde{J}N$, where $\\tilde{J}$ is the effective exchange coupling and $N$ is the number of spins (both $N-1$ and $N$ are acceptable). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$E_g \\simeq -\\tilde{J} N$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.2 + ], + "modality": "text+illustration figure", + "field": "Modern Physics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_2_c_1.png" + ] + }, + { + "id": "APhO_2025_2_C_4", + "context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part C: Phase transitions in spin chains] \n\nNext we consider the same chain made of $N$ spins from Part B, except the spin vectors are now restricted to point either up or down along the $z$-axis, so that the spin component along $z$ can be written as $S_{i, z} = s_{i} S$, where $s_{i} = \\pm 1$, see Figure C.1. In addition to the nearest neighbor interactions, we could have an external magnetic field pointing along the $z$-axis so that the total energy of the system is given by \n$E = -\\tilde{J} \\sum_i s_i s_{i+1} - h \\sum_i s_i$ \nWe assume $\\tilde{J} \\geq 0$, and $h$ is a constant dependent on the magnetic field. The spin system is at equilibrium with a heat bath at temperature $T$. Ignore the edges of the chain. \n\n[figure1] \nFigure C.1. \n\n(C.1) Assume first that $\\tilde{J} = 0$, what is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_{\\uparrow}$ to being anti-aligned to the magnetic field $p_{\\downarrow}$? Express $p_{\\uparrow} / p_{\\downarrow}$ in terms of $h, T$ and fundamental constants. \n\n(C.2) Find the average polarization of the system $\\bar{s} \\equiv \\frac{1}{N} \\sum_{i} s_{i}$ for $N \\gg 1$ in terms of $h, T$ and fundamental constants. If the magnetic field $h$ can range from $-h_{0}$ to $h_{0}$, make a sketch of $\\bar{s}$ as a function of $h$ for the cases $h_{o} \\gg k_{B} T$, $h_{o} \\approx k_{B} T$ and $h_{o} \\ll k_{B} T$. \n\nIn the remaining questions, we turn off the magnetic field, so $h = 0$, and set $\\tilde{J} > 0$. \n\n(C.3) What is the energy $E_{g}$ of the ground state (the lowest energy state)? Express your answer in terms of $\\tilde{J}$ and $N$. \n\nInstead of considering the interactions between each spin and its neighbors, we assume that each spin sees an average polarization $\\bar{s}$ from its nearest-neighbors.", + "question": "Approximate the energy of the system as a sum over all spins $E = -\\tilde{J}_{\\text{eff}} \\sum_i s_i$ and express $\\tilde{J}_{\\text{eff}}$ in terms of $\\tilde{J}$ and $\\bar{s}$.", + "marking": [ + [ + "Award 0.1 pt if the answer realizes that $s_{i+1}$ can be replaced with the average polarization $\\bar{s}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct final result $E = -\\tilde{J}_{\\text{eff}} \\sum_i s_i = -\\tilde{J} \\sum_i s_i \\bar{s}$, where $\\tilde{J}_{\\text{eff}} = \\tilde{J} \\bar{s}$ is the effective coupling constant. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\tilde{J}_{\\text{eff}} = \\tilde{J} \\bar{s}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.2 + ], + "modality": "text+illustration figure", + "field": "Modern Physics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_2_c_1.png" + ] + }, + { + "id": "APhO_2025_2_C_5", + "context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part C: Phase transitions in spin chains] \n\nNext we consider the same chain made of $N$ spins from Part B, except the spin vectors are now restricted to point either up or down along the $z$-axis, so that the spin component along $z$ can be written as $S_{i, z} = s_{i} S$, where $s_{i} = \\pm 1$, see Figure C.1. In addition to the nearest neighbor interactions, we could have an external magnetic field pointing along the $z$-axis so that the total energy of the system is given by \n$E = -\\tilde{J} \\sum_i s_i s_{i+1} - h \\sum_i s_i$ \nWe assume $\\tilde{J} \\geq 0$, and $h$ is a constant dependent on the magnetic field. The spin system is at equilibrium with a heat bath at temperature $T$. Ignore the edges of the chain. \n\n[figure1] \nFigure C.1. \n\n(C.1) Assume first that $\\tilde{J} = 0$, what is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_{\\uparrow}$ to being anti-aligned to the magnetic field $p_{\\downarrow}$? Express $p_{\\uparrow} / p_{\\downarrow}$ in terms of $h, T$ and fundamental constants. \n\n(C.2) Find the average polarization of the system $\\bar{s} \\equiv \\frac{1}{N} \\sum_{i} s_{i}$ for $N \\gg 1$ in terms of $h, T$ and fundamental constants. If the magnetic field $h$ can range from $-h_{0}$ to $h_{0}$, make a sketch of $\\bar{s}$ as a function of $h$ for the cases $h_{o} \\gg k_{B} T$, $h_{o} \\approx k_{B} T$ and $h_{o} \\ll k_{B} T$. \n\nIn the remaining questions, we turn off the magnetic field, so $h = 0$, and set $\\tilde{J} > 0$. \n\n(C.3) What is the energy $E_{g}$ of the ground state (the lowest energy state)? Express your answer in terms of $\\tilde{J}$ and $N$. \n\nInstead of considering the interactions between each spin and its neighbors, we assume that each spin sees an average polarization $\\bar{s}$ from its nearest-neighbors. \n\n(C.4) Approximate the energy of the system as a sum over all spins $E = -\\tilde{J}_{\\text{eff}} \\sum_i s_i$ and express $\\tilde{J}_{\\text{eff}}$ in terms of $\\tilde{J}$ and $\\bar{s}$.", + "question": "(1) Using your result from C.2, find an equation that the average polarization $\\bar{s}$ must satisfy. \n(2) The number of solutions to this equation depends on $T$. Find the critical temperature $T_{c}$ at which the number of solutions changes. Express your answer in terms of $\\tilde{J}$ and fundamental constants.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly states the self-consistent equation for the polarization as $\\bar{s} = \\tanh\\left( \\frac{\\tilde{J}_{\\text{eff}}}{k_B T} \\right)$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer replaces $\\tilde{J}_{\\text{eff}} = \\tilde{J} \\bar{s}$ into the result from C.2, leading to $\\bar{s} = \\tanh\\left( \\frac{\\tilde{J} \\bar{s}}{k_B T} \\right)$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer realizes that for $\\tilde{J} \\ll k_B T$, there exists only one trivial solution $\\bar{s} = 0$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer realizes that for $\\tilde{J} \\gg k_B T$, there exist two non-trivial solutions for $\\bar{s}$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer clearly states the condition when the number of solutions changes, namely when the slope condition at $\\bar{s}=0$ is satisfied: $\\frac{d}{d \\bar{s}} \\tanh\\left( \\frac{\\tilde{J} \\bar{s}}{k_B T_c} \\right) \\right|_{\\bar{s}=0} = \\frac{d}{d \\bar{s}} \\bar{s} \\right|_{\\bar{s}=0}$, which simplifies to $\\frac{\\tilde{J}}{k_B T_c} = 1$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct final critical temperature $T_c = \\frac{\\tilde{J}}{k_B}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\bar{s} = \\tanh\\left(\\frac{\\tilde{J} \\bar{s}}{k_B T}\\right)$}", + "\\boxed{$T_c = \\frac{\\tilde{J}}{k_B}$}" + ], + "answer_type": [ + "Equation", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 0.3, + 0.9 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_2_c_1.png" + ] + }, + { + "id": "APhO_2025_2_C_6", + "context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part C: Phase transitions in spin chains] \n\nNext we consider the same chain made of $N$ spins from Part B, except the spin vectors are now restricted to point either up or down along the $z$-axis, so that the spin component along $z$ can be written as $S_{i, z} = s_{i} S$, where $s_{i} = \\pm 1$, see Figure C.1. In addition to the nearest neighbor interactions, we could have an external magnetic field pointing along the $z$-axis so that the total energy of the system is given by \n$E = -\\tilde{J} \\sum_i s_i s_{i+1} - h \\sum_i s_i$ \nWe assume $\\tilde{J} \\geq 0$, and $h$ is a constant dependent on the magnetic field. The spin system is at equilibrium with a heat bath at temperature $T$. Ignore the edges of the chain. \n\n[figure1] \nFigure C.1. \n\n(C.1) Assume first that $\\tilde{J} = 0$, what is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_{\\uparrow}$ to being anti-aligned to the magnetic field $p_{\\downarrow}$? Express $p_{\\uparrow} / p_{\\downarrow}$ in terms of $h, T$ and fundamental constants. \n\n(C.2) Find the average polarization of the system $\\bar{s} \\equiv \\frac{1}{N} \\sum_{i} s_{i}$ for $N \\gg 1$ in terms of $h, T$ and fundamental constants. If the magnetic field $h$ can range from $-h_{0}$ to $h_{0}$, make a sketch of $\\bar{s}$ as a function of $h$ for the cases $h_{o} \\gg k_{B} T$, $h_{o} \\approx k_{B} T$ and $h_{o} \\ll k_{B} T$. \n\nIn the remaining questions, we turn off the magnetic field, so $h = 0$, and set $\\tilde{J} > 0$. \n\n(C.3) What is the energy $E_{g}$ of the ground state (the lowest energy state)? Express your answer in terms of $\\tilde{J}$ and $N$. \n\nInstead of considering the interactions between each spin and its neighbors, we assume that each spin sees an average polarization $\\bar{s}$ from its nearest-neighbors. \n\n(C.4) Approximate the energy of the system as a sum over all spins $E = -\\tilde{J}_{\\text{eff}} \\sum_i s_i$ and express $\\tilde{J}_{\\text{eff}}$ in terms of $\\tilde{J}$ and $\\bar{s}$. \n\n(C.5) Using your result from C.2, find an equation that the average polarization $\\bar{s}$ must satisfy. The number of solutions to this equation depends on $T$. Find the critical temperature $T_{c}$ at which the number of solutions changes. Express your answer in terms of $\\tilde{J}$ and fundamental constants.", + "question": "Find all possible values of $\\bar{s}$ when $T < T_{c}$ and $T_{c} - T \\ll T_{c}$. Express your answers in terms of $T$ and $T_{c}$. Sketch all possible values of $\\bar{s}$ for the temperature $T$ in the range $0 \\leq T \\leq 2 T_{c}$.", + "marking": [ + [ + "Award 0.1 pt if the answer uses the proper approximation $\\tanh(x) \\approx x - \\frac{1}{3}x^3$ for small $x$, leading to $\\bar{s} = \\frac{T_c}{T} \\bar{s} - \\frac{1}{3}\\left( \\frac{T_c}{T} \\bar{s} \\right)^3$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer derives a correct non-trivial solution for $\\bar{s}$, i.e., $\\bar{s} = \\sqrt{3 \\left[ \\left(\\frac{T}{T_c}\\right)^2 - \\left(\\frac{T}{T_c}\\right)^3 \\right]} = \\sqrt{3 \\left(\\frac{T}{T_c}\\right)^2 \\cdot \\left(1 - \\frac{T}{T_c}\\right) } \\approx \\sqrt{3 \\frac{T_c - T}{T_c}}$, even if not fully simplified. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer derives a correct non-trivial solution for $\\bar{s}$, i.e., $\\bar{s} = - \\sqrt{3 \\left[ \\left(\\frac{T}{T_c}\\right)^2 - \\left(\\frac{T}{T_c}\\right)^3 \\right]} = - \\sqrt{3 \\left(\\frac{T}{T_c}\\right)^2 \\cdot \\left(1 - \\frac{T}{T_c}\\right) } \\approx - \\sqrt{3 \\frac{T_c - T}{T_c}}$, even if not fully simplified. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer sketches $\\bar{s} = 0$ as the only solution for $T > T_c$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer shows that the two non-trivial solutions $\\bar{s}$ emerge vertically at $T = T_c$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer sketches $\\bar{s} = 0$ as a valid solution also for $T < T_c$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer shows that the two non-trivial solutions monotonically increase in magnitude and approach $\\pm 1$ as $T \\to 0$ (award 0.1 pt each). Otherwise, award 0 pt.", + "Award 0.1 pt if the answer states that either of the non-trivial solutions has zero slope as $T \\to 0$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\bar{s} = 0$}", + "\\boxed{$\\bar{s} = \\sqrt{3 \\cdot \\frac{T_c - T}{T_c}}$}", + "\\boxed{$\\bar{s} = -\\sqrt{3 \\cdot \\frac{T_c - T}{T_c}}$}" + ], + "answer_type": [ + "Numerical Value", + "Expression", + "Expression" + ], + "unit": [ + null, + null, + null + ], + "points": [ + 0.2, + 0.4, + 0.4 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_2_c_1.png" + ] + }, + { + "id": "APhO_2025_2_C_7", + "context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part C: Phase transitions in spin chains] \n\nNext we consider the same chain made of $N$ spins from Part B, except the spin vectors are now restricted to point either up or down along the $z$-axis, so that the spin component along $z$ can be written as $S_{i, z} = s_{i} S$, where $s_{i} = \\pm 1$, see Figure C.1. In addition to the nearest neighbor interactions, we could have an external magnetic field pointing along the $z$-axis so that the total energy of the system is given by \n$E = -\\tilde{J} \\sum_i s_i s_{i+1} - h \\sum_i s_i$ \nWe assume $\\tilde{J} \\geq 0$, and $h$ is a constant dependent on the magnetic field. The spin system is at equilibrium with a heat bath at temperature $T$. Ignore the edges of the chain. \n\n[figure1] \nFigure C.1. \n\n(C.1) Assume first that $\\tilde{J} = 0$, what is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_{\\uparrow}$ to being anti-aligned to the magnetic field $p_{\\downarrow}$? Express $p_{\\uparrow} / p_{\\downarrow}$ in terms of $h, T$ and fundamental constants. \n\n(C.2) Find the average polarization of the system $\\bar{s} \\equiv \\frac{1}{N} \\sum_{i} s_{i}$ for $N \\gg 1$ in terms of $h, T$ and fundamental constants. If the magnetic field $h$ can range from $-h_{0}$ to $h_{0}$, make a sketch of $\\bar{s}$ as a function of $h$ for the cases $h_{o} \\gg k_{B} T$, $h_{o} \\approx k_{B} T$ and $h_{o} \\ll k_{B} T$. \n\nIn the remaining questions, we turn off the magnetic field, so $h = 0$, and set $\\tilde{J} > 0$. \n\n(C.3) What is the energy $E_{g}$ of the ground state (the lowest energy state)? Express your answer in terms of $\\tilde{J}$ and $N$. \n\nInstead of considering the interactions between each spin and its neighbors, we assume that each spin sees an average polarization $\\bar{s}$ from its nearest-neighbors. \n\n(C.4) Approximate the energy of the system as a sum over all spins $E = -\\tilde{J}_{\\text{eff}} \\sum_i s_i$ and express $\\tilde{J}_{\\text{eff}}$ in terms of $\\tilde{J}$ and $\\bar{s}$. \n\n(C.5) Using your result from C.2, find an equation that the average polarization $\\bar{s}$ must satisfy. The number of solutions to this equation depends on $T$. Find the critical temperature $T_{c}$ at which the number of solutions changes. Express your answer in terms of $\\tilde{J}$ and fundamental constants. \n\n(C.6) Find all possible values of $\\bar{s}$ when $T < T_{c}$ and $T_{c} - T \\ll T_{c}$. Express your answers in terms of $T$ and $T_{c}$. Sketch all possible values of $\\bar{s}$ for the temperature $T$ in the range $0 \\leq T \\leq 2 T_{c}$.", + "question": "Write the option letter (A or B) in your answer. \n(1) What magnetic phase of matter does $T > T_{c}$ correspond to? (A) Paramagnetic (B) Ferromagnetic. \n(2) What magnetic phase of matter does $T < T_{c}$ correspond to? (A) Paramagnetic (B) Ferromagnetic.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly classifies the phase for $T > T_c$ as paramagnetic (option A). Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly classifies the phase for $T < T_c$ as ferromagnetic (option B). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{A}", + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice", + "Multiple Choice" + ], + "unit": [ + null, + null + ], + "points": [ + 0.1, + 0.1 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_2_c_1.png" + ] + }, + { + "id": "APhO_2025_3_A_1", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part A: Surface Temperature of the Earth] \n\nIn this section, we study the effect of the atmosphere on the Earth surface's temperature. Assume that Earth and its atmosphere have an albedo $a = 0.3$ for solar radiation, which is the reflected fraction of the total incident radiation. You may use this value in all parts of this problem. In addition, assume the Earth radiates as a black body.", + "question": "Express the average net solar power received by the Earth and atmosphere system $P_{0}$ in terms of $F_{s}$, $a$ and $R_{E}$, the radius of the Earth.", + "marking": [ + [ + "Award 0.1 pt if the answer identifies the correct effective cross-sectional area as $A = \\pi R_E^2$, where $R_E$ is the Earth's radius. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct final absorbed power expression $P_0 = (1 - a) \\pi R_E^2 F_s$, where $a$ is the albedo and $F_s$ is the solar flux. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$P_0 = (1 - a) \\pi R_E^2 F_s$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.2 + ], + "modality": "text-only", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [] + }, + { + "id": "APhO_2025_3_A_2", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part A: Surface Temperature of the Earth] \n\nIn this section, we study the effect of the atmosphere on the Earth surface's temperature. Assume that Earth and its atmosphere have an albedo $a = 0.3$ for solar radiation, which is the reflected fraction of the total incident radiation. You may use this value in all parts of this problem. In addition, assume the Earth radiates as a black body. \n\n(A.1) Express the average net solar power received by the Earth and atmosphere system $P_{0}$ in terms of $F_{s}$, $a$ and $R_{E}$, the radius of the Earth.", + "question": "Estimate the temperature of the Earth's surface $T_{g 0}$ assuming that it is at a steady state. Ignore the atmosphere. Express your answer in $K$.", + "marking": [ + [ + "Award 0.1 pt if the answer sets up the energy balance condition $P_{bd} = P_0$, where $P_{bd}$ is the blackbody radiation power and $P_0$ is the absorbed solar power. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer writes the correct explicit blackbody radiation formula $P_{bd} = \\sigma A T^4$ with $A = 4 \\pi R_E^2$, where $\\sigma$ is the Stefan–Boltzmann constant, $T$ is the temperature, and $R_E$ is the Earth's radius. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer obtains the correct numerical value for the temperature, $T_{g0} \\approx 255 \\text{K} \\approx -18^{\\circ}\\text{C}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{255}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "K" + ], + "points": [ + 0.3 + ], + "modality": "text-only", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [] + }, + { + "id": "APhO_2025_3_A_3", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part A: Surface Temperature of the Earth] \n\nIn this section, we study the effect of the atmosphere on the Earth surface's temperature. Assume that Earth and its atmosphere have an albedo $a = 0.3$ for solar radiation, which is the reflected fraction of the total incident radiation. You may use this value in all parts of this problem. In addition, assume the Earth radiates as a black body. \n\n(A.1) Express the average net solar power received by the Earth and atmosphere system $P_{0}$ in terms of $F_{s}$, $a$ and $R_{E}$, the radius of the Earth. \n\n(A.2) Estimate the temperature of the Earth's surface $T_{g 0}$ assuming that it is at a steady state. Ignore the atmosphere. \n\nYour answer for (A.2) should be lower than what you would expect. We now consider adding a thin atmospheric layer at temperature $T_{a}$, see Figure A.1. The atmospheric layer transmits a net fraction $t_{\\mathrm{sw}}$ of the incident solar radiation and a net fraction $t_{\\text{lw}}$ of the Earth's thermal radiation. Otherwise, you may treat the atmosphere as a black body. \n\n[figure1] \nFigure A.1", + "question": "Assuming the system is in a steady state, calculate $T_{g}$, the temperature of the ground. Use $t_{\\mathrm{sw}} = 0.9$ and $t_{\\mathrm{lw}} = 0.2$. Express your answer in $K$.", + "marking": [ + [ + "Award 0.1 pt if the answer includes a correct statement of radiation balance in the region outside the atmosphere, e.g., $t_{lw} P_E + P_{atmo} = P_0$, where $t_{lw}$ is the longwave transmission coefficient, $P_E$ is the Earth's emitted power, $P_{atmo}$ is the atmospheric radiation, and $P_0$ is the absorbed solar power. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer includes a correct statement of radiation balance in the region between the Earth's surface and the atmosphere, e.g., $P_E = P_{atmo} + t_{sw} P_0$, where $t_{sw}$ is the shortwave transmission coefficient. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer uses $t_{sw}$ correctly in the equation $P_E = P_{atmo} + t_{sw} P_0$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer uses $t_{lw}$ correctly in the equation $t_{lw} P_E + P_{atmo} = P_0$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct numerical result for the ground temperature, $T_g = \\left( \\frac{1+t_{sw}}{1+t_{lw}} \\right)^{1/4} T_{g0} \\approx 286 \\text{K} \\approx 13^{\\circ}\\text{C}$, where $T_{g0}$ is the temperature of the Earth's surface. Partial points: award 0.1 pt if only the analytic form is given in the answer. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{286}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "K" + ], + "points": [ + 0.7 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_3_a_1.png" + ] + }, + { + "id": "APhO_2025_3_B_1", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part B: The absorption spectrum of atmospheric gases] \n\nThe infrared radiation emitted by Earth has low energy, incapable of exciting electrons within the molecules, but it has the ability to excite the vibrational and rotational modes of the molecules.", + "question": "Consider a simple diatomic molecule modeled as two point masses $m_{A}$ and $m_{B}$ connected by a spring with spring constant $k$. What is the angular frequency of vibrations $\\omega_{d}$?", + "marking": [ + [ + "Award 0.1 pt if the answer writes the correct equation of motion for particle A: $\\frac{d^2 x_A}{dt^2} = +\\frac{k}{m_A}(\\ell - \\ell_0)$, where $x_A$ is the position of particle A, $m_A$ is its mass, $k$ is the spring constant, $\\ell$ is the instantaneous spring length, and $\\ell_0$ is the natural spring length. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer writes the correct equation of motion for particle B: $\\frac{d^2 x_B}{dt^2} = -\\frac{k}{m_B}(\\ell - \\ell_0)$, where $x_B$ is the position of particle B and $m_B$ is its mass. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly derives the equation of motion for the relative coordinate $\\ell = x_B - x_A$, namely $\\frac{d^2 \\ell}{dt^2} = -k\\left( \\frac{1}{m_A} + \\frac{1}{m_B} \\right)(\\ell - \\ell_0)$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer provides the correct final angular frequency of oscillation: $\\omega_d = \\sqrt{\\frac{k}{\\mu}} = \\sqrt{ k \\frac{m_A + m_B}{m_A m_B}}$, where $\\mu = \\frac{m_A m_B}{m_A + m_B}$ is the reduced mass. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\omega_d = \\sqrt{k \\frac{m_A + m_B}{m_A m_B}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.5 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "APhO_2025", + "image_question": [] + }, + { + "id": "APhO_2025_3_B_2", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part B: The absorption spectrum of atmospheric gases] \n\nThe infrared radiation emitted by Earth has low energy, incapable of exciting electrons within the molecules, but it has the ability to excite the vibrational and rotational modes of the molecules. \n\n(B.1) Consider a simple diatomic molecule modeled as two point masses $m_{A}$ and $m_{B}$ connected by a spring with spring constant $k$. What is the angular frequency of vibrations $\\omega_{d}$?", + "question": "Quantum mechanics dictates that vibrational excitations due to absorbing a photon can only raise the quantum energy level by one. What is the energy of the photon $E_{p}$ that can excite the vibration in B.1? Neglect recoil effects.", + "marking": [ + [ + "Award 0.2 pt if the answer gives the correct photon energy as $E = \\hbar \\omega_d$ (where $\\hbar$ is the reduced Planck constant and $\\omega_d$ is the angular frequency). Partial points: only award 0.1 pt if $h$ is used instead of $\\hbar$; no other numerical factors receive credit." + ] + ], + "answer": [ + "\\boxed{$E_p = \\hbar \\omega_d$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.2 + ], + "modality": "text-only", + "field": "Modern Physics", + "source": "APhO_2025", + "image_question": [] + }, + { + "id": "APhO_2025_3_B_3", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part B: The absorption spectrum of atmospheric gases] \n\nThe infrared radiation emitted by Earth has low energy, incapable of exciting electrons within the molecules, but it has the ability to excite the vibrational and rotational modes of the molecules. \n\n(B.1) Consider a simple diatomic molecule modeled as two point masses $m_{A}$ and $m_{B}$ connected by a spring with spring constant $k$. What is the angular frequency of vibrations $\\omega_{d}$? \n\n(B.2) Quantum mechanics dictates that vibrational excitations due to absorbing a photon can only raise the quantum energy level by one. What is the energy of the photon $E_{p}$ that can excite the vibration in B.1? Neglect recoil effects. \n\nQuantum mechanics forbids the vibrational modes of symmetric diatomic molecules, such as nitrogen and oxygen (the most abundant gasses in the Earth's atmosphere) to be excited by light. This explains why $N_{2}$ and $O_{2}$ do not contribute to the green house effect. In general, the absorption of light by molecules is governed by the allowed energy transitions in them. However, the energy of the light absorbed does not have to exactly match the energy gap in the molecule. Suppose that a molecule at rest has a spectral line (an allowed transition) at frequency $f_{0}$.", + "question": "What is the shift in the spectral line $f - f_{0}$ if the molecule is moving with velocity $v$ towards the emitter such that $|v| \\ll c$, where $c$ is the speed of light.", + "marking": [ + [ + "Award 0.1 pt if the answer writes down an expression for the Doppler effect, even if it is incorrect. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct frequency shift as $f - f_0 = \\frac{v}{c} f_0$, where $f$ is the observed frequency, $f_0$ is the source frequency, $v$ is the velocity of the source, and $c$ is the speed of light. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$f - f_0 = \\frac{v}{c} f_0$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.2 + ], + "modality": "text-only", + "field": "Optics", + "source": "APhO_2025", + "image_question": [] + }, + { + "id": "APhO_2025_3_B_4", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part B: The absorption spectrum of atmospheric gases] \n\nThe infrared radiation emitted by Earth has low energy, incapable of exciting electrons within the molecules, but it has the ability to excite the vibrational and rotational modes of the molecules. \n\n(B.1) Consider a simple diatomic molecule modeled as two point masses $m_{A}$ and $m_{B}$ connected by a spring with spring constant $k$. What is the angular frequency of vibrations $\\omega_{d}$? \n\n(B.2) Quantum mechanics dictates that vibrational excitations due to absorbing a photon can only raise the quantum energy level by one. What is the energy of the photon $E_{p}$ that can excite the vibration in B.1? Neglect recoil effects. \n\nQuantum mechanics forbids the vibrational modes of symmetric diatomic molecules, such as nitrogen and oxygen (the most abundant gasses in the Earth's atmosphere) to be excited by light. This explains why $N_{2}$ and $O_{2}$ do not contribute to the green house effect. In general, the absorption of light by molecules is governed by the allowed energy transitions in them. However, the energy of the light absorbed does not have to exactly match the energy gap in the molecule. Suppose that a molecule at rest has a spectral line (an allowed transition) at frequency $f_{0}$. \n\n(B.3) What is the shift in the spectral line $f - f_{0}$ if the molecule is moving with velocity $v$ towards the emitter such that $|v| \\ll c$, where $c$ is the speed of light. \n\nFor a gas at temperature $T$, the velocity of its molecules is distributed according to Maxwell's distribution. For a molecule of mass $m$, the probability to find a molecule's velocity along one dimension to be between $v$ and $v + dv$ is $p_{1}(v) d v$, where $p_{1}(v)$ is a probability distribution function given by \n$p_{1}(v) = C \\exp\\left( -\\frac{mv^{2}}{2k_{B}T} \\right)$ \n$C$ is a normalization constant ensuring the probabilities add up to one, and $k_{B}$ is the Boltzmann constant.", + "question": "Find the normalization constant $C$, assuming that the velocity $v$ could range from $-\\infty$ to $\\infty$.", + "marking": [ + [ + "Award 0.1 pt if the answer writes down the normalization condition $\\int_{-\\infty}^{\\infty} p(v) dv = 1$, even if it is incorrectly applied from 0 to $\\infty$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct result for the normalization constant as $C = \\sqrt{ \\frac{m}{2 \\pi k_B T} }$, where $m$ is the particle mass, $k_B$ is the Boltzmann constant, and $T$ is the temperature. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$C = \\sqrt{\\frac{m}{2 \\pi k_B T}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.2 + ], + "modality": "text-only", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [] + }, + { + "id": "APhO_2025_3_B_5", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part B: The absorption spectrum of atmospheric gases] \n\nThe infrared radiation emitted by Earth has low energy, incapable of exciting electrons within the molecules, but it has the ability to excite the vibrational and rotational modes of the molecules. \n\n(B.1) Consider a simple diatomic molecule modeled as two point masses $m_{A}$ and $m_{B}$ connected by a spring with spring constant $k$. What is the angular frequency of vibrations $\\omega_{d}$? \n\n(B.2) Quantum mechanics dictates that vibrational excitations due to absorbing a photon can only raise the quantum energy level by one. What is the energy of the photon $E_{p}$ that can excite the vibration in B.1? Neglect recoil effects. \n\nQuantum mechanics forbids the vibrational modes of symmetric diatomic molecules, such as nitrogen and oxygen (the most abundant gasses in the Earth's atmosphere) to be excited by light. This explains why $N_{2}$ and $O_{2}$ do not contribute to the green house effect. In general, the absorption of light by molecules is governed by the allowed energy transitions in them. However, the energy of the light absorbed does not have to exactly match the energy gap in the molecule. Suppose that a molecule at rest has a spectral line (an allowed transition) at frequency $f_{0}$. \n\n(B.3) What is the shift in the spectral line $f - f_{0}$ if the molecule is moving with velocity $v$ towards the emitter such that $|v| \\ll c$, where $c$ is the speed of light. \n\nFor a gas at temperature $T$, the velocity of its molecules is distributed according to Maxwell's distribution. For a molecule of mass $m$, the probability to find a molecule's velocity along one dimension to be between $v$ and $v + dv$ is $p_{1}(v) d v$, where $p_{1}(v)$ is a probability distribution function given by \n$p_{1}(v) = C \\exp\\left( -\\frac{mv^{2}}{2k_{B}T} \\right)$ \n$C$ is a normalization constant ensuring the probabilities add up to one, and $k_{B}$ is the Boltzmann constant. \n\n(B.4) Find the normalization constant $C$, assuming that the velocity $v$ could range from $-\\infty$ to $\\infty$.", + "question": "Find the probability distribution function $p_{2}(f)$ to find a molecule with a spectral line $f_{0}$ shifted to $f$ due to thermal motion, up to a normalization factor, in terms of $f, f_{0}, T, m$ and fundamental constants. Use $C$ to represent the normalization factor.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly replaces the velocity $v$ in the distribution using the Doppler effect relation $v = \\frac{f - f_0}{f_0} c$, where $f$ is the observed frequency, $f_0$ is the source frequency, and $c$ is the speed of light. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer writes the correct exponential dependence for the probability distribution as $p(f) \\propto \\exp \\left[ - \\frac{m c^2}{2 k_B T} \\left( \\frac{f - f_0}{f_0} \\right)^2 \\right]$, where $m$ is the particle mass, $k_B$ is the Boltzmann constant, and $T$ is the temperature. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$p_2(f) = C \\exp\\left[-\\frac{mc^2}{2k_B T} \\left(\\frac{f - f_0}{f_0}\\right)^2\\right]$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.3 + ], + "modality": "text-only", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [] + }, + { + "id": "APhO_2025_3_B_6", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part B: The absorption spectrum of atmospheric gases] \n\nThe infrared radiation emitted by Earth has low energy, incapable of exciting electrons within the molecules, but it has the ability to excite the vibrational and rotational modes of the molecules. \n\n(B.1) Consider a simple diatomic molecule modeled as two point masses $m_{A}$ and $m_{B}$ connected by a spring with spring constant $k$. What is the angular frequency of vibrations $\\omega_{d}$? \n\n(B.2) Quantum mechanics dictates that vibrational excitations due to absorbing a photon can only raise the quantum energy level by one. What is the energy of the photon $E_{p}$ that can excite the vibration in B.1? Neglect recoil effects. \n\nQuantum mechanics forbids the vibrational modes of symmetric diatomic molecules, such as nitrogen and oxygen (the most abundant gasses in the Earth's atmosphere) to be excited by light. This explains why $N_{2}$ and $O_{2}$ do not contribute to the green house effect. In general, the absorption of light by molecules is governed by the allowed energy transitions in them. However, the energy of the light absorbed does not have to exactly match the energy gap in the molecule. Suppose that a molecule at rest has a spectral line (an allowed transition) at frequency $f_{0}$. \n\n(B.3) What is the shift in the spectral line $f - f_{0}$ if the molecule is moving with velocity $v$ towards the emitter such that $|v| \\ll c$, where $c$ is the speed of light. \n\nFor a gas at temperature $T$, the velocity of its molecules is distributed according to Maxwell's distribution. For a molecule of mass $m$, the probability to find a molecule's velocity along one dimension to be between $v$ and $v + dv$ is $p_{1}(v) d v$, where $p_{1}(v)$ is a probability distribution function given by \n$p_{1}(v) = C \\exp\\left( -\\frac{mv^{2}}{2k_{B}T} \\right)$ \n$C$ is a normalization constant ensuring the probabilities add up to one, and $k_{B}$ is the Boltzmann constant. \n\n(B.4) Find the normalization constant $C$, assuming that the velocity $v$ could range from $-\\infty$ to $\\infty$. \n\n(B.5) Find the probability distribution function $p_{2}(f)$ to find a molecule with a spectral line $f_{0}$ shifted to $f$ due to thermal motion, up to a normalization factor, in terms of $f, f_{0}, T, m$ and fundamental constants. Use $C$ to represent the normalization factor.", + "question": "Sketch $p_{2}(f)$ as a function of $f - f_{0}$, and determine the shift $f^{\\star} - f_{0}$ at which $p_{2}(f^{\\star})$ is a fraction $1 / e$ of its peak value, where $e$ is the natural number.", + "marking": [ + [ + "Award 0.1 pt if the answer states that the probability distribution $p(f)$ has a single peak at zero frequency shift ($f - f_0 = 0$). Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly identifies that the distribution is symmetric about $f - f_0 = 0$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer states that the probability distribution decays to zero as $f - f_0 \\to \\pm \\infty$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct condition for the $1/e$ point of the distribution as $f^* - f_0 = f_0 \\sqrt{ \\frac{2 k_B T}{m c^2} }$, where $m$ is the particle mass, $k_B$ is the Boltzmann constant, $T$ is the temperature, and $c$ is the speed of light. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$f^{\\star} - f_0 = f_0 \\sqrt{\\frac{2k_B T}{mc^2}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.4 + ], + "modality": "text-only", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [] + }, + { + "id": "APhO_2025_3_C_1", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part C: Stability of air in the atmosphere] \n\nConsider a small cylindrical mass of air at height $z$ above the ground. The pressure and mass density of air at that height are $p(z)$ and $\\rho(z)$, respectively, see Figure C.1. Assume a uniform downward gravitational field $g$ and that the pressure on the Earth's surface is $p_{o}$. \n\n[figure1] \nFigure C.1", + "question": "Assuming that the small air mass is at hydrostatic equilibrium, derive an expression of the rate of change of pressure with respect to height, $d p / d z$ in terms of $g$ and $\\rho(z)$.", + "marking": [ + [ + "Award 0.1 pt if the answer states that the sum of forces equals zero in hydrostatic equilibrium. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly identifies the pressure force contributions above and below the thin layer, i.e. $p(z)S = p(z + \\mathrm{d}z)S + \\rho(z) g S \\mathrm{d}z$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct final hydrostatic equilibrium equation $\\frac{dp}{dz} = - \\rho(z) g$, where $\\rho(z)$ is the density and $g$ is the gravitational acceleration. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{dp}{dz} = -\\rho(z) g$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.3 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_3_c_1.png" + ] + }, + { + "id": "APhO_2025_3_C_2", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part C: Stability of air in the atmosphere] \n\nConsider a small cylindrical mass of air at height $z$ above the ground. The pressure and mass density of air at that height are $p(z)$ and $\\rho(z)$, respectively, see Figure C.1. Assume a uniform downward gravitational field $g$ and that the pressure on the Earth's surface is $p_{o}$. \n\n[figure1] \nFigure C.1 \n\n(C.1) Assuming that the small air mass is at hydrostatic equilibrium, derive an expression of the rate of change of pressure with respect to height, $d p / d z$ in terms of $g$ and $\\rho(z)$.", + "question": "Express $d p / d z$ in terms of $\\mu_{\\text{air}}, g, p(z)$ and $T(z)$, the temperature at height $z$ and fundamental constants.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly uses the ideal gas law $pV = nRT$ and rewrites it as $\\rho(z) = \\frac{p(z) \\mu_{air}}{R T(z)}$, where $\\mu_{air}$ is the molar mass of air and $R$ is the gas constant. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct final hydrostatic equilibrium expression $\\frac{dp}{dz} = - \\frac{\\mu_{air} p(z)}{R T(z)} g$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{dp}{dz} = -\\frac{\\mu_{\\text{air}} p(z)}{R T(z)} g$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.2 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_3_c_1.png" + ] + }, + { + "id": "APhO_2025_3_C_3", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part C: Stability of air in the atmosphere] \n\nConsider a small cylindrical mass of air at height $z$ above the ground. The pressure and mass density of air at that height are $p(z)$ and $\\rho(z)$, respectively, see Figure C.1. Assume a uniform downward gravitational field $g$ and that the pressure on the Earth's surface is $p_{o}$. \n\n[figure1] \nFigure C.1 \n\n(C.1) Assuming that the small air mass is at hydrostatic equilibrium, derive an expression of the rate of change of pressure with respect to height, $d p / d z$ in terms of $g$ and $\\rho(z)$. \n\n(C.2) Express $d p / d z$ in terms of $\\mu_{\\text{air}}, g, p(z)$ and $T(z)$, the temperature at height $z$ and fundamental constants.", + "question": "Assuming an isothermal atmosphere, $T(z) = T$, find an expression for $p(z)$ in terms of $z, \\mu_{\\text{air}}, g, p_{o}, T$ and fundamental constants.", + "marking": [ + [ + "Award 0.1 pt if the answer recognizes and correctly rewrites the equation as a separable differential equation $\\frac{dp}{p} = - \\frac{\\mu_{air}}{R T} g dz$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct final solution $p(z) = p_0 \\exp \\left( - \\frac{\\mu_{air}}{R T} g z \\right)$, where $p_0$ is the pressure at $z=0$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$p(z) = p_0 \\exp\\left(-\\frac{\\mu_{\\text{air}}}{RT} gz\\right)$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.2 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_3_c_1.png" + ] + }, + { + "id": "APhO_2025_3_C_4", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part C: Stability of air in the atmosphere] \n\nConsider a small cylindrical mass of air at height $z$ above the ground. The pressure and mass density of air at that height are $p(z)$ and $\\rho(z)$, respectively, see Figure C.1. Assume a uniform downward gravitational field $g$ and that the pressure on the Earth's surface is $p_{o}$. \n\n[figure1] \nFigure C.1 \n\n(C.1) Assuming that the small air mass is at hydrostatic equilibrium, derive an expression of the rate of change of pressure with respect to height, $d p / d z$ in terms of $g$ and $\\rho(z)$. \n\n(C.2) Express $d p / d z$ in terms of $\\mu_{\\text{air}}, g, p(z)$ and $T(z)$, the temperature at height $z$ and fundamental constants. \n\n(C.3) Assuming an isothermal atmosphere, $T(z) = T$, find an expression for $p(z)$ in terms of $z, \\mu_{\\text{air}}, g, p_{o}, T$ and fundamental constants. \n\nIn a real atmosphere, the temperature is not constant but changes with height. The rate of decrease of temperature with height $\\Gamma(z) = -d T / d z$ is called the lapse rate. Consider a small mass of air rising adiabatically in the atmosphere such that it remains at mechanical equilibrium with its surrounding.", + "question": "For the adiabatically rising air mass, find the adiabatic lapse rate $\\Gamma_{a}$ in terms of $c_{p}$, the molar specific heat at constant pressure, $\\mu_{\\text{air}}$ and $g$.", + "marking": [ + [ + "Award 0.1 pt if the answer writes the adiabatic relation in any correct form for an ideal gas, e.g., $p V^{\\gamma} = \\text{const.}$ or equivalently $p^{1-\\gamma} T^{\\gamma} = \\text{const.}$ where $\\gamma = c_p/c_v$, $c_p$ and $c_v$ are the molar specific heats at constant pressure and volume. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer differentiates the adiabatic relation with respect to height $z$ to relate temperature and pressure gradients as $\\frac{dT}{dz} = -\\frac{1-\\gamma}{\\gamma} \\frac{T(z)}{p(z)} \\frac{dp}{dz}$, where $T(z)$ is temperature and $p(z)$ is pressure. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer obtains the correct adiabatic lapse rate $\\Gamma_a = \\frac{\\mu_{\\text{air}}}{c_p} g$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\Gamma_a = \\frac{\\mu_{\\text{air}}}{c_p} g$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.6 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_3_c_1.png" + ] + }, + { + "id": "APhO_2025_3_C_5", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part C: Stability of air in the atmosphere] \n\nConsider a small cylindrical mass of air at height $z$ above the ground. The pressure and mass density of air at that height are $p(z)$ and $\\rho(z)$, respectively, see Figure C.1. Assume a uniform downward gravitational field $g$ and that the pressure on the Earth's surface is $p_{o}$. \n\n[figure1] \nFigure C.1 \n\n(C.1) Assuming that the small air mass is at hydrostatic equilibrium, derive an expression of the rate of change of pressure with respect to height, $d p / d z$ in terms of $g$ and $\\rho(z)$. \n\n(C.2) Express $d p / d z$ in terms of $\\mu_{\\text{air}}, g, p(z)$ and $T(z)$, the temperature at height $z$ and fundamental constants. \n\n(C.3) Assuming an isothermal atmosphere, $T(z) = T$, find an expression for $p(z)$ in terms of $z, \\mu_{\\text{air}}, g, p_{o}, T$ and fundamental constants. \n\nIn a real atmosphere, the temperature is not constant but changes with height. The rate of decrease of temperature with height $\\Gamma(z) = -d T / d z$ is called the lapse rate. Consider a small mass of air rising adiabatically in the atmosphere such that it remains at mechanical equilibrium with its surrounding. \n\n(C.4) For the adiabatically rising air mass, find the adiabatic lapse rate $\\Gamma_{a}$ in terms of $c_{p}$, the molar specific heat at constant pressure, $\\mu_{\\text{air}}$ and $g$. \n\nTo analyze the stability of an atmosphere, we imagine starting from an equilibrium state, and then perturbing a small mass of air and analyze its response. Consider a small air mass initially in equilibrium with the surrounding air at height $z$ and temperature $T$. It is then adiabatically displaced vertically by a displacement $\\delta z_{0}$. Assume that throughout the motion, the air parcel always has the same pressure as the surrounding air at the same height. The surrounding atmosphere is unaltered and has a different lapse rate $\\Gamma$. Neglect viscosity.", + "question": "(1) Find the equation of motion for $\\delta z$, the instantaneous vertical displacement. \n(2) Under what condition is the equilibrium at $z$ stable? \n(3) What is the angular frequency $\\omega$ of small oscillation? Express your answers in terms of $T, \\Gamma, g, \\mu_{\\text{air}}$ and $c_{p}$.", + "marking": [ + [ + "Award 0.2 pt if the answer obtains the gravitational force with parcel density correctly as $\\delta m g = \\rho_p \\delta V g$, where $\\delta m$ is the mass of the parcel, $\\rho_p$ is the density of the parcel, and $\\delta V$ is its volume. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer obtains the buoyancy force with air density correctly as $\\rho_a(z) g \\delta V$, where $\\rho_a$ is the surrounding air density. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer writes the correct equation of motion as $\\delta m \\frac{d^2 z}{dt^2} = \\rho_a(z) g \\delta V - \\delta m g$, and after substitution simplifies to $\\frac{d^2 z}{dt^2} = \\frac{\\rho_a(z+\\delta z) - \\rho_p(z+\\delta z)}{\\rho_p(z+\\delta z)} g$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly relates density to inverse temperature as $\\rho \\propto \\frac{1}{T}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer uses the appropriate approximation $T(z+\\delta z) = T(z) + \\Gamma \\delta z$, and simplifies to $\\frac{d^2 z}{dt^2} = \\frac{T(z) + \\Gamma \\delta z - T(z) - \\Gamma_a \\delta z}{T(z) + \\Gamma_a \\delta z} g$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer provides the correct stability requirement that motion is stable whenever $\\Gamma_a > \\Gamma$, where $\\Gamma_a = \\mu_{air} g / c_p$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer obtains the correct angular frequency of small oscillation as $\\omega = \\sqrt{\\frac{\\Gamma_a - \\Gamma}{T} g} = \\sqrt{\\frac{\\mu_{air} g / c_p - \\Gamma}{T} g}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{d^2 z}{d t^2} = \\frac{\\Gamma - \\mu_{\\text{air}} g/c_p}{T} g \\delta_z$}", + "\\boxed{$\\mu_{\\text{air}} g/c_p > \\Gamma$}", + "\\boxed{$\\omega = \\sqrt{\\frac{\\mu_{\\text{air}} g/c_p - \\Gamma}{T} g}$}" + ], + "answer_type": [ + "Equation", + "Inequality", + "Expression" + ], + "unit": [ + null, + null, + null + ], + "points": [ + 1.1, + 0.1, + 0.2 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_3_c_1.png" + ] + }, + { + "id": "APhO_2025_3_D_1", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part D: Moisture] \n\nEven though water constitutes a small portion of the atmosphere, it has a significant role in climate science. It is responsible for precipitation, and it is the most significant greenhouse gas. The phase of water depends on what temperature and pressure the water system is at, depicted on a $p-T$ phase diagram, see Figure D.1. When the pressure and temperature lie on the coexistence curve, both liquid and vapor water can be present in the system. The slope of the coexistence curve is given by the ClausiusClapeyron equation: \n$\\frac{d p_s}{d T} = \\frac{\\Delta S}{\\Delta V}$ \nwhere $p_{s}$ is the saturation pressure, the pressure at the phase transition, $\\Delta S$ and $\\Delta V$ are the changes in entropy and volume across the phase transitions, respectively. Treat water vapor as an ideal gas. \n\n[figure1] \nFigure D.1", + "question": "Express $d p_{s} / d T$ for the water liquid-vapor coexistence curve in terms of the water latent heat of evaporation $L, \\mu_{\\text{H_2O}}, p_{s}, T$ and fundamental constants.", + "marking": [ + [ + "Award 0.2 pt if the answer obtains the correct entropy change as $\\Delta S = \\frac{L m}{T}$, where $L$ is the latent heat of evaporation, $m$ is the mass of liquid water, and $T$ is the temperature. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly states that $V_{vapor} \\gg V_{liquid}$ and therefore approximates $\\Delta V \\approx V_{vapor}$, with $V_{vapor} = \\frac{nRT}{p_s(T)}$, where $n$ is the number of moles, $R$ is the gas constant, and $p_s(T)$ is the saturation vapor pressure. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer obtains the correct final relation $\\frac{dp_s}{dT} = \\frac{\\mu_{H_2O} L p_s}{R T^2}$, where $\\mu_{H_2O}$ is the molar mass of water. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{d p_s}{d T} = \\frac{\\mu_{\\text{H_2O}} L p_s}{R T^2}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.5 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_3_d_1.png" + ] + }, + { + "id": "APhO_2025_3_D_2", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part D: Moisture] \n\nEven though water constitutes a small portion of the atmosphere, it has a significant role in climate science. It is responsible for precipitation, and it is the most significant greenhouse gas. The phase of water depends on what temperature and pressure the water system is at, depicted on a $p-T$ phase diagram, see Figure D.1. When the pressure and temperature lie on the coexistence curve, both liquid and vapor water can be present in the system. The slope of the coexistence curve is given by the ClausiusClapeyron equation: \n$\\frac{d p_s}{d T} = \\frac{\\Delta S}{\\Delta V}$ \nwhere $p_{s}$ is the saturation pressure, the pressure at the phase transition, $\\Delta S$ and $\\Delta V$ are the changes in entropy and volume across the phase transitions, respectively. Treat water vapor as an ideal gas. \n\n[figure1] \nFigure D.1 \n\n(D.1) Express $d p_{s} / d T$ for the water liquid-vapor coexistence curve in terms of the water latent heat of evaporation $L, \\mu_{\\text{H_2O}}, p_{s}, T$ and fundamental constants.", + "question": "If for some reference temperature $T_{o}$, $p_{s} = p_{s o}$, find an expression for $p_{s}(T)$ in terms of $p_{s o}, \\mu_{\\text{H_2O}}, L, T, T_{o}$ and fundamental constants.", + "marking": [ + [ + "Award 0.1 pt if the answer recognizes a separable differential equation and obtains $\\ln [\\frac{p_s(T)}{p_{so}}] = -\\frac{\\mu_{\\text{H_2O}} L}{R} (\\frac{1}{T} - \\frac{1}{T_o})$, where $p_s$ is the saturation vapor pressure, $\\mu_{\\text{H_2O}}$ is the molar mass of water, $L$ is latent heat, $R$ is the gas constant, and $T$ is the temperature. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer obtains the correct final solution $p_s(T) = p_{so} \\exp \\left[- \\frac{\\mu_{\\text{H_2O}} L}{R} \\left( \\frac{1}{T} - \\frac{1}{T_0} \\right) \\right]$, where $p_{so}$ is the reference saturation vapor pressure at $T_0$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$p_s(T) = p_{s0} \\exp\\left[-\\frac{\\mu_{\\text{H_2O}} L}{R} \\left(\\frac{1}{T} - \\frac{1}{T_0}\\right)\\right]$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.2 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_3_d_1.png" + ] + }, + { + "id": "APhO_2025_3_D_3", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part D: Moisture] \n\nEven though water constitutes a small portion of the atmosphere, it has a significant role in climate science. It is responsible for precipitation, and it is the most significant greenhouse gas. The phase of water depends on what temperature and pressure the water system is at, depicted on a $p-T$ phase diagram, see Figure D.1. When the pressure and temperature lie on the coexistence curve, both liquid and vapor water can be present in the system. The slope of the coexistence curve is given by the ClausiusClapeyron equation: \n$\\frac{d p_s}{d T} = \\frac{\\Delta S}{\\Delta V}$ \nwhere $p_{s}$ is the saturation pressure, the pressure at the phase transition, $\\Delta S$ and $\\Delta V$ are the changes in entropy and volume across the phase transitions, respectively. Treat water vapor as an ideal gas. \n\n[figure1] \nFigure D.1 \n\n(D.1) Express $d p_{s} / d T$ for the water liquid-vapor coexistence curve in terms of the water latent heat of evaporation $L, \\mu_{\\text{H_2O}}, p_{s}, T$ and fundamental constants. \n\n(D.2) If for some reference temperature $T_{o}$, $p_{s} = p_{s o}$, find an expression for $p_{s}(T)$ in terms of $p_{s o}, \\mu_{\\text{H_2O}}, L, T, T_{o}$ and fundamental constants. \n\nNow we consider a `moist' air mass that rises adiabatically starting from a temperature $T_{i}$. The mass mixing ratio of water vapor (the mass of water vapor relative to the total mass) is $\\phi$. Take the air mass to have a specific molar heat at constant pressure $c_{p}$. The universal gas constant is $R = 8.31 \\mathrm{J} /(\\mathrm{mol} \\mathrm{K})$.", + "question": "Assuming that the air mass starts at $T_{i} = 17.0^{\\circ} \\mathrm{C}$ and $p_{i} = 10^{5} \\mathrm{Pa}$. Find the temperature $T_{l}$ at which liquid water starts forming in it if $\\phi = 10^{-2}$. Assume that the water content in the air mass stays constant during the rise. Use $L = 2460 \\mathrm{kJ} / \\mathrm{kg}$ and $p_{s o} = 1.94 \\times 10^{3} \\mathrm{Pa}$ at $T_{i} = 17.0^{\\circ} \\mathrm{C}$. Express your answer in $K$.", + "marking": [ + [ + "Award 0.4 pt if the answer uses Dalton's law to correctly express the partial pressure of water vapor as $p_w = \\frac{n_{H_2O}}{n_{air}} p = \\frac{m_{H_2O}/\\mu_{H_2O}}{m_{air}/\\mu_{air}} p = \\phi \\frac{\\mu_{air}}{\\mu_{H_2O}} p$, where $n$ is number of moles, $m$ is mass, $\\mu$ is molar mass, $p$ is total pressure, and $\\phi$ is mixing ratio. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly relates the mole ratio to the mass ratio via $\\frac{n_{H_2O}}{n_{air}} = \\frac{m_{H_2O}/\\mu_{H_2O}}{m_{air}/\\mu_{air}}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer states the correct adiabatic process relation for pressure: $p(T) = p_i \\left( \\frac{T}{T_i} \\right)^{c_p/R}$, where $p_i$ is the initial pressure, $T_i$ is the initial temperature, $c_p$ is the specific heat at constant pressure, and $R$ is the gas constant. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer shows that partial pressure of water needs to reach saturation for condensation to start. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer attempts to solve the transcendental equation iteratively by isolating $T$ on one side, such as rearranging to $T_l = \\frac{1}{\\frac{1}{T_i} - \\frac{R}{\\mu_{H_2O} L} \\ln \\left[ \\phi \\frac{\\mu_{air}}{\\mu_{H_2O}} \\frac{p_i}{p_{so}} \\left( \\frac{T_l}{T_i} \\right)^{c_p/R} \\right]}$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer provides the correct numerical solution $T \\approx 286.8 \\text{K} \\approx 13.7^{\\circ}\\text{C}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{286.8}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "K" + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_3_d_1.png" + ] + }, + { + "id": "APhO_2025_3_E_1", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part E: Sun halo] \n\nUnder suitable atmospheric conditions, a bright ring appears around the Sun which is called a halo. Halos are caused by ice crystals present in the upper troposphere. One interesting feature about halos is that they always appear at a specific angle relative to the direction of the Sun. \n\n[figure1] \nFigure E.1. On the left: A photograph showing a halo around the Sun. On the right: The path of a light ray passing through the prism.", + "question": "Consider a simple prism with an apex angle of $\\varphi$ and direct a light ray onto it at an incidence angle $\\alpha$, as shown in Figure E.1. Let the refractive index of the prism be $n$. Express the angle of deviation $\\delta$ of the light ray after passing through the prism in terms of $\\alpha, n$ and $\\varphi$.", + "marking": [ + [ + "Award 0.1 pt if the answer writes Snell's law correctly for the first refraction as $\\frac{\\sin \\alpha}{\\sin \\alpha'} = n$, where $\\alpha$ is the angle of incidence, $\\alpha'$ is the refracted angle, and $n$ is the refractive index. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer writes Snell's law correctly for the second refraction as $\\frac{\\sin \\beta}{\\sin \\beta'} = n$, where $\\beta$ is the angle of incidence inside the prism, $\\beta'$ is the refracted angle on exit, and $n$ is the refractive index. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer expresses the deviation angle as $\\delta = \\alpha + \\beta - \\varphi$, where $\\varphi$ is the prism angle. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer uses the geometric relation $\\alpha' + \\beta' = \\varphi$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer performs the correct calculation steps, including substituting $\\alpha' = \\arcsin \\left( \\frac{\\sin \\alpha}{n} \\right)$ and $\\beta = \\arcsin \\left\\{ n \\sin \\left[ \\varphi - \\arcsin \\left( \\frac{\\sin \\alpha}{n} \\right) \\right] \\right\\}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer provides the correct final formula for $\\delta$, such as $\\delta = \\alpha + \\arcsin \\left\\{ n \\sin \\left[ \\varphi - \\arcsin \\left( \\frac{\\sin \\alpha}{n} \\right) \\right] \\right\\} - \\varphi$, or any other equivalent expression. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\delta = \\alpha + \\arcsin\\{n \\sin[\\varphi - \\arcsin(\\frac{\\sin\\alpha}{n})]\\} - \\varphi$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.8 + ], + "modality": "text+variable figure", + "field": "Optics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_3_e_1.png" + ] + }, + { + "id": "APhO_2025_3_E_2", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part E: Sun halo] \n\nUnder suitable atmospheric conditions, a bright ring appears around the Sun which is called a halo. Halos are caused by ice crystals present in the upper troposphere. One interesting feature about halos is that they always appear at a specific angle relative to the direction of the Sun. \n\n[figure1] \nFigure E.1. On the left: A photograph showing a halo around the Sun. On the right: The path of a light ray passing through the prism. \n\n(E.1) Consider a simple prism with an apex angle of $\\varphi$ and direct a light ray onto it at an incidence angle $\\alpha$, as shown in Figure E.1. Let the refractive index of the prism be $n$. Express the angle of deviation $\\delta$ of the light ray after passing through the prism in terms of $\\alpha, n$ and $\\varphi$. \n\nThe most common type of halo forms when tiny ice crystals take the shape of regular hexagonal prisms. Light from the Sun falls onto randomly oriented ice crystals drifting in the atmosphere and scatters into various directions. However, in certain specific directions, the intensity of the refracted light is maximal, and this determines the angle at which the bright ring appears. \n\n[figure2] \nFigure E.2. \n\nConsider a hexagonal ice prism whose six-fold symmetry axis is perpendicular to the direction of the Sun's rays. Investigate a light ray that refracts through two rectangular faces of the prism indicated in Figure E.2. Due to the random orientation of the ice crystals, the light strikes the crystal faces at varying incidence angles $\\alpha$.", + "question": "Calculate the deviation angle $\\delta$ for incidence angles $\\alpha = 20^{\\circ}, 30^{\\circ}, 40^{\\circ}, 50^{\\circ}, 60^{\\circ}, 70^{\\circ}$, in that order. Output the six values in degrees ($^{\\circ}$), each with three significant figures, listed sequentially and separately. The refractive index of ice is $n = 1.31$.", + "marking": [ + [ + "Award 0.2 pt if the answer gives all six correct values (when \\alpha = 20^{\\circ}, \\delta = 27.5^{\\circ}; when \\alpha = 30^{\\circ}, \\delta = 23.0^{\\circ}; when \\alpha = 40^{\\circ}, \\delta = 21.8^{\\circ}; when \\alpha = 50^{\\circ}, \\delta = 22.5^{\\circ}; when \\alpha = 60^{\\circ}, \\delta = 24.7^{\\circ}; when \\alpha = 70^{\\circ}, \\delta = 28.7^{\\circ}). Partial points: award 0.1 pt if 3-5 values are correct; otherwise, award 0 pt.", + "Award 0.2 pt if the calculated data points for $\\delta$ are correctly plotted against $\\alpha$ on the graph, with $\\alpha$ on the horizontal axis and $\\delta$ on the vertical axis. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly identifies and shows that $\\delta$ has a local minimum (around $\\alpha \\approx 40^\\circ$). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\delta = 27.5^{\\circ}$ when $\\alpha = 20^{\\circ}$}", + "\\boxed{$\\delta = 23.0^{\\circ}$ when $\\alpha = 30^{\\circ}$}", + "\\boxed{$\\delta = 21.8^{\\circ}$ when $\\alpha = 40^{\\circ}$}", + "\\boxed{$\\delta = 22.5^{\\circ}$ when $\\alpha = 50^{\\circ}$}", + "\\boxed{$\\delta = 24.7^{\\circ}$ when $\\alpha = 60^{\\circ}$}", + "\\boxed{$\\delta = 28.7^{\\circ}$ when $\\alpha = 70^{\\circ}$}" + ], + "answer_type": [ + "Numerical Value", + "Numerical Value", + "Numerical Value", + "Numerical Value", + "Numerical Value", + "Numerical Value" + ], + "unit": [ + "degrees", + "degrees", + "degrees", + "degrees", + "degrees", + "degrees" + ], + "points": [ + 0.1, + 0.1, + 0.1, + 0.1, + 0.1, + 0.1 + ], + "modality": "text+variable figure", + "field": "Optics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_3_e_1.png", + "image_question/APhO_2025_3_e_2.png" + ] + }, + { + "id": "APhO_2025_3_E_3", + "context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part E: Sun halo] \n\nUnder suitable atmospheric conditions, a bright ring appears around the Sun which is called a halo. Halos are caused by ice crystals present in the upper troposphere. One interesting feature about halos is that they always appear at a specific angle relative to the direction of the Sun. \n\n[figure1] \nFigure E.1. On the left: A photograph showing a halo around the Sun. On the right: The path of a light ray passing through the prism. \n\n(E.1) Consider a simple prism with an apex angle of $\\varphi$ and direct a light ray onto it at an incidence angle $\\alpha$, as shown in Figure E.1. Let the refractive index of the prism be $n$. Express the angle of deviation $\\delta$ of the light ray after passing through the prism in terms of $\\alpha, n$ and $\\varphi$. \n\nThe most common type of halo forms when tiny ice crystals take the shape of regular hexagonal prisms. Light from the Sun falls onto randomly oriented ice crystals drifting in the atmosphere and scatters into various directions. However, in certain specific directions, the intensity of the refracted light is maximal, and this determines the angle at which the bright ring appears. \n\n[figure2] \nFigure E.2. \n\nConsider a hexagonal ice prism whose six-fold symmetry axis is perpendicular to the direction of the Sun's rays. Investigate a light ray that refracts through two rectangular faces of the prism indicated in Figure E.2. Due to the random orientation of the ice crystals, the light strikes the crystal faces at varying incidence angles $\\alpha$. \n\n(E.2) Calculate the deviation angle $\\delta$ for incidence angles $\\alpha = 20^{\\circ}, 30^{\\circ}, 40^{\\circ}, 50^{\\circ}, 60^{\\circ}, 70^{\\circ}$, in that order. Output the six values in degrees ($^{\\circ}$), each with three significant figures, listed sequentially and separately. The refractive index of ice is $n = 1.31$.", + "question": "Using the numerical results from the previous question (E.2), determine at what angle the halo appears the brightest relative to the direction of the Sun. Express your answer in degrees ($^{\\circ}$) with three significant figures.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly reads and states the minimal value of $\\delta$ as approximately $21.8^\\circ$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer concludes that the angular size of the halo corresponds to this minimal value of $\\delta$, i.e. about $21.8^\\circ$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{21.8}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "degrees" + ], + "points": [ + 0.2 + ], + "modality": "text+variable figure", + "field": "Optics", + "source": "APhO_2025", + "image_question": [ + "image_question/APhO_2025_3_e_1.png", + "image_question/APhO_2025_3_e_2.png" + ] + } +] \ No newline at end of file diff --git a/data/EuPhO_2024.json b/data/EuPhO_2024.json new file mode 100644 index 0000000000000000000000000000000000000000..ba2e27de46ad63cdd2655e5a45eefcf7318d50ed --- /dev/null +++ b/data/EuPhO_2024.json @@ -0,0 +1,302 @@ +[ + { + "information": "None." + }, + { + "id": "EuPhO_2024_1_1", + "context": "As shown in the figure, a puck (a small disc) with radius $r$ and uniform density is moving on a horizontal plane with the velocity $v_{0}$ without rotation. The puck meets a fixed half-circular wall with a radius $R \\gg r$ and starts to move along the wall. The coefficient of friction with the wall is $\\mu$, and friction with the horizontal plane is negligible.", + "question": "Find the velocity of the puck $v_{e}$ when it leaves the wall. Consider different possible cases.", + "marking": [ + [ + "Award 0.3 pt if the answer realizes that puck is sliding initially. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer realizes that puck may roll without sliding. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer states that sliding ends when roll condition $v = {r\\omega }$ is met. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly equates the normal force with $m v^2 / R$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer uses $$F_f = \\mu N$$ for the friction force. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer gives the correct equation for translational motion: $m \\frac{d v}{d t} = -\\mu m \\frac{v^2}{R}$. Partial points: deduct 0.2 pt for wrong sign. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the integral expression for translational motion with correct initial conditions: $\\int_{v_0}^v \\frac{d v}{v^2} = - \\frac{\\mu}{R} \\int_0^t dt$, where $$t=0$$ is the time at which the puck meets the semicircular wall and has the initial velocity $$v_0$$. Otherwise, award 0 pt.", + "Award 1.0 pt if the answer gives the expression for $$v$$ as a function of time or angle as in $v(t) = \\frac{v_0}{1 + t/\\tau}$ or $v = v_0 \\exp(-\\mu \\varphi)$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer gives the equation of motion for rotation: $$r \\frac{d \\omega}{d t} = \\frac{\\mu m r^2}{R I} v^2 = \\frac{2\\mu}{R} \\cdot \\frac{v_0^2}{(1 + t/\\tau)^2}$$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer uses $$I = \\frac{1}{2} m r^2$$ as the moment of inertia. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the integral expression for rotational motion with correct initial conditions: $r \\int_0^{\\omega} d \\omega = \\frac{2 v_0}{\\tau} \\int_0^t \\frac{d t}{(1 + t/\\tau)^2}$. Otherwise, award 0 pt.", + "Award 1.0 pt if the answer gives the expression for $$r \\omega$$ as a function of time or angle as in $r \\omega = v_0 \\frac{2t / \\tau}{1 + t/\\tau}$ or $r \\omega = 2v_0 (1 - \\exp(-\\mu \\varphi))$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly finds the time $$\\frac{R}{2 v_0 \\mu}$$ or angle $$\\frac{\\ln(3/2)}{\\mu}$$ for the transition to rolling without sliding. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer obtains the critical coefficient of friction: $$\\mu_c = \\frac{\\ln(3/2)}{\\pi}$$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer finds the final velocity $$v_e = \\frac{2 v_0}{3}$$ for rolling without sliding. Otherwise, award 0 pt.", + "Award 1.0 pt if the answer finds the velocity $$v_e = v_0 \\exp(-\\pi \\mu)$$ if the puck slides the whole time. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$v_e = \\frac{2 v_0}{3}$}", + "\\boxed{$v_e = v_0 \\exp(-\\pi \\mu)$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 4.0, + 4.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "EuPhO_2024", + "image_question": [ + "images_question/EuPhO_2024_1_1_1.png" + ] + }, + { + "id": "EuPhO_2024_2_1", + "context": "Alice and Bob are twin astronauts on a long space mission. After many years, they are finally approaching each other to reunite. Alice's spaceship is moving towards Bob's spaceship at a speed of $u = \\frac{3}{5}c$ ,where $c$ is the speed of light.\n\nDuring their approach, both Alice and Bob send gifts to each other. Alice sends gifts to Bob at regular time intervals $\\Delta t_{0}$ in her own frame of reference, with each gift travelling at a velocity $v = \\frac{4}{5}c$ (again, in her frame of reference). Similarly, Bob sends gifts to Alice at the same regular time intervals $\\Delta t_{0}$ in his own frame of reference, with each gift also travelling at a velocity $v = \\frac{4}{5} c$ in his frame of reference. Assume that the distance $L$ between Alice and Bob is so large that there are many gifts in transit at any given moment.", + "question": "In Bob's reference frame: \n\n(1) Find the distance $l_B$ between two successive gifts sent by Alice. \n(2) Find the time interval $\\Delta t_{1}$ at which these gifts from Alice arrive at Bob's spaceship.", + "marking": [ + [ + "Award 0.5 pt if the answer writes the correct formula for relativistic addition of velocities. Partial points: deduct 0.3 pt for one mistake. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the relative velocity $v_B$ of frames B and G as $v_B = \\frac{35}{37} c$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer finds $l_{A} = \\frac{4}{5} c \\Delta t_0$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer gives correct $\\gamma$ formula $\\gamma_v = \\frac{1}{\\sqrt{1 - v^2/c^2}}$. Partial points: deduct 0.2 pt for one mistake. Otherwise, award 0 pt.", + "Award 0.7 pt if the answer states that $l_{1} = l_{2} / \\gamma$ is only true in rest frame. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer uses $l_{A} = l_{G} / \\gamma_v$, where $G$ is the rest frame of the gifts. Partial points: deduct 0.1 pt for each mistake. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer uses $l_{B} = l_{G} / \\gamma_{v_B}$, where $G$ is the rest frame of the gifts. Partial points: deduct 0.1 pt for each mistake. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the correct expression for $l_{B}$: $l_B = v \\Delta t_0 \\frac{\\gamma_v}{\\gamma_{v_B}} = \\frac{16}{37} \\Delta t_0 c$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the correct numerical result $16/37 = 0.\\overline{432}$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer uses $\\Delta t_{1} = l_{B} / v_{B}$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the correct numerical result $16/35 \\approx 0.457$. Otherwise, award 0 pt." + ], + [ + "Award 0.5 pt if the answer writes the correct formula for relativistic addition of velocities. Partial points: deduct 0.3 pt for one mistake. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the relative velocity $v_B$ of frames B and G as $v_B = \\frac{35}{37} c$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer gives correct $\\gamma$ formula $\\gamma_v = \\frac{1}{\\sqrt{1 - v^2/c^2}}$. Partial points: deduct 0.2 pt for one mistake. Otherwise, award 0 pt.", + "Award 0.7 pt if the answer realizes that two subsequent gifts are sent from the same place in Alice's frame. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer finds $\\Delta {t}_{0,B} = \\gamma_{u} \\Delta t_{0}$. Partial points: deduct 0.1 pt for each mistake. Otherwise, award 0 pt.", + "Award 0.7 pt if the answer finds that in Bob's frame,second gift at position $u \\Delta t_{0,B}$ while first gift at $v_{B} \\Delta t_{0,B}$. Partial points: deduct 0.2 pt for each mistake. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the correct expression for $l_{B}$: $l_B = (v_B - u) \\Delta t_0 \\gamma_u = \\frac{16}{37} \\Delta t_0 c$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the correct numerical result $16/37 = 0.\\overline{432}$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer uses $\\Delta t_{1} = l_{B} / v_{B}$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the correct numerical result $16/35 \\approx 0.457$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$l_B = \\frac{16}{37} c \\Delta t_{0}$}", + "\\boxed{$\\Delta t_{1} = \\frac{16}{35} \\Delta t_0$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 4.0, + 1.0 + ], + "modality": "text-only", + "field": "Modern Physics", + "source": "EuPhO_2024", + "image_question": [] + }, + { + "id": "EuPhO_2024_2_2", + "context": "Alice and Bob are twin astronauts on a long space mission. After many years, they are finally approaching each other to reunite. Alice's spaceship is moving towards Bob's spaceship at a speed of $u = \\frac{3}{5}c$ ,where $c$ is the speed of light.\n\nDuring their approach, both Alice and Bob send gifts to each other. Alice sends gifts to Bob at regular time intervals $\\Delta t_{0}$ in her own frame of reference, with each gift travelling at a velocity $v = \\frac{4}{5}c$ (again, in her frame of reference). Similarly, Bob sends gifts to Alice at the same regular time intervals $\\Delta t_{0}$ in his own frame of reference, with each gift also travelling at a velocity $v = \\frac{4}{5} c$ in his frame of reference. Assume that the distance $L$ between Alice and Bob is so large that there are many gifts in transit at any given moment.", + "question": "At a given instant, Alice can see a number of gifts moving away from her and a number of gifts moving towards her. What is the ratio between these two numbers?", + "marking": [ + [ + "Award 0.3 pt if the answer identifies the distance to Bob as $d_B$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the light travel time to Bob as $t_l = d_B / c$, where $d_B$ is the distance to Bob. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer recognizes the need to correct for light travel time in Alice's frame. Otherwise, award 0 pt.", + "Award 0.9 pt if the answer gives $d_{AG} = d_B + t_l v$. Partial points: deduct 0.3 pt for each mistake. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer computes the number of gifts sent by Alice to Bob $N_{a \\rightarrow b} = d_{AG} / L_A$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer recognizes the need to correct for light travel time in the incoming direction. Otherwise, award 0 pt.", + "Award 0.9 pt if the answer gives $d_{BG} = d_B - t_l v_B$. Partial points: deduct 0.3 pt for each mistake. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer computes $N_{b \\rightarrow a} = d_{BG} / L_B$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives a symbolic expression for the ratio $N_{\\text{out}} / N_{\\text{in}}$ = $\\frac{(1+v/c) c}{(1-v_B/c)v} \\frac{16}{37}$ or $\\frac{(c+v) v_B \\Delta t_1}{v \\Delta t_0 (c-v)}$. Partial points: deduct 0.2 pt for each mistake. Otherwise, award 0 pt.", + "Award 0.5 pt if the final numerical result is correct (ratio=18). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{18}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 5.0 + ], + "modality": "text-only", + "field": "Modern Physics", + "source": "EuPhO_2024", + "image_question": [] + }, + { + "id": "EuPhO_2024_3_1", + "context": "As shown in the figure, a Fabry-Pérot interferometer consists of two identical parallel planar mirrors separated by a distance $L$. The space between and outside the mirrors is filled with air. The mirrors are partially reflective; when light is aimed towards one of these mirrors along the normal direction, the reflected beam has intensity $R < 1$ times the intensity of the incident beam. Assume that the mirrors are symmetric, meaning they interact the same way with light incident from either side, and lossless. Assume also that they are highly reflective, meaning $1 - R \\ll 1$. A monochromatic laser beam of power $P$ is aimed towards the interferometer perpendicular to the mirrors. The distance $L$ is chosen so that the back-reflected beam vanishes, i.e., all the optical power is transmitted through the interferometer. \n\n[figure1]", + "question": "Show that the laser beam must acquire a nonzero phase shift $\\phi$ when it passes through either of the mirrors.", + "marking": [ + [ + "Award 0.3 pt if the answer shows understanding that some light is initially reflected without entering the interferometer. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer shows understanding that light bounces back and forth between the mirrors. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer uses one or two travelling waves in each region. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer writes equations relating amplitudes via $r$ and $t$, such as $B = tA + rC$ and $0 = rA + tC$. Otherwise, award 0 pt.", + "Award 0.6 pt if the answer solves the system to obtain the condition $e^{-2i k L} = r^2 - t^2$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer uses the relation $|r|^2 + |t|^2 = 1$ or $R + T = 1$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer states that $|r|^2 + |t|^2 = 1$ is a consequence of conservation of energy. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer indicates that the solutions $r$ and $t$ should be complex numbers. Otherwise, award 0 pt." + ], + [ + "Award 0.3 pt if the answer shows understanding that some light is initially reflected without entering the interferometer. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer shows understanding that light bounces back and forth between the mirrors. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer refers to the idea of superposition of complex amplitudes. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly includes the effects on amplitudes from reflection, transmission, and propagation. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer sums up the complex amplitudes as a geometric series. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer derives the equation: $e^{-2i k L} = r^2 - t^2$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer uses the condition $|r|^2 + |t|^2 = 1$ or $R + T = 1$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer states that $|r|^2 + |t|^2 = 1$ is a consequence of conservation of energy. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer shows understanding that the solutions $r$ and $t$ should be complex. Otherwise, award 0 pt." + ], + [ + "Award 0.3 pt if the answer shows understanding that some light is initially reflected without entering the interferometer. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer shows understanding that light bounces back and forth between the mirrors. Otherwise, award 0 pt.", + "Award 0.7 pt if the answer uses the relation $1 + r = t$. Otherwise, award 0 pt.", + "Award 0.8 pt if the answer justifies the relation $1 + r = t$ using continuity of the electric field or thin-mirror arguments. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer uses the condition $|r|^2 + |t|^2 = 1$ or $R + T = 1$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer states that $|r|^2 + |t|^2 = 1$ is a consequence of conservation of energy. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer shows understanding that the solutions $r$ and $t$ should be complex. Otherwise, award 0 pt." + ] + ], + "answer": [ + "" + ], + "answer_type": [ + "Open-Ended" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Optics", + "source": "EuPhO_2024", + "image_question": [ + "images_question/EuPhO_2024_3_1_1.png" + ] + }, + { + "id": "EuPhO_2024_3_2", + "context": "As shown in the figure, a Fabry-Pérot interferometer consists of two identical parallel planar mirrors separated by a distance $L$. The space between and outside the mirrors is filled with air. The mirrors are partially reflective; when light is aimed towards one of these mirrors along the normal direction, the reflected beam has intensity $R < 1$ times the intensity of the incident beam. Assume that the mirrors are symmetric, meaning they interact the same way with light incident from either side, and lossless. Assume also that they are highly reflective, meaning $1 - R \\ll 1$. A monochromatic laser beam of power $P$ is aimed towards the interferometer perpendicular to the mirrors. The distance $L$ is chosen so that the back-reflected beam vanishes, i.e., all the optical power is transmitted through the interferometer. \n\n[figure1]", + "question": "The laser beam must acquire a nonzero phase shift $\\phi$ when it passes through either of the mirrors. Find the magnitude of $\\phi$ (expressed in $^{\\circ}$).", + "marking": [ + [ + "Award 0.5 pt if the answer explicitly concludes that the phase difference is $90^{\\circ}$. Otherwise, award 0 pt.", + "Award 0.7 pt if the answer takes the modulus of $e^{-2ikL} = r^2 - t^2$ to get the conditions involving only $r$ and $t$. Otherwise, award 0 pt.", + "Award 0.8 pt if the answer correctly manipulates $r, t, r^*, t^*$ to show that $r$ is an imaginary number times $t$. Otherwise, award 0 pt." + ], + [ + "Award 0.5 pt if the answer explicitly concludes that the phase difference is $90^{\\circ}$. Otherwise, award 0 pt.", + "Award 0.7 pt if the answer takes the modulus of $e^{-2ikL} = r^2 - t^2$ to get the conditions involving only $r$ and $t$. Otherwise, award 0 pt.", + "Award 0.8 pt if the answer uses a geometric argument to show that $r$ and $t$ must make a right angle. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\pm 90$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "$^{\\circ}$" + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Optics", + "source": "EuPhO_2024", + "image_question": [ + "images_question/EuPhO_2024_3_1_1.png" + ] + }, + { + "id": "EuPhO_2024_3_3", + "context": "As shown in the figure, a Fabry-Pérot interferometer consists of two identical parallel planar mirrors separated by a distance $L$. The space between and outside the mirrors is filled with air. The mirrors are partially reflective; when light is aimed towards one of these mirrors along the normal direction, the reflected beam has intensity $R < 1$ times the intensity of the incident beam. Assume that the mirrors are symmetric, meaning they interact the same way with light incident from either side, and lossless. Assume also that they are highly reflective, meaning $1 - R \\ll 1$. A monochromatic laser beam of power $P$ is aimed towards the interferometer perpendicular to the mirrors. The distance $L$ is chosen so that the back-reflected beam vanishes, i.e., all the optical power is transmitted through the interferometer. \n\n[figure1]", + "question": "At a certain moment, the incident laser beam is switched off rapidly. Find the total energy $E$ of the light that travels back from the interferometer towards the laser after the laser is switched off.", + "marking": [ + [ + "Award 0.5 pt if the answer states that since $|t| \\ll |r|$, the amplitudes $|B|$ and $|C|$ are very large and their difference is small, thus, $|B|$ and $|C|$ are approximately equal. Otherwise, award 0 pt.", + "Award 1.0 pt if the answer applies symmetry to show that $2E = U$ ($U$ is the initial energy stored inside the interferometer) or that half the energy is emitted towards the laser. Otherwise, award 0 pt.", + "Award 1.5 pt if the answer finds the relation between $P^{\\prime}$ and $P$ (suppose the power contained in each wave (forwards- and backwards-propagating) in region II is $P^{\\prime}$), i.e., $P^{\\prime} \\approx P / (1 - R)$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the correct value for the initial energy stored inside the interferometer: $U \\approx \\frac{2}{1 - R} \\cdot \\frac{L P}{c}$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the correct value for the total energy: $E \\approx \\frac{1}{1 - R} \\cdot \\frac{L P}{c}$. Otherwise, award 0 pt." + ], + [ + "Award 1.5 pt if the answer shows, even by reasoning, that the power propagating out through the first mirror decreases by a factor $R^2$ every $\\Delta t$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer multiplies by $\\Delta t$ to convert power or intensity to energy. Otherwise, award 0 pt.", + "Award 1.5 pt if the answer sums a geometric series to find the total energy $E$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the correct value for $E$, i.e., $E \\approx \\frac{1}{1 - R} \\cdot \\frac{L P}{c}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$E \\approx \\frac{1}{1 - R} \\frac{LP}{c}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 4.0 + ], + "modality": "text+illustration figure", + "field": "Optics", + "source": "EuPhO_2024", + "image_question": [ + "images_question/EuPhO_2024_3_1_1.png" + ] + }, + { + "id": "EuPhO_2024_3_4", + "context": "As shown in the figure, a Fabry-Pérot interferometer consists of two identical parallel planar mirrors separated by a distance $L$. The space between and outside the mirrors is filled with air. The mirrors are partially reflective; when light is aimed towards one of these mirrors along the normal direction, the reflected beam has intensity $R < 1$ times the intensity of the incident beam. Assume that the mirrors are symmetric, meaning they interact the same way with light incident from either side, and lossless. Assume also that they are highly reflective, meaning $1 - R \\ll 1$. A monochromatic laser beam of power $P$ is aimed towards the interferometer perpendicular to the mirrors. The distance $L$ is chosen so that the back-reflected beam vanishes, i.e., all the optical power is transmitted through the interferometer. \n\n[figure1]", + "question": "Estimate the duration $T$ of the light pulse that travels back towards the laser.", + "marking": [ + [ + "Award 0.2 pt if the answer states that energy reduces by a factor $R^2$ each time a wave is removed. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer uses the fact that the reduction of energy occurs at intervals $\\Delta t$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer provides a valid mathematical argument that leads to the correct result: $T \\approx \\frac{1}{1 - R} \\cdot \\frac{L}{c}$ when $1 - R \\ll 1$. Otherwise, award 0 pt." + ], + [ + "Award 0.2 pt if the answer states that the decay of stored energy is roughly exponential like $e^{-2 \\log(1/R) t / (\\Delta t)}$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer states the outwards energy flux. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer uses the energy decay equation to determine the decay constant and finds $T \\approx \\frac{1}{1 - R} \\cdot \\frac{L}{c}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$T \\approx \\frac{1}{1 - R} \\cdot \\frac{L}{c}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Optics", + "source": "EuPhO_2024", + "image_question": [ + "images_question/EuPhO_2024_3_1_1.png" + ] + } +] \ No newline at end of file diff --git a/data/EuPhO_2025.json b/data/EuPhO_2025.json new file mode 100644 index 0000000000000000000000000000000000000000..49aac00f9aafc9abc1280b5004141d2fa788995a --- /dev/null +++ b/data/EuPhO_2025.json @@ -0,0 +1,208 @@ +[ + { + "information": "None." + }, + { + "id": "EuPhO_2025_1_1", + "context": "You are asked to study the features of the brightly lit circle and dark rings in the figures below. Make your calculations for an idealized situation: the chair leg is strictly cylindrical of radius $a$, strictly vertical, with a perfectly smooth, cylindrical, and perfectly reflecting surface. You may make any additional model assumptions and approximations you deem reasonable that will simplify your calculations.", + "question": "Determine how the illuminance surplus $I(r, \\theta)$ inside the brightly lit circle on the floor depends on the polar coordinates $r \\gg a$ and $\\theta$. The illuminance quantifies the amount of incoming light per area. By \"surplus\" we mean the additional illuminance introduced due to the presence of the cylinder. Express the answer in terms of $I_0$ defined as the illuminance difference between points $A$ and $B$ in the figure.", + "marking": [ + [ + "Award 0.5 pt if the answer uses correct angles, $2\\alpha + \\theta > \\pi$. Partial points: award 0.2 pt if the answer only states $2\\alpha + \\theta = \\pi$ without justification. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer states the reflection law in any form. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer includes the equation $x = a \\sin \\alpha$. Otherwise, award 0 pt.", + "Award 1.0 pt if the answer derives $I_0 \\delta x = 2I l \\delta \\alpha$. Partial points: subtract 0.3 pt if the factor of 2 is missing; subtract 0.3 pt if $r$ is used instead of $l$. If the answer does not derive such formula, award 0 pt.", + "Award 0.5 pt if the answer derives $I = -\\frac{I_0 a}{2l} \\cos(\\beta)$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer justifies the approximation $l \\approx r$. Partial points: award 0.2 pt if the answer states the approximation without justification. Otherwise, award 0 pt.", + "Award 1.5 pt if the answer correctly obtains the results $I \\approx \\frac{I_{0}a}{2r} \\sin (\\theta /2)$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{I_{0}a}{2r} \\sin (\\theta /2)$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 5.0 + ], + "modality": "text+variable figure", + "field": "Optics", + "source": "EuPhO_2025", + "image_question": [ + "image_question/EuPhO_2025_1_1_1.png" + ] + }, + { + "id": "EuPhO_2025_1_2", + "context": "You are asked to study the features of the brightly lit circle and dark rings in the figures below. Make your calculations for an idealized situation: the chair leg is strictly cylindrical of radius $a$, strictly vertical, with a perfectly smooth, cylindrical, and perfectly reflecting surface. You may make any additional model assumptions and approximations you deem reasonable that will simplify your calculations.", + "question": "In the following figure, some fingers are blocking some of the light from reaching the chair leg. Let $R(\\theta)$ denote the radial distance of the middle dark ring as a function of the angle $\\theta$ and let $R_{\\min}$ be the minimal value of $R(\\theta)$. Determine $R(\\theta) - R_{\\min}$.", + "marking": [ + [ + "Award 1.0 pt if the answer includes a correct Cosine Law expression. Otherwise, award 0 pt.", + "Award 1.0 pt if the answer correctly obtains $l = l_0 + a \\cos(\\alpha) = l_0 + a |\\cos(\\beta)|$ where $l_0$ is the horizontal distance when $\\alpha = \\pi/2$. Partial points: award 0.5 pt if the answer only provides a qualitative explanation of why $R(\\theta)$ varies with $\\theta$. Otherwise, award 0 pt.", + "Award 1.0 pt if the answer justifies $l_0 \\approx R_{\\min}$ to leading order. Partial points: award 0.5 pt if the answer only states $l_0 \\approx R_{\\min}$ without justification. Otherwise, award 0 pt.", + "Award 2.0 pt if the answer correcly derives $R - R_{\\min} \\approx 2a \\sin(\\theta/2)$. Partial points: award 1.0 pt if the answer gives $R - R_{\\min} = a \\sin(\\theta/2)$; award 0.5 pt if the answer only states $R_{\\max} - R_{\\min} = 2a$; award 0.0 pt if the answer only states $R_{\\max} - R_{\\min} = a$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$R(\\theta) - R_{\\min} \\approx 2a \\sin(\\theta/2)$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 5.0 + ], + "modality": "text+variable figure", + "field": "Optics", + "source": "EuPhO_2025", + "image_question": [ + "image_question/EuPhO_2025_1_2_1.png" + ] + }, + { + "id": "EuPhO_2025_2_1", + "context": "A table is made by fastening a metal frame to a massive uniform plate (so they form a rigid body) and attaching it with chains to another frame that is fixed on the horizontal ground. The motion of the table is limited to the plane of the side view (right picture).\n\nThe masses of the chains and the frame can be neglected. The chains are frictionless, inextensible, and remain tensioned in oscillations. The grid step is $a = 0.100 m$, the acceleration of gravity $g = 9.81 m/s^2$.", + "question": "Show that in the configuration on the side view (right picture), the table is in a stable equilibrium.", + "marking": [ + [ + "Award 2.0 pt if the answer shows that the table is in equilibrium. Partial points: award 1.0 pt if the answer only gives a sketch of forces or an equation for forces. Otherwise, award 0 pt.", + "Award 2.0 pt if the answer shows that the equilibrium is stable. Partial points: award 1.0 pt if the answer gives a sketch with returning forces, but there is no proof that $\\omega_{0} = 0$; award 1.0 pt if the answer gives a statement that the stability comes from $y = k x^2$ if $k > 0$, but $k$ is not found correctly ($k = \\frac{1}{3a}$), which could potentially be negative. Otherwise, award 0 pt." + ] + ], + "answer": [ + "" + ], + "answer_type": [ + "Open-Ended" + ], + "unit": [ + null + ], + "points": [ + 4.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "EuPhO_2025", + "image_question": [ + "image_question/EuPhO_2025_2_1_1.png" + ] + }, + { + "id": "EuPhO_2025_2_2", + "context": "A table is made by fastening a metal frame to a massive uniform plate (so they form a rigid body) and attaching it with chains to another frame that is fixed on the horizontal ground. The motion of the table is limited to the plane of the side view (right picture).\n\nThe masses of the chains and the frame can be neglected. The chains are frictionless, inextensible, and remain tensioned in oscillations. The grid step is $a = 0.100 m$, the acceleration of gravity $g = 9.81 m/s^2$.", + "question": "Find the period $T$ of the small oscillations: (1) write the formula for $T$, (2) calculate the number of $T$ (keep three significant figures, and express the unit in $s$).", + "marking": [ + [ + "Award 1.0 pt if the answer shows that the table can rotate, either explicitly in the sketch or by introducing $\\varphi$ in the equations. Otherwise, award 0 pt.", + "Award 1.0 pt if the answer correctly argues that the table does not have immediate rotation, either by: geometric reasoning showing $\\varphi \\sim x^{2}$; or constraint equations leading to $\\varphi \\sim x^{2}$; or using immediate velocities to show no rotation. Otherwise, award 0 pt.", + "Award 1.0 pt if the answer outlines a valid plan to find the oscillation period using any of the following ideas: second Newton's law: $\\ddot{x} \\sim -x$; or identifying kinetic and potential energies; or using curvature of the trajectory. Otherwise, award 0 pt.", + "Award 2.0 pt if the answer includes all necessary elements: small horizontal forces; correct approximations of kinetic and potential energies; accelerations/curvatures. Partial points: award 1.0 pt if not all elements are present or if the answer contains mistakes; award 1.0 pt if the answer misses the proof of $\\omega_0 = 0$ (or does not consider the rotation), but everything else is correct; award 0.0 pt if only partial elements are present with an unrelated approach (e.g., using energy but only writing force expressions). Otherwise, award 0 pt.", + "Award 1.0 pt if the answer provides both the correct formula ($T = 2 \\pi \\sqrt{\\frac{3a}{2g}}$) and numerical value ($T = 0.777 s$) for $T$. Partial points: award 0.5 pt the answer only gives the formula or only the number; award 0.5 pt if a simple mistake is made in the answer (like inverse formula under the root). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$T = 2 \\pi \\sqrt{\\frac{3a}{2g}}$}", + "\\boxed{0.777}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null, + "s" + ], + "points": [ + 3.0, + 3.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "EuPhO_2025", + "image_question": [ + "image_question/EuPhO_2025_2_1_1.png" + ] + }, + { + "id": "EuPhO_2025_3_2", + "context": "", + "question": "Now consider two infinite, straight, thin wires (wires $X$ and $Y$), each carrying a current $I$ as shown in the figure. The $x$-axis coincides with wire $X$, while wire $Y$ is parallel to the $y$-axis and passes through the point $(0, 0, -a)$. Let $P$ be the point $(3a, 0, r)$. Assuming $r \\ll a$, calculate $d$, the distance of closest approach of the magnetic field line that passes through $P$ to the wire $X$.", + "marking": [ + [ + "Award 0.4 pt if the answer correctly states that the magnetic field around an infinite, straight, thin wire carrying a current $I$ has magnitude $\\frac{\\mu_0 I}{2 \\pi \\rho}$, where $\\rho$ is the perpendicular distance to the wire. Partial points: award 0.2 pt if the direction is unclear. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer states that the magnetic field line is locally nearly circular. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer describes that the field line resembles helix tightly wound around wire. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer mentions that the radius of the helix is changing. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer introduces the idea of considering the funnel surface $S$. Otherwise, award 0 pt.", + "Award 1.0 pt if the answer relizes and justifies that the $\\vec{B}_Y$ flux is conserved along the funnel. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly argues or shows that the radius of the flux tube is smallest at $x = 0$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer approximates $\\vec{B}_Y$ as uniform across the flux tube cross-sections. Otherwise, award 0 pt.", + "Award 0.5 pt for correctly determining flux at $x = 3a$. Partial points: award 0.3 pt for correct projection; award 0.2 pt for correct area. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer determines the flux at $x = 0$ using $\\rho$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly calculates the final result for $d$: $d = r/\\sqrt{10}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer checks the validity of approximations used in the considered region. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$d = r/\\sqrt{10}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 5.0 + ], + "modality": "text+variable figure", + "field": "Electromagnetism", + "source": "EuPhO_2025", + "image_question": [ + "image_question/EuPhO_2025_3_2_1.png" + ] + }, + { + "id": "EuPhO_2025_3_3", + "context": "Now consider two infinite, straight, thin wires (wires $X$ and $Y$), each carrying a current $I$ as shown in the figure. The $x$-axis coincides with wire $X$, while wire $Y$ is parallel to the $y$-axis and passes through the point $(0, 0, -a)$. Let $P$ be the point $(3a, 0, r)$. Assuming $r \\ll a$, calculate $d$, the distance of closest approach of the magnetic field line that passes through $P$ to the wire $X$.", + "question": "Let $L$ be the length of this field line between $P$ and its point of closest approach to wire $X$. Using values $a = 10 cm$ and $r = 1.0 mm$, calculate $L$ to within 20% relative error (express the unit in $m$).", + "marking": [ + [ + "Award 0.2 pt if the answer states or implicitly assumes that $L$ is much larger than $a$ and $r$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer correctly equates $\\mathrm{d}x$ with $\\mathrm{d}L$ using $B$-field components or an angle. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer identifies that $B_{\\prep}$ is dominated by wire $X$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer derives a correct integral expression for $L$: $L = \\int_Q^P \\mathrm{d}L = \\int_0^{3a} \\frac{a^2+x^2}{a \\rho} \\mathrm{d}x$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer provides an expression for $\\rho$ as a function of $a$, $r$, and $x$: $\\rho^2 = \\frac{r^2 (a^2 + x^2)}{10 a^2}$. Otherwise, award 0 pt.", + "Award 1.2 pt if the answer carries out reasonable numerical approximation ($L = \\int_0^{3a} \\frac{\\sqrt{10}}{r} \\sqrt{a^2 + x^2} \\mathrm{d}x = \\frac{\\sqrt{10} a^2}{r} \\int_0^3 \\sqrt{1+u^2} \\mathrm{d}u$) or rigorous calculation of integral ($\\int_0^3 \\sqrt{1+u^2} \\mathrm{d}u \\approx 6.24$). Otherwise, award 0 pt.", + "Award 1.0 pt if the final result of $L$ within the range of $140 m \\leq L \\leq 215 m$. Partial points: award 0.8 pt if the answer is within the correct range but has only 1 significant figure or more than 3. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{[140, 215]}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "m" + ], + "points": [ + 4.0 + ], + "modality": "text+variable figure", + "field": "Electromagnetism", + "source": "EuPhO_2025", + "image_question": [ + "image_question/EuPhO_2025_3_2_1.png" + ] + } +] \ No newline at end of file diff --git a/data/F=MA_2024.json b/data/F=MA_2024.json new file mode 100644 index 0000000000000000000000000000000000000000..67c328e8311ac4c93d79691ac9d8df6a5ae8500f --- /dev/null +++ b/data/F=MA_2024.json @@ -0,0 +1,565 @@ +[ + { + "information": "Use $g = 10 \\mathrm{N}/\\mathrm{kg}$ throughout, unless otherwise specified." + }, + { + "id": "F=MA_2024_01", + "context": "", + "question": "An archer fires an arrow from the ground so that it passes through two hoops, which are both a height $h$ above the ground. The arrow passes through the first hoop one second after the arrow is launched, and through the second hoop another second later. What is the value of $h$?\n\n(A) $5 m$. \n(B) $10 m$. \n(C) $12 m$. \n(D) $15 m$. \n(E) There is not enough information to decide.", + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [] + }, + { + "id": "F=MA_2024_02", + "context": "", + "question": "An amusement park ride consists of a circular, horizontal room. A rider leans against its frictionless outer walls,which are angled back at $30^{\\circ}$ with respect to the vertical, so that the rider's center of mass is $5.0 m$ from the center of the room. When the room begins to spin about its center, at what angular velocity will the rider's feet first lift off the floor?\n\n(A) $1.9 \\mathrm{rad}/\\mathrm{s}$. \n(B) $2.3 \\mathrm{rad}/\\mathrm{s}$. \n(C) $3.5 \\mathrm{rad}/\\mathrm{s}$. \n(D) $4.0 \\mathrm{rad}/\\mathrm{s}$. \n(E) $5.6 \\mathrm{rad}/\\mathrm{s}$.", + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [] + }, + { + "id": "F=MA_2024_03", + "context": "", + "question": "As shown in the figure, a simple bridge is made of five thin rods rigidly connected at four vertices. The ground is frictionless, so that it can only exert vertical normal forces at $B$ and $D$. The weight of the bridge is negligible, but a person stands at its middle, exerting a downward force $F$ at vertex $C$. In static equilibrium, each rod can be experiencing either tension or compression. Which of the following is true?\n\n(A) Only the vertical rod is in tension. \n(B) Only the horizontal rods are in tension. \n(C) Both the vertical rod and the diagonal rods are in tension. \n(D) Both the vertical rod and the horizontal rods are in tension. \n(E) All of the rods are in tension.", + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_03_1.png" + ] + }, + { + "id": "F=MA_2024_04", + "context": "", + "question": "A bouncy ball is thrown vertically upward from the ground. Air resistance is negligible, and the ball's collisions with the ground are perfectly elastic. Which of the following plots in the figure shows the kinetic energy of the ball as a function of time? Assume the collisions are too quick for their duration to be seen in the plot.", + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_04_1.png" + ] + }, + { + "id": "F=MA_2024_05", + "context": "", + "question": "As shown in the figure, a massless inclined plane with angle $30^{\\circ}$ to the horizontal is fixed to a scale. A block of mass $m$ is released from the top of the plane, which is frictionless. As the block slides down the plane, what is the reading on the scale? \n\n(A) $\\sqrt{3}{mg}/4$. \n(B) ${mg}/2$. \n(C) ${3mg}/4$. \n(D) $\\sqrt{3}{mg}/2$. \n(E) ${mg}$.", + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_05_1.png" + ] + }, + { + "id": "F=MA_2024_06", + "context": "", + "question": "As shown in the figure, a pendulum is made with a string and a bucket full of water. When the string is vertical, the bottom of the bucket is near the ground. Then, the pendulum is set swinging with a small amplitude, and a very small hole is opened at the bottom of the bucket, which leaks water at a constant rate. After a few full swings, which of the following plots in the figure best shows the amount of water that has landed on the ground as a function of position?", + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_06_1.png", + "image_question/F=MA_2024_06_2.png" + ] + }, + { + "id": "F=MA_2024_07", + "context": "", + "question": "A particle travels in a straight line. Its velocity as a function of time is shown in the figure. Which of the following plots shows the velocity as a function of distance $x$ from its initial position?", + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_07_1.png", + "image_question/F=MA_2024_07_2.png" + ] + }, + { + "id": "F=MA_2024_08", + "context": "", + "question": "As shown in the figure, a rod of length $L$ is sliding down a frictionless wall. When the rod makes an angle of $45^{\\circ}$ to the horizontal, the bottom of the rod has speed $v$. At this moment, what is the speed of the middle of the rod?\n\n(A) $v/2$. \n(B) $v/\\sqrt{2}$. \n(C) $v$. \n(D) $\\sqrt{2}v$. \n(E) ${2v}$.", + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_08_1.png" + ] + }, + { + "id": "F=MA_2024_09", + "context": "", + "question": "When a car's brakes are fully engaged, it takes $100 m$ to stop on a dry road, which has coefficient of kinetic friction $\\mu_{k} = 0.8$ with the tires. Now suppose only the first $50 m$ of the road is dry, and the rest is covered with ice, with $\\mu_{k} = 0.2$. What total distance does the car need to stop?\n\n(A) $150 m$. \n(B) $200 m$. \n(C) $250 m$. \n(D) $400 m$. \n(E) $850 m$.", + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [] + }, + { + "id": "F=MA_2024_10", + "context": "", + "question": "As shown in the figure, a block of mass $m$ is connected to the walls of a frictionless box by two massless springs with relaxed lengths $\\ell$ and $2\\ell$, and spring constants $k$ and $2k$ respectively. The length of the box is $3\\ell$. The system rotates with a constant angular velocity $\\omega$ about one of its walls. Suppose the block stays at a constant distance $r$ from the axis of rotation, without touching either of the walls. What is the value of $r$?\n\n(A) $\\frac{2k\\ell}{2k - m\\omega^2}$. \n(B) $\\frac{2k\\ell}{2k + m\\omega^2}$. \n(C) $\\frac{2k\\ell}{3k + m\\omega^2}$. \n(D) $\\frac{3k\\ell}{3k - m\\omega^2}$. \n(E) $\\frac{3k\\ell}{3k + m\\omega^2}$.", + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_10_1.png" + ] + }, + { + "id": "F=MA_2024_11", + "context": "", + "question": "Two hemispherical shells can be pressed together to form a airtight sphere of radius $40 cm$. Suppose the shells are pressed together at a high altitude, where the air pressure is half its value at sea level. The sphere is then returned to sea level, where the air pressure is $10^{5}\\mathrm{Pa}$. What force $F$, applied directly outward to each hemisphere, is required to pull them apart?\n\n(A) $25,000 N$. \n(B) $50,000 N$. \n(C) $100,000 N$. \n(D) $200,000 N$. \n(E) $400,000 N$.", + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [] + }, + { + "id": "F=MA_2024_12", + "context": "", + "question": "A space probe with mass $m$ at point P traverses through a cluster of three asteroids,at points A, B, and C. The masses and locations of the asteroids are shown in the figure. What is the torque on the probe about point C?\n\n(A) $\\frac{1}{2\\sqrt{2}} \\frac{GMm}{d}$. \n(B) $\\frac{1}{2} \\frac{GMm}{d}$. \n(C) $\\frac{1}{\\sqrt{2}}\\frac{GMm}{d}$. \n(D) $\\frac{GMm}{d}$. \n(E) $\\frac{\\sqrt{2}GMm}{d}$.", + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_12_1.png" + ] + }, + { + "id": "F=MA_2024_13", + "context": "", + "question": "Two frictionless blocks of mass $m$ are connected by a massless string which passes through a fixed massless pulley, which is a height $h$ above the ground. Suppose the blocks are initially held with horizontal separation $x$, and the length of the string is chosen so that the right block hangs in the air as shown in the figure. If the blocks are relased, the tension in the string immediately afterward will be $T$. Which of the following plots in the figure shows a plot of $T$ versus $x$?", + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_13_1.png", + "image_question/F=MA_2024_13_2.png" + ] + }, + { + "id": "F=MA_2024_14", + "context": "", + "question": "A bead of mass $m$ can slide frictionlessly on a vertical circular wire hoop of radius $20 cm$. The hoop is attached to a stand of mass $m$, which can slide frictionlessly on the ground. Initially, the bead is at the bottom of the hoop, the stand is at rest, and the bead has velocity $2 m/s$ to the right. At some point, the bead will stop moving with respect to the hoop. At that moment, through what angle along the hoop has the bead traveled?\n\n(A) $30^{\\circ}$. \n(B) $45^{\\circ}$. \n(C) $60^{\\circ}$. \n(D) $90^{\\circ}$. \n(E) $120^{\\circ}$.", + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_14_1.png" + ] + }, + { + "id": "F=MA_2024_15", + "context": "", + "question": "The viscous force between two plates of area $A$, with relative speed $v$ and separation $d$, is $F = {\\eta Av}/d$, where $\\eta$ is the viscosity. In fluid mechanics,the Ohnesorge number is a dimensionless number proportional to $\\eta$ which characterizes the importance of viscous forces,in a drop of fluid of density $\\rho$, surface tension $\\gamma$, and length scale $\\ell$. Which of the following could be the definition of the Ohnesorge number?\n\n(A) $\\frac{\\eta \\ell}{\\sqrt{\\rho \\gamma}}$. \n(B) $\\eta \\ell \\sqrt{\\frac{\\rho}{\\gamma}}$. \n(C) $\\eta \\sqrt{\\frac{\\rho}{\\gamma \\ell}}$. \n(D) $\\eta \\sqrt{\\frac{\\rho \\ell}{\\gamma}}$. \n(E) $\\frac{\\eta}{\\sqrt{\\rho \\gamma \\ell}}$.", + "answer": [ + "\\boxed{E}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [] + }, + { + "id": "F=MA_2024_16", + "context": "", + "question": "A child of mass $m$ holds onto the end of a massless rope of length $\\ell$ ,which is attached to a pivot a height $H$ above the ground. The child is released from rest when the rope is straight and horizontal. As shown in the figure, at some point, the child lets go of the rope, flies through the air, and lands on the ground a horizontal distance $d$ from the pivot. On Earth,the maximum possible value of $d$ is $d_E$. If the setup is moved to the Moon, which has $1/6$ the gravitational acceleration, what is the new maximum possible value of $d$?\n\n(A) $d_E/6$. \n(B) $d_E/\\sqrt{6}$. \n(C) $d_E$. \n(D) $\\sqrt{6}d_E$. \n(E) $6d_E$.", + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_16_1.png" + ] + }, + { + "id": "F=MA_2024_17", + "context": "", + "question": "As shown in the figure, consider the system of massless and frictionless pulleys, ropes, and springs. Initially, a block of mass $m$ is attached to the end of a rope, and the system is in equilibrium. Next the block is doubled in mass, and the system is allowed to come to equilibrium again. During the transition between these equilibria, how far does the end of the rope (where the block is suspended) move?\n\n(A) $\\frac{7}{12}\\frac{mg}{k}$. \n(B) $\\frac{11}{12}\\frac{mg}{k}$. \n(C) $\\frac{13}{12}\\frac{mg}{k}$. \n(D) $\\frac{7}{6}\\frac{mg}{k}$. \n(E) $\\frac{11}{6}\\frac{mg}{k}$.", + "answer": [ + "\\boxed{E}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_17_1.png" + ] + }, + { + "id": "F=MA_2024_18", + "context": "", + "question": "As shown in the figure, a satellite is initially in a circular orbit of radius $R$ around a planet of mass $M$. It fires its rockets to instantaneously increase its speed by $\\Delta v$, keeping the direction of its velocity the same, so that it enters an elliptical orbit whose maximum distance from the planet is $2R$. What is the value of $\\Delta v$? (Hint: when the satellite is in an elliptical orbit with semimajor axis $a$, its total energy per unit mass is $- {GM}/{2a}$.) \n\n(A) $0.08 \\sqrt{\\frac{GM}{R}}$. \n(B) $0.15 \\sqrt{\\frac{GM}{R}}$. \n(C) $0.22 \\sqrt{\\frac{GM}{R}}$. \n(D) $0.29 \\sqrt{\\frac{GM}{R}}$. \n(E) $0.41 \\sqrt{\\frac{GM}{R}}$.", + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_18_1.png" + ] + }, + { + "id": "F=MA_2024_19", + "context": "", + "question": "A wheel of radius $R$ has a thin rim and four spokes,each of which have uniform density.As shown in the figure, the entire rim has mass $m$, three of the spokes each have mass $m$, and the fourth spoke has mass $3m$. The wheel is suspended on a horizontal frictionless axle passing through its center. If the wheel is slightly rotated from its equilibrium position, what is the angular frequency of small oscillations?\n\n(A) $\\sqrt{\\frac{g}{3R}}$. \n(B) $\\sqrt{\\frac{g}{2R}}$. \n(C) $\\sqrt{\\frac{2g}{3R}}$. \n(D) $\\sqrt{\\frac{g}{R}}$. \n(E) $\\sqrt{\\frac{7g}{6R}}$.", + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_19_1.png" + ] + }, + { + "id": "F=MA_2024_20", + "context": "", + "question": "As shown in the figure, four massless rigid rods are connected into a quadrilateral by four hinges. The hinges have mass $m$, and allow the rods to freely rotate. A spring of spring constant $k$ is connected across each of the diagonals, so that the springs are at their relaxed length when the rods form a square. Assume the springs do not interfere with each other. If the square is slightly compressed along one of its diagonals, its shape will oscillate over time. What is the period of these oscillations?\n\n(A) $2\\pi \\sqrt{\\frac{m}{4k}}$. \n(B) $2\\pi \\sqrt{\\frac{m}{2k}}$. \n(C) $2\\pi \\sqrt{\\frac{m}{k}}$. \n(D) $2\\pi \\sqrt{\\frac{2m}{k}}$. \n(E) $2\\pi \\sqrt{\\frac{4m}{k}}$.", + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_20_1.png" + ] + }, + { + "id": "F=MA_2024_21", + "context": "", + "question": "A syringe is filled with water of density $\\rho$ and negligible viscosity. Its body is a cylinder of cross-sectional area $A_1$, which gradually tapers into a needle with cross-sectional area $A_2 \\ll A_1$. The syringe is held in place and its end is slowly pushed inward by a force $F$, so that it moves with constant speed $v$. Water shoots straight out of the needle's tip. What is the approximate value of $F$?\n\n(A) $\\rho {v}^2{A}_{1}$. \n(B) $\\frac{\\rho {v}^2{A}_{1}^2}{2{A}_{2}}$. \n(C) $\\frac{\\rho {v}^2{A}_{1}^2}{{A}_{2}}$. \n(D) $\\frac{\\rho {v}^2{A}_{1}^{3}}{2{A}_{2}^2}$. \n(E) $\\frac{\\rho {v}^2{A}_{1}^{3}}{{A}_{2}^2}$", + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [] + }, + { + "id": "F=MA_2024_22", + "context": "", + "question": "A spherical shell is made from a thin sheet of material with a mass per area of $\\sigma$. Consider two points, $P_1$ and $P_2$, which are close to each other, but just inside and outside the sphere, respectively. If the accelerations due to gravity at these points are $g_1$ and $g_2$, respectively, what is the value of $|g_1 - g_2|$?\n\n(A) $\\pi G\\sigma$. \n(B) $4\\pi G\\sigma /3$. \n(C) $2\\pi G\\sigma$. \n(D) $4\\pi G\\sigma$. \n(E) $8\\pi G\\sigma$.", + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [] + }, + { + "id": "F=MA_2024_23", + "context": "", + "question": "Collisions between ping pong balls and paddles are not perfectly elastic. Suppose that if a player holds a paddle still and drops a ball on top of it from any height $h$, it will bounce back up to height $h/2$. To keep the ball bouncing steadily, the player moves the paddle up and down, so that it is moving upward with speed $1.0 m/s$ whenever the ball hits it. What is the height to which the ball is bouncing?\n\n(A) $0.21 m$. \n(B) $0.45 m$. \n(C) $1.0 m$. \n(D) $1.7 m$. \n(E) There is not enough information to determine the height.", + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [] + }, + { + "id": "F=MA_2024_24", + "context": "", + "question": "When a projectile falls through a fluid, it experiences a drag force proportional to the product of its cross-sectional area, the fluid density $\\rho_{f}$, and the square of its speed. Suppose a sphere of density $\\rho_{s} \\gg \\rho_{f}$ of radius $R$ is dropped in the fluid from rest. When the projectile has reached half of its terminal velocity, which of the following is its displacement proportional to?\n\n(A) $R\\sqrt{\\rho_{s}/\\rho_{f}}$. \n(B) $R \\rho_{s}/\\rho_{f}$. \n(C) $R (\\rho_{s}/\\rho_{f})^{3/2}$. \n(D) $R (\\rho_{s}/\\rho_{f})^2$. \n(E) $R (\\rho_{s}/\\rho_{f})^{3}$", + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [] + }, + { + "id": "F=MA_2024_25", + "context": "", + "question": "A yo-yo consists of two massive uniform disks of radius $R$ connected by a thin axle. As shown in the figure, a thick string is wrapped many times around the axle, so that the end of the string is initially a distance $R$ from the axle. Then, the end of the string is held in place and the yo-yo is dropped from rest. Assume that energy losses are negligible, and that the string has negligible mass and always remains vertical. Below, we show a cross-section of the yo-yo partway through its descent. Between the moment the yo-yo is released and the moment the string completely unwinds, which of the following is true regarding the yo-yo's acceleration?\n\n(A) It is always zero.\n(B) It points downward, but decreases in magnitude over time. \n(C) It points downward and has constant magnitude. \n(D) It points downward, but increases in magnitude over time. \n(E) None of the above.", + "answer": [ + "\\boxed{E}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2024", + "image_question": [ + "image_question/F=MA_2024_25_1.png" + ] + } +] \ No newline at end of file diff --git a/data/F=MA_2025.json b/data/F=MA_2025.json new file mode 100644 index 0000000000000000000000000000000000000000..b1c92be4136354286e8a8e9dce9b7310c31dff42 --- /dev/null +++ b/data/F=MA_2025.json @@ -0,0 +1,558 @@ +[ + { + "information": "Use $g = 10 \\mathrm{N}/\\mathrm{kg}$ throughout, unless otherwise specified." + }, + { + "id": "F=MA_2025_01", + "context": "", + "question": "A particle is moving on a plane at a constant speed of $1 m/s$, but not necessarily in a straight line. Which of the following plot pairs (as shown in the figure) could describe the particle's position over time, in rectilinear coordinates?\n\n(A) I only. \n(B) II only. \n(C) III only. \n(D) II and III only. \n(E) All three plots could describe the particle's position over time.", + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [ + "image_question/F=MA_2025_01_1.png" + ] + }, + { + "id": "F=MA_2025_02", + "context": "As shown in the figure, three identical disks are placed on a frictionless table. Initially, two of the disks are at rest and in contact with each other. The third disk is launched with speed $v$ directly toward the midpoint of the two stationary disks along a path perpendicular to the line connecting their centers, as shown in the diagram. Analyze the motion of the disks after the collision, assuming all interactions are perfectly elastic and that when the disks collide, there is no friction or inelastic energy loss.", + "question": "Assume that all three disks collide simultaneously. What is the final velocity of the third disk?\n\n(A) $v/3$ in the opposite direction to the initial velocity. \n(B) $v/3$ in the same direction as the initial velocity. \n(C) $\\overrightarrow{0}$. \n(D) $v/5$ in the opposite direction to the initial velocity. \n(E) $v/5$ in the same direction as the initial velocity.", + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [ + "image_question/F=MA_2025_02_1.png" + ] + }, + { + "id": "F=MA_2025_03", + "context": "As shown in the figure, three identical disks are placed on a frictionless table. Initially, two of the disks are at rest and in contact with each other. The third disk is launched with speed $v$ directly toward the midpoint of the two stationary disks along a path perpendicular to the line connecting their centers, as shown in the diagram. Analyze the motion of the disks after the collision, assuming all interactions are perfectly elastic and that when the disks collide, there is no friction or inelastic energy loss.", + "question": "Assume that there is a little imperfection in disks' initial alignment so when the disks collide two collisions happen one at a time, rather than all three disks colliding simultaneously. What is the final speed of the third disk?\n\n (A) $v/2$. \n(B) $v/3$. \n(C) $v/4$. \n(D) $v/5$. \n(E) $0$.", + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [ + "image_question/F=MA_2025_03_1.png" + ] + }, + { + "id": "F=MA_2025_04", + "context": "", + "question": "As shown in the figure, a mouse $M$ is running from $A$ to $A^{\\prime}$ with constant speed $u_1$ . A cat $C$ is chasing the mouse with constant speed $u_2$ and direction always toward the mouse. At a certain time $MC \\perp AA^\\prime$ and the length of $MC = L$ . What is the magnitude of the acceleration of the cat $C$?\n\n(A) $0$. \n(B) $(u_1 - u_2)^2/(2\\pi L)$. \n(C) $u_1 u_2/L$. \n(D) $u_1 u_2/( 2\\pi L)$. \n(E) none of the above.", + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [ + "image_question/F=MA_2025_04_1.png" + ] + }, + { + "id": "F=MA_2025_05", + "context": "", + "question": "Three identical cylinders are used in this setup. Two of them are placed side by side on a horizontal surface, with a negligible distance between their surfaces so they do not touch. The third identical cylinder is placed on top of the first two, such that their centers form an equilateral triangle, as shown in the figure below.\n\nThe coefficients of friction are:\n\n$\\mu_1$: the coefficient of friction between the cylinders, and\n\n $\\mu_2$: the coefficient of friction between the cylinders and the ground.\n\nFor which of the following pairs $(\\mu_1, \\mu_2)$ will the system remain in equilibrium? \n\nPair 1: $(\\frac{1}{2}, \\frac{1}{12})$. Pair 2: $(\\frac{1}{3}, \\frac{1}{10})$. Pair 3: $(\\frac{1}{4}, \\frac{1}{8}})$.\n\n(A) Pair 1 only. \n(B) Pair 2 only. \n(C) Pair 3 only. \n(D) Pairs 1 and 2 only. \n(E) Pairs 1, Pair 2, and Pair 3.", + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [ + "image_question/F=MA_2025_05_1.png" + ] + }, + { + "id": "F=MA_2025_06", + "context": "", + "question": "A ball rolls without slipping down a ramp, which turns horizontal at the bottom; at the bottom of the ramp,the ball falls through the air, as in the diagram. If the ball starts from the position marked $O$, it lands $10 cm$ away from the bottom of the ramp. Which starting position will get the ball to land closest to $25 cm$ away?\n\n(A) Position A. \n(B) Position B. \n(C) Position C. \n(D) Position D. \n(E) Position E.", + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [ + "image_question/F=MA_2025_06_1.png" + ] + }, + { + "id": "F=MA_2025_07", + "context": "", + "question": "A mechanism consists of three point masses, each of mass $m$,connected by two massless rods of length $l$ and a torsion spring acting as a hinge. The potential energy of the torsion spring is given by $U_s$ . This system is designed to \"walk\" down a set of stairs, as shown in the figure. The angles $\\theta_1$ and $\\theta_2$ (see figure) represent the orientation of the rods,and their rates of change, $\\omega_1$ and $\\omega_2$, are the corresponding angular velocities. Assume that the mass on the surface is instantaneously at rest.\n\nWhich equation correctly describes the total energy of the system?\n\n(A) $E = m{l}^2\\omega_{1}^2 + \\frac{1}{2}m{l}^2\\omega_{2}^2 + m{l}^2\\omega_{1}\\omega_{2}\\cos \\left(\\theta_{1} + \\theta_{2}\\right) + {2mgl}\\sin \\theta_{1} + {mgl}\\sin \\theta_{2} + {U}_{s}$. \n(B) $E = \\frac{1}{2}m{l}^2\\omega_{1}^2 + \\frac{1}{2}m{l}^2\\omega_{2}^2 + {2mgl}\\sin \\theta_{1} + {mgl}\\sin \\theta_{2} - {U}_{s}$. \n(C) $E = m{l}^2\\omega_{1}^2 + \\frac{1}{2}m{l}^2\\omega_{2}^2 + m{l}^2\\omega_{1}\\omega_{2}\\cos \\left(\\theta_{1} - \\theta_{2}\\right) + {2mgl}\\sin \\theta_{1} + {mgl}\\sin \\theta_{2} + {U}_{s}$. \n(D) $E = m{l}^2\\omega_{1}^2 + \\frac{1}{2}m{l}^2\\omega_{2}^2 - m{l}^2\\omega_{1}\\omega_{2}\\cos \\left(\\theta_{1} - \\theta_{2}\\right) + {2mgl}\\sin \\theta_{1} + {mgl}\\sin \\theta_{2} + {U}_{s}$. \n(E) $E = \\frac{1}{2}m{l}^2\\omega_{1}^2 + \\frac{1}{2}m{l}^2\\omega_{2}^2 + {2mgl}\\sin \\theta_{1} + {mgl}\\sin \\theta_{2} + {U}_{s}$.", + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [ + "image_question/F=MA_2025_07_1.png" + ] + }, + { + "id": "F=MA_2025_08", + "context": "", + "question": "A symmetric spinning top,rotating clockwise at an angular frequency $\\omega$, is placed upright in the center of a frictionless circular plate. The plate then begins to rotate counterclockwise at a constant angular velocity $\\omega$. Assume the top's axis remains perfectly vertical and stable without any precession. From the perspective of an observer rotating with the plate, how does the top appear to rotate?\n\n(A) The top appears stationary without any rotation. \n(B) The top appears to rotate in the clockwise direction at an angular frequency $\\omega$. \n(C) The top appears to rotate in the clockwise direction at an angular frequency $2\\omega$. \n(D) The top appears to rotate in the counterclockwise direction at an angular frequency $\\omega$. \n(E) The top appears to rotate in the counterclockwise direction at an angular frequency ${2\\omega}$.", + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [] + }, + { + "id": "F=MA_2025_09", + "context": "", + "question": "$N$ circles in a plane, $C_{i}$, each rotate with frequency $\\omega$ relative to an inertial frame. The center of $C_{1}$ is fixed in the inertial frame, and the center of $C_{i}$ is fixed on $C_{i - 1}$ (for $i = 2, \\ldots, N$), as shown in the figure. Each circle has radius ${r}_{i} = \\lambda {r}_{i - 1}$, where $0 < \\lambda < 1$. A mass is fixed on $C_{N}$. The position of the mass relative to the center of $C_{1}$ is $R\\left( t\\right)$. For the $N = 4$ case shown, which of the following statements is true?\n\n During the time interval from 0 to ${2\\pi}/\\omega$, the magnitude of acceleration of mass on $C_{4}$:\n\n(A) reached its maximum and minimum more than once. \n(B) reached its maximum and minimum exactly once. \n(C) reached its maximum only once but the minimum more than once. \n(D) reached its minimum only once but the maximum more than once. \n(E) was constant.", + "answer": [ + "\\boxed{E}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [ + "image_question/F=MA_2025_09_1.png" + ] + }, + { + "id": "F=MA_2025_10", + "context": "When two objects of very different masses collide, it is difficult to transfer a substantial fraction of the energy of one to the other. Consider two objects, of mass $m$ and $M \\gg m$.", + "question": "If the lighter object is initially at rest, and the heavier object collides elastically with it, what is the approximate maximum fraction of the heavier object's kinetic energy that could be transferred to the lighter object?\n\n(A) $m/M$. \n(B) ${2m}/M$. \n(C) ${4m}/M$. \n(D) ${m}^2/{M}^2$. \n(E) $2{m}^2/{M}^2$.", + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [] + }, + { + "id": "F=MA_2025_11", + "context": "When two objects of very different masses collide, it is difficult to transfer a substantial fraction of the energy of one to the other. Consider two objects, of mass $m$ and $M \\gg m$.", + "question": "Now suppose that instead, the heavier object is initially at rest, and the lighter object collides elastically with it. What is the approximate maximum fraction of the lighter object's kinetic energy that could be transferred to the heavier object?\n\n(A) $m/M$. \n(B) ${2m}/M$. \n(C) ${4m}/M$. \n(D) ${m}^2/{M}^2$. \n(E) $2{m}^2/{M}^2$.", + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [] + }, + { + "id": "F=MA_2025_12", + "context": "", + "question": "A $50 g$ piece of clay is thrown horizontally with a velocity of $20 m/s$ striking the bob of a stationary pendulum with length $l = 1 m$ and a bob mass of $200 g$. Upon impact, the clay sticks to the pendulum weight and the pendulum starts to swing. What is the maximum change in angle of the pendulum?\n\n(A) $\\arccos(1/5)$. \n(B) $\\arcsin(7/10)$. \n(C) $\\arccos(2/3)$. \n(D) $\\arcsin(3/10)$. \n(E) $\\arctan(4/5)$.", + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [] + }, + { + "id": "F=MA_2025_13", + "context": "Angela the puppy loves chasing tennis balls, so her owners built a tennis ball launcher. It fires balls along the floor at some initial speed, applying no rotation to them. The balls initially slip along the floor, then start rolling without slipping. Ignore the potential deformation of the ball and floor during this process, as well as air resistance.", + "question": "Which of the following plot pairs (as shown in the figure) could show the linear speed $v$ and rotational speed $\\omega$ of one of the balls over time? Assume the floor has a constant roughness.", + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [ + "image_question/F=MA_2025_13_1.png" + ] + }, + { + "id": "F=MA_2025_14", + "context": "Angela the puppy loves chasing tennis balls, so her owners built a tennis ball launcher. It fires balls along the floor at some initial speed, applying no rotation to them. The balls initially slip along the floor, then start rolling without slipping. Ignore the potential deformation of the ball and floor during this process, as well as air resistance.", + "question": "There are three kinds of balls that can be launched in this set-up, all having the same radius $R$:\nI. a regular tennis ball (a thin spherical shell of rubber) of mass $m_1$. \nII. a solid wooden ball of mass $m_2$. \nIII. a solid rubber ball of mass $m_3$.\n where $m_1 < m_2 < m_3$. All three types of ball emerge from the launcher with the same velocity. For which ball will the final velocity be highest?\n\n(A) Ball I. \n(B) Ball II. \n(C) Ball III. \n(D) Balls II and III. \n(E) The final velocity will be the same for all three balls.", + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [] + }, + { + "id": "F=MA_2025_15", + "context": "", + "question": "As shown in the figure, a uniform rigid rod of mass $M$ and length $2L$ is attached to a massless rod of length $L$, which is fixed at one end to the ceiling and free to rotate in a vertical plane. The massive rod is connected to the free end of the massless rod. Suppose an impulse $J$ is applied horizontally to the bottom of the massive rod.\n\nDetermine the relationship between the magnitudes of the angular velocity $\\omega$ of the massless rod and $\\Omega$ of the massive rod immediately after the impulse is applied. The moment of inertia of a uniform rod of length $d$ and mass $m$ about its center of mass is given by $I = \\frac{1}{12}m{d}^2$.\n\n(A) $\\Omega = \\frac{3}{4}\\omega$. \n(B) $\\Omega = \\frac{2}{3}\\omega$. \n(C) $\\Omega = \\frac{4}{3}\\omega$. \n(D) $\\Omega = \\frac{1}{12}\\omega$. \n(E) $\\Omega = \\frac{3}{2}\\omega$.", + "answer": [ + "\\boxed{E}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [ + "image_question/F=MA_2025_15_1.png" + ] + }, + { + "id": "F=MA_2025_16", + "context": "", + "question": "Two soap bubbles of radii $R_1 = 1 cm$ and $R_2 = 2 cm$ conjoin together in the air,such that a narrow bridge forms between them. Assuming the system starts in equilibrium, the bubbles are extremely thin, and that air can flow freely between the bubbles through the bridge, describe the evolution and final state of the bubbles.\n\n(A) The smaller bubble will shrink and the larger bubble will grow. \n(B) The larger bubble will shrink and the smaller bubble will grow. \n(C) The bubbles will maintain their sizes. \n(D) Air will oscillate between the two bubbles. \n(E) Both bubbles will simultaneously shrink.", + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [] + }, + { + "id": "F=MA_2025_17", + "context": "", + "question": "A particle of mass $m$ moves in the ${xy}$ plane with potential energy $U(x,y) = -k\\frac{x^2 + y^2}{2}$. The closest point to the origin $(x = 0, y = 0)$ during its motion was at a distance $d$, and the particle's speed at that point was $v \\neq 0$. Which of the following statements is true regarding the path of the particle after a long time $t$ ($t \\gg d/v$)?\n\n(A) The particle's trajectory will be circular. \n(B) The particle's trajectory will be asymptotic to a straight line pointing away from the origin. \n(C) The particle will spiral outwards away from the origin. \n(D) The particle will travel on a parabolic trajectory. \n(E) The particle will spiral inwards towards the origin.", + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [] + }, + { + "id": "F=MA_2025_18", + "context": "", + "question": "A particle of mass $m$ moves in the ${xy}$ plane with potential energy $U(x,y) = {kxy}/2$. If the particle begins at the origin, then it is possible to displace it slightly in some direction, so that the particle subsequently oscillates periodically. What is the period of this motion?\n\n(A) $2\\pi\\sqrt{m/{4k}}$. \n(B) $2\\pi\\sqrt{m/{2k}}$. \n(C) $2\\pi\\sqrt{m/k}$. \n(D) $2\\pi\\sqrt{{2m}/k}$. \n(E) $2\\pi\\sqrt{{4m}/k}$.", + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [] + }, + { + "id": "F=MA_2025_19", + "context": "", + "question": "Near the ground, wind speed can be modeled as proportional to height above the ground. (This is a reasonable assumption for small heights.) A wind turbine converts a constant fraction of the available kinetic energy into electricity. The conditions are such that when operating at $10 m$ above the ground,the turbine delivers $15 kW$ of power. How much power would the same windmill deliver if it were operating at $20 m$ above the ground?\n\n(A) $15 kW$. \n(B) $21 kW$. \n(C) $30 kW$. \n(D) $60 kW$. \n(E) $120 kW$.", + "answer": [ + "\\boxed{E}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [] + }, + { + "id": "F=MA_2025_20", + "context": "", + "question": "The International Space Station orbits the Earth in a circular orbit $400 km$ above the surface,and a full revolution takes 93 minutes. An astronaut on a space walk neglects safety precautions and tosses away a spanner at a speed of $1 m/s$ directly towards the Earth. You may assume that the Earth is a sphere of uniform density. At which of the following five times will the spanner be closest to the astronaut?\n\n(A) After 139.5 minutes. \n(B) After 131.5 minutes. \n(C) After 93 minutes. \n(D) After 46.5 minutes. \n(E) After 1 minute.", + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [] + }, + { + "id": "F=MA_2025_21", + "context": "As shown in the figure, water flows through a pipe with a radius of $5 cm$ at a velocity of $10 cm/s$ before entering a narrower section of pipe with a radius of $2.5 cm$.", + "question": "What is the difference in the speed of water between the two pipes?\n\n(A) $20 cm/s$. \n(B) $30 cm/s$. \n(C) $40 cm/s$. \n(D) $50 cm/s$. \n(E) $60 cm/s$.", + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [ + "image_question/F=MA_2025_21_1.png" + ] + }, + { + "id": "F=MA_2025_22", + "context": "As shown in the figure, water flows through a pipe with a radius of $5 cm$ at a velocity of $10 cm/s$ before entering a narrower section of pipe with a radius of $2.5 cm$.", + "question": "To measure this difference, two graduated cylinders are connected to the top of the pipe (one in the broad section and the other in the narrowed section). Water then flows up each pipe and the height the water reaches is measured. Estimate the difference in height between the two cylinders.\n\n(A) $6.3 mm$. \n(B) $7.5 mm$. \n(C) $8.2 mm$. \n(D) $12.2 mm$. \n(E) $12.7 mm$.", + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [ + "image_question/F=MA_2025_22_1.png" + ] + }, + { + "id": "F=MA_2025_23", + "context": "", + "question": "A student conducted an experiment to determine the spring constant of a spring using a ruler and two different weighing scales. The measured elongation of the spring was $1.5 cm$ ,and the smallest division on the ruler was $1 mm$ . The mass of the attached weight was measured using two different scales in the school laboratory,yielding values of $198 g$ and $210 g$ . The student also found that the local acceleration due to gravity in her city is given as $(9.806 \\pm 0.001) m/s^2$. Calculate the percent error in measuring the spring constant.\n\n(A) $2\\%$. \n(B) $4\\%$. \n(C) $8\\%$. \n(D) ${11}\\%$. \n(E) ${14}\\%$.", + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [] + }, + { + "id": "F=MA_2025_24", + "context": "", + "question": "As shown in the figure, a massive bead is attached to the end of a massless rigid rod of length $L$. The other end of the rod is attached to an ideal pivot, which allows it to rotate frictionlessly in any direction. The rod is initially at angle $\\theta$ to the horizontal, and there is no gravitational force. Next, the bead receives an impulse directly into the page, giving it a speed $v$. How long does it take for the bead to return to its original position?\n\n(A) $2\\pi L / v$. \n(B) $(2 \\pi L / v) \\sin \\theta$. \n(C) $(2 \\pi L / v) \\cos \\theta$. \n(D) $(2 \\pi L / v) \\cos^2 \\theta$. \n(E) $(2 \\pi L / v) \\cos^2 (2\\theta)$.", + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [ + "image_question/F=MA_2025_24_1.png" + ] + }, + { + "id": "F=MA_2025_25", + "context": "", + "question": "As shown in the figure, a puck of mass $m$ can slide on a frictionless inclined plane (prism). The prism has a much greater mass compared to the puck and is itself sliding without friction on a horizontal surface. The velocity of the prism is $v = \\sqrt{2gh}$, where $g$ is the acceleration due to gravity and $h$ is the height from which the puck starts sliding on the prism. The transition from the prism to the horizontal surface is smooth. The puck starts from rest relative to the prism. Find the final velocity of the puck once it begins sliding on the horizontal surface.\n\n(A) $\\frac{v}{2}$. \n(B) $\\frac{v}{\\sqrt{2}}$. \n(C) $v$. \n(D) $\\sqrt{2}v$. \n(E) $2v$.", + "answer": [ + "\\boxed{E}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "F=MA_2025", + "image_question": [ + "image_question/F=MA_2025_25_1.png" + ] + } +] \ No newline at end of file diff --git a/data/IPhO_2024.json b/data/IPhO_2024.json new file mode 100644 index 0000000000000000000000000000000000000000..3b8b400f9d99afe96663f2073d9ac60c85754c74 --- /dev/null +++ b/data/IPhO_2024.json @@ -0,0 +1,1209 @@ +[ + { + "information": "None." + }, + { + "id": "IPhO_2024_1_A_1", + "context": "[The Greenhouse Effect] \n\nIn 2021, Syukuro Manabe and Klaus Hasselmann were awarded half of the Nobel Prize in Physics for their work in modeling Earth's climate and accurately predicting the global warming caused by human industrial activities. In this problem, we will examine a simple model of global warming due to the greenhouse effect. The greenhouse gases alter the optical properties of the Earth's atmosphere in transmitting or absorbing Earth's infrared radiation, resulting in a rise in the average temperature of the planet.\n\nAll objects, at different temperatures, emit thermal radiation. The quantity $u(\\lambda, T) d \\lambda$ indicates the thermal radiative power per unit area of an object at temperature $T$ between the wavelengths $\\lambda$ and $\\lambda+d \\lambda$. According to Planck's theory of blackbody radiation, we have\n\n$$u(\\lambda, T)=\\frac{2 \\pi h c^{2}}{\\lambda^{5}} \\frac{1}{\\exp \\left(\\frac{h c}{\\lambda k_{\\mathrm{B}} T}\\right)-1}$$ (Equation 1), in which $h c=1.24 \\times 10^{3} \\mathrm{eV} \\cdot \\mathrm{nm}$ and $k_{\\mathrm{B}}=8.62 \\times 10^{-5} \\mathrm{eV} / \\mathrm{K}$. The wavelength corresponding to the maximum of $u(\\lambda, T)$ comes from the relation $\\lambda_{\\max} T=b$ (Wien's displacement law). Indeed, using equation (1), it can be shown that $b=\\frac{h c}{x_{\\mathrm{m}} k_{\\mathrm{B}}}$, where the dimensionless quantity $x_{\\mathrm{m}}$ is the non-trivial root of an equation of the form $f(x)=0$; you are asked to find the function $f(x)$ in one of the following tasks. Total radiative power per unit area of a blackbody in all wavelengths is given by the Stephan-Boltzmann law as $U(T)=\\sigma T^{4}$ where $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} / \\mathrm{m}^{2} \\mathrm{K}^{4}$. Moreover, according to Kirchhoff's law of radiation, at thermal equilibrium a body absorbing a certain fraction of the incident radiation at a specific wavelength, will radiate the same fraction of the blackbody radiation at that same wavelength.\n\nThroughout this problem assume that the Sun is a blackbody at its average surface temperature of $T_{\\mathrm{S}}=5.77 \\times 10^{3} \\mathrm{K}$. The Sun's radius is $R_{\\mathrm{S}}=6.96 \\times 10^{8} \\mathrm{m}$ and the average distance between the Earth and the Sun is $d=1.50 \\times 10^{11} \\mathrm{m}$. We denote by $\\tilde{u}_{S}(\\lambda)$, the spectral solar power radiated into a unit area of the Earth normal to the direction of radiation. The integral of this quantity over all wavelengths, i.e. $S_{0}=\\int \\tilde{u}_{S}(\\lambda) d \\lambda$, is called the solar constant.\n\nIn this problem assume that the Earth is in thermal equilibrium and has the same temperature at all points on its surface. In all parts of the problem, express the desired quantity in parametric form in terms of the data given in the problem and then find its numerical value accurate to three significant figures. The required units are indicated on the answer sheet.\n\n[Earth as a Blackbody]\n\nIn this part, consider the Earth's surface as a blackbody and neglect the Earth's atmosphere.", + "question": "(1) Find the expression of the solar constant $S_0$. \n(2) Calculate the value of $S_0$ (expressed in $W/m^2$).", + "marking": [ + [ + "Award 0.4 pt if the answer gives the correct expression for the solar constant: $S_0 = \\sigma T_S^4 (\\frac{R_S}{d})^2$. Partial points: award 0.1 pt if the answer gives the incorrect expression but realizes energy conservation. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct numerical value of the solar constant: $1.35 \\times 10^{3} \\frac{W}{m^2}$. Partial points: award 0.1 pt if the numerical answer falls within the acceptable error range of the correct value but contains more than four significant figures. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$S_0 = \\sigma T_S^4 (\\frac{R_S}{d})^2$}", + "\\boxed{$1.35 \\times 10^{3}$}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null, + "$W/m^2$" + ], + "points": [ + 0.4, + 0.2 + ], + "modality": "text-only", + "field": "Thermodynamics", + "source": "IPhO_2024", + "image_question": [] + }, + { + "id": "IPhO_2024_1_A_2", + "context": "[The Greenhouse Effect] \n\nIn 2021, Syukuro Manabe and Klaus Hasselmann were awarded half of the Nobel Prize in Physics for their work in modeling Earth's climate and accurately predicting the global warming caused by human industrial activities. In this problem, we will examine a simple model of global warming due to the greenhouse effect. The greenhouse gases alter the optical properties of the Earth's atmosphere in transmitting or absorbing Earth's infrared radiation, resulting in a rise in the average temperature of the planet.\n\nAll objects, at different temperatures, emit thermal radiation. The quantity $u(\\lambda, T) d \\lambda$ indicates the thermal radiative power per unit area of an object at temperature $T$ between the wavelengths $\\lambda$ and $\\lambda+d \\lambda$. According to Planck's theory of blackbody radiation, we have\n\n$$u(\\lambda, T)=\\frac{2 \\pi h c^{2}}{\\lambda^{5}} \\frac{1}{\\exp \\left(\\frac{h c}{\\lambda k_{\\mathrm{B}} T}\\right)-1}$$ (Equation 1), in which $h c=1.24 \\times 10^{3} \\mathrm{eV} \\cdot \\mathrm{nm}$ and $k_{\\mathrm{B}}=8.62 \\times 10^{-5} \\mathrm{eV} / \\mathrm{K}$. The wavelength corresponding to the maximum of $u(\\lambda, T)$ comes from the relation $\\lambda_{\\max} T=b$ (Wien's displacement law). Indeed, using equation (1), it can be shown that $b=\\frac{h c}{x_{\\mathrm{m}} k_{\\mathrm{B}}}$, where the dimensionless quantity $x_{\\mathrm{m}}$ is the non-trivial root of an equation of the form $f(x)=0$; you are asked to find the function $f(x)$ in one of the following tasks. Total radiative power per unit area of a blackbody in all wavelengths is given by the Stephan-Boltzmann law as $U(T)=\\sigma T^{4}$ where $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} / \\mathrm{m}^{2} \\mathrm{K}^{4}$. Moreover, according to Kirchhoff's law of radiation, at thermal equilibrium a body absorbing a certain fraction of the incident radiation at a specific wavelength, will radiate the same fraction of the blackbody radiation at that same wavelength.\n\nThroughout this problem assume that the Sun is a blackbody at its average surface temperature of $T_{\\mathrm{S}}=5.77 \\times 10^{3} \\mathrm{K}$. The Sun's radius is $R_{\\mathrm{S}}=6.96 \\times 10^{8} \\mathrm{m}$ and the average distance between the Earth and the Sun is $d=1.50 \\times 10^{11} \\mathrm{m}$. We denote by $\\tilde{u}_{S}(\\lambda)$, the spectral solar power radiated into a unit area of the Earth normal to the direction of radiation. The integral of this quantity over all wavelengths, i.e. $S_{0}=\\int \\tilde{u}_{S}(\\lambda) d \\lambda$, is called the solar constant.\n\nIn this problem assume that the Earth is in thermal equilibrium and has the same temperature at all points on its surface. In all parts of the problem, express the desired quantity in parametric form in terms of the data given in the problem and then find its numerical value accurate to three significant figures. The required units are indicated on the answer sheet.\n\n[Earth as a Blackbody]\n\nIn this part, consider the Earth's surface as a blackbody and neglect the Earth's atmosphere.", + "question": "(1) Find the expression of the Earth's temperature $T_{\\mathrm{E}}$. \n(2) Calculate the value of $T_{\\mathrm{E}}$ (expressed in $\\mathrm{K}$).", + "marking": [ + [ + "Award 0.4 pt if the answer gives the correct expression for the Earth's temperature: $T_{\\mathrm{E}} = (\\frac{S_0}{4 \\sigma})^{1/4} = \\sqrt{\\frac{R_S}{2d}} T_S$. Partial points: award 0.1 pt if the answer gives the incorrect expression but realizes energy balance. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct numerical value of the Earth's temperature: $278 \\mathrm{K}$. Partial points: award 0.1 pt if the numerical answer falls within the acceptable error range of the correct value but contains more than four significant figures. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$T_{\\mathrm{E}} = (\\frac{S_0}{4 \\sigma})^{1/4} = \\sqrt{\\frac{R_S}{2d}} T_S$}", + "\\boxed{278}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null, + "$\\mathrm{K}$" + ], + "points": [ + 0.4, + 0.2 + ], + "modality": "text-only", + "field": "Thermodynamics", + "source": "IPhO_2024", + "image_question": [] + }, + { + "id": "IPhO_2024_1_A_3", + "context": "[The Greenhouse Effect] \n\nIn 2021, Syukuro Manabe and Klaus Hasselmann were awarded half of the Nobel Prize in Physics for their work in modeling Earth's climate and accurately predicting the global warming caused by human industrial activities. In this problem, we will examine a simple model of global warming due to the greenhouse effect. The greenhouse gases alter the optical properties of the Earth's atmosphere in transmitting or absorbing Earth's infrared radiation, resulting in a rise in the average temperature of the planet.\n\nAll objects, at different temperatures, emit thermal radiation. The quantity $u(\\lambda, T) d \\lambda$ indicates the thermal radiative power per unit area of an object at temperature $T$ between the wavelengths $\\lambda$ and $\\lambda+d \\lambda$. According to Planck's theory of blackbody radiation, we have\n\n$$u(\\lambda, T)=\\frac{2 \\pi h c^{2}}{\\lambda^{5}} \\frac{1}{\\exp \\left(\\frac{h c}{\\lambda k_{\\mathrm{B}} T}\\right)-1}$$ (Equation 1), in which $h c=1.24 \\times 10^{3} \\mathrm{eV} \\cdot \\mathrm{nm}$ and $k_{\\mathrm{B}}=8.62 \\times 10^{-5} \\mathrm{eV} / \\mathrm{K}$. The wavelength corresponding to the maximum of $u(\\lambda, T)$ comes from the relation $\\lambda_{\\max} T=b$ (Wien's displacement law). Indeed, using equation (1), it can be shown that $b=\\frac{h c}{x_{\\mathrm{m}} k_{\\mathrm{B}}}$, where the dimensionless quantity $x_{\\mathrm{m}}$ is the non-trivial root of an equation of the form $f(x)=0$; you are asked to find the function $f(x)$ in one of the following tasks. Total radiative power per unit area of a blackbody in all wavelengths is given by the Stephan-Boltzmann law as $U(T)=\\sigma T^{4}$ where $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} / \\mathrm{m}^{2} \\mathrm{K}^{4}$. Moreover, according to Kirchhoff's law of radiation, at thermal equilibrium a body absorbing a certain fraction of the incident radiation at a specific wavelength, will radiate the same fraction of the blackbody radiation at that same wavelength.\n\nThroughout this problem assume that the Sun is a blackbody at its average surface temperature of $T_{\\mathrm{S}}=5.77 \\times 10^{3} \\mathrm{K}$. The Sun's radius is $R_{\\mathrm{S}}=6.96 \\times 10^{8} \\mathrm{m}$ and the average distance between the Earth and the Sun is $d=1.50 \\times 10^{11} \\mathrm{m}$. We denote by $\\tilde{u}_{S}(\\lambda)$, the spectral solar power radiated into a unit area of the Earth normal to the direction of radiation. The integral of this quantity over all wavelengths, i.e. $S_{0}=\\int \\tilde{u}_{S}(\\lambda) d \\lambda$, is called the solar constant.\n\nIn this problem assume that the Earth is in thermal equilibrium and has the same temperature at all points on its surface. In all parts of the problem, express the desired quantity in parametric form in terms of the data given in the problem and then find its numerical value accurate to three significant figures. The required units are indicated on the answer sheet.\n\n[Earth as a Blackbody]\n\nIn this part, consider the Earth's surface as a blackbody and neglect the Earth's atmosphere.", + "question": "Find the function $f(x)$.", + "marking": [ + [ + "Award 0.4 pt if the answer gives the correct expression for the function $f(x)$: $f(x) = 5(1-e^{-x})-x$ (Equivalent forms are also correct, e.g., $f(x) = (5-x)e^x-5 = 5e^x -5 - x e^x$). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$f(x) = 5(1-e^{-x})-x$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.4 + ], + "modality": "text-only", + "field": "Thermodynamics", + "source": "IPhO_2024", + "image_question": [] + }, + { + "id": "IPhO_2024_1_A_4", + "context": "[The Greenhouse Effect] \n\nIn 2021, Syukuro Manabe and Klaus Hasselmann were awarded half of the Nobel Prize in Physics for their work in modeling Earth's climate and accurately predicting the global warming caused by human industrial activities. In this problem, we will examine a simple model of global warming due to the greenhouse effect. The greenhouse gases alter the optical properties of the Earth's atmosphere in transmitting or absorbing Earth's infrared radiation, resulting in a rise in the average temperature of the planet.\n\nAll objects, at different temperatures, emit thermal radiation. The quantity $u(\\lambda, T) d \\lambda$ indicates the thermal radiative power per unit area of an object at temperature $T$ between the wavelengths $\\lambda$ and $\\lambda+d \\lambda$. According to Planck's theory of blackbody radiation, we have\n\n$$u(\\lambda, T)=\\frac{2 \\pi h c^{2}}{\\lambda^{5}} \\frac{1}{\\exp \\left(\\frac{h c}{\\lambda k_{\\mathrm{B}} T}\\right)-1}$$ (Equation 1), in which $h c=1.24 \\times 10^{3} \\mathrm{eV} \\cdot \\mathrm{nm}$ and $k_{\\mathrm{B}}=8.62 \\times 10^{-5} \\mathrm{eV} / \\mathrm{K}$. The wavelength corresponding to the maximum of $u(\\lambda, T)$ comes from the relation $\\lambda_{\\max} T=b$ (Wien's displacement law). Indeed, using equation (1), it can be shown that $b=\\frac{h c}{x_{\\mathrm{m}} k_{\\mathrm{B}}}$, where the dimensionless quantity $x_{\\mathrm{m}}$ is the non-trivial root of an equation of the form $f(x)=0$; you are asked to find the function $f(x)$ in one of the following tasks. Total radiative power per unit area of a blackbody in all wavelengths is given by the Stephan-Boltzmann law as $U(T)=\\sigma T^{4}$ where $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} / \\mathrm{m}^{2} \\mathrm{K}^{4}$. Moreover, according to Kirchhoff's law of radiation, at thermal equilibrium a body absorbing a certain fraction of the incident radiation at a specific wavelength, will radiate the same fraction of the blackbody radiation at that same wavelength.\n\nThroughout this problem assume that the Sun is a blackbody at its average surface temperature of $T_{\\mathrm{S}}=5.77 \\times 10^{3} \\mathrm{K}$. The Sun's radius is $R_{\\mathrm{S}}=6.96 \\times 10^{8} \\mathrm{m}$ and the average distance between the Earth and the Sun is $d=1.50 \\times 10^{11} \\mathrm{m}$. We denote by $\\tilde{u}_{S}(\\lambda)$, the spectral solar power radiated into a unit area of the Earth normal to the direction of radiation. The integral of this quantity over all wavelengths, i.e. $S_{0}=\\int \\tilde{u}_{S}(\\lambda) d \\lambda$, is called the solar constant.\n\nIn this problem assume that the Earth is in thermal equilibrium and has the same temperature at all points on its surface. In all parts of the problem, express the desired quantity in parametric form in terms of the data given in the problem and then find its numerical value accurate to three significant figures. The required units are indicated on the answer sheet.\n\n[Earth as a Blackbody]\n\nIn this part, consider the Earth's surface as a blackbody and neglect the Earth's atmosphere.", + "question": "(1) Calculate the numerical value of $x_{\\mathrm{m}}$. \n(2) From this value $x_{\\mathrm{m}}$, find the value of $b$ (expressed in $\\mathrm{nm} \\cdot K$).", + "marking": [ + [ + "Award 0.3 pt if the answer gives the correct numerical value of $x_{\\mathrm{m}}$ within the range of $[4.96, 4.97]$. Partial points: award 0.2 pt if the answer gives a value of $x_{\\mathrm{m}}$ within the range of $[4.96, 4.97]$ but contains more than four significant figures. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct numerical value of $b$ within the range of $[2.89 \\times 10^{6}, 2.90 \\times 10^{6}]$ $\\mathrm{nm} \\cdot K$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$[4.96, 4.97]$}", + "\\boxed{$[2.89 \\times 10^{6}, 2.90 \\times 10^{6}]$}" + ], + "answer_type": [ + "Numerical Value", + "Numerical Value" + ], + "unit": [ + null, + "$\\mathrm{nm} \\cdot K$" + ], + "points": [ + 0.3, + 0.1 + ], + "modality": "text-only", + "field": "Thermodynamics", + "source": "IPhO_2024", + "image_question": [] + }, + { + "id": "IPhO_2024_1_A_5", + "context": "[The Greenhouse Effect] \n\nIn 2021, Syukuro Manabe and Klaus Hasselmann were awarded half of the Nobel Prize in Physics for their work in modeling Earth's climate and accurately predicting the global warming caused by human industrial activities. In this problem, we will examine a simple model of global warming due to the greenhouse effect. The greenhouse gases alter the optical properties of the Earth's atmosphere in transmitting or absorbing Earth's infrared radiation, resulting in a rise in the average temperature of the planet.\n\nAll objects, at different temperatures, emit thermal radiation. The quantity $u(\\lambda, T) d \\lambda$ indicates the thermal radiative power per unit area of an object at temperature $T$ between the wavelengths $\\lambda$ and $\\lambda+d \\lambda$. According to Planck's theory of blackbody radiation, we have\n\n$$u(\\lambda, T)=\\frac{2 \\pi h c^{2}}{\\lambda^{5}} \\frac{1}{\\exp \\left(\\frac{h c}{\\lambda k_{\\mathrm{B}} T}\\right)-1}$$ (Equation 1), in which $h c=1.24 \\times 10^{3} \\mathrm{eV} \\cdot \\mathrm{nm}$ and $k_{\\mathrm{B}}=8.62 \\times 10^{-5} \\mathrm{eV} / \\mathrm{K}$. The wavelength corresponding to the maximum of $u(\\lambda, T)$ comes from the relation $\\lambda_{\\max} T=b$ (Wien's displacement law). Indeed, using equation (1), it can be shown that $b=\\frac{h c}{x_{\\mathrm{m}} k_{\\mathrm{B}}}$, where the dimensionless quantity $x_{\\mathrm{m}}$ is the non-trivial root of an equation of the form $f(x)=0$; you are asked to find the function $f(x)$ in one of the following tasks. Total radiative power per unit area of a blackbody in all wavelengths is given by the Stephan-Boltzmann law as $U(T)=\\sigma T^{4}$ where $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} / \\mathrm{m}^{2} \\mathrm{K}^{4}$. Moreover, according to Kirchhoff's law of radiation, at thermal equilibrium a body absorbing a certain fraction of the incident radiation at a specific wavelength, will radiate the same fraction of the blackbody radiation at that same wavelength.\n\nThroughout this problem assume that the Sun is a blackbody at its average surface temperature of $T_{\\mathrm{S}}=5.77 \\times 10^{3} \\mathrm{K}$. The Sun's radius is $R_{\\mathrm{S}}=6.96 \\times 10^{8} \\mathrm{m}$ and the average distance between the Earth and the Sun is $d=1.50 \\times 10^{11} \\mathrm{m}$. We denote by $\\tilde{u}_{S}(\\lambda)$, the spectral solar power radiated into a unit area of the Earth normal to the direction of radiation. The integral of this quantity over all wavelengths, i.e. $S_{0}=\\int \\tilde{u}_{S}(\\lambda) d \\lambda$, is called the solar constant.\n\nIn this problem assume that the Earth is in thermal equilibrium and has the same temperature at all points on its surface. In all parts of the problem, express the desired quantity in parametric form in terms of the data given in the problem and then find its numerical value accurate to three significant figures. The required units are indicated on the answer sheet.\n\n[Earth as a Blackbody]\n\nIn this part, consider the Earth's surface as a blackbody and neglect the Earth's atmosphere.", + "question": "(1) Find $\\lambda_{\\text{max}}^{\\text{Sun}}$ for the Sun (expressed in $\\mathrm{nm}$). \n(2) Find $\\lambda_{\\text{max}}^{\\text{Earth}}$ for the Earth (expressed in $\\mathrm{nm}$).", + "marking": [ + [ + "Award 0.1 pt if the answer gives the correct numerical value of $\\lambda_{\\text{max}}^{\\text{Sun}}$ within the range of $[501, 502] \\mathrm{nm}$ (Equivalent form of $[5.01 \\times 10^{2} \\mathrm{nm}, 5.02 \\times 10^{2} \\mathrm{nm}]$ is also correct). Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct numerical value of $\\lambda_{\\text{max}}^{\\text{Earth}}$ as $1.04 \\times 10^{4} \\mathrm{nm}$ (Equivalent forms are also correct). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$[5.01 \\times 10^{2}, 5.02 \\times 10^{2}]$}", + "\\boxed{$1.04 \\times 10^{4}$}" + ], + "answer_type": [ + "Numerical Value", + "Numerical Value" + ], + "unit": [ + "$\\mathrm{nm}$", + "$\\mathrm{nm}$" + ], + "points": [ + 0.1, + 0.1 + ], + "modality": "text-only", + "field": "Thermodynamics", + "source": "IPhO_2024", + "image_question": [] + }, + { + "id": "IPhO_2024_1_A_6", + "context": "[The Greenhouse Effect] \n\nIn 2021, Syukuro Manabe and Klaus Hasselmann were awarded half of the Nobel Prize in Physics for their work in modeling Earth's climate and accurately predicting the global warming caused by human industrial activities. In this problem, we will examine a simple model of global warming due to the greenhouse effect. The greenhouse gases alter the optical properties of the Earth's atmosphere in transmitting or absorbing Earth's infrared radiation, resulting in a rise in the average temperature of the planet.\n\nAll objects, at different temperatures, emit thermal radiation. The quantity $u(\\lambda, T) d \\lambda$ indicates the thermal radiative power per unit area of an object at temperature $T$ between the wavelengths $\\lambda$ and $\\lambda+d \\lambda$. According to Planck's theory of blackbody radiation, we have\n\n$$u(\\lambda, T)=\\frac{2 \\pi h c^{2}}{\\lambda^{5}} \\frac{1}{\\exp \\left(\\frac{h c}{\\lambda k_{\\mathrm{B}} T}\\right)-1}$$ (Equation 1), in which $h c=1.24 \\times 10^{3} \\mathrm{eV} \\cdot \\mathrm{nm}$ and $k_{\\mathrm{B}}=8.62 \\times 10^{-5} \\mathrm{eV} / \\mathrm{K}$. The wavelength corresponding to the maximum of $u(\\lambda, T)$ comes from the relation $\\lambda_{\\max} T=b$ (Wien's displacement law). Indeed, using equation (1), it can be shown that $b=\\frac{h c}{x_{\\mathrm{m}} k_{\\mathrm{B}}}$, where the dimensionless quantity $x_{\\mathrm{m}}$ is the non-trivial root of an equation of the form $f(x)=0$; you are asked to find the function $f(x)$ in one of the following tasks. Total radiative power per unit area of a blackbody in all wavelengths is given by the Stephan-Boltzmann law as $U(T)=\\sigma T^{4}$ where $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} / \\mathrm{m}^{2} \\mathrm{K}^{4}$. Moreover, according to Kirchhoff's law of radiation, at thermal equilibrium a body absorbing a certain fraction of the incident radiation at a specific wavelength, will radiate the same fraction of the blackbody radiation at that same wavelength.\n\nThroughout this problem assume that the Sun is a blackbody at its average surface temperature of $T_{\\mathrm{S}}=5.77 \\times 10^{3} \\mathrm{K}$. The Sun's radius is $R_{\\mathrm{S}}=6.96 \\times 10^{8} \\mathrm{m}$ and the average distance between the Earth and the Sun is $d=1.50 \\times 10^{11} \\mathrm{m}$. We denote by $\\tilde{u}_{S}(\\lambda)$, the spectral solar power radiated into a unit area of the Earth normal to the direction of radiation. The integral of this quantity over all wavelengths, i.e. $S_{0}=\\int \\tilde{u}_{S}(\\lambda) d \\lambda$, is called the solar constant.\n\nIn this problem assume that the Earth is in thermal equilibrium and has the same temperature at all points on its surface. In all parts of the problem, express the desired quantity in parametric form in terms of the data given in the problem and then find its numerical value accurate to three significant figures. The required units are indicated on the answer sheet.\n\n[Earth as a Blackbody]\n\nIn this part, consider the Earth's surface as a blackbody and neglect the Earth's atmosphere.\n\nAs shown in the Figure 1, the functions $\\gamma \\tilde{u}_{\\mathrm{S}}(\\lambda)$ and $u\\left(\\lambda, T_{\\mathrm{E}}\\right)$ are plotted versus $\\lambda$, where $\\gamma$ is a dimensionless coefficient to rescale $\\tilde{u}_{S}(\\lambda)$ such that the values of the two peaks coincide.\n\n[figure1]\nFigure 1. The plot of $u(\\lambda, T_{\\mathrm{E}})$ (red) and $\\gamma \\tilde{u}_{S}(\\lambda)$ (blue) versus $\\lambda$.", + "question": "(1) Find the expression of $\\gamma$. \n(2) Determine the value of $\\gamma$.", + "marking": [ + [ + "Award 0.6 pt if the answer gives the correct expression for $\\gamma$: $\\gamma = (\\frac{d}{R_S})^2 \\times (\\frac{T_E}{T_S})^5 = (\\frac{d}{R_S})^2 \\times (\\frac{\\lambda_S}{\\lambda_E})^5$. Partial points: award 0.3 pt if the answer realizes that $\\tilde{u}_S = (\\frac{R_S}{d})^2 u_S(\\lambda)$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct numerical value of $\\gamma$ within the range of $[1.20 \\times 10^{-2}, 1.21 \\times 10^{-2}]$. Partial points: award 0.1 pt if the numerical answer falls within the acceptable error range of the correct value but contains more than four significant figures. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\gamma = (\\frac{d}{R_S})^2 \\times (\\frac{T_E}{T_S})^5 = (\\frac{d}{R_S})^2 \\times (\\frac{\\lambda_S}{\\lambda_E})^5$}", + "\\boxed{$[1.20 \\times 10^{-2}, 1.21 \\times 10^{-2}]$}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null, + null + ], + "points": [ + 0.6, + 0.2 + ], + "modality": "text+data figure", + "field": "Thermodynamics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_1_A_6_1.png" + ] + }, + { + "id": "IPhO_2024_1_B_1", + "context": "[The Greenhouse Effect] \n\nIn 2021, Syukuro Manabe and Klaus Hasselmann were awarded half of the Nobel Prize in Physics for their work in modeling Earth's climate and accurately predicting the global warming caused by human industrial activities. In this problem, we will examine a simple model of global warming due to the greenhouse effect. The greenhouse gases alter the optical properties of the Earth's atmosphere in transmitting or absorbing Earth's infrared radiation, resulting in a rise in the average temperature of the planet.\n\nAll objects, at different temperatures, emit thermal radiation. The quantity $u(\\lambda, T) d \\lambda$ indicates the thermal radiative power per unit area of an object at temperature $T$ between the wavelengths $\\lambda$ and $\\lambda+d \\lambda$. According to Planck's theory of blackbody radiation, we have\n\n$$u(\\lambda, T)=\\frac{2 \\pi h c^{2}}{\\lambda^{5}} \\frac{1}{\\exp \\left(\\frac{h c}{\\lambda k_{\\mathrm{B}} T}\\right)-1}$$ (Equation 1), in which $h c=1.24 \\times 10^{3} \\mathrm{eV} \\cdot \\mathrm{nm}$ and $k_{\\mathrm{B}}=8.62 \\times 10^{-5} \\mathrm{eV} / \\mathrm{K}$. The wavelength corresponding to the maximum of $u(\\lambda, T)$ comes from the relation $\\lambda_{\\max} T=b$ (Wien's displacement law). Indeed, using equation (1), it can be shown that $b=\\frac{h c}{x_{\\mathrm{m}} k_{\\mathrm{B}}}$, where the dimensionless quantity $x_{\\mathrm{m}}$ is the non-trivial root of an equation of the form $f(x)=0$; you are asked to find the function $f(x)$ in one of the following tasks. Total radiative power per unit area of a blackbody in all wavelengths is given by the Stephan-Boltzmann law as $U(T)=\\sigma T^{4}$ where $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} / \\mathrm{m}^{2} \\mathrm{K}^{4}$. Moreover, according to Kirchhoff's law of radiation, at thermal equilibrium a body absorbing a certain fraction of the incident radiation at a specific wavelength, will radiate the same fraction of the blackbody radiation at that same wavelength.\n\nThroughout this problem assume that the Sun is a blackbody at its average surface temperature of $T_{\\mathrm{S}}=5.77 \\times 10^{3} \\mathrm{K}$. The Sun's radius is $R_{\\mathrm{S}}=6.96 \\times 10^{8} \\mathrm{m}$ and the average distance between the Earth and the Sun is $d=1.50 \\times 10^{11} \\mathrm{m}$. We denote by $\\tilde{u}_{S}(\\lambda)$, the spectral solar power radiated into a unit area of the Earth normal to the direction of radiation. The integral of this quantity over all wavelengths, i.e. $S_{0}=\\int \\tilde{u}_{S}(\\lambda) d \\lambda$, is called the solar constant.\n\nIn this problem assume that the Earth is in thermal equilibrium and has the same temperature at all points on its surface. In all parts of the problem, express the desired quantity in parametric form in terms of the data given in the problem and then find its numerical value accurate to three significant figures. The required units are indicated on the answer sheet.\n\n[The Greenhouse Effect]\n\nIn this part, we introduce a simple model in which the Earth's atmosphere is modeled as a thin layer at a small distance above the Earth's surface so that the difference between the area of the atmosphere's layer and the area of the Earth's surface can be neglected (see Figure 1). In what follows assume that the major part of the thermal radiation from the Earth and the Sun are emitted at wavelengths near the $\\lambda_{\\max}$ for each one. Also assume that the \"atmosphere layer\" reflects a fraction $r_{\\mathrm{A}}=0.255$ of the visible-ultraviolet radiation incident from above or below, and completely transmits the rest. Assume that the atmosphere does not reflect any part of the infrared radiation, however, it absorbs a fraction $\\varepsilon$ of the infrared radiation and transmits the rest. This behavior, known as the greenhouse effect, changes the average temperature of the Earth. The Earth's surface, on the other hand, reflects a fraction $r_{E}$ of the visible-ultraviolet radiation and absorbs the rest of this radiation and all the infrared radiation.\n\n[figure1]\nFigure 1. Thermal flows between the Earth and the atmosphere.", + "question": "Assume that $\\varepsilon=1$ and $r_{\\mathrm{E}}=0$. (1) Find the expression of the Earth's temperature $T_{\\mathrm{E}}$; (2) Find the expression of the atmosphere's temperature $T_{\\mathrm{A}}$; (3) Calculate the numerical value of $T_{\\mathrm{E}}$ (expressed in $K$); (4) (3) Calculate the numerical value of $T_{\\mathrm{A}}$ (expressed in $K$).", + "marking": [ + [ + "Award 0.8 pt if the answer gives two correct expressions: $T_{\\mathrm{E}} = \\left( \\frac{(1-r_A) \\frac{S_0}{2}}{\\sigma} \\right)^{1/4}$, and $T_{\\mathrm{A}} = \\left( \\frac{(1-r_A) \\frac{S_0}{4}}{\\sigma} \\right)^{1/4}$. Partial points: award 0.6 pt if the answer gives only one of the two expressions correctly; or award 0.2 pt if the answer gives no correct expressions but realizes each energy balance relation. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct numerical value of $T_{\\mathrm{E}}$ as $3.07 \\times 10^{2} K$, and $T_{\\mathrm{A}}$ as $2.58 \\times 10^{2} K$. Partial points: award 0.1 pt if the numerical answer falls within the acceptable error range of the correct value but contains more than four significant figures. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$T_{\\mathrm{E}} = \\left( \\frac{(1-r_A) \\frac{S_0}{2}}{\\sigma} \\right)^{1/4}$}", + "\\boxed{$T_{\\mathrm{A}} = \\left( \\frac{(1-r_A) \\frac{S_0}{4}}{\\sigma} \\right)^{1/4}$}", + "\\boxed{$T_E = 3.07 \\times 10^2$}", + "\\boxed{$T_A = 2.58 \\times 10^2$}" + ], + "answer_type": [ + "Expression", + "Expression", + "Numerical Value", + "Numerical Value" + ], + "unit": [ + null, + null, + "K", + "K" + ], + "points": [ + 0.4, + 0.4, + 0.1, + 0.1 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_1_B_1_1.png" + ] + }, + { + "id": "IPhO_2024_1_B_2", + "context": "[The Greenhouse Effect] \n\nIn 2021, Syukuro Manabe and Klaus Hasselmann were awarded half of the Nobel Prize in Physics for their work in modeling Earth's climate and accurately predicting the global warming caused by human industrial activities. In this problem, we will examine a simple model of global warming due to the greenhouse effect. The greenhouse gases alter the optical properties of the Earth's atmosphere in transmitting or absorbing Earth's infrared radiation, resulting in a rise in the average temperature of the planet.\n\nAll objects, at different temperatures, emit thermal radiation. The quantity $u(\\lambda, T) d \\lambda$ indicates the thermal radiative power per unit area of an object at temperature $T$ between the wavelengths $\\lambda$ and $\\lambda+d \\lambda$. According to Planck's theory of blackbody radiation, we have\n\n$$u(\\lambda, T)=\\frac{2 \\pi h c^{2}}{\\lambda^{5}} \\frac{1}{\\exp \\left(\\frac{h c}{\\lambda k_{\\mathrm{B}} T}\\right)-1}$$ (Equation 1), in which $h c=1.24 \\times 10^{3} \\mathrm{eV} \\cdot \\mathrm{nm}$ and $k_{\\mathrm{B}}=8.62 \\times 10^{-5} \\mathrm{eV} / \\mathrm{K}$. The wavelength corresponding to the maximum of $u(\\lambda, T)$ comes from the relation $\\lambda_{\\max} T=b$ (Wien's displacement law). Indeed, using equation (1), it can be shown that $b=\\frac{h c}{x_{\\mathrm{m}} k_{\\mathrm{B}}}$, where the dimensionless quantity $x_{\\mathrm{m}}$ is the non-trivial root of an equation of the form $f(x)=0$; you are asked to find the function $f(x)$ in one of the following tasks. Total radiative power per unit area of a blackbody in all wavelengths is given by the Stephan-Boltzmann law as $U(T)=\\sigma T^{4}$ where $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} / \\mathrm{m}^{2} \\mathrm{K}^{4}$. Moreover, according to Kirchhoff's law of radiation, at thermal equilibrium a body absorbing a certain fraction of the incident radiation at a specific wavelength, will radiate the same fraction of the blackbody radiation at that same wavelength.\n\nThroughout this problem assume that the Sun is a blackbody at its average surface temperature of $T_{\\mathrm{S}}=5.77 \\times 10^{3} \\mathrm{K}$. The Sun's radius is $R_{\\mathrm{S}}=6.96 \\times 10^{8} \\mathrm{m}$ and the average distance between the Earth and the Sun is $d=1.50 \\times 10^{11} \\mathrm{m}$. We denote by $\\tilde{u}_{S}(\\lambda)$, the spectral solar power radiated into a unit area of the Earth normal to the direction of radiation. The integral of this quantity over all wavelengths, i.e. $S_{0}=\\int \\tilde{u}_{S}(\\lambda) d \\lambda$, is called the solar constant.\n\nIn this problem assume that the Earth is in thermal equilibrium and has the same temperature at all points on its surface. In all parts of the problem, express the desired quantity in parametric form in terms of the data given in the problem and then find its numerical value accurate to three significant figures. The required units are indicated on the answer sheet.\n\n[The Greenhouse Effect]\n\nIn this part, we introduce a simple model in which the Earth's atmosphere is modeled as a thin layer at a small distance above the Earth's surface so that the difference between the area of the atmosphere's layer and the area of the Earth's surface can be neglected (see Figure 1). In what follows assume that the major part of the thermal radiation from the Earth and the Sun are emitted at wavelengths near the $\\lambda_{\\max}$ for each one. Also assume that the \"atmosphere layer\" reflects a fraction $r_{\\mathrm{A}}=0.255$ of the visible-ultraviolet radiation incident from above or below, and completely transmits the rest. Assume that the atmosphere does not reflect any part of the infrared radiation, however, it absorbs a fraction $\\varepsilon$ of the infrared radiation and transmits the rest. This behavior, known as the greenhouse effect, changes the average temperature of the Earth. The Earth's surface, on the other hand, reflects a fraction $r_{E}$ of the visible-ultraviolet radiation and absorbs the rest of this radiation and all the infrared radiation.\n\n[figure1]\nFigure 1. Thermal flows between the Earth and the atmosphere.\n\nNow assume that $r_{\\mathrm{E}} \\neq 0$. In this case, the combined system of \"Earth + atmosphere\" reflects a different fraction of the solar radiation, called \"albedo\" and denoted by $\\alpha$.", + "question": "(1) Determine the albedo, $\\alpha$, in terms of $r_{\\mathrm{E}}$ and $r_{\\mathrm{A}}$. \n(2) Calculate the numerical value of $\\alpha$, assuming $r_{\\mathrm{E}}=0.102$ and $r_{\\mathrm{A}}=0.255$.", + "marking": [ + [ + "Award 1.4 pt if the answer gives the correct expression for $\\alpha$: $\\alpha = r_{\\mathrm{A}} + \\frac{(1-r_{\\mathrm{A}})^{2} r_{\\mathrm{E}}}{1-r_{\\mathrm{A}} r_{\\mathrm{E}}}$. If the final expression is incorrect, evaluate the following four intermediate results separately to award partial points (points from each item may be added together): \n(1) Award 0.1 pt if the answer contains an intermediate result of $\\tilde{S}_0 = r_A S_0$. \n(2) Award 0.3 pt if the answer contains an intermediate results of $\\tilde{S}_1 = (1 - r_A)^2 r_E S_0 = \\frac{(1-r_A)^2}{r_A} r_E \\tilde{S}_0$. \n(3) Award 0.5 pt if the answer contains an intermediate result of $\\tilde{S}_n = \\frac{\\tilde{S}_{n-1}}{1 - r_A} r_A r_E \\times (1 - r_A) = r_A r_E \\tilde{S}_{n-1} = (r_A r_E)^{n-1} \\tilde{S}_1$. \n(4) Award 0.3 pt if the answer contains an intermediate result of $\\tilde{S} = \\sum_{n=0}^{\\infty} \\tilde{S}_n = \\tilde{S}_0 + \\tilde{S}_1 \\sum_{n=1}^{\\infty} (r_A r_E)^{n-1}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct numerical value of $\\alpha$ as $3.13 \\times 10^{-1}$. Partial points: award 0.1 pt if the numerical answer falls within the acceptable error range of the correct value but contains more than four significant figures. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\alpha = r_{\\mathrm{A}} + \\frac{(1-r_{\\mathrm{A}})^{2} r_{\\mathrm{E}}}{1-r_{\\mathrm{A}} r_{\\mathrm{E}}}$}", + "\\boxed{$3.13 \\times 10^{-1}$}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null, + null + ], + "points": [ + 1.4, + 0.2 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_1_B_1_1.png" + ] + }, + { + "id": "IPhO_2024_1_B_3", + "context": "[The Greenhouse Effect] \n\nIn 2021, Syukuro Manabe and Klaus Hasselmann were awarded half of the Nobel Prize in Physics for their work in modeling Earth's climate and accurately predicting the global warming caused by human industrial activities. In this problem, we will examine a simple model of global warming due to the greenhouse effect. The greenhouse gases alter the optical properties of the Earth's atmosphere in transmitting or absorbing Earth's infrared radiation, resulting in a rise in the average temperature of the planet.\n\nAll objects, at different temperatures, emit thermal radiation. The quantity $u(\\lambda, T) d \\lambda$ indicates the thermal radiative power per unit area of an object at temperature $T$ between the wavelengths $\\lambda$ and $\\lambda+d \\lambda$. According to Planck's theory of blackbody radiation, we have\n\n$$u(\\lambda, T)=\\frac{2 \\pi h c^{2}}{\\lambda^{5}} \\frac{1}{\\exp \\left(\\frac{h c}{\\lambda k_{\\mathrm{B}} T}\\right)-1}$$ (Equation 1), in which $h c=1.24 \\times 10^{3} \\mathrm{eV} \\cdot \\mathrm{nm}$ and $k_{\\mathrm{B}}=8.62 \\times 10^{-5} \\mathrm{eV} / \\mathrm{K}$. The wavelength corresponding to the maximum of $u(\\lambda, T)$ comes from the relation $\\lambda_{\\max} T=b$ (Wien's displacement law). Indeed, using equation (1), it can be shown that $b=\\frac{h c}{x_{\\mathrm{m}} k_{\\mathrm{B}}}$, where the dimensionless quantity $x_{\\mathrm{m}}$ is the non-trivial root of an equation of the form $f(x)=0$; you are asked to find the function $f(x)$ in one of the following tasks. Total radiative power per unit area of a blackbody in all wavelengths is given by the Stephan-Boltzmann law as $U(T)=\\sigma T^{4}$ where $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} / \\mathrm{m}^{2} \\mathrm{K}^{4}$. Moreover, according to Kirchhoff's law of radiation, at thermal equilibrium a body absorbing a certain fraction of the incident radiation at a specific wavelength, will radiate the same fraction of the blackbody radiation at that same wavelength.\n\nThroughout this problem assume that the Sun is a blackbody at its average surface temperature of $T_{\\mathrm{S}}=5.77 \\times 10^{3} \\mathrm{K}$. The Sun's radius is $R_{\\mathrm{S}}=6.96 \\times 10^{8} \\mathrm{m}$ and the average distance between the Earth and the Sun is $d=1.50 \\times 10^{11} \\mathrm{m}$. We denote by $\\tilde{u}_{S}(\\lambda)$, the spectral solar power radiated into a unit area of the Earth normal to the direction of radiation. The integral of this quantity over all wavelengths, i.e. $S_{0}=\\int \\tilde{u}_{S}(\\lambda) d \\lambda$, is called the solar constant.\n\nIn this problem assume that the Earth is in thermal equilibrium and has the same temperature at all points on its surface. In all parts of the problem, express the desired quantity in parametric form in terms of the data given in the problem and then find its numerical value accurate to three significant figures. The required units are indicated on the answer sheet.\n\n[The Greenhouse Effect]\n\nIn this part, we introduce a simple model in which the Earth's atmosphere is modeled as a thin layer at a small distance above the Earth's surface so that the difference between the area of the atmosphere's layer and the area of the Earth's surface can be neglected (see Figure 1). In what follows assume that the major part of the thermal radiation from the Earth and the Sun are emitted at wavelengths near the $\\lambda_{\\max}$ for each one. Also assume that the \"atmosphere layer\" reflects a fraction $r_{\\mathrm{A}}=0.255$ of the visible-ultraviolet radiation incident from above or below, and completely transmits the rest. Assume that the atmosphere does not reflect any part of the infrared radiation, however, it absorbs a fraction $\\varepsilon$ of the infrared radiation and transmits the rest. This behavior, known as the greenhouse effect, changes the average temperature of the Earth. The Earth's surface, on the other hand, reflects a fraction $r_{E}$ of the visible-ultraviolet radiation and absorbs the rest of this radiation and all the infrared radiation.\n\n[figure1]\nFigure 1. Thermal flows between the Earth and the atmosphere.\n\nNow assume that $r_{\\mathrm{E}} \\neq 0$. In this case, the combined system of \"Earth + atmosphere\" reflects a different fraction of the solar radiation, called \"albedo\" and denoted by $\\alpha$.", + "question": "(1) Express the Earth's temperature $T_{\\mathrm{E}}$ in terms of $\\sigma, \\alpha, S_{0}$, and $\\varepsilon$. \n(2) Using the given data and the calculated albedo, find the numerical value of $\\varepsilon$ which leads to the current average temperature of $T_{\\mathrm{E}} = 288 \\mathrm{K}$ for the Earth.", + "marking": [ + [ + "Award 0.6 pt if the answer gives the correct expression for $T_{\\mathrm{E}}$: $T_{\\mathrm{E}} = \\left[\\frac{(1-\\alpha)}{2 \\sigma(2-\\epsilon)} S_{0}\\right]^{\\frac{1}{4}}$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer gives the correct numerical value of $T_{\\mathrm{E}}$ within the range of $[8.07 \\times 10^{-1}, 8.11 \\times 10^{-1}]$. Partial points: award 0.2 pt if the numerical answer is wrong but the expression for $\\varepsilon$ is correctly given as $\\varepsilon = \\frac{[\\sigma T_E^4 - \\frac{(1-\\alpha)}{4} S_0]}{\\sigma T_A^4} = 2 \\frac{[\\sigma T_E^4 - \\frac{(1-\\alpha)}{4} S_0]}{\\sigma T_E^4}$; or award 0.3 pt if the numerical answer falls within the acceptable error range of the correct value but contains more than four significant figures. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$T_{\\mathrm{E}} = \\left[\\frac{(1-\\alpha)}{2 \\sigma(2-\\epsilon)} S_{0}\\right]^{\\frac{1}{4}}$}", + "\\boxed{$[8.07 \\times 10^{-1}, 8.11 \\times 10^{-1}]$}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null, + null + ], + "points": [ + 0.6, + 0.4 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_1_B_1_1.png" + ] + }, + { + "id": "IPhO_2024_1_B_4", + "context": "[The Greenhouse Effect] \n\nIn 2021, Syukuro Manabe and Klaus Hasselmann were awarded half of the Nobel Prize in Physics for their work in modeling Earth's climate and accurately predicting the global warming caused by human industrial activities. In this problem, we will examine a simple model of global warming due to the greenhouse effect. The greenhouse gases alter the optical properties of the Earth's atmosphere in transmitting or absorbing Earth's infrared radiation, resulting in a rise in the average temperature of the planet.\n\nAll objects, at different temperatures, emit thermal radiation. The quantity $u(\\lambda, T) d \\lambda$ indicates the thermal radiative power per unit area of an object at temperature $T$ between the wavelengths $\\lambda$ and $\\lambda+d \\lambda$. According to Planck's theory of blackbody radiation, we have\n\n$$u(\\lambda, T)=\\frac{2 \\pi h c^{2}}{\\lambda^{5}} \\frac{1}{\\exp \\left(\\frac{h c}{\\lambda k_{\\mathrm{B}} T}\\right)-1}$$ (Equation 1), in which $h c=1.24 \\times 10^{3} \\mathrm{eV} \\cdot \\mathrm{nm}$ and $k_{\\mathrm{B}}=8.62 \\times 10^{-5} \\mathrm{eV} / \\mathrm{K}$. The wavelength corresponding to the maximum of $u(\\lambda, T)$ comes from the relation $\\lambda_{\\max} T=b$ (Wien's displacement law). Indeed, using equation (1), it can be shown that $b=\\frac{h c}{x_{\\mathrm{m}} k_{\\mathrm{B}}}$, where the dimensionless quantity $x_{\\mathrm{m}}$ is the non-trivial root of an equation of the form $f(x)=0$; you are asked to find the function $f(x)$ in one of the following tasks. Total radiative power per unit area of a blackbody in all wavelengths is given by the Stephan-Boltzmann law as $U(T)=\\sigma T^{4}$ where $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} / \\mathrm{m}^{2} \\mathrm{K}^{4}$. Moreover, according to Kirchhoff's law of radiation, at thermal equilibrium a body absorbing a certain fraction of the incident radiation at a specific wavelength, will radiate the same fraction of the blackbody radiation at that same wavelength.\n\nThroughout this problem assume that the Sun is a blackbody at its average surface temperature of $T_{\\mathrm{S}}=5.77 \\times 10^{3} \\mathrm{K}$. The Sun's radius is $R_{\\mathrm{S}}=6.96 \\times 10^{8} \\mathrm{m}$ and the average distance between the Earth and the Sun is $d=1.50 \\times 10^{11} \\mathrm{m}$. We denote by $\\tilde{u}_{S}(\\lambda)$, the spectral solar power radiated into a unit area of the Earth normal to the direction of radiation. The integral of this quantity over all wavelengths, i.e. $S_{0}=\\int \\tilde{u}_{S}(\\lambda) d \\lambda$, is called the solar constant.\n\nIn this problem assume that the Earth is in thermal equilibrium and has the same temperature at all points on its surface. In all parts of the problem, express the desired quantity in parametric form in terms of the data given in the problem and then find its numerical value accurate to three significant figures. The required units are indicated on the answer sheet.\n\n[The Greenhouse Effect]\n\nIn this part, we introduce a simple model in which the Earth's atmosphere is modeled as a thin layer at a small distance above the Earth's surface so that the difference between the area of the atmosphere's layer and the area of the Earth's surface can be neglected (see Figure 1). In what follows assume that the major part of the thermal radiation from the Earth and the Sun are emitted at wavelengths near the $\\lambda_{\\max}$ for each one. Also assume that the \"atmosphere layer\" reflects a fraction $r_{\\mathrm{A}}=0.255$ of the visible-ultraviolet radiation incident from above or below, and completely transmits the rest. Assume that the atmosphere does not reflect any part of the infrared radiation, however, it absorbs a fraction $\\varepsilon$ of the infrared radiation and transmits the rest. This behavior, known as the greenhouse effect, changes the average temperature of the Earth. The Earth's surface, on the other hand, reflects a fraction $r_{E}$ of the visible-ultraviolet radiation and absorbs the rest of this radiation and all the infrared radiation.\n\n[figure1]\nFigure 1. Thermal flows between the Earth and the atmosphere.\n\nNow assume that $r_{\\mathrm{E}} \\neq 0$. In this case, the combined system of \"Earth + atmosphere\" reflects a different fraction of the solar radiation, called \"albedo\" and denoted by $\\alpha$.", + "question": "(1) Find the expression of $\\frac{\\mathrm{d} T_{\\mathrm{E}}}{\\mathrm{d} \\varepsilon}$. \n(2) Determine the value of $\\delta T_{\\mathrm{E}}$ (expressed in $K$), i.e., the increase in Earth's temperature if $\\varepsilon$ increases by one percent.", + "marking": [ + [ + "Award 0.6 pt if the answer gives the correct expression for $\\frac{\\mathrm{d} T_{\\mathrm{E}}}{\\mathrm{d} \\varepsilon}$: $\\frac{\\mathrm{d} T_{\\mathrm{E}}}{\\mathrm{d} \\varepsilon} = \\frac{1}{4}\\left[\\frac{(1-\\alpha) S_0}{2 \\sigma(2-\\varepsilon)}\\right]^{\\frac{1}{4}} \\frac{1}{(2-\\varepsilon)}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct numerical value of $\\delta T_{\\mathrm{E}}$ within the range of $[4.87 \\times 10^{-1}, 4.92 \\times 10^{-1}] K$. Partial points: award 0.1 pt if the numerical answer falls within the acceptable error range of the correct value but contains more than four significant figures. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{\\mathrm{d} T_{\\mathrm{E}}}{\\mathrm{d} \\varepsilon} = \\frac{1}{4}\\left[\\frac{(1-\\alpha) S_0}{2 \\sigma(2-\\varepsilon)}\\right]^{\\frac{1}{4}} \\frac{1}{(2-\\varepsilon)}$}", + "\\boxed{$[4.87 \\times 10^{-1}, 4.92 \\times 10^{-1}]$}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null, + "K" + ], + "points": [ + 0.6, + 0.2 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_1_B_1_1.png" + ] + }, + { + "id": "IPhO_2024_1_B_5", + "context": "[The Greenhouse Effect] \n\nIn 2021, Syukuro Manabe and Klaus Hasselmann were awarded half of the Nobel Prize in Physics for their work in modeling Earth's climate and accurately predicting the global warming caused by human industrial activities. In this problem, we will examine a simple model of global warming due to the greenhouse effect. The greenhouse gases alter the optical properties of the Earth's atmosphere in transmitting or absorbing Earth's infrared radiation, resulting in a rise in the average temperature of the planet.\n\nAll objects, at different temperatures, emit thermal radiation. The quantity $u(\\lambda, T) d \\lambda$ indicates the thermal radiative power per unit area of an object at temperature $T$ between the wavelengths $\\lambda$ and $\\lambda+d \\lambda$. According to Planck's theory of blackbody radiation, we have\n\n$$u(\\lambda, T)=\\frac{2 \\pi h c^{2}}{\\lambda^{5}} \\frac{1}{\\exp \\left(\\frac{h c}{\\lambda k_{\\mathrm{B}} T}\\right)-1}$$ (Equation 1), in which $h c=1.24 \\times 10^{3} \\mathrm{eV} \\cdot \\mathrm{nm}$ and $k_{\\mathrm{B}}=8.62 \\times 10^{-5} \\mathrm{eV} / \\mathrm{K}$. The wavelength corresponding to the maximum of $u(\\lambda, T)$ comes from the relation $\\lambda_{\\max} T=b$ (Wien's displacement law). Indeed, using equation (1), it can be shown that $b=\\frac{h c}{x_{\\mathrm{m}} k_{\\mathrm{B}}}$, where the dimensionless quantity $x_{\\mathrm{m}}$ is the non-trivial root of an equation of the form $f(x)=0$; you are asked to find the function $f(x)$ in one of the following tasks. Total radiative power per unit area of a blackbody in all wavelengths is given by the Stephan-Boltzmann law as $U(T)=\\sigma T^{4}$ where $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} / \\mathrm{m}^{2} \\mathrm{K}^{4}$. Moreover, according to Kirchhoff's law of radiation, at thermal equilibrium a body absorbing a certain fraction of the incident radiation at a specific wavelength, will radiate the same fraction of the blackbody radiation at that same wavelength.\n\nThroughout this problem assume that the Sun is a blackbody at its average surface temperature of $T_{\\mathrm{S}}=5.77 \\times 10^{3} \\mathrm{K}$. The Sun's radius is $R_{\\mathrm{S}}=6.96 \\times 10^{8} \\mathrm{m}$ and the average distance between the Earth and the Sun is $d=1.50 \\times 10^{11} \\mathrm{m}$. We denote by $\\tilde{u}_{S}(\\lambda)$, the spectral solar power radiated into a unit area of the Earth normal to the direction of radiation. The integral of this quantity over all wavelengths, i.e. $S_{0}=\\int \\tilde{u}_{S}(\\lambda) d \\lambda$, is called the solar constant.\n\nIn this problem assume that the Earth is in thermal equilibrium and has the same temperature at all points on its surface. In all parts of the problem, express the desired quantity in parametric form in terms of the data given in the problem and then find its numerical value accurate to three significant figures. The required units are indicated on the answer sheet.\n\n[The Greenhouse Effect]\n\nIn this part, we introduce a simple model in which the Earth's atmosphere is modeled as a thin layer at a small distance above the Earth's surface so that the difference between the area of the atmosphere's layer and the area of the Earth's surface can be neglected (see Figure 1). In what follows assume that the major part of the thermal radiation from the Earth and the Sun are emitted at wavelengths near the $\\lambda_{\\max}$ for each one. Also assume that the \"atmosphere layer\" reflects a fraction $r_{\\mathrm{A}}=0.255$ of the visible-ultraviolet radiation incident from above or below, and completely transmits the rest. Assume that the atmosphere does not reflect any part of the infrared radiation, however, it absorbs a fraction $\\varepsilon$ of the infrared radiation and transmits the rest. This behavior, known as the greenhouse effect, changes the average temperature of the Earth. The Earth's surface, on the other hand, reflects a fraction $r_{E}$ of the visible-ultraviolet radiation and absorbs the rest of this radiation and all the infrared radiation.\n\n[figure1]\nFigure 1. Thermal flows between the Earth and the atmosphere.\n\nNow assume that $r_{\\mathrm{E}} \\neq 0$. In this case, the combined system of \"Earth + atmosphere\" reflects a different fraction of the solar radiation, called \"albedo\" and denoted by $\\alpha$.\n\nAssume $T_{\\mathrm{A}} = 245 \\mathrm{K}$ and $T_{\\mathrm{E}} = 288 \\mathrm{K}$. These values come from real data and may differ from the results which you have obtained in the previous tasks. Now suppose that a non-radiative (e.g. convective) thermal flow $J_{\\mathrm{NR}} = k\\left(T_{\\mathrm{E}}-T_{\\mathrm{A}}\\right)$ is maintained from the Earth to the atmosphere, where $k$ is a constant. The quantity, $J_{\\mathrm{NR}}$, is the transmitted power per unit area.", + "question": "(1) Express $\\varepsilon$ in terms of $T_{\\mathrm{E}}, T_{\\mathrm{A}}, \\sigma, \\alpha$, and $S_{0}$. \n(2) Express $k$ in terms of $T_{\\mathrm{E}}, T_{\\mathrm{A}}, \\sigma, \\alpha$, and $S_{0}$. \n(3) Calculate the value of $\\varepsilon$. \n(4) Calculate the value of $k$ (expressed in $W/(m^2 \\mathrm{K})$)", + "marking": [ + [ + "Award 0.6 pt if the answer gives the correct expression for $\\varepsilon$: $\\varepsilon = \\frac{\\sigma T_{\\mathrm{E}}^{4} - (1-\\alpha) \\frac{S_{0}}{4}}{\\sigma (T_{\\mathrm{E}}^{4} - T_{\\mathrm{A}}^{4})}$. Partial points: award 0.3 pt if the expression is incorrect but contains the correct relations for balance of energy. Otherwise, award 0 pt.", + "Award 0.6 pt if the answer gives the correct expression for $k$: $k = \\frac{(2 T_{\\mathrm{A}}^{4}-T_{\\mathrm{E}}^{4}) \\times [\\sigma T_{\\mathrm{E}}^{4} - (1-\\alpha) \\frac{S_{0}}{4}]}{(T_{\\mathrm{E}}^{4}-T_{\\mathrm{A}}^{4}) \\times (T_{\\mathrm{E}}-T_{\\mathrm{A}})}$. Partial points: award 0.3 pt if the expression is incorrect but contains the correct relations for balance of energy. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct numerical value of $\\varepsilon$ within the range of $[8.47 \\times 10^{-1}, 8.52 \\times 10^{-1}]$. Partial points: award 0.1 pt if the numerical answer falls within the acceptable error range of the correct value but contains more than four significant figures. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct numerical value of $k$ within the range of $[3.57 \\times 10^{-1}, 3.66 \\times 10^{-1}] W/(m^2 \\mathrm{K})$. Partial points: award 0.1 pt if the numerical answer falls within the acceptable error range of the correct value but contains more than four significant figures. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\varepsilon = \\frac{\\sigma T_{\\mathrm{E}}^{4} - (1-\\alpha) \\frac{S_{0}}{4}}{\\sigma (T_{\\mathrm{E}}^{4} - T_{\\mathrm{A}}^{4})}$}", + "\\boxed{$k = \\frac{(2 T_{\\mathrm{A}}^{4}-T_{\\mathrm{E}}^{4}) \\times [\\sigma T_{\\mathrm{E}}^{4}-(1-\\alpha) \\frac{S_{0}}{4}]}{(T_{\\mathrm{E}}^{4}-T_{\\mathrm{A}}^{4}) \\times (T_{\\mathrm{E}}-T_{\\mathrm{A}})}}", + "\\boxed{$[8.47 \\times 10^{-1}, 8.52 \\times 10^{-1}]$}", + "\\boxed{$[3.57 \\times 10^{-1}, 3.66 \\times 10^{-1}]$}" + ], + "answer_type": [ + "Expression", + "Expression", + "Numerical Value", + "Numerical Value" + ], + "unit": [ + null, + null, + null, + "$W/(m^2 \\mathrm{K})$" + ], + "points": [ + 0.6, + 0.6, + 0.2, + 0.2 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_1_B_1_1.png" + ] + }, + { + "id": "IPhO_2024_1_B_6", + "context": "[The Greenhouse Effect] \n\nIn 2021, Syukuro Manabe and Klaus Hasselmann were awarded half of the Nobel Prize in Physics for their work in modeling Earth's climate and accurately predicting the global warming caused by human industrial activities. In this problem, we will examine a simple model of global warming due to the greenhouse effect. The greenhouse gases alter the optical properties of the Earth's atmosphere in transmitting or absorbing Earth's infrared radiation, resulting in a rise in the average temperature of the planet.\n\nAll objects, at different temperatures, emit thermal radiation. The quantity $u(\\lambda, T) d \\lambda$ indicates the thermal radiative power per unit area of an object at temperature $T$ between the wavelengths $\\lambda$ and $\\lambda+d \\lambda$. According to Planck's theory of blackbody radiation, we have\n\n$$u(\\lambda, T)=\\frac{2 \\pi h c^{2}}{\\lambda^{5}} \\frac{1}{\\exp \\left(\\frac{h c}{\\lambda k_{\\mathrm{B}} T}\\right)-1}$$ (Equation 1), in which $h c=1.24 \\times 10^{3} \\mathrm{eV} \\cdot \\mathrm{nm}$ and $k_{\\mathrm{B}}=8.62 \\times 10^{-5} \\mathrm{eV} / \\mathrm{K}$. The wavelength corresponding to the maximum of $u(\\lambda, T)$ comes from the relation $\\lambda_{\\max} T=b$ (Wien's displacement law). Indeed, using equation (1), it can be shown that $b=\\frac{h c}{x_{\\mathrm{m}} k_{\\mathrm{B}}}$, where the dimensionless quantity $x_{\\mathrm{m}}$ is the non-trivial root of an equation of the form $f(x)=0$; you are asked to find the function $f(x)$ in one of the following tasks. Total radiative power per unit area of a blackbody in all wavelengths is given by the Stephan-Boltzmann law as $U(T)=\\sigma T^{4}$ where $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} / \\mathrm{m}^{2} \\mathrm{K}^{4}$. Moreover, according to Kirchhoff's law of radiation, at thermal equilibrium a body absorbing a certain fraction of the incident radiation at a specific wavelength, will radiate the same fraction of the blackbody radiation at that same wavelength.\n\nThroughout this problem assume that the Sun is a blackbody at its average surface temperature of $T_{\\mathrm{S}}=5.77 \\times 10^{3} \\mathrm{K}$. The Sun's radius is $R_{\\mathrm{S}}=6.96 \\times 10^{8} \\mathrm{m}$ and the average distance between the Earth and the Sun is $d=1.50 \\times 10^{11} \\mathrm{m}$. We denote by $\\tilde{u}_{S}(\\lambda)$, the spectral solar power radiated into a unit area of the Earth normal to the direction of radiation. The integral of this quantity over all wavelengths, i.e. $S_{0}=\\int \\tilde{u}_{S}(\\lambda) d \\lambda$, is called the solar constant.\n\nIn this problem assume that the Earth is in thermal equilibrium and has the same temperature at all points on its surface. In all parts of the problem, express the desired quantity in parametric form in terms of the data given in the problem and then find its numerical value accurate to three significant figures. The required units are indicated on the answer sheet.\n\n[The Greenhouse Effect]\n\nIn this part, we introduce a simple model in which the Earth's atmosphere is modeled as a thin layer at a small distance above the Earth's surface so that the difference between the area of the atmosphere's layer and the area of the Earth's surface can be neglected (see Figure 1). In what follows assume that the major part of the thermal radiation from the Earth and the Sun are emitted at wavelengths near the $\\lambda_{\\max}$ for each one. Also assume that the \"atmosphere layer\" reflects a fraction $r_{\\mathrm{A}}=0.255$ of the visible-ultraviolet radiation incident from above or below, and completely transmits the rest. Assume that the atmosphere does not reflect any part of the infrared radiation, however, it absorbs a fraction $\\varepsilon$ of the infrared radiation and transmits the rest. This behavior, known as the greenhouse effect, changes the average temperature of the Earth. The Earth's surface, on the other hand, reflects a fraction $r_{E}$ of the visible-ultraviolet radiation and absorbs the rest of this radiation and all the infrared radiation.\n\n[figure1]\nFigure 1. Thermal flows between the Earth and the atmosphere.\n\nNow assume that $r_{\\mathrm{E}} \\neq 0$. In this case, the combined system of \"Earth + atmosphere\" reflects a different fraction of the solar radiation, called \"albedo\" and denoted by $\\alpha$.\n\nAssume $T_{\\mathrm{A}} = 245 \\mathrm{K}$ and $T_{\\mathrm{E}} = 288 \\mathrm{K}$. These values come from real data and may differ from the results which you have obtained in the previous tasks. Now suppose that a non-radiative (e.g. convective) thermal flow $J_{\\mathrm{NR}} = k\\left(T_{\\mathrm{E}}-T_{\\mathrm{A}}\\right)$ is maintained from the Earth to the atmosphere, where $k$ is a constant. The quantity, $J_{\\mathrm{NR}}$, is the transmitted power per unit area.", + "question": "Express $\\varepsilon$ and $k$ in terms of $T_{\\mathrm{E}}, T_{\\mathrm{A}}, \\sigma, \\alpha$, and $S_{0}$, respectively (This is a preliminary question and do not include in the final answer). \n\n(1) Differentiating the equations obtained in part (0) with respect to $\\varepsilon$, find one algebraic equation satisfied by $\\frac{\\mathrm{d} T_{\\mathrm{A}}}{\\mathrm{d} \\varepsilon}$ and $\\frac{\\mathrm{d} T_{\\mathrm{E}}}{\\mathrm{d} \\varepsilon}$. \n(2) Differentiating the equations obtained in part (0) with respect to $\\varepsilon$, find another algebraic equation satisfied by $\\frac{\\mathrm{d} T_{\\mathrm{A}}}{\\mathrm{d} \\varepsilon}$ and $\\frac{\\mathrm{d} T_{\\mathrm{E}}}{\\mathrm{d} \\varepsilon}$. \n(3) Use these equations in parts (1) and (2) to find $\\delta T_{\\mathrm{E}}$ (expressed in $K$), i.e., the numerical value of change in the Earth's temperature as a result of a one percent increase in the value of $\\varepsilon$.", + "marking": [ + [ + "Award 0.8 pt if the answer finds two algebraic equations satisfied by $\\frac{\\mathrm{d} T_{\\mathrm{A}}}{\\mathrm{d} \\varepsilon}$ and $\\frac{\\mathrm{d} T_{\\mathrm{E}}}{\\mathrm{d} \\varepsilon}$: (1) $\\varepsilon \\left[ \\frac{1}{T_E - T_A} + \\frac{4 T_E^3}{2 T_A^4 - T_E^4} \\right] \\frac{\\mathrm{d} T_E}{\\mathrm{d} \\varepsilon} = 1 + \\varepsilon \\left[ \\frac{8 T_A^3}{2 T_A^4 - T_E^4} + \\frac{1}{T_E - T_A} \\right] \\frac{\\mathrm{d} T_A}{\\mathrm{d} \\varepsilon}$ and (2) $1 + \\varepsilon \\left[ \\frac{4 T_E^3}{T_E^4 - T_A^4} - \\frac{4 \\sigma T_E^3}{\\sigma T_E^4 - (1 - \\alpha) \\frac{S_0}{4}} \\right] \\frac{\\mathrm{d} T_E}{\\mathrm{d} \\varepsilon} = \\frac{4 T_A^3}{T_E^4 - T_A^4} \\varepsilon \\frac{\\mathrm{d} T_A}{\\mathrm{d} \\varepsilon}$. Partial points: award 0.6 pt if the answer finds only one of the equations correctly. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct numerical value of $\\delta T_E$ within the range of $[5.21 \\times 10^{-1}, 5.28 \\times 10^{-1}]$ K. Partial points: award 0.1 pt if the numerical answer falls within the acceptable error range of the correct value but contains more than four significant figures. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\varepsilon \\left[ \\frac{1}{T_E - T_A} + \\frac{4 T_E^3}{2 T_A^4 - T_E^4} \\right] \\frac{\\mathrm{d} T_E}{\\mathrm{d} \\varepsilon} = 1 + \\varepsilon \\left[ \\frac{8 T_A^3}{2 T_A^4 - T_E^4} + \\frac{1}{T_E - T_A} \\right] \\frac{\\mathrm{d} T_A}{\\mathrm{d} \\varepsilon}$}", + "\\boxed{$1 + \\varepsilon \\left[ \\frac{4 T_E^3}{T_E^4 - T_A^4} - \\frac{4 \\sigma T_E^3}{\\sigma T_E^4 - (1 - \\alpha) \\frac{S_0}{4}} \\right] \\frac{\\mathrm{d} T_E}{\\mathrm{d} \\varepsilon} = \\frac{4 T_A^3}{T_E^4 - T_A^4} \\varepsilon \\frac{\\mathrm{d} T_A}{\\mathrm{d} \\varepsilon}$}", + "$\\boxed{$[5.21 \\times 10^{-1}, 5.28 \\times 10^{-1}]$}" + ], + "answer_type": [ + "Equation", + "Equation", + "Numerical Value" + ], + "unit": [ + null, + null, + "K" + ], + "points": [ + 0.4, + 0.4, + 0.2 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_1_B_1_1.png" + ] + }, + { + "id": "IPhO_2024_2_A_1", + "context": "[Trapping Ions and Cooling Atoms] \n\nIn recent decades, trapping and cooling atoms and ions has been a fascinating topic for physicists, with several Nobel prizes awarded for work in this area. In the first part of this question, we will explore a technique for trapping ions, known as the \"Paul trap\". Wolfgang Paul and Hans Dehmelt received one half of the 1989 Nobel Prize in Physics for this work. Next, we investigate the Doppler cooling technique, one of the works cited in the press release for the 1997 Nobel Prize in Physics awarded to Steven Chu, Claude Cohen-Tannoudji, and William Daniel Phillips \"for developments of methods to cool and trap atoms with laser light\".\n\n[The Paul Trap] \n\nIt is known that with electrostatic fields, it is not possible to create a stable equilibrium for a charged particle. Therefore, creating a stable equilibrium point for ions requires more sophisticated techniques. The Paul trap is one of these techniques.\n\nAs shown in Figure 1, consider a ring of charge with a radius $R$ and a uniform positive linear charge density $\\lambda$. A positive point charge $Q$ with mass $m$ is placed at the center of the ring.\n\n[figure1]\nFigure 1. A positively charged ring with a uniform linear charge density $\\lambda$ and radius $R$: the origin of the coordinate system is at the center of the ring.", + "question": "(1) In cartesian coordinates $(x, y, z)$, obtain the electric field $\\vec{E}(x, y, z)$ due to the charged ring in the vicinity of the ring's center to the first order in $x / R$, $y / R$, and $z / R$. \n(2) Find the angular frequency $\\omega_x$ in the $x$ direction of small oscillations of the charged particle around the center of the ring in the directions for which a stable equilibrium exists. \n(3) Find the angular frequency $\\omega_y$ in the $y$ direction of small oscillations of the charged particle around the center of the ring in the directions for which a stable equilibrium exists.", + "marking": [ + [ + "Award 1.0 pt if the answer gives the correct expression for the electric field $\\vec{E}(x, y, z) = \\frac{-\\lambda x}{4 \\epsilon_0 R^2} \\hat{x} + \\frac{-\\lambda y}{4 \\epsilon_0 R^2} \\hat{y} + \\frac{\\lambda z}{2 \\epsilon_0 R^2} \\hat{z}$ in cartesian coordinates. If the final expression is not fully correct, award 0.5 pt for a correct $z$-component only, or 0.5 pt for correct $x$- and $y$-components. Deduct 0.1 pt for each incorrect coefficient and 0.2 pt for each incorrect sign in a component. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the correct expression for the angular frequency $\\omega_x = \\sqrt{\\frac{Q \\lambda}{4 \\epsilon_0 R^2 m}}$ in the $x$ direction of small oscillations and $\\omega_y = \\sqrt{\\frac{Q \\lambda}{4 \\epsilon_0 R^2 m}}$ in the $y$ direction of small oscillations. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\vec{E}(x, y, z) = \\frac{-\\lambda x}{4 \\epsilon_0 R^2} \\hat{x} + \\frac{-\\lambda y}{4 \\epsilon_0 R^2} \\hat{y} + \\frac{\\lambda z}{2 \\epsilon_0 R^2} \\hat{z}$}", + "\\boxed{$\\omega_x = \\sqrt{\\frac{Q \\lambda}{4 \\epsilon_0 R^2 m}}$}", + "\\boxed{$\\omega_y = \\sqrt{\\frac{Q \\lambda}{4 \\epsilon_0 R^2 m}}$}" + ], + "answer_type": [ + "Expression", + "Expression", + "Expression" + ], + "unit": [ + null, + null, + null + ], + "points": [ + 1.0, + 0.25, + 0.25 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_2_A_1_1.png" + ] + }, + { + "id": "IPhO_2024_2_A_2", + "context": "[Trapping Ions and Cooling Atoms] \n\nIn recent decades, trapping and cooling atoms and ions has been a fascinating topic for physicists, with several Nobel prizes awarded for work in this area. In the first part of this question, we will explore a technique for trapping ions, known as the \"Paul trap\". Wolfgang Paul and Hans Dehmelt received one half of the 1989 Nobel Prize in Physics for this work. Next, we investigate the Doppler cooling technique, one of the works cited in the press release for the 1997 Nobel Prize in Physics awarded to Steven Chu, Claude Cohen-Tannoudji, and William Daniel Phillips \"for developments of methods to cool and trap atoms with laser light\".\n\n[The Paul Trap] \n\nIt is known that with electrostatic fields, it is not possible to create a stable equilibrium for a charged particle. Therefore, creating a stable equilibrium point for ions requires more sophisticated techniques. The Paul trap is one of these techniques.\n\nAs shown in Figure 1, consider a ring of charge with a radius $R$ and a uniform positive linear charge density $\\lambda$. A positive point charge $Q$ with mass $m$ is placed at the center of the ring.\n\n[figure1]\nFigure 1. A positively charged ring with a uniform linear charge density $\\lambda$ and radius $R$: the origin of the coordinate system is at the center of the ring.\n\nIn order to trap the charge $Q$ fully, we would like to apply alternating fields to produce a dynamic equilibrium. Assume that the charge density is $\\lambda = \\lambda_{0} + u \\cos \\Omega t$ in which $\\lambda_{0}$, $u$, and $\\Omega$ are adjustable. We shall ignore radiative effects. Then the equation of motion for small displacements from the center of the ring, along the direction perpendicular to the plane of the ring will turn out to be: \n$$\\ddot{z} = (+k^{2}+ a \\Omega^{2} \\cos \\Omega t) z$$ (Equation 1).", + "question": "(1) Write $k$ in terms of the known parameters. \n(2) Write $a$ in terms of the known parameters.", + "marking": [ + [ + "Award 0.2 pt if the answer gives the correct expression for $k = \\sqrt{\\frac{Q \\lambda_{0}}{2 \\epsilon_{0} R^{2} m}}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct expression for $a = \\frac{Q u}{2 \\epsilon_{0} R^{2} m \\Omega^{2}}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$k = \\sqrt{\\frac{Q \\lambda_{0}}{2 \\epsilon_{0} R^{2} m}}$}", + "\\boxed{$a = \\frac{Q u}{2 \\epsilon_{0} R^{2} m \\Omega^{2}}$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 0.2, + 0.2 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_2_A_1_1.png" + ] + }, + { + "id": "IPhO_2024_2_A_3", + "context": "[Trapping Ions and Cooling Atoms] \n\nIn recent decades, trapping and cooling atoms and ions has been a fascinating topic for physicists, with several Nobel prizes awarded for work in this area. In the first part of this question, we will explore a technique for trapping ions, known as the \"Paul trap\". Wolfgang Paul and Hans Dehmelt received one half of the 1989 Nobel Prize in Physics for this work. Next, we investigate the Doppler cooling technique, one of the works cited in the press release for the 1997 Nobel Prize in Physics awarded to Steven Chu, Claude Cohen-Tannoudji, and William Daniel Phillips \"for developments of methods to cool and trap atoms with laser light\".\n\n[The Paul Trap] \n\nIt is known that with electrostatic fields, it is not possible to create a stable equilibrium for a charged particle. Therefore, creating a stable equilibrium point for ions requires more sophisticated techniques. The Paul trap is one of these techniques.\n\nAs shown in Figure 1, consider a ring of charge with a radius $R$ and a uniform positive linear charge density $\\lambda$. A positive point charge $Q$ with mass $m$ is placed at the center of the ring.\n\n[figure1]\nFigure 1. A positively charged ring with a uniform linear charge density $\\lambda$ and radius $R$: the origin of the coordinate system is at the center of the ring.\n\nIn order to trap the charge $Q$ fully, we would like to apply alternating fields to produce a dynamic equilibrium. Assume that the charge density is $\\lambda = \\lambda_{0} + u \\cos \\Omega t$ in which $\\lambda_{0}$, $u$, and $\\Omega$ are adjustable. We shall ignore radiative effects. Then the equation of motion for small displacements from the center of the ring, along the direction perpendicular to the plane of the ring will turn out to be: \n$$\\ddot{z} = (+k^{2}+ a \\Omega^{2} \\cos \\Omega t) z$$ (Equation 1).\n\nWe would like to obtain an approximate solution to Equation (1) by making the following simplifying assumptions: $a \\ll 1$, $\\Omega \\gg k$, and $a \\Omega^{2} \\gg k^{2}$. With these assumptions, it can be shown that the solution of this equation can be split into two parts: $z(t) = p(t) + q(t)$, where $p(t)$ is a slowly varying component and $q(t)$ is a small-amplitude rapidly-varying component with a mean value of zero. In other words, $p(t)$ may be assumed constant over a few oscillations of $q(t)$ (see Figure 2).\n\n[figure2]\nFigure 2. A typical solution for the equation of motion of the charged particle: $p(t)$ gives the overall motion, and $q(t)$ represents small oscillations around this trajectory. The ellipse on the right is a magnification of a part of this trajectory.", + "question": "(1) Using the approximations stated above, find the equation of motion for $q(t)$ in terms of $a$, $\\Omega$, and $p$. \n(2) Find the solution of this equation by considering appropriate initial conditions corresponding to the required properties of this function.", + "marking": [ + [ + "Award a total of 1.0 pt for the following: award 0.1 pt if the answer gives the correct equation of motion for $q(t)$: $\\ddot{q} = p a \\Omega^2 \\cos \\Omega t$, and award 0.3 pt for each valid approximation used: (1) $p$ is almost constant, i.e., $\\ddot{p} \\simeq 0$; (2) $k^2 \\ll a \\Omega^2$; (3) $q \\ll p$. Otherwise, award 0 pt.", + "Award a total of 0.8 pt if the answer gives the correct expression for $q = -p a \\cos \\Omega t$. Partial points: if the answer gives the incorrect expression, award 0.4 pt if the answer gives the general solution $q = -pa \\cos \\Omega t + c_1 t + c_2$, and award 0.2 pt each for correctly setting $c_1 = 0$ and $c_2 = 0$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\ddot{q} = p a \\Omega^{2} \\cos \\Omega t$}", + "\\boxed{$q = -p a \\cos \\Omega t$}" + ], + "answer_type": [ + "Equation", + "Equation" + ], + "unit": [ + null, + null + ], + "points": [ + 1.0, + 0.8 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_2_A_1_1.png", + "image_question/IPhO_2024_2_A_3_1.png" + ] + }, + { + "id": "IPhO_2024_2_A_4", + "context": "[Trapping Ions and Cooling Atoms] \n\nIn recent decades, trapping and cooling atoms and ions has been a fascinating topic for physicists, with several Nobel prizes awarded for work in this area. In the first part of this question, we will explore a technique for trapping ions, known as the \"Paul trap\". Wolfgang Paul and Hans Dehmelt received one half of the 1989 Nobel Prize in Physics for this work. Next, we investigate the Doppler cooling technique, one of the works cited in the press release for the 1997 Nobel Prize in Physics awarded to Steven Chu, Claude Cohen-Tannoudji, and William Daniel Phillips \"for developments of methods to cool and trap atoms with laser light\".\n\n[The Paul Trap] \n\nIt is known that with electrostatic fields, it is not possible to create a stable equilibrium for a charged particle. Therefore, creating a stable equilibrium point for ions requires more sophisticated techniques. The Paul trap is one of these techniques.\n\nAs shown in Figure 1, consider a ring of charge with a radius $R$ and a uniform positive linear charge density $\\lambda$. A positive point charge $Q$ with mass $m$ is placed at the center of the ring.\n\n[figure1]\nFigure 1. A positively charged ring with a uniform linear charge density $\\lambda$ and radius $R$: the origin of the coordinate system is at the center of the ring.\n\nIn order to trap the charge $Q$ fully, we would like to apply alternating fields to produce a dynamic equilibrium. Assume that the charge density is $\\lambda = \\lambda_{0} + u \\cos \\Omega t$ in which $\\lambda_{0}$, $u$, and $\\Omega$ are adjustable. We shall ignore radiative effects. Then the equation of motion for small displacements from the center of the ring, along the direction perpendicular to the plane of the ring will turn out to be: \n$$\\ddot{z} = (+k^{2}+ a \\Omega^{2} \\cos \\Omega t) z$$ (Equation 1).\n\nWe would like to obtain an approximate solution to Equation (1) by making the following simplifying assumptions: $a \\ll 1$, $\\Omega \\gg k$, and $a \\Omega^{2} \\gg k^{2}$. With these assumptions, it can be shown that the solution of this equation can be split into two parts: $z(t) = p(t) + q(t)$, where $p(t)$ is a slowly varying component and $q(t)$ is a small-amplitude rapidly-varying component with a mean value of zero. In other words, $p(t)$ may be assumed constant over a few oscillations of $q(t)$ (see Figure 2).\n\n[figure2]\nFigure 2. A typical solution for the equation of motion of the charged particle: $p(t)$ gives the overall motion, and $q(t)$ represents small oscillations around this trajectory. The ellipse on the right is a magnification of a part of this trajectory.", + "question": "(1) Using the mean effect of the rapidly varying component and obtain an effective equation of motion for $p(t)$. \n(2) Investigate the stability of the equilibrium point and find the condition for a stable equilibrium.", + "marking": [ + [ + "Award a total of 1.2 pt for the following: award 0.6 pt if the answer gives the correct equation of motion for $p(t)$: $\\ddot{p}(t) = \\left(k^{2} - \\frac{a^{2} \\Omega^{2}}{2}\\right) p$, and award 0.6 pt if the answer applies the correct approach (e.g., averaging over one period). Otherwise, award 0 pt.", + "Award 0.3 pt if the answer finds the correct condition for a stable equilibrium, $\\Omega > \\sqrt{2} \\frac{k}{a}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\ddot{p}(t) = \\left(k^{2} - \\frac{a^{2} \\Omega^{2}}{2}\\right) p$}", + "\\boxed{$\\Omega > \\sqrt{2} \\frac{k}{a}$}" + ], + "answer_type": [ + "Equation", + "Inequality" + ], + "unit": [ + null, + null + ], + "points": [ + 1.2, + 0.3 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_2_A_1_1.png", + "image_question/IPhO_2024_2_A_3_1.png" + ] + }, + { + "id": "IPhO_2024_2_A_5", + "context": "[Trapping Ions and Cooling Atoms] \n\nIn recent decades, trapping and cooling atoms and ions has been a fascinating topic for physicists, with several Nobel prizes awarded for work in this area. In the first part of this question, we will explore a technique for trapping ions, known as the \"Paul trap\". Wolfgang Paul and Hans Dehmelt received one half of the 1989 Nobel Prize in Physics for this work. Next, we investigate the Doppler cooling technique, one of the works cited in the press release for the 1997 Nobel Prize in Physics awarded to Steven Chu, Claude Cohen-Tannoudji, and William Daniel Phillips \"for developments of methods to cool and trap atoms with laser light\".\n\n[The Paul Trap] \n\nIt is known that with electrostatic fields, it is not possible to create a stable equilibrium for a charged particle. Therefore, creating a stable equilibrium point for ions requires more sophisticated techniques. The Paul trap is one of these techniques.\n\nAs shown in Figure 1, consider a ring of charge with a radius $R$ and a uniform positive linear charge density $\\lambda$. A positive point charge $Q$ with mass $m$ is placed at the center of the ring.\n\n[figure1]\nFigure 1. A positively charged ring with a uniform linear charge density $\\lambda$ and radius $R$: the origin of the coordinate system is at the center of the ring.\n\nIn order to trap the charge $Q$ fully, we would like to apply alternating fields to produce a dynamic equilibrium. Assume that the charge density is $\\lambda = \\lambda_{0} + u \\cos \\Omega t$ in which $\\lambda_{0}$, $u$, and $\\Omega$ are adjustable. We shall ignore radiative effects. Then the equation of motion for small displacements from the center of the ring, along the direction perpendicular to the plane of the ring will turn out to be: \n$$\\ddot{z} = (+k^{2}+ a \\Omega^{2} \\cos \\Omega t) z$$ (Equation 1).\n\nWe would like to obtain an approximate solution to Equation (1) by making the following simplifying assumptions: $a \\ll 1$, $\\Omega \\gg k$, and $a \\Omega^{2} \\gg k^{2}$. With these assumptions, it can be shown that the solution of this equation can be split into two parts: $z(t) = p(t) + q(t)$, where $p(t)$ is a slowly varying component and $q(t)$ is a small-amplitude rapidly-varying component with a mean value of zero. In other words, $p(t)$ may be assumed constant over a few oscillations of $q(t)$ (see Figure 2).\n\n[figure2]\nFigure 2. A typical solution for the equation of motion of the charged particle: $p(t)$ gives the overall motion, and $q(t)$ represents small oscillations around this trajectory. The ellipse on the right is a magnification of a part of this trajectory.\n\nAssume that $\\lambda_{0} = 8 \\times 10^{-9} \\mathrm{C} / \\mathrm{m}$ and $R = 10 \\mathrm{cm}$. We would like to use this device to trap a singly ionized atom 100 times heavier than a hydrogen atom. \n\nPhysical constants: mass of hydrogen atom $m_H = 1.674 \\times 10^{-27} kg$, charge of an electron $e = 1.602 \\times 10^{-19} C$, permittivity of free space $\\epsilon_0 = 8.854 \\times 10^{-12} F/m$, Boltzmann constant $k_B = 1.381 \\times 10^{-23} J/K$, Planck constant $\\hbar = 1.055 \\times 10^{-34} J \\cdot s$.", + "question": "(1) Calculate $k$ (expressed in $\\mathrm{rad/s}$). \n(2) Assume $a = 0.04$ and estimate the smallest frequency $\\Omega_{\\text{min}}$ required to stabilize the motion of this ion (expressed in $\\mathrm{rad/s}$). Use the data given at the end of the context.", + "marking": [ + [ + "Award 0.2 pt if the answer gives the correct value for $k = 2 \\times 10^{5} \\mathrm{rad/s}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct value for $\\Omega_{\\text{min}} \\approx 7 \\times 10^{6} \\mathrm{rad/s}$. Partial points: award 0.1 pt if the answer falls within the acceptable error range of the correct value but contains inappropriate number of significant figures. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$2 \\times 10^{5} \\mathrm{rad/s}$}", + "\\boxed{$7 \\times 10^{6} \\mathrm{rad/s}$}" + ], + "answer_type": [ + "Numerical Value", + "Numerical Value" + ], + "unit": [ + "$\\mathrm{rad/s}$", + "$\\mathrm{rad/s}$" + ], + "points": [ + 0.2, + 0.2 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_2_A_1_1.png", + "image_question/IPhO_2024_2_A_3_1.png" + ] + }, + { + "id": "IPhO_2024_2_B_1", + "context": "[Trapping Ions and Cooling Atoms] \n\nIn recent decades, trapping and cooling atoms and ions has been a fascinating topic for physicists, with several Nobel prizes awarded for work in this area. In the first part of this question, we will explore a technique for trapping ions, known as the \"Paul trap\". Wolfgang Paul and Hans Dehmelt received one half of the 1989 Nobel Prize in Physics for this work. Next, we investigate the Doppler cooling technique, one of the works cited in the press release for the 1997 Nobel Prize in Physics awarded to Steven Chu, Claude Cohen-Tannoudji, and William Daniel Phillips \"for developments of methods to cool and trap atoms with laser light\".\n\n[Doppler Cooling] \n\nIt may be necessary to cool a trapped atom or ion. Assume that a trapped atom of mass $m$, has two energy levels with an energy difference of $E_{0} = \\hbar \\omega_{\\mathrm{A}}$. Electrons in the lower level may absorb a photon and jump to the higher level, but after a period $\\tau$ they will return to the lower level and emit a photon with a frequency predominantly within $[\\omega_{\\mathrm{A}}-\\Gamma, \\omega_{\\mathrm{A}}+\\Gamma]$.", + "question": "Use the Heisenberg's uncertainty principle to find $\\Gamma$", + "marking": [ + [ + "Award 0.5 pt if the answer gives the correct expression for $\\Gamma = \\frac{1}{\\tau}$. Answers with different numerical coefficients should be considered as correct answers. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\Gamma = \\frac{1}{\\tao}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.5 + ], + "modality": "text-only", + "field": "Modern Physics", + "source": "IPhO_2024", + "image_question": [] + }, + { + "id": "IPhO_2024_2_B_2", + "context": "[Trapping Ions and Cooling Atoms] \n\nIn recent decades, trapping and cooling atoms and ions has been a fascinating topic for physicists, with several Nobel prizes awarded for work in this area. In the first part of this question, we will explore a technique for trapping ions, known as the \"Paul trap\". Wolfgang Paul and Hans Dehmelt received one half of the 1989 Nobel Prize in Physics for this work. Next, we investigate the Doppler cooling technique, one of the works cited in the press release for the 1997 Nobel Prize in Physics awarded to Steven Chu, Claude Cohen-Tannoudji, and William Daniel Phillips \"for developments of methods to cool and trap atoms with laser light\".\n\n[Doppler Cooling] \n\nIt may be necessary to cool a trapped atom or ion. Assume that a trapped atom of mass $m$, has two energy levels with an energy difference of $E_{0} = \\hbar \\omega_{\\mathrm{A}}$. Electrons in the lower level may absorb a photon and jump to the higher level, but after a period $\\tau$ they will return to the lower level and emit a photon with a frequency predominantly within $[\\omega_{\\mathrm{A}}-\\Gamma, \\omega_{\\mathrm{A}}+\\Gamma]$.\n\nWith a similar reasoning, when we shine a laser light on the trapped atom, if the angular frequency of the laser, $\\omega_{\\mathrm{L}}$, falls in the interval $[\\omega_{\\mathrm{A}}-\\Gamma, \\omega_{\\mathrm{A}}+\\Gamma]$, the atom may absorb the photon. Assume that the frequency $\\omega_{\\mathrm{L}}$ of the laser light is slightly lower than $\\omega_{\\mathrm{A}}$. For a particular device, the rate of photon absorption by an atom in the reference frame of the atom is given in Figure 1. The absorbed photon is then re-emitted in a random direction. To make things simple, we consider the problem in one dimension, i.e. we assume that the atoms can only move in the $x$ direction and the laser light shines on them both from the left and from the right. In the atom's reference frame, the light has a higher or lower frequency due to the motion of the atoms. Since the velocity $v$ of the atoms is very small, we only include terms of order $v / c$ and ignore all the higher-order terms. Moreover, we have $m \\gg \\hbar \\omega_{\\mathrm{A}} / c^{2}$ so that the velocity of the atom nearly does not change after absorbing the photon. Also, the change in frequency due to the Doppler effect is so small compared to $\\omega_{\\mathrm{A}}-\\omega_{\\mathrm{L}}$, that the function for $s$ in the diagram of Figure 1 may be approximated by the following linear function:\n\n$s(\\omega) = s_{\\mathrm{L}} + \\alpha (\\omega - \\omega_{\\mathrm{L}})$\nwhere $s$ is the number of absorbed photons per unit of time, $s_{\\mathrm{L}}$ is the value of $s$ for $\\omega = \\omega_{\\mathrm{L}}$, and $\\alpha$ is the slope of the line tangent to the curve at $\\omega_{\\mathrm{L}}$. The frequency of the re-emitted photon is almost equal to the frequency of the incident photon, but it is emitted with equal probability in the positive or negative $x$-direction. In fact, up to the order considered here the two frequencies are identical. Note that we are considering the whole process in the atom's reference frame.\n\n[figure1]\nFigure 1. The rate of photon absorption as a function of the frequency for a particular trap: the frequency corresponding to the energy difference between the two atomic levels is indicated by $\\omega_{\\mathrm{A}}$ and the slighlty smaller frequency of the laser is indicated by $\\omega_{\\mathrm{L}}$.", + "question": "Assume that the trapped atom is moving with a velocity, $v = v_{\\mathrm{x}}$ in the lab frame. In the frame of reference of the atom: \n(1) Calculate the collision rate of the photons, incident from one direction, with the atoms (denoted by $s_{+}$). \n(2) Calculate the collision rate of the photons, incident from another direction, with the atoms (denoted by $s_{-}$). \n(3) Calculate the rate of absorption of momentum in one direction (denoted by $\\pi_{+}$). \n(4) Calculate the rate of absorption of momentum in another direction (denoted by $\\pi_{-}$). \n(5) Determine the effective force $F$ on the atom as a function of $v$, $k_{\\mathrm{L}} = \\omega_{\\mathrm{L}} / c$, $\\hbar$, and $\\alpha$, in the reference frame of the laboratory. Assume $s_{\\mathrm{L}} \\ll \\alpha \\omega_{\\mathrm{L}}$.", + "marking": [ + [ + "Award a total of 0.5 pt for the following: award 0.3 pt if the answer gives the correct Doppler shift formulas for $\\omega_{+} = \\omega_{L} \\left(1 + \\frac{v}{c}\\right)$ and award 0.2 pt if the answer finds the correct expression for $s_{+} = s_{L} + \\alpha \\omega_{L} \\frac{v}{c}$. Otherwise, award 0 pt.", + "Award a total of 0.5 pt for the following: award 0.3 pt if the answer gives the correct Doppler shift formulas for $\\omega_{-} = \\omega_{L} \\left(1 - \\frac{v}{c}\\right)$ and award 0.2 pt if the answer finds the correct expression for $s_{-} = s_{L} - \\alpha \\omega_{L} \\frac{v}{c}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct expression for $\\pi_{+} = s_{+} \\times (-\\hbar k_{+})$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct expression for $\\pi_{-} = s_{-} \\times (+\\hbar k_{-})$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the correct expression for the force $F = -(2\\alpha\\hbar k_{L}^2) v$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$s_{+} = s_L + \\alpha \\omega_L \\frac{v}{c}$}", + "\\boxed{$s_{-} = s_L - \\alpha \\omega_L \\frac{v}{c}$}", + "\\boxed{$\\pi_{+} = s_{+} \\times (-\\hbar k_{+})$}", + "\\boxed{$\\pi_{-} = s_{-} \\times (+\\hbar k_{-})$}", + "\\boxed{$F = -(2\\alpha\\hbar k_L^2) v$}" + ], + "answer_type": [ + "Expression", + "Expression", + "Expression", + "Expression", + "Expression" + ], + "unit": [ + null, + null, + null, + null, + null + ], + "points": [ + 0.5, + 0.5, + 0.1, + 0.1, + 0.5 + ], + "modality": "text+variable figure", + "field": "Modern Physics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_2_B_2_1.png" + ] + }, + { + "id": "IPhO_2024_2_B_3", + "context": "[Trapping Ions and Cooling Atoms] \n\nIn recent decades, trapping and cooling atoms and ions has been a fascinating topic for physicists, with several Nobel prizes awarded for work in this area. In the first part of this question, we will explore a technique for trapping ions, known as the \"Paul trap\". Wolfgang Paul and Hans Dehmelt received one half of the 1989 Nobel Prize in Physics for this work. Next, we investigate the Doppler cooling technique, one of the works cited in the press release for the 1997 Nobel Prize in Physics awarded to Steven Chu, Claude Cohen-Tannoudji, and William Daniel Phillips \"for developments of methods to cool and trap atoms with laser light\".\n\n[Doppler Cooling] \n\nIt may be necessary to cool a trapped atom or ion. Assume that a trapped atom of mass $m$, has two energy levels with an energy difference of $E_{0} = \\hbar \\omega_{\\mathrm{A}}$. Electrons in the lower level may absorb a photon and jump to the higher level, but after a period $\\tau$ they will return to the lower level and emit a photon with a frequency predominantly within $[\\omega_{\\mathrm{A}}-\\Gamma, \\omega_{\\mathrm{A}}+\\Gamma]$.\n\nWith a similar reasoning, when we shine a laser light on the trapped atom, if the angular frequency of the laser, $\\omega_{\\mathrm{L}}$, falls in the interval $[\\omega_{\\mathrm{A}}-\\Gamma, \\omega_{\\mathrm{A}}+\\Gamma]$, the atom may absorb the photon. Assume that the frequency $\\omega_{\\mathrm{L}}$ of the laser light is slightly lower than $\\omega_{\\mathrm{A}}$. For a particular device, the rate of photon absorption by an atom in the reference frame of the atom is given in Figure 1. The absorbed photon is then re-emitted in a random direction. To make things simple, we consider the problem in one dimension, i.e. we assume that the atoms can only move in the $x$ direction and the laser light shines on them both from the left and from the right. In the atom's reference frame, the light has a higher or lower frequency due to the motion of the atoms. Since the velocity $v$ of the atoms is very small, we only include terms of order $v / c$ and ignore all the higher-order terms. Moreover, we have $m \\gg \\hbar \\omega_{\\mathrm{A}} / c^{2}$ so that the velocity of the atom nearly does not change after absorbing the photon. Also, the change in frequency due to the Doppler effect is so small compared to $\\omega_{\\mathrm{A}}-\\omega_{\\mathrm{L}}$, that the function for $s$ in the diagram of Figure 1 may be approximated by the following linear function:\n\n$s(\\omega) = s_{\\mathrm{L}} + \\alpha (\\omega - \\omega_{\\mathrm{L}})$\nwhere $s$ is the number of absorbed photons per unit of time, $s_{\\mathrm{L}}$ is the value of $s$ for $\\omega = \\omega_{\\mathrm{L}}$, and $\\alpha$ is the slope of the line tangent to the curve at $\\omega_{\\mathrm{L}}$. The frequency of the re-emitted photon is almost equal to the frequency of the incident photon, but it is emitted with equal probability in the positive or negative $x$-direction. In fact, up to the order considered here the two frequencies are identical. Note that we are considering the whole process in the atom's reference frame.\n\n[figure1]\nFigure 1. The rate of photon absorption as a function of the frequency for a particular trap: the frequency corresponding to the energy difference between the two atomic levels is indicated by $\\omega_{\\mathrm{A}}$ and the slighlty smaller frequency of the laser is indicated by $\\omega_{\\mathrm{L}}$.\n\nWe would like to find the lowest temperature that can be achieved using this technique. Assume that the velocity of a particular atom has been reduced to zero exactly, and at this very moment it absorbs a photon (incident from any of the two directions), and re-emits the photon randomly in any of the two directions, with almost the same frequency. Assume that this process happens once every $\\tau$ units of time.", + "question": "Considering the momentum of the atom after such a process for the two possible outcomes, calculate the average power absorbed by the atom.", + "marking": [ + [ + "Award 0.5 pt if the answer considers two equally likely outcomes for the final momentum: (1) The photon is emitted in the positive $x$-direction which causes the atom's momentum to become $p=0$, (2) The photon is emitted in the negative $x$-direction which causes the atom's momentum to become $P_{\\mathrm{f}}=+2 \\hbar k_{\\mathrm{L}}$. Partial points: award 0.3 pt if the answer considers only one of the two outcomes. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the correct expression for the average power absorbed by the atom, $P_{\\mathrm{in}} = \\frac{\\hbar^{2} k_{\\mathrm{L}}^{2}}{m \\tau}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{\\hbar^{2} k_{\\mathrm{L}}^{2}}{m \\tau}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+variable figure", + "field": "Modern Physics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_2_B_2_1.png" + ] + }, + { + "id": "IPhO_2024_2_B_4", + "context": "[Trapping Ions and Cooling Atoms] \n\nIn recent decades, trapping and cooling atoms and ions has been a fascinating topic for physicists, with several Nobel prizes awarded for work in this area. In the first part of this question, we will explore a technique for trapping ions, known as the \"Paul trap\". Wolfgang Paul and Hans Dehmelt received one half of the 1989 Nobel Prize in Physics for this work. Next, we investigate the Doppler cooling technique, one of the works cited in the press release for the 1997 Nobel Prize in Physics awarded to Steven Chu, Claude Cohen-Tannoudji, and William Daniel Phillips \"for developments of methods to cool and trap atoms with laser light\".\n\n[Doppler Cooling] \n\nIt may be necessary to cool a trapped atom or ion. Assume that a trapped atom of mass $m$, has two energy levels with an energy difference of $E_{0} = \\hbar \\omega_{\\mathrm{A}}$. Electrons in the lower level may absorb a photon and jump to the higher level, but after a period $\\tau$ they will return to the lower level and emit a photon with a frequency predominantly within $[\\omega_{\\mathrm{A}}-\\Gamma, \\omega_{\\mathrm{A}}+\\Gamma]$.\n\nWith a similar reasoning, when we shine a laser light on the trapped atom, if the angular frequency of the laser, $\\omega_{\\mathrm{L}}$, falls in the interval $[\\omega_{\\mathrm{A}}-\\Gamma, \\omega_{\\mathrm{A}}+\\Gamma]$, the atom may absorb the photon. Assume that the frequency $\\omega_{\\mathrm{L}}$ of the laser light is slightly lower than $\\omega_{\\mathrm{A}}$. For a particular device, the rate of photon absorption by an atom in the reference frame of the atom is given in Figure 1. The absorbed photon is then re-emitted in a random direction. To make things simple, we consider the problem in one dimension, i.e. we assume that the atoms can only move in the $x$ direction and the laser light shines on them both from the left and from the right. In the atom's reference frame, the light has a higher or lower frequency due to the motion of the atoms. Since the velocity $v$ of the atoms is very small, we only include terms of order $v / c$ and ignore all the higher-order terms. Moreover, we have $m \\gg \\hbar \\omega_{\\mathrm{A}} / c^{2}$ so that the velocity of the atom nearly does not change after absorbing the photon. Also, the change in frequency due to the Doppler effect is so small compared to $\\omega_{\\mathrm{A}}-\\omega_{\\mathrm{L}}$, that the function for $s$ in the diagram of Figure 1 may be approximated by the following linear function:\n\n$s(\\omega) = s_{\\mathrm{L}} + \\alpha (\\omega - \\omega_{\\mathrm{L}})$\nwhere $s$ is the number of absorbed photons per unit of time, $s_{\\mathrm{L}}$ is the value of $s$ for $\\omega = \\omega_{\\mathrm{L}}$, and $\\alpha$ is the slope of the line tangent to the curve at $\\omega_{\\mathrm{L}}$. The frequency of the re-emitted photon is almost equal to the frequency of the incident photon, but it is emitted with equal probability in the positive or negative $x$-direction. In fact, up to the order considered here the two frequencies are identical. Note that we are considering the whole process in the atom's reference frame.\n\n[figure1]\nFigure 1. The rate of photon absorption as a function of the frequency for a particular trap: the frequency corresponding to the energy difference between the two atomic levels is indicated by $\\omega_{\\mathrm{A}}$ and the slighlty smaller frequency of the laser is indicated by $\\omega_{\\mathrm{L}}$.\n\nWe would like to find the lowest temperature that can be achieved using this technique. Assume that the velocity of a particular atom has been reduced to zero exactly, and at this very moment it absorbs a photon (incident from any of the two directions), and re-emits the photon randomly in any of the two directions, with almost the same frequency. Assume that this process happens once every $\\tau$ units of time.", + "question": "Determine the effective force $F$ on the atom as a function of $v$, $k_{\\mathrm{L}} = \\omega_{\\mathrm{L}} / c$, $\\hbar$, and $\\alpha$, in the reference frame of the laboratory. Assume $s_{\\mathrm{L}} \\ll \\alpha \\omega_{\\mathrm{L}}$ (This is a preliminary question and do not include in the final answer). \n\n(1) Consider the calculated force $F$ and calculate the output power. \n(2) Calculate the average value of $v^{2}$ at equilibrium. \n(3) Using your knowledge of the kinetic theory of gases estimate the temperature of the atoms $T$.", + "marking": [ + [ + "Award 0.3 pt if the answer gives the correct expression for the output power, $P_{\\text{out}} = -2 \\alpha \\hbar k_{L}^2 v^2$. Answers with different numerical coefficients should be considered as correct answers. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer gives the correct expression for the average value of $v^{2}$ at equilibrium, $\\bar{v^{2}} = \\frac{\\hbar \\Gamma}{2 \\alpha m}$. Answers with different numerical coefficients should be considered as correct answers. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct expression for the temperature of the atoms, $T = \\frac{\\hbar \\Gamma}{2 \\alpha k_{B}}$. Answers with different numerical coefficients should be considered as correct answers. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$-2 \\alpha \\hbar k_L^2 v^2$}", + "\\boxed{$\\frac{\\hbar \\Gamma}{2\\alpha m}$}", + "\\boxed{$\\frac{\\hbar \\Gamma}{2\\alpha k_B}$}" + ], + "answer_type": [ + "Expression", + "Expression", + "Expression" + ], + "unit": [ + null, + null, + null + ], + "points": [ + 0.3, + 0.3, + 0.2 + ], + "modality": "text+variable figure", + "field": "Modern Physics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_2_B_2_1.png" + ] + }, + { + "id": "IPhO_2024_2_B_5", + "context": "[Trapping Ions and Cooling Atoms] \n\nIn recent decades, trapping and cooling atoms and ions has been a fascinating topic for physicists, with several Nobel prizes awarded for work in this area. In the first part of this question, we will explore a technique for trapping ions, known as the \"Paul trap\". Wolfgang Paul and Hans Dehmelt received one half of the 1989 Nobel Prize in Physics for this work. Next, we investigate the Doppler cooling technique, one of the works cited in the press release for the 1997 Nobel Prize in Physics awarded to Steven Chu, Claude Cohen-Tannoudji, and William Daniel Phillips \"for developments of methods to cool and trap atoms with laser light\".\n\n[Doppler Cooling] \n\nIt may be necessary to cool a trapped atom or ion. Assume that a trapped atom of mass $m$, has two energy levels with an energy difference of $E_{0} = \\hbar \\omega_{\\mathrm{A}}$. Electrons in the lower level may absorb a photon and jump to the higher level, but after a period $\\tau$ they will return to the lower level and emit a photon with a frequency predominantly within $[\\omega_{\\mathrm{A}}-\\Gamma, \\omega_{\\mathrm{A}}+\\Gamma]$.\n\nWith a similar reasoning, when we shine a laser light on the trapped atom, if the angular frequency of the laser, $\\omega_{\\mathrm{L}}$, falls in the interval $[\\omega_{\\mathrm{A}}-\\Gamma, \\omega_{\\mathrm{A}}+\\Gamma]$, the atom may absorb the photon. Assume that the frequency $\\omega_{\\mathrm{L}}$ of the laser light is slightly lower than $\\omega_{\\mathrm{A}}$. For a particular device, the rate of photon absorption by an atom in the reference frame of the atom is given in Figure 1. The absorbed photon is then re-emitted in a random direction. To make things simple, we consider the problem in one dimension, i.e. we assume that the atoms can only move in the $x$ direction and the laser light shines on them both from the left and from the right. In the atom's reference frame, the light has a higher or lower frequency due to the motion of the atoms. Since the velocity $v$ of the atoms is very small, we only include terms of order $v / c$ and ignore all the higher-order terms. Moreover, we have $m \\gg \\hbar \\omega_{\\mathrm{A}} / c^{2}$ so that the velocity of the atom nearly does not change after absorbing the photon. Also, the change in frequency due to the Doppler effect is so small compared to $\\omega_{\\mathrm{A}}-\\omega_{\\mathrm{L}}$, that the function for $s$ in the diagram of Figure 1 may be approximated by the following linear function:\n\n$s(\\omega) = s_{\\mathrm{L}} + \\alpha (\\omega - \\omega_{\\mathrm{L}})$\nwhere $s$ is the number of absorbed photons per unit of time, $s_{\\mathrm{L}}$ is the value of $s$ for $\\omega = \\omega_{\\mathrm{L}}$, and $\\alpha$ is the slope of the line tangent to the curve at $\\omega_{\\mathrm{L}}$. The frequency of the re-emitted photon is almost equal to the frequency of the incident photon, but it is emitted with equal probability in the positive or negative $x$-direction. In fact, up to the order considered here the two frequencies are identical. Note that we are considering the whole process in the atom's reference frame.\n\n[figure1]\nFigure 1. The rate of photon absorption as a function of the frequency for a particular trap: the frequency corresponding to the energy difference between the two atomic levels is indicated by $\\omega_{\\mathrm{A}}$ and the slighlty smaller frequency of the laser is indicated by $\\omega_{\\mathrm{L}}$.\n\nWe would like to find the lowest temperature that can be achieved using this technique. Assume that the velocity of a particular atom has been reduced to zero exactly, and at this very moment it absorbs a photon (incident from any of the two directions), and re-emits the photon randomly in any of the two directions, with almost the same frequency. Assume that this process happens once every $\\tau$ units of time.", + "question": "Estimate the temperature of the atoms $T$ (expressed in $K$), for an atom 100 times heavier than a hydrogen atom. Assume that $\\omega_{\\mathrm{L}} = 2 \\times 10^{16} \\mathrm{rad} / \\mathrm{s}, \\tau = 5 \\times 10^{-9} \\mathrm{s}$, and $\\alpha=4$.", + "marking": [ + [ + "Award 0.4 pt if the answer gives the correct expression for the temperature of the atoms, $T = 2 \\times 10^{-4} K$. If the answer falls within the acceptable error range of the correct value, the answer should be considered correct. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$2 \\times 10^{-4}$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "$K$" + ], + "points": [ + 0.4 + ], + "modality": "text+variable figure", + "field": "Modern Physics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_2_B_2_1.png" + ] + }, + { + "id": "IPhO_2024_3_A_1", + "context": "[Black Widow Pulsar] \n\nA significant number of the observed stars are binaries. One or both of the stars may be neutron stars rotating with a high angular velocity and emitting electromagnetic waves; such stars are called pulsars. Sometimes a companion star is an expansive mass of gas that gradually falls down onto the neutron star and causes its mass to increase (Figure 1-a). In this way, a neutron star gradually swallows up a portion of the mass of its companion star. For this reason, the neutron star has been compared to a black widow (or redback spider), a female spider which eats its mate after mating. The heating of the gas falling down onto the black widow generates radiation which can be observed. The heaviest neutron stars often are black widows and they serve as natural laboratories for testing fundamental physics. Figure 1-b shows the picture of the companion of the neutron star PSR J2215+5135, taken by the 3.4-meter optical telescope of the Iranian National Observatory. No neutron star can be seen in this image and the observed light is due to its companion.\n\n[figure1]\nFigure 1. (a) The falling gases of the companion star onto the neutron star. (b) The companion of the neutron star PSR J2215+5135. \n\n[A Binary System]\n\nConsider a simple model in which the black widow and its companion star, are represented by two point masses $M_{1}$ and $M_{2}$ moving on a circular orbit around their center of mass. To investigate the dynamics of this system, consider a rotating coordinate system in which the two bodies are stationary. Take the center of mass to be the origin of the coordinate system. Assume that the two point masses lie on the $x$-axis on both sides of the origin at a distance $a$ from each other, and that $M_{1}$ lies on the negative $x$-axis. At an arbitrary point $(x, y)$ in the plane of motion, the effective potential $\\varphi(x, y)$ for a unit test mass is the sum of the gravitational potentials of the two point masses plus the centrifugal potential.", + "question": "Write $\\varphi(x, y)$ in terms of $M_{1}$, $M_{2}$, $G$, and $a$.", + "marking": [ + [ + "Award 1.0 pt if the answer gives the correct expression for $\\varphi(x, y)$: $\\varphi(x, y) = -\\frac{G M_{1}}{\\sqrt{\\left(x + \\frac{M_2}{M_1+M_2} a \\right)^{2}+y^{2}}} - \\frac{G M_{2}}{\\sqrt{\\left(x-\\frac{M_{1}}{M_1+M_2} a \\right)^{2}+y^{2}}} - \\frac{1}{2} \\frac{G (M_1+M_2)}{a^{3}}(x^{2}+y^{2})$. Partial points: if the answer gives the correct expression for the gravitational part, award 0.5 pt; if the answer gives the correct expression for the centrifugal part, award 0.5 pt. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\varphi(x, y) = -\\frac{G M_{1}}{\\sqrt{\\left(x + \\frac{M_2}{M_1+M_2} a \\right)^{2}+y^{2}}} - \\frac{G M_{2}}{\\sqrt{\\left(x-\\frac{M_{1}}{M_1+M_2} a \\right)^{2}+y^{2}}} - \\frac{1}{2} \\frac{G (M_1+M_2)}{a^{3}}(x^{2}+y^{2})$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_3_A_1_1.png" + ] + }, + { + "id": "IPhO_2024_3_A_3", + "context": "[Black Widow Pulsar] \n\nA significant number of the observed stars are binaries. One or both of the stars may be neutron stars rotating with a high angular velocity and emitting electromagnetic waves; such stars are called pulsars. Sometimes a companion star is an expansive mass of gas that gradually falls down onto the neutron star and causes its mass to increase (Figure 1-a). In this way, a neutron star gradually swallows up a portion of the mass of its companion star. For this reason, the neutron star has been compared to a black widow (or redback spider), a female spider which eats its mate after mating. The heating of the gas falling down onto the black widow generates radiation which can be observed. The heaviest neutron stars often are black widows and they serve as natural laboratories for testing fundamental physics. Figure 1-b shows the picture of the companion of the neutron star PSR J2215+5135, taken by the 3.4-meter optical telescope of the Iranian National Observatory. No neutron star can be seen in this image and the observed light is due to its companion.\n\n[figure1]\nFigure 1. (a) The falling gases of the companion star onto the neutron star. (b) The companion of the neutron star PSR J2215+5135. \n\n[A Binary System]\n\nConsider a simple model in which the black widow and its companion star, are represented by two point masses $M_{1}$ and $M_{2}$ moving on a circular orbit around their center of mass. To investigate the dynamics of this system, consider a rotating coordinate system in which the two bodies are stationary. Take the center of mass to be the origin of the coordinate system. Assume that the two point masses lie on the $x$-axis on both sides of the origin at a distance $a$ from each other, and that $M_{1}$ lies on the negative $x$-axis. At an arbitrary point $(x, y)$ in the plane of motion, the effective potential $\\varphi(x, y)$ for a unit test mass is the sum of the gravitational potentials of the two point masses plus the centrifugal potential.\n\nSuppose $M_{2} = M_{1} / 3$ and assume that $M_{2}$ is surrounded by a rarefied gas of very low density. The mass of this gas is insignificant and we ignore its gravitational effects. If the size of this gas envelope becomes greater than a specific limit, the gas will overflow onto $M_{1}$. Suppose the overflow occurs through $x = x_{0}$ on the $x$-axis.", + "question": "Find the numerical value of $\\frac{x_{0}}{a}$, up to two significant figures.", + "marking": [ + [ + "Award 0.5 pt if the answer gives the correct numerical value of $\\frac{x_{0}}{a}$, which is approximately 0.36. Partial points: if the answer obtains correct equation of $f(\\bar{x}_0) = \\frac{a}{G M} \\frac{d \\varphi}{d \\bar{x}} = 0$ with $\\varphi(\\bar{x}, 0) = \\frac{GM}{a} \\left[-\\frac{3/4}{(\\bar{x}+1/4)} + \\frac{1/4}{(\\bar{x}-3/4)} - \\frac{1}{2} \\bar{x}^2 \\right]$, but does not solve it correctly, award 0.2 pt; if the answer gives a numerical value of $\\frac{x_{0}}{a}$ that is within the acceptable error range of the correct value but rounded to only one decimal figure, award 0.3 pt. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{0.36}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 0.5 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_3_A_1_1.png" + ] + }, + { + "id": "IPhO_2024_3_A_4", + "context": "[Black Widow Pulsar] \n\nA significant number of the observed stars are binaries. One or both of the stars may be neutron stars rotating with a high angular velocity and emitting electromagnetic waves; such stars are called pulsars. Sometimes a companion star is an expansive mass of gas that gradually falls down onto the neutron star and causes its mass to increase (Figure 1-a). In this way, a neutron star gradually swallows up a portion of the mass of its companion star. For this reason, the neutron star has been compared to a black widow (or redback spider), a female spider which eats its mate after mating. The heating of the gas falling down onto the black widow generates radiation which can be observed. The heaviest neutron stars often are black widows and they serve as natural laboratories for testing fundamental physics. Figure 1-b shows the picture of the companion of the neutron star PSR J2215+5135, taken by the 3.4-meter optical telescope of the Iranian National Observatory. No neutron star can be seen in this image and the observed light is due to its companion.\n\n[figure1]\nFigure 1. (a) The falling gases of the companion star onto the neutron star. (b) The companion of the neutron star PSR J2215+5135. \n\n[A Binary System]\n\nConsider a simple model in which the black widow and its companion star, are represented by two point masses $M_{1}$ and $M_{2}$ moving on a circular orbit around their center of mass. To investigate the dynamics of this system, consider a rotating coordinate system in which the two bodies are stationary. Take the center of mass to be the origin of the coordinate system. Assume that the two point masses lie on the $x$-axis on both sides of the origin at a distance $a$ from each other, and that $M_{1}$ lies on the negative $x$-axis. At an arbitrary point $(x, y)$ in the plane of motion, the effective potential $\\varphi(x, y)$ for a unit test mass is the sum of the gravitational potentials of the two point masses plus the centrifugal potential.\n\nTake the rotational period of the stars around their center of mass to be $P$. Assume that mass flows from $M_{2}$ to $M_{1}$ at a very small rate of $\\mathrm{d} M_{1} / \\mathrm{d} t = \\beta$. This rate is so small that in each period of rotation, the distance between the two stars can be assumed to be constant. However, after a long period of time, the distance between the two stars changes, while the motion remains circular.", + "question": "(1) Calculate $\\dot{a}$, i.e., the rate of change of $a$ in terms of $\\beta$, $M_{1}$, $M_{2}$, $G$, and $a$. \n(2) Calculate $\\dot{P}$, i.e., the rate of change of $P$ in terms of $\\beta$, $M_{1}$, $M_{2}$, $G$, and $a$.", + "marking": [ + [ + "Award 0.6 pt if the answer gives both correct expressions for $\\dot{a}$ and $\\dot{P}$: $\\dot{a} = -2 \\beta a \\left(\\frac{1}{M_{1}} - \\frac{1}{M_{2}}\\right)$ and $\\dot{P} = -6 \\pi \\sqrt{\\frac{a^{3}}{G M}} \\beta \\left(\\frac{1}{M_{1}} - \\frac{1}{M_{2}}\\right)$. Partial points: if the answer gives only the correct expression for $\\dot{a}$, award 0.3 pt; if the answer gives only the correct expression for $\\dot{P}$, award 0.3 pt. If the answer gives both incorrect expressions but uses the correct approach of conservation of momentum, award 0.2 pt. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\dot{a} = -2 \\beta a \\left(\\frac{1}{M_{1}} - \\frac{1}{M_{2}}\\right)$}", + "\\boxed{$\\dot{P} = -6 \\pi \\sqrt{\\frac{a^{3}}{G M}} \\beta \\left(\\frac{1}{M_{1}} - \\frac{1}{M_{2}}\\right)$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 0.3, + 0.3 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_3_A_1_1.png" + ] + }, + { + "id": "IPhO_2024_3_A_5", + "context": "[Black Widow Pulsar] \n\nA significant number of the observed stars are binaries. One or both of the stars may be neutron stars rotating with a high angular velocity and emitting electromagnetic waves; such stars are called pulsars. Sometimes a companion star is an expansive mass of gas that gradually falls down onto the neutron star and causes its mass to increase (Figure 1-a). In this way, a neutron star gradually swallows up a portion of the mass of its companion star. For this reason, the neutron star has been compared to a black widow (or redback spider), a female spider which eats its mate after mating. The heating of the gas falling down onto the black widow generates radiation which can be observed. The heaviest neutron stars often are black widows and they serve as natural laboratories for testing fundamental physics. Figure 1-b shows the picture of the companion of the neutron star PSR J2215+5135, taken by the 3.4-meter optical telescope of the Iranian National Observatory. No neutron star can be seen in this image and the observed light is due to its companion.\n\n[figure1]\nFigure 1. (a) The falling gases of the companion star onto the neutron star. (b) The companion of the neutron star PSR J2215+5135. \n\n[A Binary System]\n\nConsider a simple model in which the black widow and its companion star, are represented by two point masses $M_{1}$ and $M_{2}$ moving on a circular orbit around their center of mass. To investigate the dynamics of this system, consider a rotating coordinate system in which the two bodies are stationary. Take the center of mass to be the origin of the coordinate system. Assume that the two point masses lie on the $x$-axis on both sides of the origin at a distance $a$ from each other, and that $M_{1}$ lies on the negative $x$-axis. At an arbitrary point $(x, y)$ in the plane of motion, the effective potential $\\varphi(x, y)$ for a unit test mass is the sum of the gravitational potentials of the two point masses plus the centrifugal potential.\n\nTake the rotational period of the stars around their center of mass to be $P$. Assume that mass flows from $M_{2}$ to $M_{1}$ at a very small rate of $\\mathrm{d} M_{1} / \\mathrm{d} t = \\beta$. This rate is so small that in each period of rotation, the distance between the two stars can be assumed to be constant. However, after a long period of time, the distance between the two stars changes, while the motion remains circular.\n\nThe gas separated from $M_{2}$ forms a disk rotating around $M_{1}$ and heats up due to friction (Figure 1 -a). As the gas loses energy, it spirals inward toward $M_{1}$ and finally falls onto it. In the steady state, the mass flows at the constant rate of $\\beta$, from $M_{2}$ to the disc and from the disc onto $M_{1}$. At the same time, the heated disk emits thermal radiation as a blackbody. This disk forms very close to the neutron star so the gravitational pull of the $M_{2}$ star can be ignored for the analysis of the disk's motion. Also, ignore the heat capacity of the gas.", + "question": "Determine the temperature $T$ of the disc at distance $r$ from the center of the star $M_{1}$ in terms of $\\beta$, $M_{1}$, $G$, and $\\sigma$ (Stefan-Boltzmann constant).", + "marking": [ + [ + "Award 0.5 pt if the answer uses the correct approach of energy relations. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer gives the correct expression for the temperature $T$: $T = \\left(\\frac{G M_{1} \\beta}{8 \\pi \\sigma r^{3}}\\right)^{\\frac{1}{4}}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$T = \\left(\\frac{G M_{1} \\beta}{8 \\pi \\sigma r^{3}}\\right)^{\\frac{1}{4}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_3_A_1_1.png" + ] + }, + { + "id": "IPhO_2024_3_A_6", + "context": "[Black Widow Pulsar] \n\nA significant number of the observed stars are binaries. One or both of the stars may be neutron stars rotating with a high angular velocity and emitting electromagnetic waves; such stars are called pulsars. Sometimes a companion star is an expansive mass of gas that gradually falls down onto the neutron star and causes its mass to increase (Figure 1-a). In this way, a neutron star gradually swallows up a portion of the mass of its companion star. For this reason, the neutron star has been compared to a black widow (or redback spider), a female spider which eats its mate after mating. The heating of the gas falling down onto the black widow generates radiation which can be observed. The heaviest neutron stars often are black widows and they serve as natural laboratories for testing fundamental physics. Figure 1-b shows the picture of the companion of the neutron star PSR J2215+5135, taken by the 3.4-meter optical telescope of the Iranian National Observatory. No neutron star can be seen in this image and the observed light is due to its companion.\n\n[figure1]\nFigure 1. (a) The falling gases of the companion star onto the neutron star. (b) The companion of the neutron star PSR J2215+5135. \n\n[A Binary System]\n\nConsider a simple model in which the black widow and its companion star, are represented by two point masses $M_{1}$ and $M_{2}$ moving on a circular orbit around their center of mass. To investigate the dynamics of this system, consider a rotating coordinate system in which the two bodies are stationary. Take the center of mass to be the origin of the coordinate system. Assume that the two point masses lie on the $x$-axis on both sides of the origin at a distance $a$ from each other, and that $M_{1}$ lies on the negative $x$-axis. At an arbitrary point $(x, y)$ in the plane of motion, the effective potential $\\varphi(x, y)$ for a unit test mass is the sum of the gravitational potentials of the two point masses plus the centrifugal potential.\n\nTake the rotational period of the stars around their center of mass to be $P$. Assume that mass flows from $M_{2}$ to $M_{1}$ at a very small rate of $\\mathrm{d} M_{1} / \\mathrm{d} t = \\beta$. This rate is so small that in each period of rotation, the distance between the two stars can be assumed to be constant. However, after a long period of time, the distance between the two stars changes, while the motion remains circular.\n\nThe gas separated from $M_{2}$ forms a disk rotating around $M_{1}$ and heats up due to friction (Figure 1 -a). As the gas loses energy, it spirals inward toward $M_{1}$ and finally falls onto it. In the steady state, the mass flows at the constant rate of $\\beta$, from $M_{2}$ to the disc and from the disc onto $M_{1}$. At the same time, the heated disk emits thermal radiation as a blackbody. This disk forms very close to the neutron star so the gravitational pull of the $M_{2}$ star can be ignored for the analysis of the disk's motion. Also, ignore the heat capacity of the gas.\n\nIn the binary system PSR J2215+5135, the mass of the neutron star is $M_{\\mathrm{NS}} = 2.27 M_{\\odot}$ and the mass of its companion star is $M_{\\mathrm{S}} = 0.33 M_{\\odot}$, where $M_{\\odot} = 1.98 \\times 10^{30} \\mathrm{kg}$ is the mass of the Sun. The rotational period is $P = 4.14 \\mathrm{hr}$, and the Stefan-Boltzmann constant is $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} / (\\mathrm{m}^{2} \\mathrm{K}^{4})$, and the gravitational constant is $G = 6.67 \\times 10^{-11} \\mathrm{m}^{3} / (\\mathrm{kg} s)^{2}$. Assume that the mass flow rate to the neutron star is $\\beta = \\dot{M}_{\\mathrm{NS}} = 9 \\times 10^{-10} M_{\\odot} \\mathrm{yr}^{-1}$.", + "question": "Calculate the temperature $T$ of the disc at the radius $r = \\frac{a}{10}$ in kelvins.", + "marking": [ + [ + "Award 0.5 pt if the answer gives the correct value of the temperature, $T = 9 \\times 10^{3} K$. Partial points: if the numerical answer is incorrect, award 0.3 pt for the correct expression for $a = \\left[\\frac{P^{2} G\\left(M_{\\mathrm{S}}+M_{\\mathrm{NS}}\\right)}{4 \\pi^{2}}\\right]^{\\frac{1}{3}}$ and 0.1 pt for the correct expression for $T = \\left(\\frac{500 \\pi M_{\\mathrm{NS}} \\beta}{\\sigma P^{2}\\left(M_{\\mathrm{S}}+M_{\\mathrm{NS}}\\right)}\\right)^{\\frac{1}{4}}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$9 \\times 10^{3}$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "K" + ], + "points": [ + 0.5 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_3_A_1_1.png" + ] + }, + { + "id": "IPhO_2024_3_A_7", + "context": "[Black Widow Pulsar] \n\nA significant number of the observed stars are binaries. One or both of the stars may be neutron stars rotating with a high angular velocity and emitting electromagnetic waves; such stars are called pulsars. Sometimes a companion star is an expansive mass of gas that gradually falls down onto the neutron star and causes its mass to increase (Figure 1-a). In this way, a neutron star gradually swallows up a portion of the mass of its companion star. For this reason, the neutron star has been compared to a black widow (or redback spider), a female spider which eats its mate after mating. The heating of the gas falling down onto the black widow generates radiation which can be observed. The heaviest neutron stars often are black widows and they serve as natural laboratories for testing fundamental physics. Figure 1-b shows the picture of the companion of the neutron star PSR J2215+5135, taken by the 3.4-meter optical telescope of the Iranian National Observatory. No neutron star can be seen in this image and the observed light is due to its companion.\n\n[figure1]\nFigure 1. (a) The falling gases of the companion star onto the neutron star. (b) The companion of the neutron star PSR J2215+5135. \n\n[A Binary System]\n\nConsider a simple model in which the black widow and its companion star, are represented by two point masses $M_{1}$ and $M_{2}$ moving on a circular orbit around their center of mass. To investigate the dynamics of this system, consider a rotating coordinate system in which the two bodies are stationary. Take the center of mass to be the origin of the coordinate system. Assume that the two point masses lie on the $x$-axis on both sides of the origin at a distance $a$ from each other, and that $M_{1}$ lies on the negative $x$-axis. At an arbitrary point $(x, y)$ in the plane of motion, the effective potential $\\varphi(x, y)$ for a unit test mass is the sum of the gravitational potentials of the two point masses plus the centrifugal potential.\n\nTake the rotational period of the stars around their center of mass to be $P$. Assume that mass flows from $M_{2}$ to $M_{1}$ at a very small rate of $\\mathrm{d} M_{1} / \\mathrm{d} t = \\beta$. This rate is so small that in each period of rotation, the distance between the two stars can be assumed to be constant. However, after a long period of time, the distance between the two stars changes, while the motion remains circular.\n\nThe gas separated from $M_{2}$ forms a disk rotating around $M_{1}$ and heats up due to friction (Figure 1 -a). As the gas loses energy, it spirals inward toward $M_{1}$ and finally falls onto it. In the steady state, the mass flows at the constant rate of $\\beta$, from $M_{2}$ to the disc and from the disc onto $M_{1}$. At the same time, the heated disk emits thermal radiation as a blackbody. This disk forms very close to the neutron star so the gravitational pull of the $M_{2}$ star can be ignored for the analysis of the disk's motion. Also, ignore the heat capacity of the gas.\n\nIn the binary system PSR J2215+5135, the mass of the neutron star is $M_{\\mathrm{NS}} = 2.27 M_{\\odot}$ and the mass of its companion star is $M_{\\mathrm{S}} = 0.33 M_{\\odot}$, where $M_{\\odot} = 1.98 \\times 10^{30} \\mathrm{kg}$ is the mass of the Sun. The rotational period is $P = 4.14 \\mathrm{hr}$, and the Stefan-Boltzmann constant is $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} / (\\mathrm{m}^{2} \\mathrm{K}^{4})$, and the gravitational constant is $G = 6.67 \\times 10^{-11} \\mathrm{m}^{3} / (\\mathrm{kg} s)^{2}$. Assume that the mass flow rate to the neutron star is $\\beta = \\dot{M}_{\\mathrm{NS}} = 9 \\times 10^{-10} M_{\\odot} \\mathrm{yr}^{-1}$.\n\nAssume that after a sudden explosion, the $M_{1}$ star ejects a part of its mass out of the binary system at a very high speed, and its mass becomes $M_{1}^{\\prime}$. Take the magnitude of the velocity of $M_{1}^{\\prime}$ relative to $M_{2}$ to be $v^{\\prime}$ after the explosion.", + "question": "(1) Determine the maximum value of $v^{\\prime}$, in terms of $M_{1}^{\\prime}$, $M_{2}$, $G$, and $a$, that allows the new binary system to stay bounded. \n(2) Assuming that the explosion is isotropic, what is the minimum value of $M_{1}^{\\prime}$ for the binary system to remain bounded?", + "marking": [ + [ + "Award 0.2 pt if the answer contains $E^{\\prime} = \\frac{1}{2} \\mu^{\\prime} v^{\\prime 2} - \\frac{G M_{1}^{\\prime} M_{2}}{a} < 0$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer contains $v^{\\prime}_{\\text{max}} = \\sqrt{\\frac{2 G (M_{1}^{\\prime} + M_{2})}{a}}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer contains $v^{\\prime} = v$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the minimum value of $M_{1}^{\\prime}$ as $\\frac{M_{1} - M_{2}}{2}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\sqrt{\\frac{2 G (M_{1}^{\\prime} + M_{2})}{a}}$}", + "\\boxed{$\\frac{M_{1} - M_{2}}{2}$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 0.4, + 0.3 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_3_A_1_1.png" + ] + }, + { + "id": "IPhO_2024_3_B_1", + "context": "[Black Widow Pulsar] \n\nA significant number of the observed stars are binaries. One or both of the stars may be neutron stars rotating with a high angular velocity and emitting electromagnetic waves; such stars are called pulsars. Sometimes a companion star is an expansive mass of gas that gradually falls down onto the neutron star and causes its mass to increase (Figure 1-a). In this way, a neutron star gradually swallows up a portion of the mass of its companion star. For this reason, the neutron star has been compared to a black widow (or redback spider), a female spider which eats its mate after mating. The heating of the gas falling down onto the black widow generates radiation which can be observed. The heaviest neutron stars often are black widows and they serve as natural laboratories for testing fundamental physics. Figure 1-b shows the picture of the companion of the neutron star PSR J2215+5135, taken by the 3.4-meter optical telescope of the Iranian National Observatory. No neutron star can be seen in this image and the observed light is due to its companion.\n\n[figure1]\nFigure 1. (a) The falling gases of the companion star onto the neutron star. (b) The companion of the neutron star PSR J2215+5135. \n\nIn this part we study the stability of a single star. Consider a star containing a specific kind of matter with the equation of state $p = K \\rho^{\\gamma}$ where $K$ and $\\gamma$ are constants. Let $p(r)$ and $\\rho(r)$ be the pressure and density at a distance $r$ from the center of the star, respectively. The pressure and density at the center of the star are $p_{c}$ and $\\rho_{c}$, respectively. In all questions of this part, take all outward vectors to be positive.", + "question": "Determine the gravitational acceleration $g(r)$ near the center of the star in terms of $r$ and the constants $G$ and $\\rho_{c}$.", + "marking": [ + [ + "Award 0.2 pt if the answer gives the correct expression for $g(r) = -\\frac{4 \\pi G \\rho_{c} r}{3}$. If the answer misses the minus sign and gives $g(r) = \\frac{4 \\pi G \\rho_{c} r}{3}$, award 0.1 pt. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$g(r) = -\\frac{4 \\pi G \\rho_{c} r}{3}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.2 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_3_A_1_1.png" + ] + }, + { + "id": "IPhO_2024_3_B_2", + "context": "[Black Widow Pulsar] \n\nA significant number of the observed stars are binaries. One or both of the stars may be neutron stars rotating with a high angular velocity and emitting electromagnetic waves; such stars are called pulsars. Sometimes a companion star is an expansive mass of gas that gradually falls down onto the neutron star and causes its mass to increase (Figure 1-a). In this way, a neutron star gradually swallows up a portion of the mass of its companion star. For this reason, the neutron star has been compared to a black widow (or redback spider), a female spider which eats its mate after mating. The heating of the gas falling down onto the black widow generates radiation which can be observed. The heaviest neutron stars often are black widows and they serve as natural laboratories for testing fundamental physics. Figure 1-b shows the picture of the companion of the neutron star PSR J2215+5135, taken by the 3.4-meter optical telescope of the Iranian National Observatory. No neutron star can be seen in this image and the observed light is due to its companion.\n\n[figure1]\nFigure 1. (a) The falling gases of the companion star onto the neutron star. (b) The companion of the neutron star PSR J2215+5135. \n\nIn this part we study the stability of a single star. Consider a star containing a specific kind of matter with the equation of state $p = K \\rho^{\\gamma}$ where $K$ and $\\gamma$ are constants. Let $p(r)$ and $\\rho(r)$ be the pressure and density at a distance $r$ from the center of the star, respectively. The pressure and density at the center of the star are $p_{c}$ and $\\rho_{c}$, respectively. In all questions of this part, take all outward vectors to be positive.", + "question": "Derive a (differential) equation for determining $\\rho(r)$ at equilibrium, and write it in the following form: $\\frac{d}{d r}\\left[h_{1}(\\rho, r) \\frac{d \\rho}{d r} \\right] + h_{2}(r) \\rho = 0$. \n(1) Find the function $h_{1}$. \n(2) Find the function $h_{2}$.", + "marking": [ + [ + "Award 0.6 pt if the answer finds both the function $h_{1}(\\rho, r) = r^{2} \\rho^{\\gamma-2}$ and the function $h_{2}(r) = \\frac{4 \\pi G r^{2}}{K \\gamma}$. Partial points: if the answer gives only one correct expression for $h_{1}$ or $h_{2}$, award 0.3 pt; if the answer gives the incorrect expression for both $h_{1}$ and $h_{2}$, but contains the correct form of the equation $\\vec{F} = -\\frac{G M(\\vec{r}) \\rho}{r^{2}} \\mathrm{A} \\Delta r - \\Delta p A = 0$, award 0.3 pt. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$h_{1} = r^2 \\rho^{\\gamma-2}$}", + "\\boxed{$h_{2} = \\frac{4 \\pi G r^{2}}{K \\gamma}$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 0.3, + 0.3 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_3_A_1_1.png" + ] + }, + { + "id": "IPhO_2024_3_B_3", + "context": "[Black Widow Pulsar] \n\nA significant number of the observed stars are binaries. One or both of the stars may be neutron stars rotating with a high angular velocity and emitting electromagnetic waves; such stars are called pulsars. Sometimes a companion star is an expansive mass of gas that gradually falls down onto the neutron star and causes its mass to increase (Figure 1-a). In this way, a neutron star gradually swallows up a portion of the mass of its companion star. For this reason, the neutron star has been compared to a black widow (or redback spider), a female spider which eats its mate after mating. The heating of the gas falling down onto the black widow generates radiation which can be observed. The heaviest neutron stars often are black widows and they serve as natural laboratories for testing fundamental physics. Figure 1-b shows the picture of the companion of the neutron star PSR J2215+5135, taken by the 3.4-meter optical telescope of the Iranian National Observatory. No neutron star can be seen in this image and the observed light is due to its companion.\n\n[figure1]\nFigure 1. (a) The falling gases of the companion star onto the neutron star. (b) The companion of the neutron star PSR J2215+5135. \n\nIn this part we study the stability of a single star. Consider a star containing a specific kind of matter with the equation of state $p = K \\rho^{\\gamma}$ where $K$ and $\\gamma$ are constants. Let $p(r)$ and $\\rho(r)$ be the pressure and density at a distance $r$ from the center of the star, respectively. The pressure and density at the center of the star are $p_{c}$ and $\\rho_{c}$, respectively. In all questions of this part, take all outward vectors to be positive.", + "question": "Construct a quantity $r_{0}$ of the form $r_{0} = G^{l} p_{c}^{m} \\rho_{c}^{n}$ with the dimension of length.", + "marking": [ + [ + "Award 0.4 pt if the answer gives the correct expression for $r_{0} = G^{-\\frac{1}{2}} p_{c}^{\\frac{1}{2}} \\rho_{c}^{-1}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$r_{0} = G^{-\\frac{1}{2}} p_{c}^{\\frac{1}{2}} \\rho_{c}^{-1}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.4 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_3_A_1_1.png" + ] + }, + { + "id": "IPhO_2024_3_B_4", + "context": "[Black Widow Pulsar] \n\nA significant number of the observed stars are binaries. One or both of the stars may be neutron stars rotating with a high angular velocity and emitting electromagnetic waves; such stars are called pulsars. Sometimes a companion star is an expansive mass of gas that gradually falls down onto the neutron star and causes its mass to increase (Figure 1-a). In this way, a neutron star gradually swallows up a portion of the mass of its companion star. For this reason, the neutron star has been compared to a black widow (or redback spider), a female spider which eats its mate after mating. The heating of the gas falling down onto the black widow generates radiation which can be observed. The heaviest neutron stars often are black widows and they serve as natural laboratories for testing fundamental physics. Figure 1-b shows the picture of the companion of the neutron star PSR J2215+5135, taken by the 3.4-meter optical telescope of the Iranian National Observatory. No neutron star can be seen in this image and the observed light is due to its companion.\n\n[figure1]\nFigure 1. (a) The falling gases of the companion star onto the neutron star. (b) The companion of the neutron star PSR J2215+5135. \n\nIn this part we study the stability of a single star. Consider a star containing a specific kind of matter with the equation of state $p = K \\rho^{\\gamma}$ where $K$ and $\\gamma$ are constants. Let $p(r)$ and $\\rho(r)$ be the pressure and density at a distance $r$ from the center of the star, respectively. The pressure and density at the center of the star are $p_{c}$ and $\\rho_{c}$, respectively. In all questions of this part, take all outward vectors to be positive.", + "question": "Derive a (differential) equation for determining $\\rho(r)$ at equilibrium, and write it in the following form: $\\frac{d}{d x}\\left[A_{1}(u, x) \\frac{d u}{d x}\\right] + A_{2}(x) u(x) = 0$, where $x = \\frac{r}{r_{0}}$ and $u = \\frac{\\rho}{\\rho_{c}}$. \n(1) Find the function $A_{1}(u, x)$. \n(2) Find the function $A_{2}(x)$.", + "marking": [ + [ + "Award 0.15 pt if the answer gives the correct expression for $A_1(u, x) = x^{2} u^{\\gamma-2}$, up to a constant coefficient. Otherwise, award 0 pt.", + "Award 0.15 pt if the answer gives the correct expression for $A_2(x) = \\frac{4 \\pi x^{2}}{\\gamma}$, up to a constant coefficient. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$A_1(u, x) = x^2 u^{\\gamma-2}$}", + "\\boxed{$A_2(x) = \\frac{4\\pi x^2}{\\gamma}$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 0.15, + 0.15 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_3_A_1_1.png" + ] + }, + { + "id": "IPhO_2024_3_B_5", + "context": "[Black Widow Pulsar] \n\nA significant number of the observed stars are binaries. One or both of the stars may be neutron stars rotating with a high angular velocity and emitting electromagnetic waves; such stars are called pulsars. Sometimes a companion star is an expansive mass of gas that gradually falls down onto the neutron star and causes its mass to increase (Figure 1-a). In this way, a neutron star gradually swallows up a portion of the mass of its companion star. For this reason, the neutron star has been compared to a black widow (or redback spider), a female spider which eats its mate after mating. The heating of the gas falling down onto the black widow generates radiation which can be observed. The heaviest neutron stars often are black widows and they serve as natural laboratories for testing fundamental physics. Figure 1-b shows the picture of the companion of the neutron star PSR J2215+5135, taken by the 3.4-meter optical telescope of the Iranian National Observatory. No neutron star can be seen in this image and the observed light is due to its companion.\n\n[figure1]\nFigure 1. (a) The falling gases of the companion star onto the neutron star. (b) The companion of the neutron star PSR J2215+5135. \n\nIn this part we study the stability of a single star. Consider a star containing a specific kind of matter with the equation of state $p = K \\rho^{\\gamma}$ where $K$ and $\\gamma$ are constants. Let $p(r)$ and $\\rho(r)$ be the pressure and density at a distance $r$ from the center of the star, respectively. The pressure and density at the center of the star are $p_{c}$ and $\\rho_{c}$, respectively. In all questions of this part, take all outward vectors to be positive.", + "question": "For $\\gamma=2$ one finds $u(x) = \\frac{f(x)}{x}$. Determine $f(x)$.", + "marking": [ + [ + "Award 0.6 pt if the answer gives the correct expression for $f(x) = \\frac{\\sin (\\sqrt{2 \\pi} x)}{\\sqrt{2 \\pi}}$. Partial points: if the answer gives the equivalent form of the function $f(x) = A \\sin (\\sqrt{2 \\pi} x) + B \\cos (\\sqrt{2 \\pi} x)$, where $A$ and $B$ are constants, award 0.3 pt; if the answer gives the correct form of $A = \\frac{1}{\\sqrt{2 \\pi}}$, award 0.2 pt; if the answer gives the correct form of $B = 0$, award 0.1 pt. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$f(x) = \\frac{\\sin (\\sqrt{2 \\pi} x)}{\\sqrt{2 \\pi}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.6 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_3_A_1_1.png" + ] + }, + { + "id": "IPhO_2024_3_B_6", + "context": "[Black Widow Pulsar] \n\nA significant number of the observed stars are binaries. One or both of the stars may be neutron stars rotating with a high angular velocity and emitting electromagnetic waves; such stars are called pulsars. Sometimes a companion star is an expansive mass of gas that gradually falls down onto the neutron star and causes its mass to increase (Figure 1-a). In this way, a neutron star gradually swallows up a portion of the mass of its companion star. For this reason, the neutron star has been compared to a black widow (or redback spider), a female spider which eats its mate after mating. The heating of the gas falling down onto the black widow generates radiation which can be observed. The heaviest neutron stars often are black widows and they serve as natural laboratories for testing fundamental physics. Figure 1-b shows the picture of the companion of the neutron star PSR J2215+5135, taken by the 3.4-meter optical telescope of the Iranian National Observatory. No neutron star can be seen in this image and the observed light is due to its companion.\n\n[figure1]\nFigure 1. (a) The falling gases of the companion star onto the neutron star. (b) The companion of the neutron star PSR J2215+5135. \n\nIn this part we study the stability of a single star. Consider a star containing a specific kind of matter with the equation of state $p = K \\rho^{\\gamma}$ where $K$ and $\\gamma$ are constants. Let $p(r)$ and $\\rho(r)$ be the pressure and density at a distance $r$ from the center of the star, respectively. The pressure and density at the center of the star are $p_{c}$ and $\\rho_{c}$, respectively. In all questions of this part, take all outward vectors to be positive.\n\nAssume that for a particular star $\\frac{d u}{d x}$, as a function of $x$, is given by the curve given in Figure 2.\n\n[figure2]\nFigure 2. The plot of $\\frac{d u}{d x}$.", + "question": "Use the behavior of the curve in Figure 2, in the vicinity of the point $x = 0$, to find $\\gamma$ up to 3 significant figures.", + "marking": [ + [ + "Award 0.1 pt if the answer contains $u^{\\prime} = 0$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer contains $\\lim_{x \\rightarrow 0} \\frac{u^{\\prime}(x)}{x} = u^{\\prime \\prime}(0)$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer contains $\\gamma = -\\frac{4 \\pi}{3 u^{\\prime \\prime}(0)}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct value of $\\gamma$ within the range of $[1.64, 1.70]$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$[1.64, 1.70]$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 0.8 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_3_A_1_1.png", + "image_question/IPhO_2024_3_B_6_1.png" + ] + }, + { + "id": "IPhO_2024_3_B_7", + "context": "[Black Widow Pulsar] \n\nA significant number of the observed stars are binaries. One or both of the stars may be neutron stars rotating with a high angular velocity and emitting electromagnetic waves; such stars are called pulsars. Sometimes a companion star is an expansive mass of gas that gradually falls down onto the neutron star and causes its mass to increase (Figure 1-a). In this way, a neutron star gradually swallows up a portion of the mass of its companion star. For this reason, the neutron star has been compared to a black widow (or redback spider), a female spider which eats its mate after mating. The heating of the gas falling down onto the black widow generates radiation which can be observed. The heaviest neutron stars often are black widows and they serve as natural laboratories for testing fundamental physics. Figure 1-b shows the picture of the companion of the neutron star PSR J2215+5135, taken by the 3.4-meter optical telescope of the Iranian National Observatory. No neutron star can be seen in this image and the observed light is due to its companion.\n\n[figure1]\nFigure 1. (a) The falling gases of the companion star onto the neutron star. (b) The companion of the neutron star PSR J2215+5135. \n\nIn this part we study the stability of a single star. Consider a star containing a specific kind of matter with the equation of state $p = K \\rho^{\\gamma}$ where $K$ and $\\gamma$ are constants. Let $p(r)$ and $\\rho(r)$ be the pressure and density at a distance $r$ from the center of the star, respectively. The pressure and density at the center of the star are $p_{c}$ and $\\rho_{c}$, respectively. In all questions of this part, take all outward vectors to be positive.\n\nTo analyze the stability of the system, we assume that the star deviates slightly from its equilibrium state: we assume that the spherical shell, which was in equilibrium at radius $r$, now has a radius $\\tilde{r}$, similarly the parameters $g$, $p$, and $\\rho$ have changed to $\\tilde{g}$, $\\tilde{p}$, and $\\tilde{\\rho}$ respectively. For convenience, we shall only consider small $r^{\\prime}$s near the center of the star, for which we can assume that $\\tilde{r} = r(1 + \\varepsilon(t))$, where $\\varepsilon(t) \\ll 1$.", + "question": "(1) Find $\\tilde{\\rho}$ in terms of $\\rho$ and $g$ to the first order in $\\varepsilon$. \n(2) Find $\\tilde{g}$ in terms of $\\rho$ and $g$ to the first order in $\\varepsilon$.", + "marking": [ + [ + "Award 0.6 pt if the answer gives the correct expression for $\\tilde{\\rho} \\simeq \\rho(1 - 3\\epsilon)$. Partial points: if the answer gives the expression for $\\tilde{\\rho} = \\rho(1 + \\epsilon)^{-3}$, award 0.4 pt. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer gives the correct expression for $\\tilde{g} \\simeq g(1 - 2\\epsilon)$. Partial points: if the answer gives the expression for $\\tilde{g} = g(1 + \\epsilon)^{-2}$, award 0.2 pt. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\tilde{\\rho} = \\rho(1 - 3\\epsilon)$}", + "\\boxed{$\\tilde{g} = g(1 - 2\\epsilon)$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 0.6, + 0.3 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_3_A_1_1.png" + ] + }, + { + "id": "IPhO_2024_3_B_8", + "context": "[Black Widow Pulsar] \n\nA significant number of the observed stars are binaries. One or both of the stars may be neutron stars rotating with a high angular velocity and emitting electromagnetic waves; such stars are called pulsars. Sometimes a companion star is an expansive mass of gas that gradually falls down onto the neutron star and causes its mass to increase (Figure 1-a). In this way, a neutron star gradually swallows up a portion of the mass of its companion star. For this reason, the neutron star has been compared to a black widow (or redback spider), a female spider which eats its mate after mating. The heating of the gas falling down onto the black widow generates radiation which can be observed. The heaviest neutron stars often are black widows and they serve as natural laboratories for testing fundamental physics. Figure 1-b shows the picture of the companion of the neutron star PSR J2215+5135, taken by the 3.4-meter optical telescope of the Iranian National Observatory. No neutron star can be seen in this image and the observed light is due to its companion.\n\n[figure1]\nFigure 1. (a) The falling gases of the companion star onto the neutron star. (b) The companion of the neutron star PSR J2215+5135. \n\nIn this part we study the stability of a single star. Consider a star containing a specific kind of matter with the equation of state $p = K \\rho^{\\gamma}$ where $K$ and $\\gamma$ are constants. Let $p(r)$ and $\\rho(r)$ be the pressure and density at a distance $r$ from the center of the star, respectively. The pressure and density at the center of the star are $p_{c}$ and $\\rho_{c}$, respectively. In all questions of this part, take all outward vectors to be positive.\n\nTo analyze the stability of the system, we assume that the star deviates slightly from its equilibrium state: we assume that the spherical shell, which was in equilibrium at radius $r$, now has a radius $\\tilde{r}$, similarly the parameters $g$, $p$, and $\\rho$ have changed to $\\tilde{g}$, $\\tilde{p}$, and $\\tilde{\\rho}$ respectively. For convenience, we shall only consider small $r^{\\prime}$s near the center of the star, for which we can assume that $\\tilde{r} = r(1 + \\varepsilon(t))$, where $\\varepsilon(t) \\ll 1$.", + "question": "Using Newton's equation of motion for the spherical layer with the equilibrium radius of $r$, find $\\frac{d^{2} \\tilde{r}}{d t^{2}}$ in terms of $\\tilde{g}$, $\\tilde{\\rho}$, $K$, $\\gamma$, and $\\frac{\\partial \\tilde{\\rho}}{\\partial \\tilde{r}}$ (By $\\frac{\\partial \\tilde{\\rho}}{\\partial \\tilde{r}}$ we mean derivative of $\\tilde{\\rho}$ with respect to $\\tilde{r}$ at constant $t$.)", + "marking": [ + [ + "Award 0.6 pt if the answer gives the correct expression for $\\frac{d^{2} \\tilde{r}}{d t^{2}} = \\tilde{g} - K \\gamma \\tilde{\\rho}^{\\gamma-2} \\frac{\\partial \\tilde{\\rho}}{\\partial \\tilde{r}}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{d^{2} \\tilde{r}}{d t^{2}} = \\tilde{g} - K \\gamma \\tilde{\\rho}^{\\gamma-2} \\frac{\\partial \\tilde{\\rho}}{\\partial \\tilde{r}}$}" + ], + "answer_type": [ + "Equation" + ], + "unit": [ + null + ], + "points": [ + 0.6 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_3_A_1_1.png" + ] + }, + { + "id": "IPhO_2024_3_B_9", + "context": "[Black Widow Pulsar] \n\nA significant number of the observed stars are binaries. One or both of the stars may be neutron stars rotating with a high angular velocity and emitting electromagnetic waves; such stars are called pulsars. Sometimes a companion star is an expansive mass of gas that gradually falls down onto the neutron star and causes its mass to increase (Figure 1-a). In this way, a neutron star gradually swallows up a portion of the mass of its companion star. For this reason, the neutron star has been compared to a black widow (or redback spider), a female spider which eats its mate after mating. The heating of the gas falling down onto the black widow generates radiation which can be observed. The heaviest neutron stars often are black widows and they serve as natural laboratories for testing fundamental physics. Figure 1-b shows the picture of the companion of the neutron star PSR J2215+5135, taken by the 3.4-meter optical telescope of the Iranian National Observatory. No neutron star can be seen in this image and the observed light is due to its companion.\n\n[figure1]\nFigure 1. (a) The falling gases of the companion star onto the neutron star. (b) The companion of the neutron star PSR J2215+5135. \n\nIn this part we study the stability of a single star. Consider a star containing a specific kind of matter with the equation of state $p = K \\rho^{\\gamma}$ where $K$ and $\\gamma$ are constants. Let $p(r)$ and $\\rho(r)$ be the pressure and density at a distance $r$ from the center of the star, respectively. The pressure and density at the center of the star are $p_{c}$ and $\\rho_{c}$, respectively. In all questions of this part, take all outward vectors to be positive.\n\nTo analyze the stability of the system, we assume that the star deviates slightly from its equilibrium state: we assume that the spherical shell, which was in equilibrium at radius $r$, now has a radius $\\tilde{r}$, similarly the parameters $g$, $p$, and $\\rho$ have changed to $\\tilde{g}$, $\\tilde{p}$, and $\\tilde{\\rho}$ respectively. For convenience, we shall only consider small $r^{\\prime}$s near the center of the star, for which we can assume that $\\tilde{r} = r(1 + \\varepsilon(t))$, where $\\varepsilon(t) \\ll 1$.", + "question": "(1) Obtain $\\frac{d^{2} \\varepsilon}{d t^{2}}$ in terms of $\\varepsilon$ and the constants given in the problem. \n(2) Find the minimum value of $\\gamma$ for a stable equilibrium. \n(3) Find the oscillation's angular frequency $\\omega$ of the star.", + "marking": [ + [ + "Award 0.4 pt if the answer gives the correct expression for $\\frac{d^{2} \\varepsilon}{d t^{2}} = -\\frac{4 \\pi G \\rho_{c}}{3} (3 \\gamma - 4) \\varepsilon$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the minimum value of $\\gamma = \\frac{4}{3}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct expression for $\\omega = \\sqrt{\\frac{4 \\pi G \\rho_{c}}{3} (3 \\gamma - 4)}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{d^{2} \\varepsilon}{d t^{2}} = -\\frac{4 \\pi G \\rho_{c}}{3} (3 \\gamma - 4) \\varepsilon$}", + "\\boxed{$\\frac{4}{3}$}", + "\\boxed{$\\omega = \\sqrt{\\frac{4 \\pi G \\rho_{c}}{3} (3 \\gamma - 4)}$}" + ], + "answer_type": [ + "Equation", + "Numerical Value", + "Expression" + ], + "unit": [ + null, + null, + null + ], + "points": [ + 0.4, + 0.1, + 0.1 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2024", + "image_question": [ + "image_question/IPhO_2024_3_A_1_1.png" + ] + } +] \ No newline at end of file diff --git a/data/IPhO_2025.json b/data/IPhO_2025.json new file mode 100644 index 0000000000000000000000000000000000000000..96bb18d8326e063c93d1daac2f1a37fcd4518f07 --- /dev/null +++ b/data/IPhO_2025.json @@ -0,0 +1,1541 @@ +[ + { + "information": "General Data Sheet: \n\nSpeed of light in vacuum: $c = 2.99792458 \\times 10^8 \\mathrm{m \\cdot s^{-1}}$ \nPlanck constant: $h = 6.62607015 \\times 10^{-34} \\mathrm{kg \\cdot m^2 \\cdot s^{-1}}$ \nReduced Planck constant: $\\hbar = \\frac{h}{2\\pi} = 1.054571818 \\times 10^{-34} \\mathrm{kg \\cdot m^2 \\cdot s^{-1}}$ \nBoltzmann constant: $k_B = 1.380649 \\times 10^{-23} \\mathrm{kg \\cdot m^2 \\cdot s^{-2} \\cdot K^{-1}}$ \nAvogadro constant: $N_A = 6.02214076 \\times 10^{23} \\mathrm{mol}^{-1}$ \nMolar gas constant: $R = 8.31446261815324 \\mathrm{kg \\cdot m^2 \\cdot s^{-2} \\cdot mol^{-1} \\cdot K^{-1}}$ \nElementary charge: $e = 1.602176634 \\times 10^{-19} \\mathrm{A \\cdot s}$ \nUniversal constant of gravitation: $G = 6.67430(15) \\times 10^{-11} \\mathrm{m^3 \\cdot kg^{-1} \\cdot s^{-2}}$ \nStandard acceleration due to gravity: $g = 9.80665 \\mathrm{m \\cdot s^{-2}}$ \nStefan Boltzmann constant: $\\sigma = 5.670374419 \\times 10^{-8} \\mathrm{kg \\cdot s^{-3} \\cdot K^{-4}}$ \nVacuum permeability (magnetic constant): $\\mu_0 = 1.25663706127(20) \\times 10^{-6} \\mathrm{kg \\cdot m \\cdot A^{-2} \\cdot s^{-2}}$ \nVacuum permittivity (electrical constant): $\\varepsilon_0 = 8.8541878188(14) \\times 10^{-12} \\mathrm{A^2 \\cdot s^4 \\cdot kg^{-1} \\cdot m^{-3}}$ \nRydberg constant: $R_{\\infty} = 1.0973731568157(12) \\times 10^7 \\mathrm{m^{-1}}$ \nMass of the electron: $m_e = 9.1093837139(28) \\times 10^{-31} \\mathrm{kg}$ \nMass of the proton: $m_p = 1.67262192595(52) \\times 10^{-27} \\mathrm{kg}$ \nMass of the neutron: $m_n = 1.67492750056(85) \\times 10^{-27} \\mathrm{kg}$ \nAtomic mass constant: $m_u = 1.66053906892(52) \\times 10^{-27} \\mathrm{kg}$ \nElectronvolt: $\\mathrm{eV} = 1.602176634 \\times 10^{-19} \\mathrm{kg \\cdot m^2 \\cdot s^{-2}}$" + }, + { + "id": "IPhO_2025_1_A_1", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part A - Introduction] \n[Bohr model] \n\nWe assume that the hydrogen atom consists of a non-relativistic electron, with mass $m_e$, orbiting a fixed proton. Throughout this part, we assume its motion is on a circular orbit.", + "question": "Determine the electron's velocity $v$ in a circular orbit of radius $r$.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly uses Newton's second law on the electron in the electrical field of the proton for a circular orbit and projected on $\\vec{u}_r$: $-m_e \\frac{v^2}{r} = -\\frac{e^2}{4\\pi \\epsilon_0 r^2}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct expression for the electron's velocity: $v = \\sqrt{\\frac{e^2}{4\\pi\\varepsilon_0 m_e r}}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$v = \\sqrt{\\frac{e^2}{4 \\pi \\varepsilon_0 m_e r}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.2 + ], + "modality": "text-only", + "field": "Modern Physics", + "source": "IPhO_2025", + "image_question": [] + }, + { + "id": "IPhO_2025_1_A_2", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part A - Introduction] \n[Bohr model] \n\nWe assume that the hydrogen atom consists of a non-relativistic electron, with mass $m_e$, orbiting a fixed proton. Throughout this part, we assume its motion is on a circular orbit. \n\n(A.1) Determine the electron's velocity $v$ in a circular orbit of radius $r$. \n\nIn the Bohr model, we assume the magnitude of the electron's angular momentum $L$ is quantized, $L = n \\hbar$ where $n > 0$ is an integer. We define $\\alpha = \\frac{e^2}{4 \\pi \\varepsilon_0 \\hbar c} \\approx 7.27 \\times 10^{-3}$.", + "question": "(1) Show that the radius of each orbit is given by $r_n = n^2 r_1$, where $r_1$ is called the Bohr radius. (2) Express $r_1$ in terms of $\\alpha$, $m_e$, $c$ and $\\hbar$ and (3) calculate its numerical value in $m$ with 3 digits. (4) Express $v_1$, the velocity on the orbit of radius $r_1$, in terms of $\\alpha$ and $c$.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly derives the expression for $r_n$ as $r_n = \\frac{\\hbar^2 n^2}{\\alpha m_e c}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly derives the expression for $r_1$ as $r_1 = \\frac{\\hbar}{\\alpha m_e c}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the numerical value for $r_1$ as $5.31 \\times 10^{-11} \\mathrm{m}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly gives the expression for $v_1$ as $v_1 = \\alpha c$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$r_n = \\frac{\\hbar^2 n^2}{\\alpha m_e c}$}", + "\\boxed{$r_1 = \\frac{\\hbar}{\\alpha m_e c}$}", + "\\boxed{$5.31 \\times 10^{-11}$}", + "\\boxed{$v_1 = \\alpha c$}" + ], + "answer_type": [ + "Expression", + "Expression", + "Numerical Value", + "Expression" + ], + "unit": [ + null, + null, + "m", + null + ], + "points": [ + 0.1, + 0.1, + 0.1, + 0.2 + ], + "modality": "text-only", + "field": "Modern Physics", + "source": "IPhO_2025", + "image_question": [] + }, + { + "id": "IPhO_2025_1_A_3", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part A - Introduction] \n[Bohr model] \n\nWe assume that the hydrogen atom consists of a non-relativistic electron, with mass $m_e$, orbiting a fixed proton. Throughout this part, we assume its motion is on a circular orbit. \n\n(A.1) Determine the electron's velocity $v$ in a circular orbit of radius $r$. \n\nIn the Bohr model, we assume the magnitude of the electron's angular momentum $L$ is quantized, $L = n \\hbar$ where $n > 0$ is an integer. We define $\\alpha = \\frac{e^2}{4 \\pi \\varepsilon_0 \\hbar c} \\approx 7.27 \\times 10^{-3}$. \n\n(A.2) Show that the radius of each orbit is given by $r_n = n^2 r_1$, where $r_1$ is called the Bohr radius. Express $r_1$ in terms of $\\alpha$, $m_e$, $c$ and $\\hbar$ and calculate its numerical value in $m$ with 3 digits. Express $v_1$, the velocity on the orbit of radius $r_1$, in terms of $\\alpha$ and $c$.", + "question": "(1) Determine the electron's mechanical energy $E_n$ on an orbit of radius $r_n$ in terms of $e$, $\\varepsilon_0$, $r_1$ and $n$. \n(2) Determine $E_1$ in the ground state in terms of $\\alpha$, $m_e$ and $c$. \n(3)Compute the numerical value of $E_1$ in eV.", + "marking": [ + [ + "Award 0.2 pt if the answer gives the correct expression for the electron's mechanical energy $E_n$ as $E_n = -\\frac{e^2}{8\\pi\\varepsilon_0 n^2 r_1}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the correct expression for $E_1$ in the ground state using $\\alpha$ as $E_1 = -\\frac{1}{2} \\alpha^2 m_e c^2$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct numerical value for $E_1$ as $-13.6 \\mathrm{eV}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$E_n = -\\frac{e^2}{8 \\pi \\varepsilon_0 n^2 r_1}$}", + "\\boxed{$E_1 = -\\frac{1}{2} \\alpha^2 m_e c^2$}", + "\\boxed{-13.6}" + ], + "answer_type": [ + "Expression", + "Expression", + "Numerical Value" + ], + "unit": [ + null, + null, + "eV" + ], + "points": [ + 0.2, + 0.2, + 0.1 + ], + "modality": "text-only", + "field": "Modern Physics", + "source": "IPhO_2025", + "image_question": [] + }, + { + "id": "IPhO_2025_1_A_4", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part A - Introduction] \n[Bohr model] \n\nWe assume that the hydrogen atom consists of a non-relativistic electron, with mass $m_e$, orbiting a fixed proton. Throughout this part, we assume its motion is on a circular orbit. \n\n(A.1) Determine the electron's velocity $v$ in a circular orbit of radius $r$. \n\nIn the Bohr model, we assume the magnitude of the electron's angular momentum $L$ is quantized, $L = n \\hbar$ where $n > 0$ is an integer. We define $\\alpha = \\frac{e^2}{4 \\pi \\varepsilon_0 \\hbar c} \\approx 7.27 \\times 10^{-3}$. \n\n(A.2) Show that the radius of each orbit is given by $r_n = n^2 r_1$, where $r_1$ is called the Bohr radius. Express $r_1$ in terms of $\\alpha$, $m_e$, $c$ and $\\hbar$ and calculate its numerical value in $m$ with 3 digits. Express $v_1$, the velocity on the orbit of radius $r_1$, in terms of $\\alpha$ and $c$. \n\n(A.3) Determine the electron's mechanical energy $E_n$ on an orbit of radius $r_n$ in terms of $e$, $\\varepsilon_0$, $r_1$ and $n$. Determine $E_1$ in the ground state in terms of $\\alpha$, $m_e$ and $c$. Compute its numerical value in eV. \n\n[Hydrogen fine and hyperfine structures] \n\nThe rare spontaneous inversion of the electron's spin causes a photon to be emitted on average once per 10 million years per hydrogen atom. This emission serves as a hydrogen tracer in the universe and is thus fundamental in astrophysics. We will study the transition responsible for this emission in two steps. \n\nFirst, consider the interaction between the electron spin and the relative motion of the electron and the proton. Working in the electron's frame of reference, the proton orbits the electron at a distance $r_1$. This produces a magnetic field $\\vec{B}_1$.", + "question": "Determine the magnitude $B_1$ of $\\vec{B}_1$ at the position of the electron in terms of $\\mu_0$, $e$, $\\alpha$, $c$ and $r_1$.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly expresses the period as $T = \\frac{2\\pi r_1}{v_1}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly expresses the current $i$ corresponding to the orbit of the proton as $i = \\frac{e \\alpha c}{2\\pi r_1}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the general formula for the magnetic field $B$ created by a loop with current $i$ and radius $R$ as $B = \\frac{\\mu_0 i}{2R}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the final expression for $B_1$ as $B_1 = \\frac{\\mu_0 e \\alpha c}{4 \\pi r_1^2}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$B_1 = \\frac{\\mu_0 e \\alpha c}{4 \\pi r_1^2}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.5 + ], + "modality": "text-only", + "field": "Electromagnetism", + "source": "IPhO_2025", + "image_question": [] + }, + { + "id": "IPhO_2025_1_A_5", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part A - Introduction] \n[Bohr model] \n\nWe assume that the hydrogen atom consists of a non-relativistic electron, with mass $m_e$, orbiting a fixed proton. Throughout this part, we assume its motion is on a circular orbit. \n\n(A.1) Determine the electron's velocity $v$ in a circular orbit of radius $r$. \n\nIn the Bohr model, we assume the magnitude of the electron's angular momentum $L$ is quantized, $L = n \\hbar$ where $n > 0$ is an integer. We define $\\alpha = \\frac{e^2}{4 \\pi \\varepsilon_0 \\hbar c} \\approx 7.27 \\times 10^{-3}$. \n\n(A.2) Show that the radius of each orbit is given by $r_n = n^2 r_1$, where $r_1$ is called the Bohr radius. Express $r_1$ in terms of $\\alpha$, $m_e$, $c$ and $\\hbar$ and calculate its numerical value in $m$ with 3 digits. Express $v_1$, the velocity on the orbit of radius $r_1$, in terms of $\\alpha$ and $c$. \n\n(A.3) Determine the electron's mechanical energy $E_n$ on an orbit of radius $r_n$ in terms of $e$, $\\varepsilon_0$, $r_1$ and $n$. Determine $E_1$ in the ground state in terms of $\\alpha$, $m_e$ and $c$. Compute its numerical value in eV. \n\n[Hydrogen fine and hyperfine structures] \n\nThe rare spontaneous inversion of the electron's spin causes a photon to be emitted on average once per 10 million years per hydrogen atom. This emission serves as a hydrogen tracer in the universe and is thus fundamental in astrophysics. We will study the transition responsible for this emission in two steps. \n\nFirst, consider the interaction between the electron spin and the relative motion of the electron and the proton. Working in the electron's frame of reference, the proton orbits the electron at a distance $r_1$. This produces a magnetic field $\\vec{B}_1$. \n\n(A.4) Determine the magnitude $B_1$ of $\\vec{B}_1$ at the position of the electron in terms of $\\mu_0$, $e$, $\\alpha$, $c$ and $r_1$. \n\nSecond, the electron spin creates a magnetic moment $\\vec{\\mathcal{M}}_s$. Its magnitude is roughly $\\mathcal{M}_s = \\frac{e}{m_e} \\hbar$. The fine (F) structure is related to the energy difference $\\Delta E_F$ between an electron with $\\vec{\\mathcal{M}}_s$ parallel to $\\vec{B}_1$ and that of an electron with $\\vec{\\mathcal{M}}_s$ anti-parallel to $\\vec{B}_1$. Similarly, the hyperfine (HF) structure is related to the energy difference $\\Delta E_{HF}$, due to the interaction between parallel and anti-parallel magnetic moments of the electron and the proton. It is known to be approximately $\\Delta E_{HF} \\approx 3.72 \\frac{m_e}{m_p} \\Delta E_F$ where $m_p$ is the proton mass.", + "question": "(1) Express $\\Delta E_F$ as a function of $\\alpha$ and $E_1$. \n(2) Express the wavelength $\\lambda_{HF}$ of a photon emitted during a transition between the two states of the hyperfine structure and (3) give its numerical value in $cm$ with two digits.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly derives the expression for potential energy corresponding to the interaction between the spin magnetic moment $\\vec{\\mathcal{M}}_s$ and the nuclear magnetic field as $E_p = -\\vec{\\mathcal{M}}_s \\cdot \\vec{B}_1$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly derives the expression for the difference $\\Delta E_F$ between the energy of two electrons with a spin parallel and antiparallel to $\\vec{B}_1$ as $\\Delta E_F = -4 \\alpha^2 E_1$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly expresses $\\Delta E_{HF}$ in terms of $\\alpha$: $\\Delta E_{HF} = -3.72 \\cdot \\frac{m_e}{m_p} \\cdot 4 \\alpha^2 E_1$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly expresses the wavelength $\\lambda_{HF}$ as $\\lambda_{HF} = \\frac{hc}{\\Delta E_{HF}} = -\\frac{hc}{3.72 \\cdot \\frac{m_e}{m_p} \\cdot 4 \\alpha^2 E_1}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer provides the correct numerical value of $\\lambda_{HF}$ as $$\\lambda_{HF}$ = 21 \\mathrm{cm}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\Delta E_F = -4 \\alpha^2 E_1$}", + "\\boxed{$\\lambda_{\\text{HF}} = -\\frac{hc}{3.72 \\cdot \\frac{m_e}{m_p} 4 \\alpha^2 E_1}$}", + "\\boxed{21}" + ], + "answer_type": [ + "Expression", + "Expression", + "Numerical Value" + ], + "unit": [ + null, + null, + "cm" + ], + "points": [ + 0.2, + 0.2, + 0.1 + ], + "modality": "text-only", + "field": "Modern Physics", + "source": "IPhO_2025", + "image_question": [] + }, + { + "id": "IPhO_2025_1_B_1", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part B - Rotation curves of galaxies] \n\nData: \nKiloparsec: $1 \\text{kpc} = 3.09 \\times 10^{19} m$ \nSolar mass: $1 M_\\odot = 1.99 \\times 10^{30} \\text{kg}$ \nWe consider a spherical galaxy centered around a fixed point $O$. At any point $P$, let $\\rho = \\rho(P)$ be the volumetric mass density and $\\varphi = \\varphi(P)$ the associated gravitational potential (i.e. potential energy per unit mass). Both $\\rho$ and $\\varphi$ depend only on $r = \\|\\vec{\\text{OP}}\\|$. The motion of a mass $m$ located at $P$, due to the field $\\varphi$, is restricted to a plane containing $O$.", + "question": "In the case of a circular orbit, determine the velocity $v_c$ of an object on a circular orbit passing through $P$ in terms of $r$ and $\\frac{d \\varphi}{d r}$.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly uses the Newton's second law for a circular orbit as $m \\frac{v_c^2}{r} = m \\frac{d \\phi}{d r}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly derives the final expression for the velocity $v_c$ as $v_c = \\sqrt{r \\frac{d \\varphi}{d r}}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$v_c = \\sqrt{r \\frac{d \\varphi}{d r}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.2 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [] + }, + { + "id": "IPhO_2025_1_B_2", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part B - Rotation curves of galaxies] \n\nData: \nKiloparsec: $1 \\text{kpc} = 3.09 \\times 10^{19} m$ \nSolar mass: $1 M_\\odot = 1.99 \\times 10^{30} \\text{kg}$ \nWe consider a spherical galaxy centered around a fixed point $O$. At any point $P$, let $\\rho = \\rho(P)$ be the volumetric mass density and $\\varphi = \\varphi(P)$ the associated gravitational potential (i.e. potential energy per unit mass). Both $\\rho$ and $\\varphi$ depend only on $r = \\|\\vec{\\text{OP}}\\|$. The motion of a mass $m$ located at $P$, due to the field $\\varphi$, is restricted to a plane containing $O$. \n\n(B.1) In the case of a circular orbit, determine the velocity $v_c$ of an object on a circular orbit passing through $P$ in terms of $r$ and $\\frac{d \\varphi}{d r}$. \n\nFig. 1(A) is a picture of the spiral galaxy NGC 6946 in the visible band (from the $0.8 m$ Schulman Telescope at the Mount Lemmon Sky Center in Arizona). The little ellipses in Fig. 1(B) show experimental measurements of $v_c$ for this galaxy. The central region ($r < 1 \\text{kpc}$) is named the bulge. In this region, the mass distribution is roughly homogeneous. The red curve is a prediction for $v_c$ if the system were homogeneous in the bulge and keplerian ($\\varphi(r) = -\\beta / r$ with $\\beta > 0$) outside it, i.e. considering that the total mass of the galaxy is concentrated in the bulge. \n\n[figure1]\nFig. 1: NGC 6946 galaxy: Picture (A) and rotation curve (B).", + "question": "Deduce the mass $M_b$ of the bulge of NGC 6946 from the red rotation curve in Fig. 1(B), in solar mass units ($M_{\\odot}$).", + "marking": [ + [ + "Award 0.1 pt if the answer gives $g(r) = G M_{\\text{int}}(r) / r^2$ via Gauss' Theorem or an equivalent method, where $M_{\\text{int}}(r)$ is the interior mass. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the expression for $\\vec{g}(r > r_b)$ as $\\vec{g}(r > r_b) = -\\frac{G M_b}{r^2} \\vec{u}_r$ with $r > r_b$ and $M_{\\text{int}}(r) = M_b$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the correct expression for $M_b$ as $M_b = v_c^2 R / G$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly identifies the value of $v_c$ from the figure as $v_c = 20 \\mathrm{km/s}$ at $R = 10 \\mathrm{kpc}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer provides the final numerical value for $M_b$ within the range $[6.75 \\times 10^8 M_{\\odot}, 11.25 \\times 10^8 M_{\\odot}]$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{[6.75 \\times 10^8, 11.25 \\times 10^8]}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "$M_{\\odot}$" + ], + "points": [ + 0.5 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_1_b_1.png" + ] + }, + { + "id": "IPhO_2025_1_B_3", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part B - Rotation curves of galaxies] \n\nData: \nKiloparsec: $1 \\text{kpc} = 3.09 \\times 10^{19} m$ \nSolar mass: $1 M_\\odot = 1.99 \\times 10^{30} \\text{kg}$ \nWe consider a spherical galaxy centered around a fixed point $O$. At any point $P$, let $\\rho = \\rho(P)$ be the volumetric mass density and $\\varphi = \\varphi(P)$ the associated gravitational potential (i.e. potential energy per unit mass). Both $\\rho$ and $\\varphi$ depend only on $r = \\|\\vec{\\text{OP}}\\|$. The motion of a mass $m$ located at $P$, due to the field $\\varphi$, is restricted to a plane containing $O$. \n\n(B.1) In the case of a circular orbit, determine the velocity $v_c$ of an object on a circular orbit passing through $P$ in terms of $r$ and $\\frac{d \\varphi}{d r}$. \n\nFig. 1(A) is a picture of the spiral galaxy NGC 6946 in the visible band (from the $0.8 m$ Schulman Telescope at the Mount Lemmon Sky Center in Arizona). The little ellipses in Fig. 1(B) show experimental measurements of $v_c$ for this galaxy. The central region ($r < 1 \\text{kpc}$) is named the bulge. In this region, the mass distribution is roughly homogeneous. The red curve is a prediction for $v_c$ if the system were homogeneous in the bulge and keplerian ($\\varphi(r) = -\\beta / r$ with $\\beta > 0$) outside it, i.e. considering that the total mass of the galaxy is concentrated in the bulge. \n\n[figure1]\nFig. 1: NGC 6946 galaxy: Picture (A) and rotation curve (B). \n\n(B.2) Deduce the mass $M_b$ of the bulge of NGC 6946 from the red rotation curve in Fig. 1(B), in solar mass units. \n\nComparing the keplerian model and the experimental data makes astronomers confident that part of the mass is invisible in the picture. They thus suppose that the galaxy's actual mass density is given by \n$\\rho_{m}(r) = \\frac{C_{m}}{r_{m}^{2} + r^{2}}$ (Equation 1) \nwhere $C_{m} > 0$ and $r_{m} > 0$ are constants.", + "question": "Show that the velocity profile $v_{c,m}(r)$, corresponding to the mass density in Eq. 1, can be written $v_{c,m}(r) = \\sqrt{k_1 - \\frac{k_2 \\cdot \\arctan(\\frac{r}{r_m})}{r}}$. \n(1) Express $k_1$ in terms of $C_m$, $r_m$ and $G$. \n(2) Express $k_2$ in terms of $C_m$, $r_m$ and $G$. \n(Hints: $\\int_0^r \\frac{x^2}{a^2 + x^2} dx = r - a \\arctan(r/a)$, and: $\\arctan(x) \\simeq x - x^3 / 3$ for $x \\ll 1$.) \n(3) Simplify $v_{c,m}(r)$ when $r \\ll r_m$. \n(4) Simplify $v_{c,m}(r)$ when $r \\gg r_m$. \n(5) Show that if $r \\gg r_m$, the mass $M_m(r)$ embedded in a sphere of radius $r$ with the mass density given by Eq. 1 simplifies and depends only on $C_m$ and $r$. \n(6) Estimate the mass of the galaxy NGC 6946 actually present in the picture in Fig. 1(A) in solar mass units ($M_{\\odot}$).", + "marking": [ + [ + "Award 0.2 pt if the answer gives $g(r) = G M_{\\text{int}}(r) / r^2$ via Gauss' Theorem or an equivalent method, where $M_{\\text{int}}(r)$ is the interior mass. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer derives the correct expression for the interior mass $M_{\\text{int}}$ as $$M_{\\text{int}} = 4 \\pi C_m \\left[r - r_m \\arctan(\\frac{r}{r_m}) \\right]$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer derives the correct expression for $g(r)$ as $g_m(r) = -\\frac{4 \\pi C_m \\left[r - r_m \\arctan(\\frac{r}{r_m}) \\right]}{r^2}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly applies Newton's second law, deriving $v_{c,m} = \\sqrt{r g_m(r)}$ from $-m \\frac{v_{c,m}^2}{r} = -m g_m(r)$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly derives the expression for $k_1$ as $k_1 = 4 \\pi C_m G$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly derives the expression for $k_2$ as $k_2 = 4 \\pi C_m G r_m$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly simplifies $v_{c,m}$ in the case $r \\ll r_m$ as $v_{c,m} \\simeq \\sqrt{\\frac{4 \\pi C_{m} G r^{2}}{3 r_{m}^{2}}}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly simplifies $v_{c,m}$ in the case $r \\gg r_m$ as $v_{c,m} \\simeq \\sqrt{4 \\pi C_{m} G}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer provides the correct value of $C_m$ as $C_m \\simeq 3 \\times 10^{19} \\mathrm{kg/m}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly gives the expression for $M_m$ in the case $r \\gg r_m$ as $M_m(r) \\simeq 4 \\pi C_m r$. Otherwise, award 0 pt.", + "Award 0.1 pt if the the answer correctly gives the value of the mass in the figure within the range $[10^{10.5} M_{\\odot}, 10^{11.5} M_{\\odot}]$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$k_1 = 4 \\pi C_m G$}", + "\\boxed{$k_2 = 4 \\pi C_m G r_m$}", + "\\boxed{$v_{c,m} \\simeq \\sqrt{\\frac{4 \\pi C_{m} G r^{2}}{3 r_{m}^{2}}}$ if $r \\ll r_m$}", + "\\boxed{$v_{c,m} \\simeq \\sqrt{4 \\pi C_{m} G}$ if $r \\gg r_m$}", + "\\boxed{$M_m(r) \\simeq 4 \\pi C_m r$}", + "\\boxed{[10^{10.5}, 10^{11.5}]}" + ], + "answer_type": [ + "Expression", + "Expression", + "Expression", + "Expression", + "Expression", + "Numerical Value" + ], + "unit": [ + null, + null, + null, + null, + null, + "$M_{\\odot}$" + ], + "points": [ + 0.45, + 0.45, + 0.2, + 0.2, + 0.4, + 0.1 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_1_b_1.png" + ] + }, + { + "id": "IPhO_2025_1_C_1", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part B - Rotation curves of galaxies] \n\nData: \nKiloparsec: $1 \\text{kpc} = 3.09 \\times 10^{19} m$ \nSolar mass: $1 M_\\odot = 1.99 \\times 10^{30} \\text{kg}$ \nWe consider a spherical galaxy centered around a fixed point $O$. At any point $P$, let $\\rho = \\rho(P)$ be the volumetric mass density and $\\varphi = \\varphi(P)$ the associated gravitational potential (i.e. potential energy per unit mass). Both $\\rho$ and $\\varphi$ depend only on $r = \\|\\vec{\\text{OP}}\\|$. The motion of a mass $m$ located at $P$, due to the field $\\varphi$, is restricted to a plane containing $O$. \n\n(B.1) In the case of a circular orbit, determine the velocity $v_c$ of an object on a circular orbit passing through $P$ in terms of $r$ and $\\frac{d \\varphi}{d r}$. \n\nFig. 1(A) is a picture of the spiral galaxy NGC 6946 in the visible band (from the $0.8 m$ Schulman Telescope at the Mount Lemmon Sky Center in Arizona). The little ellipses in Fig. 1(B) show experimental measurements of $v_c$ for this galaxy. The central region ($r < 1 \\text{kpc}$) is named the bulge. In this region, the mass distribution is roughly homogeneous. The red curve is a prediction for $v_c$ if the system were homogeneous in the bulge and keplerian ($\\varphi(r) = -\\beta / r$ with $\\beta > 0$) outside it, i.e. considering that the total mass of the galaxy is concentrated in the bulge. \nFig. 1: NGC 6946 galaxy: Picture (A) and rotation curve (B). \n\n(B.2) Deduce the mass $M_b$ of the bulge of NGC 6946 from the red rotation curve in Fig. 1(B), in solar mass units. \n\nComparing the keplerian model and the experimental data makes astronomers confident that part of the mass is invisible in the picture. They thus suppose that the galaxy's actual mass density is given by \n$\\rho_{m}(r) = \\frac{C_{m}}{r_{m}^{2} + r^{2}}$ (Equation 1) \nwhere $C_{m} > 0$ and $r_{m} > 0$ are constants. \n\n(B.3) Show that the velocity profile $v_{c,m}(r)$, corresponding to the mass density in Eq. 1, can be written $v_{c,m}(r) = \\sqrt{k_1 - \\frac{k_2 \\cdot \\arctan(\\frac{r}{r_m})}{r}}$. Express $k_1$ and $k_2$ in terms of $C_m$, $r_m$ and $G$. (Hints: $\\int_0^r \\frac{x^2}{a^2 + x^2} dx = r - a \\arctan(r/a)$, and: $\\arctan(x) \\simeq x - x^3 / 3$ for $x \\ll 1$.) Simplify $v_{c,m}(r)$ when $r \\ll r_m$ and when $r \\gg r_m$. Show that if $r \\gg r_m$, the mass $M_m(r)$ embedded in a sphere of radius $r$ with the mass density given by Eq. 1 simplifies and depends only on $C_m$ and $r$. Estimate the mass of the galaxy NGC 6946 actually present in the picture in Fig. 1(A). \n\n[Part C - Mass distribution in our galaxy] \n\nFor a spiral galaxy, the model for Eq. 1 is modified and one usually considers the gravitational potential: $\\varphi_G(r, z) = \\varphi_0 \\ln(\\frac{r}{r_0}) \\exp\\left[-(\\frac{z}{z_0})^2\\right]$, where $z$ is the distance to the galactic plane (defined by $z = 0$), and $r < r_0$ is now the axial radius and $\\varphi_0 > 0$ a constant to be determined. $r_0$ and $z_0$ are constant values.", + "question": "(1) Find the equation of motion on $z$ for the vertical motion of a point mass $m$ in such a potential, assuming $r$ is constant. \n(2) Show that, if $r < r_0$, the galactic plane is a stable equilibrium state by giving the angular frequency $\\omega_0$ of small oscillations around it.", + "marking": [ + [ + "Award 0.1 pt if the answer applies the Newton's second law to get $m \\vec{a} = \\vec{F} = -m \\vec{\\nabla} \\varphi$ or an equivalent method is used. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer projects the Newton's second law $m \\vec{a} = \\vec{F} = -m \\vec{\\nabla} \\varphi$ on $z$-asis and gives $m \\ddot{z} = -m \\frac{\\partial \\varphi}{\\partial z}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly provides the equation of motion as $\\ddot{z} = \\frac{2z}{z_{0}^{2}} \\varphi_{0} \\ln(\\frac{r}{r_{0}}) \\exp[-(\\frac{z}{z_{0}})^{2}]$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer simplifies the equation of motion near the galactic plane ($z=0$) as $\\ddot{z} \\simeq \\frac{2z}{z_0^2} \\varphi_0 \\ln(\\frac{r}{r_0})$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the expression for $\\omega_0$ as $\\omega_0 = \\sqrt{\\frac{2 \\varphi_0}{z_0^2} \\left| \\ln(\\frac{r}{r_0}) \\right|}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\ddot{z} = \\frac{2z}{z_{0}^{2}} \\varphi_{0} \\ln(\\frac{r}{r_{0}}) \\exp[-(\\frac{z}{z_{0}})^{2}]$}", + "\\boxed{$\\omega_0 = \\sqrt{\\frac{2 \\varphi_0}{z_0^2} \\left| \\ln(\\frac{r}{r_0}) \\right|}$}" + ], + "answer_type": [ + "Equation", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 0.3, + 0.2 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [] + }, + { + "id": "IPhO_2025_1_C_2", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part B - Rotation curves of galaxies] \n\nData: \nKiloparsec: $1 \\text{kpc} = 3.09 \\times 10^{19} m$ \nSolar mass: $1 M_\\odot = 1.99 \\times 10^{30} \\text{kg}$ \nWe consider a spherical galaxy centered around a fixed point $O$. At any point $P$, let $\\rho = \\rho(P)$ be the volumetric mass density and $\\varphi = \\varphi(P)$ the associated gravitational potential (i.e. potential energy per unit mass). Both $\\rho$ and $\\varphi$ depend only on $r = \\|\\vec{\\text{OP}}\\|$. The motion of a mass $m$ located at $P$, due to the field $\\varphi$, is restricted to a plane containing $O$. \n\n(B.1) In the case of a circular orbit, determine the velocity $v_c$ of an object on a circular orbit passing through $P$ in terms of $r$ and $\\frac{d \\varphi}{d r}$. \n\nFig. 1(A) is a picture of the spiral galaxy NGC 6946 in the visible band (from the $0.8 m$ Schulman Telescope at the Mount Lemmon Sky Center in Arizona). The little ellipses in Fig. 1(B) show experimental measurements of $v_c$ for this galaxy. The central region ($r < 1 \\text{kpc}$) is named the bulge. In this region, the mass distribution is roughly homogeneous. The red curve is a prediction for $v_c$ if the system were homogeneous in the bulge and keplerian ($\\varphi(r) = -\\beta / r$ with $\\beta > 0$) outside it, i.e. considering that the total mass of the galaxy is concentrated in the bulge. \nFig. 1: NGC 6946 galaxy: Picture (A) and rotation curve (B). \n\n(B.2) Deduce the mass $M_b$ of the bulge of NGC 6946 from the red rotation curve in Fig. 1(B), in solar mass units. \n\nComparing the keplerian model and the experimental data makes astronomers confident that part of the mass is invisible in the picture. They thus suppose that the galaxy's actual mass density is given by \n$\\rho_{m}(r) = \\frac{C_{m}}{r_{m}^{2} + r^{2}}$ (Equation 1) \nwhere $C_{m} > 0$ and $r_{m} > 0$ are constants. \n\n(B.3) Show that the velocity profile $v_{c,m}(r)$, corresponding to the mass density in Eq. 1, can be written $v_{c,m}(r) = \\sqrt{k_1 - \\frac{k_2 \\cdot \\arctan(\\frac{r}{r_m})}{r}}$. Express $k_1$ and $k_2$ in terms of $C_m$, $r_m$ and $G$. (Hints: $\\int_0^r \\frac{x^2}{a^2 + x^2} dx = r - a \\arctan(r/a)$, and: $\\arctan(x) \\simeq x - x^3 / 3$ for $x \\ll 1$.) Simplify $v_{c,m}(r)$ when $r \\ll r_m$ and when $r \\gg r_m$. Show that if $r \\gg r_m$, the mass $M_m(r)$ embedded in a sphere of radius $r$ with the mass density given by Eq. 1 simplifies and depends only on $C_m$ and $r$. Estimate the mass of the galaxy NGC 6946 actually present in the picture in Fig. 1(A). \n\n[Part C - Mass distribution in our galaxy] \n\nFor a spiral galaxy, the model for Eq. 1 is modified and one usually considers the gravitational potential: $\\varphi_G(r, z) = \\varphi_0 \\ln(\\frac{r}{r_0}) \\exp\\left[-(\\frac{z}{z_0})^2\\right]$, where $z$ is the distance to the galactic plane (defined by $z = 0$), and $r < r_0$ is now the axial radius and $\\varphi_0 > 0$ a constant to be determined. $r_0$ and $z_0$ are constant values. \n\n(C.1) Find the equation of motion on $z$ for the vertical motion of a point mass $m$ in such a potential, assuming $r$ is constant. Show that, if $r < r_0$, the galactic plane is a stable equilibrium state by giving the angular frequency $\\omega_0$ of small oscillations around it. \n\nFrom here on, we set $z = 0$.", + "question": "(1) Identify the regime, either $r \\gg r_m$ or $r \\ll r_m$, in which the model of Eq. 1 recovers a potential of the form $\\varphi_G(r, 0)$ with a suitable definition of $\\varphi_0$. \n(2) Write down the definition of $\\varphi_0$. \n(3) Under this condition $v_c(r)$ no longer depends on $r$. Express it in terms of $\\varphi_0$.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly identifies the regime as $r \\gg r_m$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly gives the expression for gravitational potential as $\\varphi(r) = +4 \\pi C_{m} G \\ln(r) + constant$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correcly gives the expression for $\\varphi_0$ as $\\varphi_{0} = +4 \\pi C_{m} G$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer applies the Newton's second law $-m\\frac{v_c^2}{r} = -m g_m(r)$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the expression for $v_c$ as $v_c = \\sqrt{\\varphi_0}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$r \\gg r_m$}", + "\\boxed{$\\varphi_{0} = +4 \\pi C_{m} G$}", + "\\boxed{$v_c = \\sqrt{\\varphi_0}$}" + ], + "answer_type": [ + "Inequality", + "Expression", + "Expression" + ], + "unit": [ + null, + null, + null + ], + "points": [ + 0.1, + 0.3, + 0.2 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [] + }, + { + "id": "IPhO_2025_1_C_3", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part B - Rotation curves of galaxies] \n\nData: \nKiloparsec: $1 \\text{kpc} = 3.09 \\times 10^{19} m$ \nSolar mass: $1 M_\\odot = 1.99 \\times 10^{30} \\text{kg}$ \nWe consider a spherical galaxy centered around a fixed point $O$. At any point $P$, let $\\rho = \\rho(P)$ be the volumetric mass density and $\\varphi = \\varphi(P)$ the associated gravitational potential (i.e. potential energy per unit mass). Both $\\rho$ and $\\varphi$ depend only on $r = \\|\\vec{\\text{OP}}\\|$. The motion of a mass $m$ located at $P$, due to the field $\\varphi$, is restricted to a plane containing $O$. \n\n(B.1) In the case of a circular orbit, determine the velocity $v_c$ of an object on a circular orbit passing through $P$ in terms of $r$ and $\\frac{d \\varphi}{d r}$. \n\nFig. 1(A) is a picture of the spiral galaxy NGC 6946 in the visible band (from the $0.8 m$ Schulman Telescope at the Mount Lemmon Sky Center in Arizona). The little ellipses in Fig. 1(B) show experimental measurements of $v_c$ for this galaxy. The central region ($r < 1 \\text{kpc}$) is named the bulge. In this region, the mass distribution is roughly homogeneous. The red curve is a prediction for $v_c$ if the system were homogeneous in the bulge and keplerian ($\\varphi(r) = -\\beta / r$ with $\\beta > 0$) outside it, i.e. considering that the total mass of the galaxy is concentrated in the bulge. \nFig. 1: NGC 6946 galaxy: Picture (A) and rotation curve (B). \n\n(B.2) Deduce the mass $M_b$ of the bulge of NGC 6946 from the red rotation curve in Fig. 1(B), in solar mass units. \n\nComparing the keplerian model and the experimental data makes astronomers confident that part of the mass is invisible in the picture. They thus suppose that the galaxy's actual mass density is given by \n$\\rho_{m}(r) = \\frac{C_{m}}{r_{m}^{2} + r^{2}}$ (Equation 1) \nwhere $C_{m} > 0$ and $r_{m} > 0$ are constants. \n\n(B.3) Show that the velocity profile $v_{c,m}(r)$, corresponding to the mass density in Eq. 1, can be written $v_{c,m}(r) = \\sqrt{k_1 - \\frac{k_2 \\cdot \\arctan(\\frac{r}{r_m})}{r}}$. Express $k_1$ and $k_2$ in terms of $C_m$, $r_m$ and $G$. (Hints: $\\int_0^r \\frac{x^2}{a^2 + x^2} dx = r - a \\arctan(r/a)$, and: $\\arctan(x) \\simeq x - x^3 / 3$ for $x \\ll 1$.) Simplify $v_{c,m}(r)$ when $r \\ll r_m$ and when $r \\gg r_m$. Show that if $r \\gg r_m$, the mass $M_m(r)$ embedded in a sphere of radius $r$ with the mass density given by Eq. 1 simplifies and depends only on $C_m$ and $r$. Estimate the mass of the galaxy NGC 6946 actually present in the picture in Fig. 1(A). \n\n[Part C - Mass distribution in our galaxy] \n\nFor a spiral galaxy, the model for Eq. 1 is modified and one usually considers the gravitational potential: $\\varphi_G(r, z) = \\varphi_0 \\ln(\\frac{r}{r_0}) \\exp\\left[-(\\frac{z}{z_0})^2\\right]$, where $z$ is the distance to the galactic plane (defined by $z = 0$), and $r < r_0$ is now the axial radius and $\\varphi_0 > 0$ a constant to be determined. $r_0$ and $z_0$ are constant values. \n\n(C.1) Find the equation of motion on $z$ for the vertical motion of a point mass $m$ in such a potential, assuming $r$ is constant. Show that, if $r < r_0$, the galactic plane is a stable equilibrium state by giving the angular frequency $\\omega_0$ of small oscillations around it. \n\nFrom here on, we set $z = 0$. \n\n(C.2) Identify the regime, either $r \\gg r_m$ or $r \\ll r_m$, in which the model of Eq. 1 recovers a potential of the form $\\varphi_G(r, 0)$ with a suitable definition of $\\varphi_0$. Under this condition $v_c(r)$ no longer depends on $r$. Express it in terms of $\\varphi_0$. \n\nTherefore, outside the bulge the velocity modulus $v_{c}$ does not depend on the distance to the galactic center. We will use this fact, as astronomers do, to measure the galaxy's mass distribution from the inside. \n\nAll galactic objects considered here for astronomical observations, such as stars or nebulae, are primarily composed of hydrogen. Outside the bulge, we assume that they rotate on circular orbits around the galactic center $C$. $S$ is the sun's position and $E$ that of a given galactic object emitting in the hydrogen spectrum. In the galactic plane, we consider a line of sight $\\text{SE}$ corresponding to the orientation of an observation, on the unit vector $\\hat{u}_{v}$ (see Fig. 2). \n\n[figure2]\nFig. 2: Geometry of the measurement \n\nLet $\\ell$ be the galactic longitude, measuring the angle between $\\text{SC}$ and the $\\text{SE}$. The sun's velocity on its circular orbit of radius $R_\\odot = 8.00 \\text{kpc}$ is denoted $\\vec{v}_\\odot$. A galactic object in $E$ orbits on another circle of radius $R$ at velocity $\\vec{v}_E$. Using a Doppler effect on the previously studied 21 cm line, one can obtain the relative radial velocity $v_{rE/S}$ of the emitter $E$ with respect to the sun $S$: it is the projection of $\\vec{v}_E - \\vec{v}_\\odot$ on the line of sight.", + "question": "(1) Determine $v_{rE/S}$ in terms of $\\ell$, $R$, $R_\\odot$ and $v_\\odot$. \n(2) Then, express $R$ in terms of $R_\\odot$, $v_\\odot$, $\\ell$ and $v_{rE/S}$.", + "marking": [ + [ + "Award 0.1 pt if the answer gives the expression for $\\vec{Ss}$ in the figure as $\\vec{Ss} = v_{\\odot} \\sin(\\alpha) \\hat{u}_v$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the expression for $\\vec{Ee}$ in the figure as $\\vec{Ee} = v_E \\cos(\\beta) \\hat{u}_v$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer derives $\\alpha = \\ell$ in the figure. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly derives the expression for $\\cos(\\beta)$ as $\\cos(\\beta) = \\frac{R_{\\odot}}{R} \\sin(\\ell)$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly derives the expression for $v_{rE/S}$ as $v_{rE/S} = v_{\\odot} \\left(\\frac{R_{\\odot}}{R} - 1\\right) \\sin(\\ell)$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correcly derives the expression for $R$ as $R = \\frac{R_{\\odot}}{1 + \\frac{v_{rE/S}}{v_{\\odot} \\sin(\\ell)}}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$v_{rE/S} = v_{\\odot} \\left(\\frac{R_{\\odot}}{R} - 1\\right) \\sin(\\ell)$}", + "\\boxed{$R = \\frac{R_{\\odot}}{1 + \\frac{v_{rE/S}}{v_{\\odot} \\sin(\\ell)}}$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 0.6, + 0.1 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_1_c_1.png" + ] + }, + { + "id": "IPhO_2025_1_C_4", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part B - Rotation curves of galaxies] \n\nData: \nKiloparsec: $1 \\text{kpc} = 3.09 \\times 10^{19} m$ \nSolar mass: $1 M_\\odot = 1.99 \\times 10^{30} \\text{kg}$ \nWe consider a spherical galaxy centered around a fixed point $O$. At any point $P$, let $\\rho = \\rho(P)$ be the volumetric mass density and $\\varphi = \\varphi(P)$ the associated gravitational potential (i.e. potential energy per unit mass). Both $\\rho$ and $\\varphi$ depend only on $r = \\|\\vec{\\text{OP}}\\|$. The motion of a mass $m$ located at $P$, due to the field $\\varphi$, is restricted to a plane containing $O$. \n\n(B.1) In the case of a circular orbit, determine the velocity $v_c$ of an object on a circular orbit passing through $P$ in terms of $r$ and $\\frac{d \\varphi}{d r}$. \n\nFig. 1(A) is a picture of the spiral galaxy NGC 6946 in the visible band (from the $0.8 m$ Schulman Telescope at the Mount Lemmon Sky Center in Arizona). The little ellipses in Fig. 1(B) show experimental measurements of $v_c$ for this galaxy. The central region ($r < 1 \\text{kpc}$) is named the bulge. In this region, the mass distribution is roughly homogeneous. The red curve is a prediction for $v_c$ if the system were homogeneous in the bulge and keplerian ($\\varphi(r) = -\\beta / r$ with $\\beta > 0$) outside it, i.e. considering that the total mass of the galaxy is concentrated in the bulge. \nFig. 1: NGC 6946 galaxy: Picture (A) and rotation curve (B). \n\n(B.2) Deduce the mass $M_b$ of the bulge of NGC 6946 from the red rotation curve in Fig. 1(B), in solar mass units. \n\nComparing the keplerian model and the experimental data makes astronomers confident that part of the mass is invisible in the picture. They thus suppose that the galaxy's actual mass density is given by \n$\\rho_{m}(r) = \\frac{C_{m}}{r_{m}^{2} + r^{2}}$ (Equation 1) \nwhere $C_{m} > 0$ and $r_{m} > 0$ are constants. \n\n(B.3) Show that the velocity profile $v_{c,m}(r)$, corresponding to the mass density in Eq. 1, can be written $v_{c,m}(r) = \\sqrt{k_1 - \\frac{k_2 \\cdot \\arctan(\\frac{r}{r_m})}{r}}$. Express $k_1$ and $k_2$ in terms of $C_m$, $r_m$ and $G$. (Hints: $\\int_0^r \\frac{x^2}{a^2 + x^2} dx = r - a \\arctan(r/a)$, and: $\\arctan(x) \\simeq x - x^3 / 3$ for $x \\ll 1$.) Simplify $v_{c,m}(r)$ when $r \\ll r_m$ and when $r \\gg r_m$. Show that if $r \\gg r_m$, the mass $M_m(r)$ embedded in a sphere of radius $r$ with the mass density given by Eq. 1 simplifies and depends only on $C_m$ and $r$. Estimate the mass of the galaxy NGC 6946 actually present in the picture in Fig. 1(A). \n\n[Part C - Mass distribution in our galaxy] \n\nFor a spiral galaxy, the model for Eq. 1 is modified and one usually considers the gravitational potential: $\\varphi_G(r, z) = \\varphi_0 \\ln(\\frac{r}{r_0}) \\exp\\left[-(\\frac{z}{z_0})^2\\right]$, where $z$ is the distance to the galactic plane (defined by $z = 0$), and $r < r_0$ is now the axial radius and $\\varphi_0 > 0$ a constant to be determined. $r_0$ and $z_0$ are constant values. \n\n(C.1) Find the equation of motion on $z$ for the vertical motion of a point mass $m$ in such a potential, assuming $r$ is constant. Show that, if $r < r_0$, the galactic plane is a stable equilibrium state by giving the angular frequency $\\omega_0$ of small oscillations around it. \n\nFrom here on, we set $z = 0$. \n\n(C.2) Identify the regime, either $r \\gg r_m$ or $r \\ll r_m$, in which the model of Eq. 1 recovers a potential of the form $\\varphi_G(r, 0)$ with a suitable definition of $\\varphi_0$. Under this condition $v_c(r)$ no longer depends on $r$. Express it in terms of $\\varphi_0$. \n\nTherefore, outside the bulge the velocity modulus $v_{c}$ does not depend on the distance to the galactic center. We will use this fact, as astronomers do, to measure the galaxy's mass distribution from the inside. \n\nAll galactic objects considered here for astronomical observations, such as stars or nebulae, are primarily composed of hydrogen. Outside the bulge, we assume that they rotate on circular orbits around the galactic center $C$. $S$ is the sun's position and $E$ that of a given galactic object emitting in the hydrogen spectrum. In the galactic plane, we consider a line of sight $\\text{SE}$ corresponding to the orientation of an observation,on the unit vector $\\hat{u}_{v}$ (see Fig. 2). \n\n[figure2]\nFig. 2: Geometry of the measurement \n\nLet $\\ell$ be the galactic longitude, measuring the angle between $\\text{SC}$ and the $\\text{SE}$. The sun's velocity on its circular orbit of radius $R_\\odot = 8.00 \\text{kpc}$ is denoted $\\vec{v}_\\odot$. A galactic object in $E$ orbits on another circle of radius $R$ at velocity $\\vec{v}_E$. Using a Doppler effect on the previously studied 21 cm line, one can obtain the relative radial velocity $v_{rE/S}$ of the emitter $E$ with respect to the sun $S$: it is the projection of $\\vec{v}_E - \\vec{v}_\\odot$ on the line of sight. \n\n(C.3) Determine $v_{rE/S}$ in terms of $\\ell$, $R$, $R_\\odot$ and $v_\\odot$. Then, express $R$ in terms of $R_\\odot$, $v_\\odot$, $\\ell$ and $v_{rE/S}$. \n\nUsing a radio telescope, we make observations in the plane of our galaxy toward a longitude $\\ell = 30^{\\circ}$. The frequency band used contains the $21 \\mathrm{cm}$ line, whose frequency is $f_{0} = 1.42 \\mathrm{GHz}$. The results are reported in Fig. 3. \n\n[figure3]\nFig. 3: Electromagnetic signal as a function of the frequency shift, measured in the radio frequency band at $\\ell = 30^{\\circ}$ using EU-HOU RadioAstronomy.", + "question": "In our galaxy, $v_{\\odot} = 220 \\mathrm{km} \\cdot \\mathrm{s}^{-1}$. \nDetermine the values of the relative radial velocity (with 3 significant digits) of the 3 sources observed in Fig. 3: \n(1) $v_{r,1}$, (2) $v_{r,2}$, (3) $v_{r,3}$, expressed in $\\mathrm{km}/\\mathrm{s}$. \nDetermine the values of the distances from the galactic center (with 2 significant digits) of the 3 sources: \n(4) $R_1$, (5) $R_2$, (6) $R_3$, expressed as multiples of $R_{\\odot}$.", + "marking": [ + [ + "Award 0.1 pt if the answer applies the Doppler formula for $v_r$: $v_{r,i} = c \\delta f_i / f_0$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer obtains 3 numerical values for $\\Delta f$ from the figure: $\\delta f_1 = 0.03 \\mathrm{MHz}$, $\\delta f_2 = 0.15 \\mathrm{MHz}$, and $\\delta f_3 = 0.26 \\mathrm{MHz}$. Partial points: award 0.1 pt if the answer provides only two correct values. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correcly gives three numerical values of $v_{r,1}$, $v_{r,2}$, and $v_{r,3}$ in $\\mathrm{km/s}$: $v_{r,1} \\in [6.32, 6.34]$, $v_{r,2} \\in [31.6, 31.8]$, and $v_{r,3} \\in [54.8, 55.0]$. Partial points: award 0.1 pt if the answer provides only two correct values. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives three numerical values of $R_1$, $R_2$, and $R_3$ in $R_\\odot$: $R_1 \\in [0.94, 0.96]$, $R_2 \\in [0.77, 0.79]$, and $R_3 \\in [0.66, 0.68]$. Partial points: award 0.05 pt if the answer provides only two correct values. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{[6.32, 6.34] for $v_{r,1}$}", + "\\boxed{[31.6, 31.8] for $v_{r,2}$}", + "\\boxed{[54.8, 55.0] for $v_{r,3}$}", + "\\boxed{[0.94, 0.96] for $R_1$}", + "\\boxed{[0.77, 0.79] for $R_2$}", + "\\boxed{[0.66, 0.68] for $R_3$}" + ], + "answer_type": [ + "Numerical Value", + "Numerical Value", + "Numerical Value", + "Numerical Value", + "Numerical Value", + "Numerical Value" + ], + "unit": [ + "km/s", + "km/s", + "km/s", + "$R_\\odot$", + "$R_\\odot$", + "$R_\\odot$" + ], + "points": [ + 0.15, + 0.15, + 0.15, + 0.05, + 0.05, + 0.05 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_1_c_1.png", + "image_question/IPhO_2025_1_c_2.png" + ] + }, + { + "id": "IPhO_2025_1_D_1", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part B - Rotation curves of galaxies] \n\nData: \nKiloparsec: $1 \\text{kpc} = 3.09 \\times 10^{19} m$ \nSolar mass: $1 M_\\odot = 1.99 \\times 10^{30} \\text{kg}$ \nWe consider a spherical galaxy centered around a fixed point $O$. At any point $P$, let $\\rho = \\rho(P)$ be the volumetric mass density and $\\varphi = \\varphi(P)$ the associated gravitational potential (i.e. potential energy per unit mass). Both $\\rho$ and $\\varphi$ depend only on $r = \\|\\vec{\\text{OP}}\\|$. The motion of a mass $m$ located at $P$, due to the field $\\varphi$, is restricted to a plane containing $O$. \n\n(B.1) In the case of a circular orbit, determine the velocity $v_c$ of an object on a circular orbit passing through $P$ in terms of $r$ and $\\frac{d \\varphi}{d r}$. \n\nFig. 1(A) is a picture of the spiral galaxy NGC 6946 in the visible band (from the $0.8 m$ Schulman Telescope at the Mount Lemmon Sky Center in Arizona). The little ellipses in Fig. 1(B) show experimental measurements of $v_c$ for this galaxy. The central region ($r < 1 \\text{kpc}$) is named the bulge. In this region, the mass distribution is roughly homogeneous. The red curve is a prediction for $v_c$ if the system were homogeneous in the bulge and keplerian ($\\varphi(r) = -\\beta / r$ with $\\beta > 0$) outside it, i.e. considering that the total mass of the galaxy is concentrated in the bulge. \n\n[figure1]\nFig. 1: NGC 6946 galaxy: Picture (A) and rotation curve (B). \n\n(B.2) Deduce the mass $M_b$ of the bulge of NGC 6946 from the red rotation curve in Fig. 1(B), in solar mass units. \n\nComparing the keplerian model and the experimental data makes astronomers confident that part of the mass is invisible in the picture. They thus suppose that the galaxy's actual mass density is given by \n$\\rho_{m}(r) = \\frac{C_{m}}{r_{m}^{2} + r^{2}}$ (Equation 1) \nwhere $C_{m} > 0$ and $r_{m} > 0$ are constants. \n\n(B.3) Show that the velocity profile $v_{c,m}(r)$, corresponding to the mass density in Eq. 1, can be written $v_{c,m}(r) = \\sqrt{k_1 - \\frac{k_2 \\cdot \\arctan(\\frac{r}{r_m})}{r}}$. Express $k_1$ and $k_2$ in terms of $C_m$, $r_m$ and $G$. (Hints: $\\int_0^r \\frac{x^2}{a^2 + x^2} dx = r - a \\arctan(r/a)$, and: $\\arctan(x) \\simeq x - x^3 / 3$ for $x \\ll 1$.) Simplify $v_{c,m}(r)$ when $r \\ll r_m$ and when $r \\gg r_m$. Show that if $r \\gg r_m$, the mass $M_m(r)$ embedded in a sphere of radius $r$ with the mass density given by Eq. 1 simplifies and depends only on $C_m$ and $r$. Estimate the mass of the galaxy NGC 6946 actually present in the picture in Fig. 1(A). \n\n[Part D - Tully-Fisher relation and MOND theory] \n\nThe flat external velocity curve of NGC 6946 in Fig. 1 is a common property of spiral galaxies, as can be seen in Fig. 4 (left). Plotting the external constant velocity value $v_{c,\\infty}$ as a function of the measured total mass $M_{\\text{tot}}$ of each galaxy gives an interesting correlation called the Tully-Fischer relation, see Fig. 4 (right). \n\n[figure4]\nFig. 4. Left: Rotation curves for typical spiral galaxies - Right: $\\log_{10}(M_{\\text{tot}})$ as a function of $\\log_{10}(v_{c,\\infty})$ on linear scales. Colored dots correspond to different galaxies and different surveys. The green line is the Tully-Fischer relation which is in very good agreement with the best fit line of the data (in black).", + "question": "Assuming that the radius $R$ of a galaxy doesn't depend on its mass, show that the model of Eq. 1 (part B) gives a relation of the form $M_{\\text{tot}} = \\eta v_{c,\\infty}^\\gamma$ where $\\gamma$ and $\\eta$ should be specified. \nCompare this expression to the Tully-Fischer relation by computing $\\gamma_{TF}$. \nWrite down the expressions of (1) $\\eta$, (2) $\\gamma$, and (3) $\\gamma_{TF}$.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly gives the expression for $v_{c,\\infty}$ as $v_{c,\\infty} = 4 \\pi C_m G$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the expression for $\\eta$ as $\\eta = R/G$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the expression for $\\gamma$ as $\\gamma = 2$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the numerical value for $\\gamma_{TF}$ within the range [3.5, 4.0]. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\eta = \\frac{R}{G}$}", + "\\boxed{$\\gamma = 2$}", + "\\boxed{[3.5, 4.0]}" + ], + "answer_type": [ + "Expression", + "Numerical Value", + "Numerical Value" + ], + "unit": [ + null, + null, + null + ], + "points": [ + 0.2, + 0.1, + 0.1 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_1_b_1.png", + "image_question/IPhO_2025_1_d_1.png" + ] + }, + { + "id": "IPhO_2025_1_D_2", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part B - Rotation curves of galaxies] \n\nData: \nKiloparsec: $1 \\text{kpc} = 3.09 \\times 10^{19} m$ \nSolar mass: $1 M_\\odot = 1.99 \\times 10^{30} \\text{kg}$ \nWe consider a spherical galaxy centered around a fixed point $O$. At any point $P$, let $\\rho = \\rho(P)$ be the volumetric mass density and $\\varphi = \\varphi(P)$ the associated gravitational potential (i.e. potential energy per unit mass). Both $\\rho$ and $\\varphi$ depend only on $r = \\|\\vec{\\text{OP}}\\|$. The motion of a mass $m$ located at $P$, due to the field $\\varphi$, is restricted to a plane containing $O$. \n\n(B.1) In the case of a circular orbit, determine the velocity $v_c$ of an object on a circular orbit passing through $P$ in terms of $r$ and $\\frac{d \\varphi}{d r}$. \n\nFig. 1(A) is a picture of the spiral galaxy NGC 6946 in the visible band (from the $0.8 m$ Schulman Telescope at the Mount Lemmon Sky Center in Arizona). The little ellipses in Fig. 1(B) show experimental measurements of $v_c$ for this galaxy. The central region ($r < 1 \\text{kpc}$) is named the bulge. In this region, the mass distribution is roughly homogeneous. The red curve is a prediction for $v_c$ if the system were homogeneous in the bulge and keplerian ($\\varphi(r) = -\\beta / r$ with $\\beta > 0$) outside it, i.e. considering that the total mass of the galaxy is concentrated in the bulge. \n\n[figure1]\nFig. 1: NGC 6946 galaxy: Picture (A) and rotation curve (B). \n\n(B.2) Deduce the mass $M_b$ of the bulge of NGC 6946 from the red rotation curve in Fig. 1(B), in solar mass units. \n\nComparing the keplerian model and the experimental data makes astronomers confident that part of the mass is invisible in the picture. They thus suppose that the galaxy's actual mass density is given by \n$\\rho_{m}(r) = \\frac{C_{m}}{r_{m}^{2} + r^{2}}$ (Equation 1) \nwhere $C_{m} > 0$ and $r_{m} > 0$ are constants. \n\n(B.3) Show that the velocity profile $v_{c,m}(r)$, corresponding to the mass density in Eq. 1, can be written $v_{c,m}(r) = \\sqrt{k_1 - \\frac{k_2 \\cdot \\arctan(\\frac{r}{r_m})}{r}}$. Express $k_1$ and $k_2$ in terms of $C_m$, $r_m$ and $G$. (Hints: $\\int_0^r \\frac{x^2}{a^2 + x^2} dx = r - a \\arctan(r/a)$, and: $\\arctan(x) \\simeq x - x^3 / 3$ for $x \\ll 1$.) Simplify $v_{c,m}(r)$ when $r \\ll r_m$ and when $r \\gg r_m$. Show that if $r \\gg r_m$, the mass $M_m(r)$ embedded in a sphere of radius $r$ with the mass density given by Eq. 1 simplifies and depends only on $C_m$ and $r$. Estimate the mass of the galaxy NGC 6946 actually present in the picture in Fig. 1(A). \n\n[Part D - Tully-Fisher relation and MOND theory] \n\nThe flat external velocity curve of NGC 6946 in Fig. 1 is a common property of spiral galaxies, as can be seen in Fig. 4 (left). Plotting the external constant velocity value $v_{c,\\infty}$ as a function of the measured total mass $M_{\\text{tot}}$ of each galaxy gives an interesting correlation called the Tully-Fischer relation, see Fig. 4 (right). \n\n[figure4]\nFig. 4. Left: Rotation curves for typical spiral galaxies - Right: $\\log_{10}(M_{\\text{tot}})$ as a function of $\\log_{10}(v_{c,\\infty})$ on linear scales. Colored dots correspond to different galaxies and different surveys. The green line is the Tully-Fischer relation which is in very good agreement with the best fit line of the data (in black). \n\n(D.1) Assuming that the radius $R$ of a galaxy doesn't depend on its mass, show that the model of Eq. 1 (part B) gives a relation of the form $M_{\\text{tot}} = \\eta v_{c,\\infty}^\\gamma$ where $\\gamma$ and $\\eta$ should be specified. \nCompare this expression to the Tully-Fischer relation by computing $\\gamma_{TF}$. \n\nIn the extremely low acceleration regime, of the order of $a_0 = 10^{-10} \\mathrm{m} \\cdot \\mathrm{s}^{-2}$, the MOdified Newtonian Dynamics (MOND) theory suggests that one can modify Newton's second law using: $\\vec{F} = m \\mu \\left(\\frac{a}{a_0}\\right) \\vec{a}$ where $a = \\| \\vec{a} \\|$ is the modulus of the acceleration and the $\\mu$ function is defined by: $\\mu(x) = \\frac{x}{1 + x}$.", + "question": "Using data for NGC 6946 in Fig. 1, estimate, within Newton's theory, the modulus of the acceleration $a_m$ of a mass in the outer regions of NGC 6946. \n(1) Write down the expression for $a_m$. \n(2) Estimate the numerical value for $a_m$ in $m/s^2$.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly gives the expression for $a_m$ as $a_m = v_c^2 / R$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the numerical value for $a_m$ within the range $[10^{-10.5} \\mathrm{m/s^2}, 10^{-9.5} \\mathrm{m/s^2}]$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$a_m \\simeq v_c^2 / R$}", + "\\boxed{[10^{-10.5}, 10^{-9.5}]}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null, + "m/s^2" + ], + "points": [ + 0.1, + 0.1 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_1_b_1.png", + "image_question/IPhO_2025_1_d_1.png" + ] + }, + { + "id": "IPhO_2025_1_D_3", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part B - Rotation curves of galaxies] \n\nData: \nKiloparsec: $1 \\text{kpc} = 3.09 \\times 10^{19} m$ \nSolar mass: $1 M_\\odot = 1.99 \\times 10^{30} \\text{kg}$ \nWe consider a spherical galaxy centered around a fixed point $O$. At any point $P$, let $\\rho = \\rho(P)$ be the volumetric mass density and $\\varphi = \\varphi(P)$ the associated gravitational potential (i.e. potential energy per unit mass). Both $\\rho$ and $\\varphi$ depend only on $r = \\|\\vec{\\text{OP}}\\|$. The motion of a mass $m$ located at $P$, due to the field $\\varphi$, is restricted to a plane containing $O$. \n\n(B.1) In the case of a circular orbit, determine the velocity $v_c$ of an object on a circular orbit passing through $P$ in terms of $r$ and $\\frac{d \\varphi}{d r}$. \n\nFig. 1(A) is a picture of the spiral galaxy NGC 6946 in the visible band (from the $0.8 m$ Schulman Telescope at the Mount Lemmon Sky Center in Arizona). The little ellipses in Fig. 1(B) show experimental measurements of $v_c$ for this galaxy. The central region ($r < 1 \\text{kpc}$) is named the bulge. In this region, the mass distribution is roughly homogeneous. The red curve is a prediction for $v_c$ if the system were homogeneous in the bulge and keplerian ($\\varphi(r) = -\\beta / r$ with $\\beta > 0$) outside it, i.e. considering that the total mass of the galaxy is concentrated in the bulge. \n\n[figure1]\nFig. 1: NGC 6946 galaxy: Picture (A) and rotation curve (B). \n\n(B.2) Deduce the mass $M_b$ of the bulge of NGC 6946 from the red rotation curve in Fig. 1(B), in solar mass units. \n\nComparing the keplerian model and the experimental data makes astronomers confident that part of the mass is invisible in the picture. They thus suppose that the galaxy's actual mass density is given by \n$\\rho_{m}(r) = \\frac{C_{m}}{r_{m}^{2} + r^{2}}$ (Equation 1) \nwhere $C_{m} > 0$ and $r_{m} > 0$ are constants. \n\n(B.3) Show that the velocity profile $v_{c,m}(r)$, corresponding to the mass density in Eq. 1, can be written $v_{c,m}(r) = \\sqrt{k_1 - \\frac{k_2 \\cdot \\arctan(\\frac{r}{r_m})}{r}}$. Express $k_1$ and $k_2$ in terms of $C_m$, $r_m$ and $G$. (Hints: $\\int_0^r \\frac{x^2}{a^2 + x^2} dx = r - a \\arctan(r/a)$, and: $\\arctan(x) \\simeq x - x^3 / 3$ for $x \\ll 1$.) Simplify $v_{c,m}(r)$ when $r \\ll r_m$ and when $r \\gg r_m$. Show that if $r \\gg r_m$, the mass $M_m(r)$ embedded in a sphere of radius $r$ with the mass density given by Eq. 1 simplifies and depends only on $C_m$ and $r$. Estimate the mass of the galaxy NGC 6946 actually present in the picture in Fig. 1(A). \n\n[Part D - Tully-Fisher relation and MOND theory] \n\nThe flat external velocity curve of NGC 6946 in Fig. 1 is a common property of spiral galaxies, as can be seen in Fig. 4 (left). Plotting the external constant velocity value $v_{c,\\infty}$ as a function of the measured total mass $M_{\\text{tot}}$ of each galaxy gives an interesting correlation called the Tully-Fischer relation, see Fig. 4 (right). \n\n[figure4]\nFig. 4. Left: Rotation curves for typical spiral galaxies - Right: $\\log_{10}(M_{\\text{tot}})$ as a function of $\\log_{10}(v_{c,\\infty})$ on linear scales. Colored dots correspond to different galaxies and different surveys. The green line is the Tully-Fischer relation which is in very good agreement with the best fit line of the data (in black). \n\n(D.1) Assuming that the radius $R$ of a galaxy doesn't depend on its mass, show that the model of Eq. 1 (part B) gives a relation of the form $M_{\\text{tot}} = \\eta v_{c,\\infty}^\\gamma$ where $\\gamma$ and $\\eta$ should be specified. \nCompare this expression to the Tully-Fischer relation by computing $\\gamma_{TF}$. \n\nIn the extremely low acceleration regime, of the order of $a_0 = 10^{-10} \\mathrm{m} \\cdot \\mathrm{s}^{-2}$, the MOdified Newtonian Dynamics (MOND) theory suggests that one can modify Newton's second law using: $\\vec{F} = m \\mu \\left(\\frac{a}{a_0}\\right) \\vec{a}$ where $a = \\| \\vec{a} \\|$ is the modulus of the acceleration and the $\\mu$ function is defined by: $\\mu(x) = \\frac{x}{1 + x}$. \n\n(D.2) Using data for NGC 6946 in Fig. 1, estimate, within Newton's theory, the modulus of the acceleration $a_m$ of a mass in the outer regions of NGC 6946.", + "question": "Let $m$ be a mass on a circular orbit of radius $r$ with velocity $v_{c,\\infty}$ in the gravity field of a fixed mass $M$. \n(1) Within the MOND theory, with $a \\ll a_0$, determine the Tully-Fischer exponent $\\gamma_{\\text{MOND}}$. \nUsing data for NGC 6946 and/or Tully-Fischer law, calculate $a_0$ to show that MOND operates in the correct regime. \n(2) Write down the expression for $a_0$. \n(3) Calculate the numerical value of $a_0$ in $m/s^2$.", + "marking": [ + [ + "Award 0.1 pt if the answer considers the hypothesis $a \\ll a_0$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer applies the Newton's second law and derives $G \\frac{M}{r^2} m = m \\frac{a^2}{a_0}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the expression for $v_{c,\\infty}$ as $v_{c,\\infty} = (a_0 G M)^{1/4}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the numerical value for $\\gamma_{\\text{MOND}}$ as $\\gamma_{\\text{MOND}} = 4$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the numerical value for $\\log_{10}(v_{c,\\infty} / {1 \\mathrm{km/s}}) = 2.2$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the numerical value for $\\log_{10}(M_{\\text{tot}} / M_{\\odot}) = 10.5$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the expression for $a_0$ as $a_0 = \\frac{v_{c,\\infty}^4}{G M_{\\text{tot}}}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the numerical value for $a_0$ within the range [10^{-10.5}, 10^{-9.5}]. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\gamma_{\\text{MOND}} = 4$}", + "\\boxed{$a_0 = \\frac{v_{c,\\infty}^4}{G M_{\\text{tot}}}$}", + "\\boxed{[10^{-10.5}, 10^{-9.5}]}" + ], + "answer_type": [ + "Numerical Value", + "Expression", + "Numerical Value" + ], + "unit": [ + null, + null, + "m/s^2" + ], + "points": [ + 0.4, + 0.3, + 0.1 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_1_b_1.png", + "image_question/IPhO_2025_1_d_1.png" + ] + }, + { + "id": "IPhO_2025_1_D_4", + "context": "[Hydrogen and galaxies] \n\nThis problem aims to study the peculiar physics of galaxies, such as their dynamics and structure. In particular, we explain how to measure the mass distribution of our galaxy from the inside. For this we will focus on hydrogen, its main constituent. \nThroughout this problem we will only use $\\hbar$, defined as $\\hbar = h / {2\\pi}$. \n\n[Part B - Rotation curves of galaxies] \n\nData: \nKiloparsec: $1 \\text{kpc} = 3.09 \\times 10^{19} m$ \nSolar mass: $1 M_\\odot = 1.99 \\times 10^{30} \\text{kg}$ \nWe consider a spherical galaxy centered around a fixed point $O$. At any point $P$, let $\\rho = \\rho(P)$ be the volumetric mass density and $\\varphi = \\varphi(P)$ the associated gravitational potential (i.e. potential energy per unit mass). Both $\\rho$ and $\\varphi$ depend only on $r = \\|\\vec{\\text{OP}}\\|$. The motion of a mass $m$ located at $P$, due to the field $\\varphi$, is restricted to a plane containing $O$. \n\n(B.1) In the case of a circular orbit, determine the velocity $v_c$ of an object on a circular orbit passing through $P$ in terms of $r$ and $\\frac{d \\varphi}{d r}$. \n\nFig. 1(A) is a picture of the spiral galaxy NGC 6946 in the visible band (from the $0.8 m$ Schulman Telescope at the Mount Lemmon Sky Center in Arizona). The little ellipses in Fig. 1(B) show experimental measurements of $v_c$ for this galaxy. The central region ($r < 1 \\text{kpc}$) is named the bulge. In this region, the mass distribution is roughly homogeneous. The red curve is a prediction for $v_c$ if the system were homogeneous in the bulge and keplerian ($\\varphi(r) = -\\beta / r$ with $\\beta > 0$) outside it, i.e. considering that the total mass of the galaxy is concentrated in the bulge. \n\n[figure1]\nFig. 1: NGC 6946 galaxy: Picture (A) and rotation curve (B). \n\n(B.2) Deduce the mass $M_b$ of the bulge of NGC 6946 from the red rotation curve in Fig. 1(B), in solar mass units. \n\nComparing the keplerian model and the experimental data makes astronomers confident that part of the mass is invisible in the picture. They thus suppose that the galaxy's actual mass density is given by \n$\\rho_{m}(r) = \\frac{C_{m}}{r_{m}^{2} + r^{2}}$ (Equation 1) \nwhere $C_{m} > 0$ and $r_{m} > 0$ are constants. \n\n(B.3) Show that the velocity profile $v_{c,m}(r)$, corresponding to the mass density in Eq. 1, can be written $v_{c,m}(r) = \\sqrt{k_1 - \\frac{k_2 \\cdot \\arctan(\\frac{r}{r_m})}{r}}$. Express $k_1$ and $k_2$ in terms of $C_m$, $r_m$ and $G$. (Hints: $\\int_0^r \\frac{x^2}{a^2 + x^2} dx = r - a \\arctan(r/a)$, and: $\\arctan(x) \\simeq x - x^3 / 3$ for $x \\ll 1$.) Simplify $v_{c,m}(r)$ when $r \\ll r_m$ and when $r \\gg r_m$. Show that if $r \\gg r_m$, the mass $M_m(r)$ embedded in a sphere of radius $r$ with the mass density given by Eq. 1 simplifies and depends only on $C_m$ and $r$. Estimate the mass of the galaxy NGC 6946 actually present in the picture in Fig. 1(A). \n\n[Part D - Tully-Fisher relation and MOND theory] \n\nThe flat external velocity curve of NGC 6946 in Fig. 1 is a common property of spiral galaxies, as can be seen in Fig. 4 (left). Plotting the external constant velocity value $v_{c,\\infty}$ as a function of the measured total mass $M_{\\text{tot}}$ of each galaxy gives an interesting correlation called the Tully-Fischer relation, see Fig. 4 (right). \n\n[figure4]\nFig. 4. Left: Rotation curves for typical spiral galaxies - Right: $\\log_{10}(M_{\\text{tot}})$ as a function of $\\log_{10}(v_{c,\\infty})$ on linear scales. Colored dots correspond to different galaxies and different surveys. The green line is the Tully-Fischer relation which is in very good agreement with the best fit line of the data (in black). \n\n(D.1) Assuming that the radius $R$ of a galaxy doesn't depend on its mass, show that the model of Eq. 1 (part B) gives a relation of the form $M_{\\text{tot}} = \\eta v_{c,\\infty}^\\gamma$ where $\\gamma$ and $\\eta$ should be specified. \nCompare this expression to the Tully-Fischer relation by computing $\\gamma_{TF}$. \n\nIn the extremely low acceleration regime, of the order of $a_0 = 10^{-10} \\mathrm{m} \\cdot \\mathrm{s}^{-2}$, the MOdified Newtonian Dynamics (MOND) theory suggests that one can modify Newton's second law using: $\\vec{F} = m \\mu \\left(\\frac{a}{a_0}\\right) \\vec{a}$ where $a = \\| \\vec{a} \\|$ is the modulus of the acceleration and the $\\mu$ function is defined by: $\\mu(x) = \\frac{x}{1 + x}$. \n\n(D.2) Using data for NGC 6946 in Fig. 1, estimate, within Newton's theory, the modulus of the acceleration $a_m$ of a mass in the outer regions of NGC 6946. \n\n(D.3) Let $m$ be a mass on a circular orbit of radius $r$ with velocity $v_{c,\\infty}$ in the gravity field of a fixed mass $M$. Within the MOND theory, with $a \\ll a_0$, determine the Tully-Fischer exponent $\\gamma_{\\text{MOND}}$. Using data for NGC 6946 and/or Tully-Fischer law, calculate $a_0$ to show that MOND operates in the correct regime.", + "question": "Considering relevant cases, determine $v_c(r)$ for all values of $r$ in the MOND theory in the case of a gravitational field due to a homogeneously distributed mass $M$ with radius $R_b$.", + "marking": [ + [ + "Award 0.1 pt if the answer gives the modified Newton's second law with circular velocity $v_c$ at radius $r$ as $\\mathcal{g}(r) m = -m \\frac{\\frac{v_f^2}{a_0 r}}{1 + \\frac{v_f^2}{a_0 r}} \\frac{v_f^2}{r}$, where $\\mathcal{g}(r)$ is the gravitational field of the homogeneous ball of mass $M$ and with radius $R_b$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the gravitational field for $r > R_b$ as $\\mathcal{g}(r) = -G M / r^2$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the gravitational field for $r \\leq R_b$ as $\\mathcal{g}(r) = -G M r / {R_b}^3$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly derives the bi-quadratic equation in the case $r > R_b$: $v_c^4 - \\frac{GM}{r} v_c^2 - a_0 GM = 0$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly obtains the expression for $v_c$ in the case $r > R_b$ as $v_c(r) = \\sqrt{\\frac{GM}{2r} \\left(1 + \\sqrt{1 + \\frac{4 a_0 r^2}{GM}}\\right)}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly states the behavior in the limit $r \\to \\infty$: $v_c$ is asymptotically constant and $M \\to \\frac{v_{c,\\infty}^4}{a_0 G}$$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly derives the bi-quadratic equation in the case $r < R_b$: $v_c^4 - \\frac{GM}{R_b} (\\frac{r}{R_b})^3 v_c^2 - a_0 GM (\\frac{r}{R_b})^3 = 0$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly obtains the expression for $v_c$ in the case $r < R_b$ as $v_c(r) = \\sqrt{\\frac{GM}{2r} (\\frac{r}{R_b})^3 \\left[1 + \\sqrt{1 + \\frac{4 a_0 r^2}{GM} (\\frac{R_b}{r})^3}\\right]}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly states the behavior in the limit $r \\to 0$: $v_c \\to 0$, consistent with the experimental data. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$v_c(r) = \\sqrt{\\frac{GM}{2r} \\left(1 + \\sqrt{1 + \\frac{4 a_0 r^2}{GM}}\\right)}$ if $r > R_b$.}", + "\\boxed{$v_c(r) = \\sqrt{\\frac{GM}{2r} \\left(\\frac{r}{R_b}\\right)^3 \\left[1 + \\sqrt{1 + \\frac{4 a_0 r^2}{GM} \\left(\\frac{R_b}{r}\\right)^3}\\right]}$ if $r \\leq R_b$.}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 0.45, + 0.45 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_1_b_1.png", + "image_question/IPhO_2025_1_d_1.png" + ] + }, + { + "id": "IPhO_2025_2_A_1", + "context": "[Cox's Timepiece] \n\nIn 1765, British clockmaker James Cox invented a clock whose only source of energy is the fluctuations in atmospheric pressure. Cox's clock used two vessels containing mercury. Changes in atmospheric pressure caused mercury to move between the vessels, and the two vessels to move relative to each other. This movement acted as an energy source for the actual clock. \n\nWe propose an analysis of this device. Throughout, we assume that \n- the Earth's gravitational field $\\vec{g} = -g\\vec{u_{z}}$ is uniform with $g = 9.8 \\mathrm{m} \\cdot \\mathrm{s}^{-2}$ and $\\vec{u_{z}}$ a unit vector; \n- all liquids are incompressible and their density is denoted $\\rho$; \n- no surface tension effects will be considered; \n- the variations of atmospheric pressure with altitude are neglected; \n- the surrounding temperature $T_{\\mathrm{a}}$ is uniform and all transformations are isothermal. \n\n[figure1] \nFig. 1. Artistic view of Cox's clock \n\n[Part A - Pulling on a submerged tube] \n\nWe first consider a bath of water that occupies the semi-infinite space $z \\leq 0$. The air above it is at a pressure $P_{\\mathrm{a}} = P_{0}$. A cylindrical vertical tube of length $H = 1 \\mathrm{m}$, cross-sectional area $S = 10 \\mathrm{cm}^{2}$ and mass $m = 0.5 \\mathrm{kg}$ is dipped into the bath. The bottom end of the tube is open, and the top end of the tube is closed. We denote $h$ the altitude of the top of the tube and $z_{\\ell}$ that of the water inside the tube. The thickness of the tube walls is neglected. \n\n[figure2] \nFig. 2. Sketch of the tube in different configurations. \n\nWe start from the situation where the tube in Fig. 2 contains no gas and its top is at the bath level: in other words, $h = 0$ and $z_{\\ell} = 0$ (case a). The tube is then slowly lifted until its bottom end reaches the bath level. The pulling force exerted on the tube is denoted $\\vec{F} = F \\vec{u_{z}}$.", + "question": "For the configuration shown in Fig. 2 (case b): \n(1) Express the pressure $P_{\\mathrm{w}}$ in the water at the top of the tube. \n(2) Express the force $\\vec{F}$ necessary to maintain the tube at this position. Expressions must be written in terms of $P_{0}, \\rho, m, S, h, g$ and $\\vec{u_{z}}$.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly gives the expression of $P_w$ as a function of $P_a$ or $P_0$: $P_w = P_a - \\rho g h = P_0 - \\rho g h$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly derives the expression of $\\vec{F}$ as $\\vec{F} = [m + \\rho S h] g \\vec{u_{z}}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$P_{\\mathrm{w}} = P_{0} - \\rho g h$}", + "\\boxed{$\\vec{F} = [m + \\rho S h] g \\vec{u_{z}}$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 0.1, + 0.1 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_2_a_1.png", + "image_question/IPhO_2025_2_a_2.png" + ] + }, + { + "id": "IPhO_2025_2_A_2", + "context": "[Cox's Timepiece] \n\nIn 1765, British clockmaker James Cox invented a clock whose only source of energy is the fluctuations in atmospheric pressure. Cox's clock used two vessels containing mercury. Changes in atmospheric pressure caused mercury to move between the vessels, and the two vessels to move relative to each other. This movement acted as an energy source for the actual clock. \n\nWe propose an analysis of this device. Throughout, we assume that \n- the Earth's gravitational field $\\vec{g} = -g\\vec{u_{z}}$ is uniform with $g = 9.8 \\mathrm{m} \\cdot \\mathrm{s}^{-2}$ and $\\vec{u_{z}}$ a unit vector; \n- all liquids are incompressible and their density is denoted $\\rho$; \n- no surface tension effects will be considered; \n- the variations of atmospheric pressure with altitude are neglected; \n- the surrounding temperature $T_{\\mathrm{a}}$ is uniform and all transformations are isothermal. \n\n[figure1] \nFig. 1. Artistic view of Cox's clock \n\n[Part A - Pulling on a submerged tube] \n\nWe first consider a bath of water that occupies the semi-infinite space $z \\leq 0$. The air above it is at a pressure $P_{\\mathrm{a}} = P_{0}$. A cylindrical vertical tube of length $H = 1 \\mathrm{m}$, cross-sectional area $S = 10 \\mathrm{cm}^{2}$ and mass $m = 0.5 \\mathrm{kg}$ is dipped into the bath. The bottom end of the tube is open, and the top end of the tube is closed. We denote $h$ the altitude of the top of the tube and $z_{\\ell}$ that of the water inside the tube. The thickness of the tube walls is neglected. \n\n[figure2] \nFig. 2. Sketch of the tube in different configurations. \n\nWe start from the situation where the tube in Fig. 2 contains no gas and its top is at the bath level: in other words, $h = 0$ and $z_{\\ell} = 0$ (case a). The tube is then slowly lifted until its bottom end reaches the bath level. The pulling force exerted on the tube is denoted $\\vec{F} = F \\vec{u_{z}}$. \n\n(A.1) For the configuration shown in Fig. 2 (case b): express the pressure $P_{\\mathrm{w}}$ in the water at the top of the tube. Also express the force $\\vec{F}$ necessary to maintain the tube at this position. Expressions must be written in terms of $P_{0}, \\rho, m, S, h, g$ and $\\vec{u_{z}}$. \n\nThree experiments are performed. In each, the tube is lifted from the initial state shown in Fig. 2(a) under the conditions specified in Table 1. \n\n|Experiment|Liquid|$T_{\\mathrm{a}} ({}^{\\circ}\\mathrm{C})$|$\\rho (\\mathrm{kg} \\cdot \\mathrm{m}^{-3})$|$P_{\\text{sat}} (\\mathrm{Pa})$| \n-|-|-|-|- \n1|Water|20|$1.00 \\times 10^{3}$|$2.34 \\times 10^{3}$ \n2|Water|80|$0.97 \\times 10^{3}$|$47.4 \\times 10^{3}$ \n3|Water|99|$0.96 \\times 10^{3}$|$99.8 \\times 10^{3}$ \nTable 1. Experimental conditions and numerical values of physical quantities for each experiment \n($P_{\\text{sat}}$ designates the saturated vapour pressure of the pure fluid) \n\nIn each case, we study the evolution of the force $F$ that must be applied in order to maintain the tube in equilibrium at an altitude $h$, the external pressure being fixed at $P_{\\mathrm{a}} = P_{0} = 1.000 \\times 10^{5} \\mathrm{Pa}$. Two different behaviours are possible.\n\n[figure3]", + "question": "For each experiment, complete the table below to indicate the expected behaviour and the numerical values for $F_{\\max}$ and for $h^{\\star}$ (when pertinent), where $F_{\\max}$ and $h^{\\star}$ are defined in the figures illustrating the two behaviours. \n\n|Experiment|Behaviour (A or B?)|$h^{\\star}$ (cm)|$F_{\\max} (N)$| \n-|-|-|- \n1|(1)|(2)|(3) \n2|(4)|(5)|(6) \n3|(7)|(8)|(9) \n\nFor each field (1)–(9), write your answers in the final answer in the same order. If a field does not need to be filled, write 'None'.", + "marking": [ + [ + "Award 0.2 pt if the answer correctly describes the behavior for all three experiments: Behavior A for Experiment 1, Behavior A for Experiment 2, and Behavior B for Experiment 3. If any answer is incorrect, award 0 pt.", + "Award 0.1 pt if the answer gives the numerical value of $F_{\\max}$ for Experiment 1 within the range [14.6, 15] (N). Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives the numerical value of $F_{\\max}$ for Experiment 2 within the range [14, 14.5] (N). Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the numerical value of $h^{\\star}$ for Experiment 3 within the range [2, 2.2] (cm). Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the numerical value of $F_{\\max}$ for Experiment 3 within the range [5, 5.2] (N). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{A}", + "\\boxed{None}", + "\\boxed{[14.6, 15]}", + "\\boxed{A}", + "\\boxed{None}", + "\\boxed{[14, 14.5]}", + "\\boxed{B}", + "\\boxed{[2, 2.2]}", + "\\boxed{[5, 5.2]}" + ], + "answer_type": [ + "Multiple Choice", + "Numerical Value", + "Numerical Value", + "Multiple Choice", + "Numerical Value", + "Numerical Value", + "Multiple Choice", + "Numerical Value", + "Numerical Value" + ], + "unit": [ + null, + "cm", + "N", + null, + "cm", + "N", + null, + "cm", + "N" + ], + "points": [ + 0.06, + 0.0, + 0.1, + 0.07, + 0.0, + 0.1, + 0.07, + 0.2, + 0.2 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_2_a_1.png", + "image_question/IPhO_2025_2_a_2.png", + "image_question/IPhO_2025_2_a_3.png" + ] + }, + { + "id": "IPhO_2025_2_A_3", + "context": "[Cox's Timepiece] \n\nIn 1765, British clockmaker James Cox invented a clock whose only source of energy is the fluctuations in atmospheric pressure. Cox's clock used two vessels containing mercury. Changes in atmospheric pressure caused mercury to move between the vessels, and the two vessels to move relative to each other. This movement acted as an energy source for the actual clock. \n\nWe propose an analysis of this device. Throughout, we assume that \n- the Earth's gravitational field $\\vec{g} = -g\\vec{u_{z}}$ is uniform with $g = 9.8 \\mathrm{m} \\cdot \\mathrm{s}^{-2}$ and $\\vec{u_{z}}$ a unit vector; \n- all liquids are incompressible and their density is denoted $\\rho$; \n- no surface tension effects will be considered; \n- the variations of atmospheric pressure with altitude are neglected; \n- the surrounding temperature $T_{\\mathrm{a}}$ is uniform and all transformations are isothermal. \n\n[figure1] \nFig. 1. Artistic view of Cox's clock \n\n[Part A - Pulling on a submerged tube] \n\nWe first consider a bath of water that occupies the semi-infinite space $z \\leq 0$. The air above it is at a pressure $P_{\\mathrm{a}} = P_{0}$. A cylindrical vertical tube of length $H = 1 \\mathrm{m}$, cross-sectional area $S = 10 \\mathrm{cm}^{2}$ and mass $m = 0.5 \\mathrm{kg}$ is dipped into the bath. The bottom end of the tube is open, and the top end of the tube is closed. We denote $h$ the altitude of the top of the tube and $z_{\\ell}$ that of the water inside the tube. The thickness of the tube walls is neglected. \n\n[figure2] \nFig. 2. Sketch of the tube in different configurations. \n\nWe start from the situation where the tube in Fig. 2 contains no gas and its top is at the bath level: in other words, $h = 0$ and $z_{\\ell} = 0$ (case a). The tube is then slowly lifted until its bottom end reaches the bath level. The pulling force exerted on the tube is denoted $\\vec{F} = F \\vec{u_{z}}$. \n\n(A.1) For the configuration shown in Fig. 2 (case b): express the pressure $P_{\\mathrm{w}}$ in the water at the top of the tube. Also express the force $\\vec{F}$ necessary to maintain the tube at this position. Expressions must be written in terms of $P_{0}, \\rho, m, S, h, g$ and $\\vec{u_{z}}$. \n\nThree experiments are performed. In each, the tube is lifted from the initial state shown in Fig. 2(a) under the conditions specified in Table 1. \n\n|Experiment|Liquid|$T_{\\mathrm{a}} ({}^{\\circ}\\mathrm{C})$|$\\rho (\\mathrm{kg} \\cdot \\mathrm{m}^{-3})$|$P_{\\text{sat}} (\\mathrm{Pa})$| \n-|-|-|-|- \n1|Water|20|$1.00 \\times 10^{3}$|$2.34 \\times 10^{3}$ \n2|Water|80|$0.97 \\times 10^{3}$|$47.4 \\times 10^{3}$ \n3|Water|99|$0.96 \\times 10^{3}$|$99.8 \\times 10^{3}$ \nTable 1. Experimental conditions and numerical values of physical quantities for each experiment \n($P_{\\text{sat}}$ designates the saturated vapour pressure of the pure fluid) \n\nIn each case, we study the evolution of the force $F$ that must be applied in order to maintain the tube in equilibrium at an altitude $h$, the external pressure being fixed at $P_{\\mathrm{a}} = P_{0} = 1.000 \\times 10^{5} \\mathrm{Pa}$. Two different behaviours are possible.\n\n[figure3] \n\n(A.2) For each experiment, complete the table below to indicate the expected behaviour and the numerical values for $F_{\\max}$ and for $h^{\\star}$ (when pertinent), where $F_{\\max}$ and $h^{\\star}$ are defined in the figures illustrating the two behaviours. \n\n|Experiment|Behaviour (A or B?)|$h^{\\star}$ (cm)|$F_{\\max} (N)$| \n-|-|-|- \n1|(1)|(2)|(3) \n2|(4)|(5)|(6) \n3|(7)|(8)|(9) \n\nWhen we replace the water with liquid mercury (whose properties are given below), behaviour B is observed. \n\n|Liquid|$T_{\\mathrm{a}} ({}^{\\circ}\\mathrm{C})$|$\\rho$ ($\\mathrm{kg} \\cdot \\mathrm{m}^{-3}$)|$P_{\\text{sat}}$ ($\\mathrm{Pa}$)| \n-|-|-|- \nMercury|20|$13.5 \\times 10^{3}$|0.163", + "question": "(1) Express the relative error, denoted $\\varepsilon$, committed when we evaluate the maximal force $F_{\\max}$ neglecting $P_{\\text{sat}}$ compared to $P_{0}$. \n(2) Give the numerical value of $\\varepsilon$.", + "marking": [ + [ + "Award 0.2 pt if the answer correctly derives the expression for $\\varepsilon$ as $\\varepsilon = \\frac{P_{\\text{sat}}}{P_{0} + mg/S}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly obtains the numerical value of $\\varepsilon$ within the range [1 \\times 10^{-6}, 2 \\times 10^{-6}]. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\varepsilon = \\frac{P_{\\text{sat}}}{P_{0} + mg/S}$}", + "\\boxed{[1 \\times 10^{-6}, 2 \\times 10^{-6}]}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null, + null + ], + "points": [ + 0.2, + 0.1 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_2_a_1.png", + "image_question/IPhO_2025_2_a_2.png", + "image_question/IPhO_2025_2_a_3.png" + ] + }, + { + "id": "IPhO_2025_2_B_1", + "context": "[Cox's Timepiece] \n\nIn 1765, British clockmaker James Cox invented a clock whose only source of energy is the fluctuations in atmospheric pressure. Cox's clock used two vessels containing mercury. Changes in atmospheric pressure caused mercury to move between the vessels, and the two vessels to move relative to each other. This movement acted as an energy source for the actual clock. \n\nWe propose an analysis of this device. Throughout, we assume that \n- the Earth's gravitational field $\\vec{g} = -g\\vec{u_{z}}$ is uniform with $g = 9.8 \\mathrm{m} \\cdot \\mathrm{s}^{-2}$ and $\\vec{u_{z}}$ a unit vector; \n- all liquids are incompressible and their density is denoted $\\rho$; \n- no surface tension effects will be considered; \n- the variations of atmospheric pressure with altitude are neglected; \n- the surrounding temperature $T_{\\mathrm{a}}$ is uniform and all transformations are isothermal. \n\n[figure1] \nFig. 1. Artistic view of Cox's clock \n\n[Part B - Two-part barometric tube] \n\nFrom now on, we work with mercury (density $\\rho = 13.5 \\times 10^{3} \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$) at the ambient temperature $T_{\\mathrm{a}} = 20^{\\circ}\\mathrm{C}$ and we take $P_{\\text{sat}} = 0$. \n\nLet us consider a tube with a reservoir on top, modeled as two superposed cylinders of different dimensions, as shown in Fig. 3. \n- the bottom part (still called the tube) has cross-sectional area $S_{\\mathrm{t}}$ and height $H_{\\mathrm{t}} = 80 \\mathrm{cm}$; \n- the top part (called the bulb) has cross-sectional area $S_{\\mathrm{b}} > S_{\\mathrm{t}}$ and height $H_{\\mathrm{b}} = 20 \\mathrm{cm}$. \nThis two-part tube is dipped into a semi-infinite liquid bath. \n\n[figure3]\nFig. 3. Sketch of the two-part barometric tube \n\nAs in Part A, the system is prepared such that the tube contains no air. We identify the vertical position of the tube by the altitude $h_{\\mathrm{t}}$ of the junction between the tube and the bulb. The height of the column of mercury is again denoted $z_{\\ell}$. The force $\\vec{F}$ that must be exerted to maintain the tube in equilibrium in the configuration shown in Fig. 3 can now be written as \n$\\vec{F} = \\left(m_{\\mathrm{tb}} + m_{\\mathrm{add}} \\right) g \\vec{u_{z}}$ (Equation 1) \nwhere $m_{\\mathrm{tb}}$ is the total mass of the two-part tube (when empty of mercury).", + "question": "Describe the area corresponding to the volume of liquid mercury that is responsible for the term $m_{\\text{add}}$ appearing in equation (1). Write down the expression of $m_{\\text{add}}$.", + "marking": [ + [ + "Award 0.2 pt if the answer correctly states that the mass $m_{\\text{add}}$ corresponds to the liquid mass in the two-part tube which is above the outside surface of the liquid bath. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly states the expression of $m_{\\text{add}}$ as $m_{\\text{add}} = \\rho(S_t h_t + S_b(z_{\\ell} - h_t))$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$m_{\\text{add}} = \\rho(S_t h_t + S_b(z_{\\ell} - h_t))$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.3 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_2_a_1.png", + "image_question/IPhO_2025_2_b_1.png" + ] + }, + { + "id": "IPhO_2025_2_B_2", + "context": "[Cox's Timepiece] \n\nIn 1765, British clockmaker James Cox invented a clock whose only source of energy is the fluctuations in atmospheric pressure. Cox's clock used two vessels containing mercury. Changes in atmospheric pressure caused mercury to move between the vessels, and the two vessels to move relative to each other. This movement acted as an energy source for the actual clock. \n\nWe propose an analysis of this device. Throughout, we assume that \n- the Earth's gravitational field $\\vec{g} = -g\\vec{u_{z}}$ is uniform with $g = 9.8 \\mathrm{m} \\cdot \\mathrm{s}^{-2}$ and $\\vec{u_{z}}$ a unit vector; \n- all liquids are incompressible and their density is denoted $\\rho$; \n- no surface tension effects will be considered; \n- the variations of atmospheric pressure with altitude are neglected; \n- the surrounding temperature $T_{\\mathrm{a}}$ is uniform and all transformations are isothermal. \n\n[figure1] \nFig. 1. Artistic view of Cox's clock \n\n[Part B - Two-part barometric tube] \n\nFrom now on, we work with mercury (density $\\rho = 13.5 \\times 10^{3} \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$) at the ambient temperature $T_{\\mathrm{a}} = 20^{\\circ}\\mathrm{C}$ and we take $P_{\\text{sat}} = 0$. \n\nLet us consider a tube with a reservoir on top, modeled as two superposed cylinders of different dimensions, as shown in Fig. 3. \n- the bottom part (still called the tube) has cross-sectional area $S_{\\mathrm{t}}$ and height $H_{\\mathrm{t}} = 80 \\mathrm{cm}$; \n- the top part (called the bulb) has cross-sectional area $S_{\\mathrm{b}} > S_{\\mathrm{t}}$ and height $H_{\\mathrm{b}} = 20 \\mathrm{cm}$. \nThis two-part tube is dipped into a semi-infinite liquid bath. \n\n[figure3]\nFig. 3. Sketch of the two-part barometric tube \n\nAs in Part A, the system is prepared such that the tube contains no air. We identify the vertical position of the tube by the altitude $h_{\\mathrm{t}}$ of the junction between the tube and the bulb. The height of the column of mercury is again denoted $z_{\\ell}$. The force $\\vec{F}$ that must be exerted to maintain the tube in equilibrium in the configuration shown in Fig. 3 can now be written as \n$\\vec{F} = \\left(m_{\\mathrm{tb}} + m_{\\mathrm{add}} \\right) g \\vec{u_{z}}$ (Equation 1) \nwhere $m_{\\mathrm{tb}}$ is the total mass of the two-part tube (when empty of mercury). \n\n(B.1) Describe the area corresponding to the volume of liquid mercury that is responsible for the term $m_{\\text{add}}$ appearing in equation (1). \n\nThe mass $m_{\\text{add}}$ depends both on the height $h_{\\mathrm{t}}$ and the atmospheric pressure $P_{\\mathrm{a}}$. For the next question, assume that the atmospheric pressure is fixed at $P_{\\mathrm{a}} = P_{0} = 1.000 \\times 10^{5} \\mathrm{Pa}$. Starting from the situation where the system is completely submerged, the tube is slowly lifted until its base is flush with the liquid bath.", + "question": "Sketch the evolution of the mass $m_{\\text{add}}$ as a function of $h_{\\mathrm{t}}$ for $h_{\\mathrm{t}} \\in [-H_{\\mathrm{b}}, H_{\\mathrm{t}}]$. (1) Write down, in order, the expressions for the slopes of the different segments, (2) as well as the $h_{\\mathrm{t}}$ analytical value of any angular points, in terms of $P_{0}, \\rho, g, S_{\\mathrm{b}}, S_{\\mathrm{t}}$, $H_{\\mathrm{b}}$ and $H_{\\mathrm{t}}$.", + "marking": [ + [ + "Award 0.2 pt if the answer correctly states the plotted function of $h_{\\mathrm{t}}$ has 4 straight pieces/segments. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly states (1) the first and second pieces of the plotted function of $h_{\\mathrm{t}}$ have positive slopes and (2) the slope of the second piece is less than that of the first piece. If any one statement is incorrect, award 0 pt.", + "Award 0.2 pt if the answer correctly states the third piece of the plotted function has a negative slope. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly states the forth piece of the plotted function has a null slope. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the first two slopes: slope = $\\rho S_{\\mathrm{b}}$ for Segment 1, and slope = $\\rho S_{\\mathrm{t}}$ for Segment 2. If any one slope is incorrect, award 0 pt.", + "Award 0.2 pt if the answer correctly gives the negative slope: slope = $-\\rho (S_{\\mathrm{b}} - S_{\\mathrm{t}})$ for Segment 3. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly provides the analytical values for all three intermediate angular points: Angular point 1: 0, Angular point 2: $z_{\\ell}^{\\star} - H_{\\mathrm{b}}$, and Angular point 3: $z_{\\ell}^{\\star}$. Partial points: award 0.2 points for two correct answers, 0.1 point for one correct answer, and 0 points for no correct answers." + ] + ], + "answer": [ + "\\boxed{Segment 1: slope = $\\rho S_{\\mathrm{b}}$}", + "\\boxed{Segment 2: slope = $\\rho S_{\\mathrm{t}}$}", + "\\boxed{Segment 3: slope = $-\\rho (S_{\\mathrm{b}} - S_{\\mathrm{t}})$}", + "\\boxed{Segment 4: slope = 0}", + "\\boxed{Angular point 1: 0}", + "\\boxed{Angular point 2: $z_{\\ell}^{\\star} - H_{\\mathrm{b}}$}", + "\\boxed{Angular point 3: $z_{\\ell}^{\\star}$}" + ], + "answer_type": [ + "Expression", + "Expression", + "Expression", + "Numerical Value", + "Numerical Value", + "Expression", + "Expression" + ], + "unit": [ + null, + null, + null, + null, + "cm", + null, + null + ], + "points": [ + 0.2, + 0.2, + 0.2, + 0.2, + 0.2, + 0.2, + 0.2 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_2_a_1.png", + "image_question/IPhO_2025_2_b_1.png" + ] + }, + { + "id": "IPhO_2025_2_B_3", + "context": "[Cox's Timepiece] \n\nIn 1765, British clockmaker James Cox invented a clock whose only source of energy is the fluctuations in atmospheric pressure. Cox's clock used two vessels containing mercury. Changes in atmospheric pressure caused mercury to move between the vessels, and the two vessels to move relative to each other. This movement acted as an energy source for the actual clock. \n\nWe propose an analysis of this device. Throughout, we assume that \n- the Earth's gravitational field $\\vec{g} = -g\\vec{u_{z}}$ is uniform with $g = 9.8 \\mathrm{m} \\cdot \\mathrm{s}^{-2}$ and $\\vec{u_{z}}$ a unit vector; \n- all liquids are incompressible and their density is denoted $\\rho$; \n- no surface tension effects will be considered; \n- the variations of atmospheric pressure with altitude are neglected; \n- the surrounding temperature $T_{\\mathrm{a}}$ is uniform and all transformations are isothermal. \n\n[figure1] \nFig. 1. Artistic view of Cox's clock \n\n[Part B - Two-part barometric tube] \n\nFrom now on, we work with mercury (density $\\rho = 13.5 \\times 10^{3} \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$) at the ambient temperature $T_{\\mathrm{a}} = 20^{\\circ}\\mathrm{C}$ and we take $P_{\\text{sat}} = 0$. \n\nLet us consider a tube with a reservoir on top, modeled as two superposed cylinders of different dimensions, as shown in Fig. 3. \n- the bottom part (still called the tube) has cross-sectional area $S_{\\mathrm{t}}$ and height $H_{\\mathrm{t}} = 80 \\mathrm{cm}$; \n- the top part (called the bulb) has cross-sectional area $S_{\\mathrm{b}} > S_{\\mathrm{t}}$ and height $H_{\\mathrm{b}} = 20 \\mathrm{cm}$. \nThis two-part tube is dipped into a semi-infinite liquid bath. \n\n[figure3]\nFig. 3. Sketch of the two-part barometric tube \n\nAs in Part A, the system is prepared such that the tube contains no air. We identify the vertical position of the tube by the altitude $h_{\\mathrm{t}}$ of the junction between the tube and the bulb. The height of the column of mercury is again denoted $z_{\\ell}$. The force $\\vec{F}$ that must be exerted to maintain the tube in equilibrium in the configuration shown in Fig. 3 can now be written as \n$\\vec{F} = \\left(m_{\\mathrm{tb}} + m_{\\mathrm{add}} \\right) g \\vec{u_{z}}$ (Equation 1) \nwhere $m_{\\mathrm{tb}}$ is the total mass of the two-part tube (when empty of mercury). \n\n(B.1) Describe the area corresponding to the volume of liquid mercury that is responsible for the term $m_{\\text{add}}$ appearing in equation (1). \n\nThe mass $m_{\\text{add}}$ depends both on the height $h_{\\mathrm{t}}$ and the atmospheric pressure $P_{\\mathrm{a}}$. For the question (B.2), assume that the atmospheric pressure is fixed at $P_{\\mathrm{a}} = P_{0} = 1.000 \\times 10^{5} \\mathrm{Pa}$. Starting from the situation where the system is completely submerged, the tube is slowly lifted until its base is flush with the liquid bath. \n\n(B.2) Sketch the evolution of the mass $m_{\\text{add}}$ as a function of $h_{\\mathrm{t}}$ for $h_{\\mathrm{t}} \\in [-H_{\\mathrm{b}}, H_{\\mathrm{t}}]$. Write down, in order, the expressions for the slopes of the different segments, as well as the $h_{\\mathrm{t}}$ analytical value of any angular points, in terms of $P_{0}, \\rho, g, S_{\\mathrm{b}}, S_{\\mathrm{t}}$, $H_{\\mathrm{b}}$ and $H_{\\mathrm{t}}$. \n\nAs the system is lifted while $P_{\\mathrm{a}} = P_{0} = 10^{5} \\mathrm{Pa}$, we stop when the free surface of the liquid is in the middle of the bulb. The value of $h_{\\mathrm{t}}$ is fixed and then we observe variations in the mass $m_{\\text{add}}$ due to variations in the atmospheric pressure described by \n$P_{\\mathrm{a}}(t) = P_{0} + P_{1}(t)$ (Equation 2) \nwhere $P_{0}$ designates the average value and $P_{1}$ is a perturbative term. We model $P_{1}$ by a periodic triangular function of amplitude $A = 5 \\times 10^{2} \\mathrm{Pa}$ and period $\\tau_{1}$ of 1 week. \n\n[figure4]\nFig. 4. Simplified model of the perturbative term $P_{1}(t)$.", + "question": "Given that $S_{\\mathrm{t}} = 5 \\mathrm{cm}^{2}$ and $S_{\\mathrm{b}} = 200 \\mathrm{cm}^{2}$, (1) express the amplitude $\\Delta m_{\\text{add}}$ of the variations of the mass $m_{\\text{add}}$ over time, (2) then give its numerical value in $\\mathrm{kg}$. Assume that the liquid surface always stays in the bulb.", + "marking": [ + [ + "Award 0.2 pt if the answer correctly provides the expression of $\\Delta m_{\\text{add}}$ as $\\Delta m_{\\text{add}} = \\frac{S_{\\mathrm{b}} A}{g}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly obtains the numerical value of $\\Delta m_{\\text{add}}$ within the range [1kg, 1.1kg]. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\Delta m_{\\text{add}} = \\frac{S_{\\mathrm{b}} A}{g}$}", + "\\boxed{[1, 1.1]}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null, + "kg" + ], + "points": [ + 0.2, + 0.1 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_2_a_1.png", + "image_question/IPhO_2025_2_b_1.png", + "image_question/IPhO_2025_2_b_2.png" + ] + }, + { + "id": "IPhO_2025_2_C_1", + "context": "[Cox's Timepiece] \n\nIn 1765, British clockmaker James Cox invented a clock whose only source of energy is the fluctuations in atmospheric pressure. Cox's clock used two vessels containing mercury. Changes in atmospheric pressure caused mercury to move between the vessels, and the two vessels to move relative to each other. This movement acted as an energy source for the actual clock. \n\nWe propose an analysis of this device. Throughout, we assume that \n- the Earth's gravitational field $\\vec{g} = -g\\vec{u_{z}}$ is uniform with $g = 9.8 \\mathrm{m} \\cdot \\mathrm{s}^{-2}$ and $\\vec{u_{z}}$ a unit vector; \n- all liquids are incompressible and their density is denoted $\\rho$; \n- no surface tension effects will be considered; \n- the variations of atmospheric pressure with altitude are neglected; \n- the surrounding temperature $T_{\\mathrm{a}}$ is uniform and all transformations are isothermal. \n\n[figure1] \nFig. 1. Artistic view of Cox's clock \n\n[Part B - Two-part barometric tube] \n\n[figure4]\nFig. 4. Simplified model of the perturbative term $P_{1}(t)$ \n\n[Part C - Cox's timepiece] \n\nThe real mechanism developed by Cox is complex (Fig. 5). We study a simplified version, depicted in Fig. 6, and described below: \n- a cylindrical bottom cistern containing a mercury bath; \n- a two-part barometric tube identical to that studied in part B, which is still completely emptied of any air, is dipped into the bath; \n- the cistern and the two-part tube are each suspended by a cable. Both cables (assumed to be inextensible and of negligible mass) pass through a system of ideal pullies and finish attached to either side of the same mass $M$, which can slide on a horizontal surface; \n- the total volume of liquid mercury contained in the system is $V_{\\ell} = 5 \\mathrm{L}$. \n\nThe height, cross-section and masses of each part are given in Table 2. The position of mass $M$ is referenced by the coordinate $x$ of its center of mass. We consider solid friction between the horizontal support and the mass $M$, without distinction between static and dynamic coefficients; the magnitude of this force when sliding occurs is denoted $F_{\\mathrm{s}}$. \n\nTwo stops limit the displacement of the mass $M$ such that $-X \\leq x \\leq X$ (with $X > 0$). Assume that the value of $X$ guarantees that the bottom of the two-part tube never touches the bottom of the cistern nor comes out of the liquid bath; and the altitude $z_{\\ell}$ of the mercury column is always in the upper bulb. \n\n[figure5]\nFig. 5. Real Cox's timepiece (without mercury) \n\n[figure6]\nFig. 6. Sketch of the system modeling the timepiece \n\n|Reference|Name|Height|Cross section area|Empty mass|\n|-|-|-|-|-|\n|1|cistern|$H_{\\mathrm{c}} = 30 \\mathrm{cm}$|$S_{\\mathrm{c}} = 210 \\mathrm{cm}^{2}$|$m_{\\mathrm{c}}$|\n|2|tubular part of the barometric tube|$H_{\\mathrm{t}} = 80 \\mathrm{cm}$|$S_{\\mathrm{t}} = 5 \\mathrm{cm}^{2}$|rowspan=\"2\" total mass of the barometric tube : $m_{\\mathrm{tb}}$|\n|$2^{\\prime}$|bulb of the barometric tube|$H_{\\mathrm{b}} = 20 \\mathrm{cm}$|$S_{\\mathrm{b}} = 200 \\mathrm{cm}^{2}$| | \nTable 2. Dimensions and notations for the model system \n\nThe system evolves in contact with the atmosphere, whose pressure fluctuates as in Fig. 4 (still with amplitude $A = 5 \\times 10^{2} \\mathrm{Pa}$ and period $\\tau_{1} = 1$ week). At the start $t = 0$, the mass $M$ is at rest at $x = 0$ and the tensions exerted by the two cables on either side of the mass $M$ are in balance while $P_{1}(0) = 0$. We define \n$\\xi = \\frac{S_{\\mathrm{b}} + S_{\\mathrm{c}} - S_{\\mathrm{t}}}{S_{\\mathrm{b}} S_{\\mathrm{c}}} \\frac{F_{\\mathrm{s}}}{A} \\simeq \\frac{S_{\\mathrm{b}} + S_{\\mathrm{c}}}{S_{\\mathrm{b}} S_{\\mathrm{c}}} \\frac{F_{\\mathrm{s}}}{A}$ (Equation 3) \nwhere the last expression uses that $S_{\\mathrm{t}} \\ll S_{\\mathrm{b}}, S_{\\mathrm{c}}$ (which we will assume is valid until the end of the problem).", + "question": "Determine the threshold $\\xi^{\\star}$ such that $M$ remains indefinitely at rest when $\\xi > \\xi^{\\star}$.", + "marking": [ + [ + "Award 0.1 pt if the answer introduces geometric parameters to locate the positions of the fluid surfaces in both vessels (barometric tube and cistern). Otherwise, award 0 pt.", + "Award 0.1 pt if the answer provides an expression for mass or volume variation of fluid in at least one vessel in terms of those geometric parameters. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly states the conservation law of the total mass or volume: $S_b \\delta_b = -[S_c - S_t] \\delta_c$ or simplified form. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer provides the hydrostatic relationship: $\\delta_b - \\delta_c = \\frac{P_1}{\\rho g}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer derives the expression of the friction force at equilibrium: $\\vec{R}_t = 2 m_{1,c} g \\vec{u}_x$ or equivalent. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer applies Coulomb's law of friction: $\\frac{2 S_b S_c}{S_b + S_c - S_t} A < F_s$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly concludes with $\\xi^{\\star} = 2$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{2}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_2_a_1.png", + "image_question/IPhO_2025_2_b_2.png", + "image_question/IPhO_2025_2_c_1.png", + "image_question/IPhO_2025_2_c_2.png" + ] + }, + { + "id": "IPhO_2025_2_C_2", + "context": "[Cox's Timepiece] \n\nIn 1765, British clockmaker James Cox invented a clock whose only source of energy is the fluctuations in atmospheric pressure. Cox's clock used two vessels containing mercury. Changes in atmospheric pressure caused mercury to move between the vessels, and the two vessels to move relative to each other. This movement acted as an energy source for the actual clock. \n\nWe propose an analysis of this device. Throughout, we assume that \n- the Earth's gravitational field $\\vec{g} = -g\\vec{u_{z}}$ is uniform with $g = 9.8 \\mathrm{m} \\cdot \\mathrm{s}^{-2}$ and $\\vec{u_{z}}$ a unit vector; \n- all liquids are incompressible and their density is denoted $\\rho$; \n- no surface tension effects will be considered; \n- the variations of atmospheric pressure with altitude are neglected; \n- the surrounding temperature $T_{\\mathrm{a}}$ is uniform and all transformations are isothermal. \n\n[figure1] \nFig. 1. Artistic view of Cox's clock \n\n[Part B - Two-part barometric tube] \n\n[figure4]\nFig. 4. Simplified model of the perturbative term $P_{1}(t)$ \n\n[Part C - Cox's timepiece] \n\nThe real mechanism developed by Cox is complex (Fig. 5). We study a simplified version, depicted in Fig. 6, and described below: \n- a cylindrical bottom cistern containing a mercury bath; \n- a two-part barometric tube identical to that studied in part B, which is still completely emptied of any air, is dipped into the bath; \n- the cistern and the two-part tube are each suspended by a cable. Both cables (assumed to be inextensible and of negligible mass) pass through a system of ideal pullies and finish attached to either side of the same mass $M$, which can slide on a horizontal surface; \n- the total volume of liquid mercury contained in the system is $V_{\\ell} = 5 \\mathrm{L}$. \n\nThe height, cross-section and masses of each part are given in Table 2. The position of mass $M$ is referenced by the coordinate $x$ of its center of mass. We consider solid friction between the horizontal support and the mass $M$, without distinction between static and dynamic coefficients; the magnitude of this force when sliding occurs is denoted $F_{\\mathrm{s}}$. \n\nTwo stops limit the displacement of the mass $M$ such that $-X \\leq x \\leq X$ (with $X > 0$). Assume that the value of $X$ guarantees that the bottom of the two-part tube never touches the bottom of the cistern nor comes out of the liquid bath; and the altitude $z_{\\ell}$ of the mercury column is always in the upper bulb. \n\n[figure5]\nFig. 5. Real Cox's timepiece (without mercury) \n\n[figure6]\nFig. 6. Sketch of the system modeling the timepiece \n\n|Reference|Name|Height|Cross section area|Empty mass|\n|-|-|-|-|-|\n|1|cistern|$H_{\\mathrm{c}} = 30 \\mathrm{cm}$|$S_{\\mathrm{c}} = 210 \\mathrm{cm}^{2}$|$m_{\\mathrm{c}}$|\n|2|tubular part of the barometric tube|$H_{\\mathrm{t}} = 80 \\mathrm{cm}$|$S_{\\mathrm{t}} = 5 \\mathrm{cm}^{2}$|rowspan=\"2\" total mass of the barometric tube : $m_{\\mathrm{tb}}$|\n|$2^{\\prime}$|bulb of the barometric tube|$H_{\\mathrm{b}} = 20 \\mathrm{cm}$|$S_{\\mathrm{b}} = 200 \\mathrm{cm}^{2}$| | \nTable 2. Dimensions and notations for the model system \n\nThe system evolves in contact with the atmosphere, whose pressure fluctuates as in Fig. 4 (still with amplitude $A = 5 \\times 10^{2} \\mathrm{Pa}$ and period $\\tau_{1} = 1$ week). At the start $t = 0$, the mass $M$ is at rest at $x = 0$ and the tensions exerted by the two cables on either side of the mass $M$ are in balance while $P_{1}(0) = 0$. We define \n$\\xi = \\frac{S_{\\mathrm{b}} + S_{\\mathrm{c}} - S_{\\mathrm{t}}}{S_{\\mathrm{b}} S_{\\mathrm{c}}} \\frac{F_{\\mathrm{s}}}{A} \\simeq \\frac{S_{\\mathrm{b}} + S_{\\mathrm{c}}}{S_{\\mathrm{b}} S_{\\mathrm{c}}} \\frac{F_{\\mathrm{s}}}{A}$ (Equation 3) \nwhere the last expression uses that $S_{\\mathrm{t}} \\ll S_{\\mathrm{b}}, S_{\\mathrm{c}}$ (which we will assume is valid until the end of the problem). \n\n(C.1) Determine the threshold $\\xi^{\\star}$ such that $M$ remains indefinitely at rest when $\\xi > \\xi^{\\star}$. \n\nFor the next question only, suppose that the mass $M$ is temporarily blocked at $x = X$.", + "question": "Give an expression for the total tension force $\\vec{T} = T \\vec{u_{x}}$ acting on the mass $M$ due to the tension in two cables at this position, when $P_{1} = 0$, in terms of $\\rho, g, X$ and pertinent cross-sections.", + "marking": [ + [ + "Award 0.1 pt if the answer introduces geometric parameters ($\\delta_b$, $\\delta_c$) to locate fluid surfaces in both the barometric tube and cistern. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer provides expressions of mass or volume variations of fluid in one of the vessels in terms of $X$ and those geometric parameters (e.g., $m_{1,c} = \\rho S_c (\\delta_c - X)$), with or without approximation $S_t \\ll S_b, S_c$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer states the conservation law of the total mass or volume: $S_b \\delta_b + (S_c - S_t) \\delta_c = (S_c - S_b + S_t) X$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer provides the expression of barometric difference of heights between the two surfaces: $\\delta_b = \\delta_c = \\frac{S_c - S_b + S_t}{S_b + S_c - S_t} X$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer derives the total tension force: $\\vec{T} = \\frac{4 S_c(S_b-S_t)}{S_b + S_c - S_t} \\rho g X \\vec{u}_x$ (approximated form $\\frac{4 S_b S_c}{S_b + S_c}\\rho g X \\vec{u}_x$ also accepted). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\vec{T} = \\frac{4 S_{\\mathrm{c}} (S_{\\mathrm{b}} - S_{\\mathrm{t}}) }{S_{\\mathrm{b}} + S_{\\mathrm{c}} - S_{\\mathrm{t}}} \\rho g X \\vec{u_{x}} \\simeq \\frac{4 S_{\\mathrm{b}} S_{\\mathrm{c}}}{S_{\\mathrm{b}} + S_{\\mathrm{c}}} \\rho g X \\vec{u_{x}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_2_a_1.png", + "image_question/IPhO_2025_2_b_2.png", + "image_question/IPhO_2025_2_c_1.png", + "image_question/IPhO_2025_2_c_2.png" + ] + }, + { + "id": "IPhO_2025_2_C_3", + "context": "[Cox's Timepiece] \n\nIn 1765, British clockmaker James Cox invented a clock whose only source of energy is the fluctuations in atmospheric pressure. Cox's clock used two vessels containing mercury. Changes in atmospheric pressure caused mercury to move between the vessels, and the two vessels to move relative to each other. This movement acted as an energy source for the actual clock. \n\nWe propose an analysis of this device. Throughout, we assume that \n- the Earth's gravitational field $\\vec{g} = -g\\vec{u_{z}}$ is uniform with $g = 9.8 \\mathrm{m} \\cdot \\mathrm{s}^{-2}$ and $\\vec{u_{z}}$ a unit vector; \n- all liquids are incompressible and their density is denoted $\\rho$; \n- no surface tension effects will be considered; \n- the variations of atmospheric pressure with altitude are neglected; \n- the surrounding temperature $T_{\\mathrm{a}}$ is uniform and all transformations are isothermal. \n\n[figure1] \nFig. 1. Artistic view of Cox's clock \n\n[Part B - Two-part barometric tube] \n\n[figure4]\nFig. 4. Simplified model of the perturbative term $P_{1}(t)$ \n\n[Part C - Cox's timepiece] \n\nThe real mechanism developed by Cox is complex (Fig. 5). We study a simplified version, depicted in Fig. 6, and described below: \n- a cylindrical bottom cistern containing a mercury bath; \n- a two-part barometric tube identical to that studied in part B, which is still completely emptied of any air, is dipped into the bath; \n- the cistern and the two-part tube are each suspended by a cable. Both cables (assumed to be inextensible and of negligible mass) pass through a system of ideal pullies and finish attached to either side of the same mass $M$, which can slide on a horizontal surface; \n- the total volume of liquid mercury contained in the system is $V_{\\ell} = 5 \\mathrm{L}$. \n\nThe height, cross-section and masses of each part are given in Table 2. The position of mass $M$ is referenced by the coordinate $x$ of its center of mass. We consider solid friction between the horizontal support and the mass $M$, without distinction between static and dynamic coefficients; the magnitude of this force when sliding occurs is denoted $F_{\\mathrm{s}}$. \n\nTwo stops limit the displacement of the mass $M$ such that $-X \\leq x \\leq X$ (with $X > 0$). Assume that the value of $X$ guarantees that the bottom of the two-part tube never touches the bottom of the cistern nor comes out of the liquid bath; and the altitude $z_{\\ell}$ of the mercury column is always in the upper bulb. \n\n[figure5]\nFig. 5. Real Cox's timepiece (without mercury) \n\n[figure6]\nFig. 6. Sketch of the system modeling the timepiece \n\n|Reference|Name|Height|Cross section area|Empty mass|\n|-|-|-|-|-|\n|1|cistern|$H_{\\mathrm{c}} = 30 \\mathrm{cm}$|$S_{\\mathrm{c}} = 210 \\mathrm{cm}^{2}$|$m_{\\mathrm{c}}$|\n|2|tubular part of the barometric tube|$H_{\\mathrm{t}} = 80 \\mathrm{cm}$|$S_{\\mathrm{t}} = 5 \\mathrm{cm}^{2}$|rowspan=\"2\" total mass of the barometric tube : $m_{\\mathrm{tb}}$|\n|$2^{\\prime}$|bulb of the barometric tube|$H_{\\mathrm{b}} = 20 \\mathrm{cm}$|$S_{\\mathrm{b}} = 200 \\mathrm{cm}^{2}$| | \nTable 2. Dimensions and notations for the model system \n\nThe system evolves in contact with the atmosphere, whose pressure fluctuates as in Fig. 4 (still with amplitude $A = 5 \\times 10^{2} \\mathrm{Pa}$ and period $\\tau_{1} = 1$ week). At the start $t = 0$, the mass $M$ is at rest at $x = 0$ and the tensions exerted by the two cables on either side of the mass $M$ are in balance while $P_{1}(0) = 0$. We define \n$\\xi = \\frac{S_{\\mathrm{b}} + S_{\\mathrm{c}} - S_{\\mathrm{t}}}{S_{\\mathrm{b}} S_{\\mathrm{c}}} \\frac{F_{\\mathrm{s}}}{A} \\simeq \\frac{S_{\\mathrm{b}} + S_{\\mathrm{c}}}{S_{\\mathrm{b}} S_{\\mathrm{c}}} \\frac{F_{\\mathrm{s}}}{A}$ (Equation 3) \nwhere the last expression uses that $S_{\\mathrm{t}} \\ll S_{\\mathrm{b}}, S_{\\mathrm{c}}$ (which we will assume is valid until the end of the problem). \n\n(C.1) Determine the threshold $\\xi^{\\star}$ such that $M$ remains indefinitely at rest when $\\xi > \\xi^{\\star}$. \n\nFor the question (C.2) only, suppose that the mass $M$ is temporarily blocked at $x = X$. \n\n(C.2) Give an expression for the total tension force $\\vec{T} = T \\vec{u_{x}}$ acting on the mass $M$ due to the tension in two cables at this position, when $P_{1} = 0$, in terms of $\\rho, g, X$ and pertinent cross-sections. \n\nWhen $\\xi < \\xi^{\\star}$, starting again from $x = 0$ and $P_{1} = 0$, two different behaviours can be observed for $t \\geq 0$. To distinguish them, we need to introduce another parameter \n$\\lambda = \\frac{2 (S_{\\mathrm{b}} - S_{\\mathrm{t}}) }{S_{\\mathrm{b}}} \\frac{\\rho g X}{A} \\simeq \\frac{2 \\rho g X}{A}$ (Equation 4)", + "question": "Complete the table in the answer sheet (Fig. 7) to indicate the condition under which each regime is obtained. Conditions must be expressed as inequalities on $\\xi$ and/or $\\lambda$. In addition, sketch the variations of $x(t) / X$ for $t \\in [0, 3 \\tau_{1}]$ that are consistent with the variations of $P_{1}(t) / A$ already present. Specification of remarkable points coordinates is not required. \n\n[figure7]", + "marking": [ + [ + "Award 0.2 pt if the answer provides the general expression for $\\vec{T}$ containing both $P_1$ and $X$ terms: $\\vec{T} = \\left[\\frac{4 S_c(S_b - S_t)}{S_b + S_c - S_t}\\rho g X + \\frac{2 S_b S_c}{S_b + S_c - S_t} P_1\\right]\\vec{u}_x$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer states at least one correct inequality condition ($\\xi + 2\\lambda > 2$ or $\\xi + 2\\lambda < 2$). Otherwise, award 0 pt.", + "Award 0.1 pt if the answer states both correct inequalities (without considering strict or large): $\\xi + 2\\lambda > 2$ for Regime 1 and $\\xi + 2\\lambda < 2$ for Regime 2. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer shows both graphs with correct global appearance: one aperiodic and one periodic. If any one appearance is correct, award 0 pt.", + "Award 0.2 pt if each graph's appearance matches the sign of its corresponding inequality ($>$ for aperiodic, $<$ for periodic). Otherwise, award 0 pt.", + "Award 0.2 pt if either graph shows the first switch from $x=0$ to $x=X$ beginning within $t \\in (0, \\frac{\\tau_1}{4}]$. Otherwise, award 0 pt.", + "Award 0.2 pt if either graph shows instantaneous switching. Otherwise, award 0 pt.", + "Award 0.1 pt if the aperiodic graph shows $x = X$ for all times after first switch. Otherwise, award 0 pt.", + "Award 0.1 pt if the periodic graph shows behavior with period $\\tau_1$ (except first switch). Otherwise, award 0 pt.", + "Award 0.2 pt if the periodic graph shows similar positive and negative parts. Otherwise, award 0 pt.", + "Award 0.2 pt if the periodic graph shows $x(t) / X$ is described by a rectangular function of magnitude 1 and duty cycle 50% in steady state. Otherwise, award 0 pt.", + "Award 0.1 pt if the periodic graph shows first step at $x = X$ lasting longer than others. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\xi + 2\\lambda > 2$}", + "\\boxed{$\\xi + 2\\lambda < 2$}" + ], + "answer_type": [ + "Inequality", + "Inequality" + ], + "unit": [ + null, + null + ], + "points": [ + 1.0, + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_2_a_1.png", + "image_question/IPhO_2025_2_b_2.png", + "image_question/IPhO_2025_2_c_1.png", + "image_question/IPhO_2025_2_c_2.png", + "image_question/IPhO_2025_2_c_3.png" + ] + }, + { + "id": "IPhO_2025_2_C_4", + "context": "[Cox's Timepiece] \n\nIn 1765, British clockmaker James Cox invented a clock whose only source of energy is the fluctuations in atmospheric pressure. Cox's clock used two vessels containing mercury. Changes in atmospheric pressure caused mercury to move between the vessels, and the two vessels to move relative to each other. This movement acted as an energy source for the actual clock. \n\nWe propose an analysis of this device. Throughout, we assume that \n- the Earth's gravitational field $\\vec{g} = -g\\vec{u_{z}}$ is uniform with $g = 9.8 \\mathrm{m} \\cdot \\mathrm{s}^{-2}$ and $\\vec{u_{z}}$ a unit vector; \n- all liquids are incompressible and their density is denoted $\\rho$; \n- no surface tension effects will be considered; \n- the variations of atmospheric pressure with altitude are neglected; \n- the surrounding temperature $T_{\\mathrm{a}}$ is uniform and all transformations are isothermal. \n\n[figure1] \nFig. 1. Artistic view of Cox's clock \n\n[Part B - Two-part barometric tube] \n\n[figure4]\nFig. 4. Simplified model of the perturbative term $P_{1}(t)$ \n\n[Part C - Cox's timepiece] \n\nThe real mechanism developed by Cox is complex (Fig. 5). We study a simplified version, depicted in Fig. 6, and described below: \n- a cylindrical bottom cistern containing a mercury bath; \n- a two-part barometric tube identical to that studied in part B, which is still completely emptied of any air, is dipped into the bath; \n- the cistern and the two-part tube are each suspended by a cable. Both cables (assumed to be inextensible and of negligible mass) pass through a system of ideal pullies and finish attached to either side of the same mass $M$, which can slide on a horizontal surface; \n- the total volume of liquid mercury contained in the system is $V_{\\ell} = 5 \\mathrm{L}$. \n\nThe height, cross-section and masses of each part are given in Table 2. The position of mass $M$ is referenced by the coordinate $x$ of its center of mass. We consider solid friction between the horizontal support and the mass $M$, without distinction between static and dynamic coefficients; the magnitude of this force when sliding occurs is denoted $F_{\\mathrm{s}}$. \n\nTwo stops limit the displacement of the mass $M$ such that $-X \\leq x \\leq X$ (with $X > 0$). Assume that the value of $X$ guarantees that the bottom of the two-part tube never touches the bottom of the cistern nor comes out of the liquid bath; and the altitude $z_{\\ell}$ of the mercury column is always in the upper bulb. \n\n[figure5]\nFig. 5. Real Cox's timepiece (without mercury) \n\n[figure6]\nFig. 6. Sketch of the system modeling the timepiece \n\n|Reference|Name|Height|Cross section area|Empty mass|\n|-|-|-|-|-|\n|1|cistern|$H_{\\mathrm{c}} = 30 \\mathrm{cm}$|$S_{\\mathrm{c}} = 210 \\mathrm{cm}^{2}$|$m_{\\mathrm{c}}$|\n|2|tubular part of the barometric tube|$H_{\\mathrm{t}} = 80 \\mathrm{cm}$|$S_{\\mathrm{t}} = 5 \\mathrm{cm}^{2}$|rowspan=\"2\" total mass of the barometric tube : $m_{\\mathrm{tb}}$|\n|$2^{\\prime}$|bulb of the barometric tube|$H_{\\mathrm{b}} = 20 \\mathrm{cm}$|$S_{\\mathrm{b}} = 200 \\mathrm{cm}^{2}$| | \nTable 2. Dimensions and notations for the model system \n\nThe system evolves in contact with the atmosphere, whose pressure fluctuates as in Fig. 4 (still with amplitude $A = 5 \\times 10^{2} \\mathrm{Pa}$ and period $\\tau_{1} = 1$ week). At the start $t = 0$, the mass $M$ is at rest at $x = 0$ and the tensions exerted by the two cables on either side of the mass $M$ are in balance while $P_{1}(0) = 0$. We define \n$\\xi = \\frac{S_{\\mathrm{b}} + S_{\\mathrm{c}} - S_{\\mathrm{t}}}{S_{\\mathrm{b}} S_{\\mathrm{c}}} \\frac{F_{\\mathrm{s}}}{A} \\simeq \\frac{S_{\\mathrm{b}} + S_{\\mathrm{c}}}{S_{\\mathrm{b}} S_{\\mathrm{c}}} \\frac{F_{\\mathrm{s}}}{A}$ (Equation 3) \nwhere the last expression uses that $S_{\\mathrm{t}} \\ll S_{\\mathrm{b}}, S_{\\mathrm{c}}$ (which we will assume is valid until the end of the problem). \n\n(C.1) Determine the threshold $\\xi^{\\star}$ such that $M$ remains indefinitely at rest when $\\xi > \\xi^{\\star}$. \n\nFor the question (C.2) only, suppose that the mass $M$ is temporarily blocked at $x = X$. \n\n(C.2) Give an expression for the total tension force $\\vec{T} = T \\vec{u_{x}}$ acting on the mass $M$ due to the tension in two cables at this position, when $P_{1} = 0$, in terms of $\\rho, g, X$ and pertinent cross-sections. \n\nWhen $\\xi < \\xi^{\\star}$, starting again from $x = 0$ and $P_{1} = 0$, two different behaviours can be observed for $t \\geq 0$. To distinguish them, we need to introduce another parameter \n$\\lambda = \\frac{2 (S_{\\mathrm{b}} - S_{\\mathrm{t}}) }{S_{\\mathrm{b}}} \\frac{\\rho g X}{A} \\simeq \\frac{2 \\rho g X}{A}$ (Equation 4) \n\n(C.3) Complete the table in the answer sheet to indicate the condition under which each regime is obtained. Conditions must be expressed as inequalities on $\\xi$ and/or $\\lambda$. In addition, sketch the variations of $x(t) / X$ for $t \\in [0, 3 \\tau_{1}]$ that are consistent with the variations of $P_{1}(t) / A$ already present. Specification of remarkable points coordinates is not required. \n\nIn the real Cox's timepiece, energy provided by the mechanism is stored using a system of ratchets and used to raise a counterweight, like in a traditional clock. In the simplified model studied here, the energy recovered by the clock corresponds to the energy dissipated by the friction force exerted by the horizontal surface on the mass $M$. From now on, we assume that the system is dimensioned such that to work in the regime that allows the clock to recuperate energy. We also assume that the permanent regime is established. We denote $W$ the energy dissipated by the solid friction force during a period $\\tau_{1}$, which can be expressed only in terms of $F_{\\mathrm{s}}$ and $X$. \n\nAll else equal, $F_{\\mathrm{s}}$ and $X$ can be adjusted to maximize the energy $W$; we denote $F_{\\mathrm{s}}^{\\star}$ and $X^{\\star}$ their respective values in the optimal situation.", + "question": "Considering $S_{\\mathrm{b}} \\simeq S_{\\mathrm{c}}$ and $S_{\\mathrm{t}} \\ll S_{\\mathrm{b}}$, determine the expressions for (1) $F_{\\mathrm{s}}^{\\star}$ and (2) $X^{\\star}$ as functions of $\\rho, g, S_{\\mathrm{c}}$ and $A$. (3) Express the corresponding maximum energy $W^{\\star}$, (4) then calculate its numerical value in $\\mathrm{mJ}$ with $A = 5 \\times 10^{2} \\mathrm{Pa}$.", + "marking": [ + [ + "Award 0.2 pt if the answer states the total friction work expression $W = 4 F_S X$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer provides either the constraint condition $\\xi + 2\\lambda = 2$ or the equivalent expression $F_s = S_c(A - 2\\rho g X)$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer derives the optimal displacement $X^{\\star} = \\frac{A}{4\\rho g}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer derives the optimal friction force $F_s^{\\star} = \\frac{A S_c}{2}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer derives the optimal work expression $W^{\\star} = \\frac{A^2 S_c}{2\\rho g}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer provides the correct numerical value for $W^{\\star}$ within the range [19 mJ, 21 mJ]. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$F_{\\mathrm{s}}^{\\star} = \\frac{A S_{\\mathrm{c}}}{2}$}", + "\\boxed{$X^{\\star} = \\frac{A}{4 \\rho g}$}", + "\\boxed{$W^{\\star} = \\frac{A^{2} S_{\\mathrm{c}}}{2 \\rho g}$}", + "\\boxed{[19, 21]}" + ], + "answer_type": [ + "Expression", + "Expression", + "Expression", + "Numerical Value" + ], + "unit": [ + null, + null, + null, + "mJ" + ], + "points": [ + 0.3, + 0.3, + 0.3, + 0.1 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_2_a_1.png", + "image_question/IPhO_2025_2_b_2.png", + "image_question/IPhO_2025_2_c_1.png", + "image_question/IPhO_2025_2_c_2.png" + ] + }, + { + "id": "IPhO_2025_2_C_5", + "context": "[Cox's Timepiece] \n\nIn 1765, British clockmaker James Cox invented a clock whose only source of energy is the fluctuations in atmospheric pressure. Cox's clock used two vessels containing mercury. Changes in atmospheric pressure caused mercury to move between the vessels, and the two vessels to move relative to each other. This movement acted as an energy source for the actual clock. \n\nWe propose an analysis of this device. Throughout, we assume that \n- the Earth's gravitational field $\\vec{g} = -g\\vec{u_{z}}$ is uniform with $g = 9.8 \\mathrm{m} \\cdot \\mathrm{s}^{-2}$ and $\\vec{u_{z}}$ a unit vector; \n- all liquids are incompressible and their density is denoted $\\rho$; \n- no surface tension effects will be considered; \n- the variations of atmospheric pressure with altitude are neglected; \n- the surrounding temperature $T_{\\mathrm{a}}$ is uniform and all transformations are isothermal. \n\n[figure1] \nFig. 1. Artistic view of Cox's clock \n\n[Part B - Two-part barometric tube] \n\n[figure4]\nFig. 4. Simplified model of the perturbative term $P_{1}(t)$ \n\n[Part C - Cox's timepiece] \n\nThe real mechanism developed by Cox is complex (Fig. 5). We study a simplified version, depicted in Fig. 6, and described below: \n- a cylindrical bottom cistern containing a mercury bath; \n- a two-part barometric tube identical to that studied in part B, which is still completely emptied of any air, is dipped into the bath; \n- the cistern and the two-part tube are each suspended by a cable. Both cables (assumed to be inextensible and of negligible mass) pass through a system of ideal pullies and finish attached to either side of the same mass $M$, which can slide on a horizontal surface; \n- the total volume of liquid mercury contained in the system is $V_{\\ell} = 5 \\mathrm{L}$. \n\nThe height, cross-section and masses of each part are given in Table 2. The position of mass $M$ is referenced by the coordinate $x$ of its center of mass. We consider solid friction between the horizontal support and the mass $M$, without distinction between static and dynamic coefficients; the magnitude of this force when sliding occurs is denoted $F_{\\mathrm{s}}$. \n\nTwo stops limit the displacement of the mass $M$ such that $-X \\leq x \\leq X$ (with $X > 0$). Assume that the value of $X$ guarantees that the bottom of the two-part tube never touches the bottom of the cistern nor comes out of the liquid bath; and the altitude $z_{\\ell}$ of the mercury column is always in the upper bulb. \n\n[figure5]\nFig. 5. Real Cox's timepiece (without mercury) \n\n[figure6]\nFig. 6. Sketch of the system modeling the timepiece \n\n|Reference|Name|Height|Cross section area|Empty mass|\n|-|-|-|-|-|\n|1|cistern|$H_{\\mathrm{c}} = 30 \\mathrm{cm}$|$S_{\\mathrm{c}} = 210 \\mathrm{cm}^{2}$|$m_{\\mathrm{c}}$|\n|2|tubular part of the barometric tube|$H_{\\mathrm{t}} = 80 \\mathrm{cm}$|$S_{\\mathrm{t}} = 5 \\mathrm{cm}^{2}$|rowspan=\"2\" total mass of the barometric tube : $m_{\\mathrm{tb}}$|\n|$2^{\\prime}$|bulb of the barometric tube|$H_{\\mathrm{b}} = 20 \\mathrm{cm}$|$S_{\\mathrm{b}} = 200 \\mathrm{cm}^{2}$| | \nTable 2. Dimensions and notations for the model system \n\nThe system evolves in contact with the atmosphere, whose pressure fluctuates as in Fig. 4 (still with amplitude $A = 5 \\times 10^{2} \\mathrm{Pa}$ and period $\\tau_{1} = 1$ week). At the start $t = 0$, the mass $M$ is at rest at $x = 0$ and the tensions exerted by the two cables on either side of the mass $M$ are in balance while $P_{1}(0) = 0$. We define \n$\\xi = \\frac{S_{\\mathrm{b}} + S_{\\mathrm{c}} - S_{\\mathrm{t}}}{S_{\\mathrm{b}} S_{\\mathrm{c}}} \\frac{F_{\\mathrm{s}}}{A} \\simeq \\frac{S_{\\mathrm{b}} + S_{\\mathrm{c}}}{S_{\\mathrm{b}} S_{\\mathrm{c}}} \\frac{F_{\\mathrm{s}}}{A}$ (Equation 3) \nwhere the last expression uses that $S_{\\mathrm{t}} \\ll S_{\\mathrm{b}}, S_{\\mathrm{c}}$ (which we will assume is valid until the end of the problem). \n\n(C.1) Determine the threshold $\\xi^{\\star}$ such that $M$ remains indefinitely at rest when $\\xi > \\xi^{\\star}$. \n\nFor the question (C.2) only, suppose that the mass $M$ is temporarily blocked at $x = X$. \n\n(C.2) Give an expression for the total tension force $\\vec{T} = T \\vec{u_{x}}$ acting on the mass $M$ due to the tension in two cables at this position, when $P_{1} = 0$, in terms of $\\rho, g, X$ and pertinent cross-sections. \n\nWhen $\\xi < \\xi^{\\star}$, starting again from $x = 0$ and $P_{1} = 0$, two different behaviours can be observed for $t \\geq 0$. To distinguish them, we need to introduce another parameter \n$\\lambda = \\frac{2 (S_{\\mathrm{b}} - S_{\\mathrm{t}}) }{S_{\\mathrm{b}}} \\frac{\\rho g X}{A} \\simeq \\frac{2 \\rho g X}{A}$ (Equation 4) \n\n(C.3) Complete the table in the answer sheet to indicate the condition under which each regime is obtained. Conditions must be expressed as inequalities on $\\xi$ and/or $\\lambda$. In addition, sketch the variations of $x(t) / X$ for $t \\in [0, 3 \\tau_{1}]$ that are consistent with the variations of $P_{1}(t) / A$ already present. Specification of remarkable points coordinates is not required. \n\nIn the real Cox's timepiece, energy provided by the mechanism is stored using a system of ratchets and used to raise a counterweight, like in a traditional clock. In the simplified model studied here, the energy recovered by the clock corresponds to the energy dissipated by the friction force exerted by the horizontal surface on the mass $M$. From now on, we assume that the system is dimensioned such that to work in the regime that allows the clock to recuperate energy. We also assume that the permanent regime is established. We denote $W$ the energy dissipated by the solid friction force during a period $\\tau_{1}$, which can be expressed only in terms of $F_{\\mathrm{s}}$ and $X$. \n\nAll else equal, $F_{\\mathrm{s}}$ and $X$ can be adjusted to maximize the energy $W$; we denote $F_{\\mathrm{s}}^{\\star}$ and $X^{\\star}$ their respective values in the optimal situation. \n\n(C.4) Considering $S_{\\mathrm{b}} \\simeq S_{\\mathrm{c}}$ and $S_{\\mathrm{t}} \\ll S_{\\mathrm{b}}$, determine the expressions for $F_{\\mathrm{s}}^{\\star}$ and $X^{\\star}$ as functions of $\\rho, g, S_{\\mathrm{c}}$ and $A$. Express the corresponding maximum energy $W^{\\star}$, then calculate its numerical value with $A = 5 \\times 10^{2} \\mathrm{Pa}$. \n\nWe denote $W_{\\mathrm{pr}}^{\\star}$ the work of atmospheric pressure forces received by the system in the optimal situation during a period $\\tau_{1}$.", + "question": "(1) Express $W_{\\mathrm{pr}}^{\\star}$, (2) then calculate the ratio $W^{\\star} / W_{\\mathrm{pr}}^{\\star}$. It could be useful to represent the evolution of the system in a $(P, V)$ diagram, where $V$ is the system's volume.", + "marking": [ + [ + "Award 0.1 pt if the answer states that in the optimal case, mass $M$ switches between two positions $x = \\pm X$ when $P_1 = \\pm A$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer describes the cycle as consisting of 2 isometric ($x$ constant) and 2 isobaric ($P$ constant) transformations (through sketch, table or other description). Otherwise, award 0 pt.", + "Award 0.2 pt if the answer provides the correct sequence of the successive states and/or direction of the cycle using $x$ and $P$ parameters. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer gives the general expression of the volume of the system in an $(P, x)$ state: $V = -S_c \\left[x + \\frac{P_1}{2 \\rho g}\\right]$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly lists all four volume states: $-3 S_c X^{*}\\rightarrow S_c X^{*} \\rightarrow 3 S_c X^{*} \\rightarrow -S_c X^{*}$. If any one volume state is incorrect, award 0 pt.", + "Award 0.2 pt if the answer uses the correct method to calculate the work of atmospheric pressure forces: $W_{pr} = -\\oint_{1\\text{period}} P_a \\mathrm{d}V$ (via explicit integral, area of the cycle in ($P$, $V$) diagram, or equivalent method). Otherwise, award 0 pt.", + "Award 0.2 pt if the answer derives the optimal work expression: $W_{pr}^{\\star} = 4 S_c X^{\\star} A = \\frac{S_c A^2}{\\rho g}$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer obtains the final result: $\\frac{W^{\\star}}{W_{pr}^{\\star}} = \\frac{1}{2}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$W_{\\mathrm{pr}}^{\\star} = \\frac{S_{\\mathrm{c}} A^{2}}{\\rho g}$}", + "\\boxed{$W^{\\star} / W_{\\mathrm{pr}}^{\\star} = \\frac{1}{2}$}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null, + null + ], + "points": [ + 1.4, + 0.3 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_2_a_1.png", + "image_question/IPhO_2025_2_b_2.png", + "image_question/IPhO_2025_2_c_1.png", + "image_question/IPhO_2025_2_c_2.png" + ] + }, + { + "id": "IPhO_2025_3_A_1", + "context": "[Champagne] \n\nChampagne is a French sparkling wine. Fermentation of sugars produces carbon dioxide $(\\text{CO}_2)$ in the bottle. The molar concentration of $(\\text{CO}_2)$ in the liquid phase $c_{\\ell}$ and the partial pressure $P_{\\text{CO}_2}$ in the gas phase are related by $c_{\\ell} = k_{\\mathrm{H}} P_{\\text{CO}_2}$, known as Henry's law and where $k_{\\mathrm{H}}$ is called Henry's constant.\n\nData: \nSurface tension of champagne $\\sigma = 47 \\times 10^{-3} \\mathrm{J} \\cdot \\mathrm{m}^{-2}$ \nDensity of the liquid $\\rho_{\\ell} = 1.0 \\times 10^{3} \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$ \nHenry's constant at $T_{0} = 20^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(20^{\\circ}\\mathrm{C}) = 3.3 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nHenry's constant at $T_{0} = 6^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(6^{\\circ}\\mathrm{C}) = 5.4 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nAtmospheric pressure $P_{0} = 1 \\mathrm{bar} = 1.0 \\times 10^{5} \\mathrm{Pa}$ \nGases are ideal with an adiabatic coefficient $\\gamma = 1.3$ \n\n[figure1] \nFig. 1. A glass filled with champagne. \n\n[Part A - Nucleation, growth and rise of bubbles] \n\nImmediately after opening a bottle of champagne at temperature $T_{0} = 20^{\\circ}\\mathrm{C}$, we fill a glass. The pressure in the liquid is $P_{0}$ and its temperature stays constant at $T_{0}$. The concentration $c_{\\ell}$ of dissolved $\\text{CO}_2$ exceeds the equilibrium concentration and we study the nucleation of a $\\text{CO}_2$ bubble. We note $a$ its radius and $P_{\\mathrm{b}}$ its inner pressure.", + "question": "Express the pressure $P_{\\mathrm{b}}$ in terms of $P_{0}$, $a$ and $\\sigma$.", + "marking": [ + [ + "Award 0.2 pt if the answer gives the correct expression of $P_{\\mathrm{b}} = P_{0} + \\frac{2 \\sigma}{a}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$P_{\\mathrm{b}} = P_{0} + \\frac{2 \\sigma}{a}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.2 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_3_a_1.png" + ] + }, + { + "id": "IPhO_2025_3_A_2", + "context": "[Champagne] \n\nChampagne is a French sparkling wine. Fermentation of sugars produces carbon dioxide $(\\text{CO}_2)$ in the bottle. The molar concentration of $(\\text{CO}_2)$ in the liquid phase $c_{\\ell}$ and the partial pressure $P_{\\text{CO}_2}$ in the gas phase are related by $c_{\\ell} = k_{\\mathrm{H}} P_{\\text{CO}_2}$, known as Henry's law and where $k_{\\mathrm{H}}$ is called Henry's constant.\n\nData: \nSurface tension of champagne $\\sigma = 47 \\times 10^{-3} \\mathrm{J} \\cdot \\mathrm{m}^{-2}$ \nDensity of the liquid $\\rho_{\\ell} = 1.0 \\times 10^{3} \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$ \nHenry's constant at $T_{0} = 20^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(20^{\\circ}\\mathrm{C}) = 3.3 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nHenry's constant at $T_{0} = 6^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(6^{\\circ}\\mathrm{C}) = 5.4 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nAtmospheric pressure $P_{0} = 1 \\mathrm{bar} = 1.0 \\times 10^{5} \\mathrm{Pa}$ \nGases are ideal with an adiabatic coefficient $\\gamma = 1.3$ \n\n[figure1] \nFig. 1. A glass filled with champagne. \n\n[Part A - Nucleation, growth and rise of bubbles] \n\nImmediately after opening a bottle of champagne at temperature $T_{0} = 20^{\\circ}\\mathrm{C}$, we fill a glass. The pressure in the liquid is $P_{0}$ and its temperature stays constant at $T_{0}$. The concentration $c_{\\ell}$ of dissolved $\\text{CO}_2$ exceeds the equilibrium concentration and we study the nucleation of a $\\text{CO}_2$ bubble. We note $a$ its radius and $P_{\\mathrm{b}}$ its inner pressure. \n\n(A.1) Express the pressure $P_{\\mathrm{b}}$ in terms of $P_{0}$, $a$ and $\\sigma$. \n\nIn the liquid, the concentration of dissolved $\\text{CO}_2$ depends on the distance to the bubble. At long distance we recover the value $c_{\\ell}$ and we note $c_{\\mathrm{b}}$ the concentration close to the bubble surface. According to Henry's law, $c_{\\mathrm{b}} = k_{\\mathrm{H}} P_{\\mathrm{b}}$. We furthermore assume in all the problem that bubbles contain only $\\text{CO}_2$. \n\nSince $c_{\\ell} \\neq c_{\\mathrm{b}}$,$\\text{CO}_2$ molecules diffuse from areas of high to low concentration. We assume also that any molecule from the liquid phase reaching the bubble surface is transferred to the vapour.", + "question": "(1) Express the critical radius $a_{\\mathrm{c}}$ above which a bubble is expected to grow in terms of $P_{0}, \\sigma, c_{\\ell}$ and $c_{0}$ where $c_{0} = k_{\\mathrm{H}} P_{0}$. \n(2) Calculate numerically $a_{\\mathrm{c}}$ for $c_{\\ell} = 4 c_{0}$. Express your answer in $\\mathrm{\\mu m}$.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly states $c_b = c_{\\ell}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly provides the expression of $a_c = \\frac{2 \\sigma}{P_0 (c_{\\ell}/c_0 - 1)}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly obtains the numerical value of $a_c = 0.3 \\mathrm{\\mu m}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$a_c = \\frac{2 \\sigma}{P_0 (c_{\\ell}/c_0 - 1)}$}", + "\\boxed{0.3}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null, + "$\\mathrm{\\mu m}$" + ], + "points": [ + 0.3, + 0.2 + ], + "modality": "text+illustration figure", + "field": "Thermodynamics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_3_a_1.png" + ] + }, + { + "id": "IPhO_2025_3_A_3", + "context": "[Champagne] \n\nChampagne is a French sparkling wine. Fermentation of sugars produces carbon dioxide $(\\text{CO}_2)$ in the bottle. The molar concentration of $(\\text{CO}_2)$ in the liquid phase $c_{\\ell}$ and the partial pressure $P_{\\text{CO}_2}$ in the gas phase are related by $c_{\\ell} = k_{\\mathrm{H}} P_{\\text{CO}_2}$, known as Henry's law and where $k_{\\mathrm{H}}$ is called Henry's constant.\n\nData: \nSurface tension of champagne $\\sigma = 47 \\times 10^{-3} \\mathrm{J} \\cdot \\mathrm{m}^{-2}$ \nDensity of the liquid $\\rho_{\\ell} = 1.0 \\times 10^{3} \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$ \nHenry's constant at $T_{0} = 20^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(20^{\\circ}\\mathrm{C}) = 3.3 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nHenry's constant at $T_{0} = 6^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(6^{\\circ}\\mathrm{C}) = 5.4 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nAtmospheric pressure $P_{0} = 1 \\mathrm{bar} = 1.0 \\times 10^{5} \\mathrm{Pa}$ \nGases are ideal with an adiabatic coefficient $\\gamma = 1.3$ \n\n[figure1] \nFig. 1. A glass filled with champagne. \n\n[Part A - Nucleation, growth and rise of bubbles] \n\nImmediately after opening a bottle of champagne at temperature $T_{0} = 20^{\\circ}\\mathrm{C}$, we fill a glass. The pressure in the liquid is $P_{0}$ and its temperature stays constant at $T_{0}$. The concentration $c_{\\ell}$ of dissolved $\\text{CO}_2$ exceeds the equilibrium concentration and we study the nucleation of a $\\text{CO}_2$ bubble. We note $a$ its radius and $P_{\\mathrm{b}}$ its inner pressure. \n\n(A.1) Express the pressure $P_{\\mathrm{b}}$ in terms of $P_{0}$, $a$ and $\\sigma$. \n\nIn the liquid, the concentration of dissolved $\\text{CO}_2$ depends on the distance to the bubble. At long distance we recover the value $c_{\\ell}$ and we note $c_{\\mathrm{b}}$ the concentration close to the bubble surface. According to Henry's law, $c_{\\mathrm{b}} = k_{\\mathrm{H}} P_{\\mathrm{b}}$. We furthermore assume in all the problem that bubbles contain only $\\text{CO}_2$. \n\nSince $c_{\\ell} \\neq c_{\\mathrm{b}}$,$\\text{CO}_2$ molecules diffuse from areas of high to low concentration. We assume also that any molecule from the liquid phase reaching the bubble surface is transferred to the vapour. \n\n(A.2) Express the critical radius $a_{\\mathrm{c}}$ above which a bubble is expected to grow in terms of $P_{0}, \\sigma, c_{\\ell}$ and $c_{0}$ where $c_{0} = k_{\\mathrm{H}} P_{0}$. Calculate numerically $a_{\\mathrm{c}}$ for $c_{\\ell} = 4 c_{0}$. \n\nIn practice, bubbles mainly grow from pre-existing gas cavities. Consider then a bubble with initial radius $a_{0} \\approx 40 \\mathrm{\\mu m}$. The number of moles of $\\text{CO}_2$ transferred at the bubble's surface per unit area and time is noted $j$. Two models are possible for $j$. \n\nmodel (1) $j = \\frac{D}{a} \\left(c_{\\ell} - c_{\\mathrm{b}}\\right)$ where $D$ is the diffusion coefficient of $\\text{CO}_2$ in the liquid. \nmodel (2) $j = K \\left(c_{\\ell} - c_{\\mathrm{b}}\\right)$ where $K$ is a constant here. \n\nExperimentally, the bubble radius $a(t)$ is found to depend on time as shown in Fig. 2. Here $c_{\\ell} \\approx 4{c}_{0}$, and since bubbles are large enough to be visible, the excess pressure due to surface tension can be neglected and $P_{\\mathrm{b}} \\approx P_{0}$. \n\n[figure2]\nFig. 2. Time evolution of $\\text{CO}_2$ bubble radius in a glass of champagne (adapted from [1]).", + "question": "(1) Express the number of $\\text{CO}_2$ moles in the bubble $n_{\\mathrm{c}}$ in terms of $a, P_{0}, T_{0}$ and ideal gas constant $R$. \n(2) Find $a(t)$ for model 1. \n(3) Find $a(t)$ for model 2. \n(4) Indicate which model explains the experimental results in Fig. 2: (A) model 1, (B) model 2. \n(5) Depending on your answer, calculate numerically $K$ or $D$.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly obtains the number of moles of $\\text{CO}_2$ (ideal gas) inside the bubble is $n_{\\mathrm{c}} = \\frac{4}{3} \\pi {a}^{3} \\frac{P_{0}}{R T_{0}}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer states any equation that that can be interpreted as a balance of $\\text{CO}_2$ in the bubble. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer provides either the equation between $\\dot{a}$ and $j$: $\\frac{d n_{\\mathrm{c}}}{d t} = j 4 \\pi a^{2}$, or the equation between $\\dot{n}_c$ and $j$: $\\frac{d a}{d t} = j \\frac{RT}{P_{0}}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly derives $a(t)$ for Model 1: $a^{2} = a_{0}^{2} + \\frac{2DR T_{0}}{P_{0}} (c_{\\ell} - c_{0}) t$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly derives $a(t)$ for Model 1: $a = a_{0} + \\frac{KR T_{0}}{P_{0}} (c_{\\ell} - c_{0}) t$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly states that Model 2 explains the experimental results in Fig. 2. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly obtains the numerical value for the slope $\\frac{d a}{d t}$ within the range [210, 250] $\\mathrm{\\mu m / s}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly obtains the numerical value for $K$ within the range $[0.9 \\times 10^{-4} \\mathrm{m/s}, 1.1 \\times 10^{-4} \\mathrm{m/s}]$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$n_{\\mathrm{c}} = \\frac{4}{3}\\pi a^{3} \\frac{P_{0}}{R T_{0}}$}", + "\\boxed{$a^{2} = a_{0}^{2} + \\frac{2DR T_{0}}{P_{0}} (c_{\\ell} - c_{0}) t$}", + "\\boxed{$a = a_{0} + \\frac{KR T_{0}}{P_{0}} (c_{\\ell} - c_{0}) t$}", + "\\boxed{B}", + "\\boxed{$[0.9 \\times 10^{-4}, 1.1 \\times 10^{-4}]$}" + ], + "answer_type": [ + "Expression", + "Expression", + "Expression", + "Multiple Choice", + "Numerical Value" + ], + "unit": [ + null, + null, + null, + null, + "m/s" + ], + "points": [ + 0.1, + 0.35, + 0.35, + 0.1, + 0.3 + ], + "modality": "text+data figure", + "field": "Thermodynamics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_3_a_1.png", + "image_question/IPhO_2025_3_a_2.png" + ] + }, + { + "id": "IPhO_2025_3_A_4", + "context": "[Champagne] \n\nChampagne is a French sparkling wine. Fermentation of sugars produces carbon dioxide $(\\text{CO}_2)$ in the bottle. The molar concentration of $(\\text{CO}_2)$ in the liquid phase $c_{\\ell}$ and the partial pressure $P_{\\text{CO}_2}$ in the gas phase are related by $c_{\\ell} = k_{\\mathrm{H}} P_{\\text{CO}_2}$, known as Henry's law and where $k_{\\mathrm{H}}$ is called Henry's constant.\n\nData: \nSurface tension of champagne $\\sigma = 47 \\times 10^{-3} \\mathrm{J} \\cdot \\mathrm{m}^{-2}$ \nDensity of the liquid $\\rho_{\\ell} = 1.0 \\times 10^{3} \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$ \nHenry's constant at $T_{0} = 20^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(20^{\\circ}\\mathrm{C}) = 3.3 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nHenry's constant at $T_{0} = 6^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(6^{\\circ}\\mathrm{C}) = 5.4 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nAtmospheric pressure $P_{0} = 1 \\mathrm{bar} = 1.0 \\times 10^{5} \\mathrm{Pa}$ \nGases are ideal with an adiabatic coefficient $\\gamma = 1.3$ \n\n[figure1] \nFig. 1. A glass filled with champagne. \n\n[Part A - Nucleation, growth and rise of bubbles] \n\nImmediately after opening a bottle of champagne at temperature $T_{0} = 20^{\\circ}\\mathrm{C}$, we fill a glass. The pressure in the liquid is $P_{0}$ and its temperature stays constant at $T_{0}$. The concentration $c_{\\ell}$ of dissolved $\\text{CO}_2$ exceeds the equilibrium concentration and we study the nucleation of a $\\text{CO}_2$ bubble. We note $a$ its radius and $P_{\\mathrm{b}}$ its inner pressure. \n\n(A.1) Express the pressure $P_{\\mathrm{b}}$ in terms of $P_{0}$, $a$ and $\\sigma$. \n\nIn the liquid, the concentration of dissolved $\\text{CO}_2$ depends on the distance to the bubble. At long distance we recover the value $c_{\\ell}$ and we note $c_{\\mathrm{b}}$ the concentration close to the bubble surface. According to Henry's law, $c_{\\mathrm{b}} = k_{\\mathrm{H}} P_{\\mathrm{b}}$. We furthermore assume in all the problem that bubbles contain only $\\text{CO}_2$. \n\nSince $c_{\\ell} \\neq c_{\\mathrm{b}}$,$\\text{CO}_2$ molecules diffuse from areas of high to low concentration. We assume also that any molecule from the liquid phase reaching the bubble surface is transferred to the vapour. \n\n(A.2) Express the critical radius $a_{\\mathrm{c}}$ above which a bubble is expected to grow in terms of $P_{0}, \\sigma, c_{\\ell}$ and $c_{0}$ where $c_{0} = k_{\\mathrm{H}} P_{0}$. Calculate numerically $a_{\\mathrm{c}}$ for $c_{\\ell} = 4 c_{0}$. \n\nIn practice, bubbles mainly grow from pre-existing gas cavities. Consider then a bubble with initial radius $a_{0} \\approx 40 \\mathrm{\\mu m}$. The number of moles of $\\text{CO}_2$ transferred at the bubble's surface per unit area and time is noted $j$. Two models are possible for $j$. \n\nmodel (1) $j = \\frac{D}{a} \\left(c_{\\ell} - c_{\\mathrm{b}}\\right)$ where $D$ is the diffusion coefficient of $\\text{CO}_2$ in the liquid. \nmodel (2) $j = K \\left(c_{\\ell} - c_{\\mathrm{b}}\\right)$ where $K$ is a constant here. \n\nExperimentally, the bubble radius $a(t)$ is found to depend on time as shown in Fig. 2. Here $c_{\\ell} \\approx 4{c}_{0}$, and since bubbles are large enough to be visible, the excess pressure due to surface tension can be neglected and $P_{\\mathrm{b}} \\approx P_{0}$. \n\n(A.3) Express the number of $\\text{CO}_2$ moles in the bubble $n_{\\mathrm{c}}$ in terms of $a, P_{0}, T_{0}$ and ideal gas constant $R$. Find $a(t)$ for both models. Indicate which model explains the experimental results in Fig. 2. Depending on your answer, calculate numerically $K$ or $D$. \n\n[figure2]\nFig. 2. Time evolution of $\\text{CO}_2$ bubble radius in a glass of champagne (adapted from [1]). \n\nEventually bubbles detach from the bottom of the glass and continue to grow while rising. Fig. 3. shows a train of bubbles. The bubbles of the train have the same initial radius and are emitted at a constant frequency $f_{\\mathrm{b}} = 20 \\mathrm{Hz}}$. \n\n[figure3]\nFig. 3. A train of bubbles. The photo is rotated horizontally for the page layout (adapted from [1]). \n\nFor the range of velocities studied here, the drag force $F$ on a bubble of radius $a$ moving at velocity $v$ in a liquid of dynamic viscosity $\\eta$ is given by Stokes' law $F = 6\\pi \\eta a v$. Measurements show that at any moment in time, the bubble can be assumed to be travelling at its terminal velocity.", + "question": "(1) Give the expression of the main forces exerted on a vertically rising bubble. \n(2) Obtain the expression of $v(a)$. \n(3) Give a numerical estimate of $\\eta$ using $\\rho_{\\ell}$, $g_{0}$ and quantities measured on Fig. 3. Express your answer in $\\mathrm{Pa} \\cdot s$.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly identifies buoyancy ($\\frac{4}{3}\\pi a^{3}\\rho_{\\ell}g_{0}$) and drag force ($6\\pi\\eta av$) as main forces. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer derives $v = \\frac{2}{9\\eta} a^{2} \\rho_{\\ell} g_{0}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly states $\\Delta t = 1/f_{b}$ for time between bubbles. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly states any one coherent value of the radius measured on Fig. 3: last bubble in $[0.20, 0.30] \\mathrm{mm}$, penultimate bubble in $[0.16, 0.24] \\mathrm{mm}$, or antepenultimate bubble in $[0.14, 0.22] \\mathrm{mm}$. If any one answer is incorrect, award 0 pt.", + "Award 0.1 pt if the answer correctly states any one coherent value of the velocity measured on Fig. 3: last bubble $v \\in [4.3, 4.8] \\mathrm{cm / s}$, penultimate bubble $v \\in [4.2, 4.6] \\mathrm{cm / s}$, or antepenultimate bubble $v \\in [3.7, 4.2] \\mathrm{cm / s}$. If any one answer is incorrect, award 0 pt.", + "Award 0.2 pt if the answer correctly obtains the numerical value of $\\eta$ within the range $[1.0, 4.0] \\times 10^{-3} \\text{Pa} \\cdot \\text{s}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$F_{\\text{buoyancy}} = \\frac{4}{3} \\pi a^{3}\\rho_{\\ell} g_{0}$}", + "\\boxed{$F_{\\text{drag}} = 6 \\pi \\eta a v$}", + "\\boxed{$v = \\frac{2}{9\\eta} a^2 \\rho_{\\ell} g_0$}", + "\\boxed{$[1.0 \\times 10^{-3}, 4.0 \\times 10^{-3}]$}" + ], + "answer_type": [ + "Expression", + "Expression", + "Expression", + "Numerical Value" + ], + "unit": [ + null, + null, + null, + "$\\mathrm{Pa} \\cdot s$" + ], + "points": [ + 0.05, + 0.05, + 0.2, + 0.5 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_3_a_1.png", + "image_question/IPhO_2025_3_a_2.png", + "image_question/IPhO_2025_3_a_3.png" + ] + }, + { + "id": "IPhO_2025_3_A_5", + "context": "[Champagne] \n\nChampagne is a French sparkling wine. Fermentation of sugars produces carbon dioxide $(\\text{CO}_2)$ in the bottle. The molar concentration of $(\\text{CO}_2)$ in the liquid phase $c_{\\ell}$ and the partial pressure $P_{\\text{CO}_2}$ in the gas phase are related by $c_{\\ell} = k_{\\mathrm{H}} P_{\\text{CO}_2}$, known as Henry's law and where $k_{\\mathrm{H}}$ is called Henry's constant.\n\nData: \nSurface tension of champagne $\\sigma = 47 \\times 10^{-3} \\mathrm{J} \\cdot \\mathrm{m}^{-2}$ \nDensity of the liquid $\\rho_{\\ell} = 1.0 \\times 10^{3} \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$ \nHenry's constant at $T_{0} = 20^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(20^{\\circ}\\mathrm{C}) = 3.3 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nHenry's constant at $T_{0} = 6^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(6^{\\circ}\\mathrm{C}) = 5.4 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nAtmospheric pressure $P_{0} = 1 \\mathrm{bar} = 1.0 \\times 10^{5} \\mathrm{Pa}$ \nGases are ideal with an adiabatic coefficient $\\gamma = 1.3$ \n\n[figure1] \nFig. 1. A glass filled with champagne. \n\n[Part A - Nucleation, growth and rise of bubbles] \n\nImmediately after opening a bottle of champagne at temperature $T_{0} = 20^{\\circ}\\mathrm{C}$, we fill a glass. The pressure in the liquid is $P_{0}$ and its temperature stays constant at $T_{0}$. The concentration $c_{\\ell}$ of dissolved $\\text{CO}_2$ exceeds the equilibrium concentration and we study the nucleation of a $\\text{CO}_2$ bubble. We note $a$ its radius and $P_{\\mathrm{b}}$ its inner pressure. \n\n(A.1) Express the pressure $P_{\\mathrm{b}}$ in terms of $P_{0}$, $a$ and $\\sigma$. \n\nIn the liquid, the concentration of dissolved $\\text{CO}_2$ depends on the distance to the bubble. At long distance we recover the value $c_{\\ell}$ and we note $c_{\\mathrm{b}}$ the concentration close to the bubble surface. According to Henry's law, $c_{\\mathrm{b}} = k_{\\mathrm{H}} P_{\\mathrm{b}}$. We furthermore assume in all the problem that bubbles contain only $\\text{CO}_2$. \n\nSince $c_{\\ell} \\neq c_{\\mathrm{b}}$,$\\text{CO}_2$ molecules diffuse from areas of high to low concentration. We assume also that any molecule from the liquid phase reaching the bubble surface is transferred to the vapour. \n\n(A.2) Express the critical radius $a_{\\mathrm{c}}$ above which a bubble is expected to grow in terms of $P_{0}, \\sigma, c_{\\ell}$ and $c_{0}$ where $c_{0} = k_{\\mathrm{H}} P_{0}$. Calculate numerically $a_{\\mathrm{c}}$ for $c_{\\ell} = 4 c_{0}$. \n\nIn practice, bubbles mainly grow from pre-existing gas cavities. Consider then a bubble with initial radius $a_{0} \\approx 40 \\mathrm{\\mu m}$. The number of moles of $\\text{CO}_2$ transferred at the bubble's surface per unit area and time is noted $j$. Two models are possible for $j$. \n\nmodel (1) $j = \\frac{D}{a} \\left(c_{\\ell} - c_{\\mathrm{b}}\\right)$ where $D$ is the diffusion coefficient of $\\text{CO}_2$ in the liquid. \nmodel (2) $j = K \\left(c_{\\ell} - c_{\\mathrm{b}}\\right)$ where $K$ is a constant here. \n\nExperimentally, the bubble radius $a(t)$ is found to depend on time as shown in Fig. 2. Here $c_{\\ell} \\approx 4{c}_{0}$, and since bubbles are large enough to be visible, the excess pressure due to surface tension can be neglected and $P_{\\mathrm{b}} \\approx P_{0}$. \n\n(A.3) Express the number of $\\text{CO}_2$ moles in the bubble $n_{\\mathrm{c}}$ in terms of $a, P_{0}, T_{0}$ and ideal gas constant $R$. Find $a(t)$ for both models. Indicate which model explains the experimental results in Fig. 2. Depending on your answer, calculate numerically $K$ or $D$. \n\n[figure2]\nFig. 2. Time evolution of $\\text{CO}_2$ bubble radius in a glass of champagne (adapted from [1]). \n\nEventually bubbles detach from the bottom of the glass and continue to grow while rising. Fig. 3. shows a train of bubbles. The bubbles of the train have the same initial radius and are emitted at a constant frequency $f_{\\mathrm{b}} = 20 \\mathrm{Hz}}$. \n\n[figure3]\nFig. 3. A train of bubbles. The photo is rotated horizontally for the page layout (adapted from [1]). \n\nFor the range of velocities studied here, the drag force $F$ on a bubble of radius $a$ moving at velocity $v$ in a liquid of dynamic viscosity $\\eta$ is given by Stokes' law $F = 6\\pi \\eta a v$. Measurements show that at any moment in time, the bubble can be assumed to be travelling at its terminal velocity. \n\n(A.4) Give the expression of the main forces exerted on a vertically rising bubble. Obtain the expression of $v(a)$. Give a numerical estimate of $\\eta$ using $\\rho_{\\ell}$, $g_{0}$ and quantities measured on Fig. 3. \n\nThe quasi-stationary growth of bubbles with rate $q_{a} = \\frac{\\mathrm{d} a}{\\mathrm{d} t}$ still applies during bubble rise.", + "question": "(1) Express the radius $a_{H_{\\ell}}$ of a bubble reaching the free surface in terms of height travelled $H_{\\ell}$, growth rate $q_{a} = \\frac{\\mathrm{d} a}{\\mathrm{d} t}$, and any constants you may need. \n(2) Assume $a_{H_{\\ell}} \\gg a_{0}$ and $q_{a}$ constant, and give the numerical value of $a_{H_{\\ell}}$ with $H_{\\ell} = 10 \\mathrm{cm}$ and $q_{a}$ corresponding to Fig. 2. Express your answer in $mm$.", + "marking": [ + [ + "Award 0.3 pt if the answer correctly derives the expression of $a_{H_{\\ell}}$ as $a_{H_{\\ell}} = \\left( \\frac{27 q_{a} \\eta H_{\\ell}}{2 \\rho_{\\ell} g_{0}} \\right)^{1/3}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly obtains the numerical value of $a_{H_{\\ell}}$ within the range [0.36 mm, 0.49 mm]. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$a_{H_{\\ell}} = \\left( \\frac{27 q_{a} \\eta H_{\\ell}}{2 \\rho_{\\ell} g_{0}} \\right)^{1/3}$}", + "\\boxed{[0.36, 0.49]}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null, + "mm" + ], + "points": [ + 0.3, + 0.2 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_3_a_1.png", + "image_question/IPhO_2025_3_a_2.png", + "image_question/IPhO_2025_3_a_3.png" + ] + }, + { + "id": "IPhO_2025_3_A_6", + "context": "[Champagne] \n\nChampagne is a French sparkling wine. Fermentation of sugars produces carbon dioxide $(\\text{CO}_2)$ in the bottle. The molar concentration of $(\\text{CO}_2)$ in the liquid phase $c_{\\ell}$ and the partial pressure $P_{\\text{CO}_2}$ in the gas phase are related by $c_{\\ell} = k_{\\mathrm{H}} P_{\\text{CO}_2}$, known as Henry's law and where $k_{\\mathrm{H}}$ is called Henry's constant.\n\nData: \nSurface tension of champagne $\\sigma = 47 \\times 10^{-3} \\mathrm{J} \\cdot \\mathrm{m}^{-2}$ \nDensity of the liquid $\\rho_{\\ell} = 1.0 \\times 10^{3} \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$ \nHenry's constant at $T_{0} = 20^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(20^{\\circ}\\mathrm{C}) = 3.3 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nHenry's constant at $T_{0} = 6^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(6^{\\circ}\\mathrm{C}) = 5.4 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nAtmospheric pressure $P_{0} = 1 \\mathrm{bar} = 1.0 \\times 10^{5} \\mathrm{Pa}$ \nGases are ideal with an adiabatic coefficient $\\gamma = 1.3$ \n\n[figure1] \nFig. 1. A glass filled with champagne. \n\n[Part A - Nucleation, growth and rise of bubbles] \n\nImmediately after opening a bottle of champagne at temperature $T_{0} = 20^{\\circ}\\mathrm{C}$, we fill a glass. The pressure in the liquid is $P_{0}$ and its temperature stays constant at $T_{0}$. The concentration $c_{\\ell}$ of dissolved $\\text{CO}_2$ exceeds the equilibrium concentration and we study the nucleation of a $\\text{CO}_2$ bubble. We note $a$ its radius and $P_{\\mathrm{b}}$ its inner pressure. \n\n(A.1) Express the pressure $P_{\\mathrm{b}}$ in terms of $P_{0}$, $a$ and $\\sigma$. \n\nIn the liquid, the concentration of dissolved $\\text{CO}_2$ depends on the distance to the bubble. At long distance we recover the value $c_{\\ell}$ and we note $c_{\\mathrm{b}}$ the concentration close to the bubble surface. According to Henry's law, $c_{\\mathrm{b}} = k_{\\mathrm{H}} P_{\\mathrm{b}}$. We furthermore assume in all the problem that bubbles contain only $\\text{CO}_2$. \n\nSince $c_{\\ell} \\neq c_{\\mathrm{b}}$,$\\text{CO}_2$ molecules diffuse from areas of high to low concentration. We assume also that any molecule from the liquid phase reaching the bubble surface is transferred to the vapour. \n\n(A.2) Express the critical radius $a_{\\mathrm{c}}$ above which a bubble is expected to grow in terms of $P_{0}, \\sigma, c_{\\ell}$ and $c_{0}$ where $c_{0} = k_{\\mathrm{H}} P_{0}$. Calculate numerically $a_{\\mathrm{c}}$ for $c_{\\ell} = 4 c_{0}$. \n\nIn practice, bubbles mainly grow from pre-existing gas cavities. Consider then a bubble with initial radius $a_{0} \\approx 40 \\mathrm{\\mu m}$. The number of moles of $\\text{CO}_2$ transferred at the bubble's surface per unit area and time is noted $j$. Two models are possible for $j$. \n\nmodel (1) $j = \\frac{D}{a} \\left(c_{\\ell} - c_{\\mathrm{b}}\\right)$ where $D$ is the diffusion coefficient of $\\text{CO}_2$ in the liquid. \nmodel (2) $j = K \\left(c_{\\ell} - c_{\\mathrm{b}}\\right)$ where $K$ is a constant here. \n\nExperimentally, the bubble radius $a(t)$ is found to depend on time as shown in Fig. 2. Here $c_{\\ell} \\approx 4{c}_{0}$, and since bubbles are large enough to be visible, the excess pressure due to surface tension can be neglected and $P_{\\mathrm{b}} \\approx P_{0}$. \n\n(A.3) Express the number of $\\text{CO}_2$ moles in the bubble $n_{\\mathrm{c}}$ in terms of $a, P_{0}, T_{0}$ and ideal gas constant $R$. Find $a(t)$ for both models. Indicate which model explains the experimental results in Fig. 2. Depending on your answer, calculate numerically $K$ or $D$. \n\n[figure2]\nFig. 2. Time evolution of $\\text{CO}_2$ bubble radius in a glass of champagne (adapted from [1]). \n\nEventually bubbles detach from the bottom of the glass and continue to grow while rising. Fig. 3. shows a train of bubbles. The bubbles of the train have the same initial radius and are emitted at a constant frequency $f_{\\mathrm{b}} = 20 \\mathrm{Hz}}$. \n\n[figure3]\nFig. 3. A train of bubbles. The photo is rotated horizontally for the page layout (adapted from [1]). \n\nFor the range of velocities studied here, the drag force $F$ on a bubble of radius $a$ moving at velocity $v$ in a liquid of dynamic viscosity $\\eta$ is given by Stokes' law $F = 6\\pi \\eta a v$. Measurements show that at any moment in time, the bubble can be assumed to be travelling at its terminal velocity. \n\n(A.4) Give the expression of the main forces exerted on a vertically rising bubble. Obtain the expression of $v(a)$. Give a numerical estimate of $\\eta$ using $\\rho_{\\ell}$, $g_{0}$ and quantities measured on Fig. 3. \n\nThe quasi-stationary growth of bubbles with rate $q_{a} = \\frac{\\mathrm{d} a}{\\mathrm{d} t}$ still applies during bubble rise. \n\n(A.5) Express the radius $a_{H_{\\ell}}$ of a bubble reaching the free surface in terms of height travelled $H_{\\ell}$, growth rate $q_{a} = \\frac{\\mathrm{d} a}{\\mathrm{d} t}$, and any constants you may need. Assume $a_{H_{\\ell}} \\gg a_{0}$ and $q_{a}$ constant, and give the numerical value of $a_{H_{\\ell}}$ with $H_{\\ell} = 10 \\mathrm{cm}$ and $q_{a}$ corresponding to Fig. 2. \n\nThere are $N_{\\mathrm{b}}$ nucleation sites of bubbles. Assume that the bubbles are nucleated at a constant frequency $f_{\\mathrm{b}}$ at the bottom of a glass of champagne (height $H_{\\ell}$ for a volume $V_{\\ell}$), with $a_{0}$ still negligible. Neglect diffusion of $\\text{CO}_2$ at the free surface.", + "question": "(1) Write the differential equation for $c_{\\ell}(t)$. \n(2) Obtain from this equation the characteristic time $\\tau$ for the decay of the concentration of dissolved $\\text{CO}_2$ in the liquid.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly states that the rate of bubbles reaching the free surface by unit time is $N_b f_b$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly derives that the volume of $\\text{CO}_2$ released per unit time at the free surface is $\\frac{d V}{d t} = \\frac{4}{3} \\pi a_{H_{\\ell}}^{3} N_{b} f_{b}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly derives the exact expression of $\\frac{d V}{d t}t = \\frac{18 \\pi N_{b} f_{b} \\eta H_{\\ell}}{\\rho_{\\ell} g_{0}} q_{a}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly derives the expression of $q_a$ as $q_a = \\frac{d a}{d t} = \\frac{R T_{0}}{P_{0}} K \\left(c_{\\ell} - c_{0}\\right)$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly derives the first order linear differential equation $\\frac{\\mathrm{d} c_{\\ell}}{\\mathrm{d} t} + \\frac{18 \\pi N_{\\mathrm{b}} f_{\\mathrm{b}} \\eta K H_{\\ell}}{\\rho_{\\ell} g_{0} V_{\\ell}} (c_{\\ell} - c_{0}) = 0$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly derives the expression of the characteristic time $\\tau$ as $\\tau = \\frac{\\rho_{\\ell} g_{0} V_{\\ell}}{18 \\pi N_{\\mathrm{b}} f_{\\mathrm{b}} \\eta K H_{\\ell}}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{\\mathrm{d} c_{\\ell}}{\\mathrm{d} t} + \\frac{18 \\pi N_{\\mathrm{b}} f_{\\mathrm{b}} \\eta K H_{\\ell}}{\\rho_{\\ell} g_{0} V_{\\ell}} (c_{\\ell} - c_{0}) = 0$}", + "\\boxed{$\\tau = \\frac{\\rho_{\\ell} g_{0} V_{\\ell}}{18 \\pi N_{\\mathrm{b}} f_{\\mathrm{b}} \\eta K H_{\\ell}}$}" + ], + "answer_type": [ + "Equation", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 0.9, + 0.2 + ], + "modality": "text+data figure", + "field": "Thermodynamics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_3_a_1.png", + "image_question/IPhO_2025_3_a_2.png", + "image_question/IPhO_2025_3_a_3.png" + ] + }, + { + "id": "IPhO_2025_3_B_1", + "context": "[Champagne] \n\nChampagne is a French sparkling wine. Fermentation of sugars produces carbon dioxide $(\\text{CO}_2)$ in the bottle. The molar concentration of $(\\text{CO}_2)$ in the liquid phase $c_{\\ell}$ and the partial pressure $P_{\\text{CO}_2}$ in the gas phase are related by $c_{\\ell} = k_{\\mathrm{H}} P_{\\text{CO}_2}$, known as Henry's law and where $k_{\\mathrm{H}}$ is called Henry's constant.\n\nData: \nSurface tension of champagne $\\sigma = 47 \\times 10^{-3} \\mathrm{J} \\cdot \\mathrm{m}^{-2}$ \nDensity of the liquid $\\rho_{\\ell} = 1.0 \\times 10^{3} \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$ \nHenry's constant at $T_{0} = 20^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(20^{\\circ}\\mathrm{C}) = 3.3 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nHenry's constant at $T_{0} = 6^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(6^{\\circ}\\mathrm{C}) = 5.4 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nAtmospheric pressure $P_{0} = 1 \\mathrm{bar} = 1.0 \\times 10^{5} \\mathrm{Pa}$ \nGases are ideal with an adiabatic coefficient $\\gamma = 1.3$ \n\n[figure1] \nFig. 1. A glass filled with champagne. \n\n[Part B - Acoustic emission of a bursting bubble] \n\nSmall bubbles are nearly spherical as they reach the free surface. Once the liquid film separating the bubble from the air thins out sufficiently, a circular hole of radius $r$ forms in the film and,driven by surface tension, opens very quickly (Fig. 4. left). The hole opens at constant speed $v_{\\mathrm{f}}$ (Fig. 4. right). The film outside the rim remains still, with constant thickness $h$. \n\n[figure4]\nFig. 4. (Left) (a) Bubble at the surface: (1) liquid, (2) air at pressure $P_{0}$ and (3), $\\text{CO}_2$ at pressure $P_{\\mathrm{b}}$, ($\\beta$) and ($\\gamma$) retraction of the liquid film, where the rim is in dark blue, ($\\delta$) bubble collapse. (Right) Retraction of the liquid film at time $t$. Top: sketch of the pierced film seen from above. Bottom: cross-section of the rim and the retracting film. During $\\mathrm{d}t$ the rim accumulates nearby liquid (dotted). \n\nDue to dissipative processes, only half of the difference of the surface energy between $t$ and $t + \\mathrm{d}t$ of the rim and the accumulated liquid is transformed into kinetic energy. We further assume that the variation of the surface of the rim is negligible compared to that of the film.", + "question": "Express $v_{\\mathrm{f}}$ in terms of $\\rho_{\\ell}$, $\\sigma$ and $h$.", + "marking": [ + [ + "Award 0.1 pt if the answer contains the expression of kinetic energy. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer includes the variation of kinetic energy: during $\\mathrm{d}t$, the volume $\\delta V$ gains kinetic energy $\\mathrm{d}E_{\\mathrm{c}} = \\frac{1}{2} \\rho_{\\ell} \\delta V v_{f}^{2} = \\frac{1}{2} \\rho_{\\ell} h \\ell v_{f} \\mathrm{d}t = \\pi r \\rho_{\\ell} h v_{f} \\mathrm{d}t$ (other differential or finite variations are also correct). Otherwise, award 0 pt.", + "Award 0.1 pt if the answer includes the exact expression of a surface energy $E_s = \\sigma S$ for a surface $S$ (a variation of expression is also correct). Otherwise, award 0 pt.", + "Award 0.3 pt if the answer derives the exact expression of $\\delta E_s = - 2 \\sigma \\ell v_{f} \\mathrm{d}t = -4 \\sigma \\pi r v_{f} \\mathrm{d}t$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly applies the kinetic energy theorem: the lost energy satisfies $\\delta E_{s}/2 < 0$, leading to $\\mathrm{d}E_{c} + \\delta E_{s} = \\delta E_{s}/2$. Partial points: if the answer neglects to account for energy loss, award 0.1 pt. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer derives the exact expression of $v_f$: $v_{f} = \\sqrt{\\frac{2\\sigma}{\\rho_{\\ell}h}}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$v_{f} = \\sqrt{\\frac{2\\sigma}{\\rho_{\\ell}h}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.1 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_3_a_1.png", + "image_question/IPhO_2025_3_b_1.png" + ] + }, + { + "id": "IPhO_2025_3_B_2", + "context": "[Champagne] \n\nChampagne is a French sparkling wine. Fermentation of sugars produces carbon dioxide $(\\text{CO}_2)$ in the bottle. The molar concentration of $(\\text{CO}_2)$ in the liquid phase $c_{\\ell}$ and the partial pressure $P_{\\text{CO}_2}$ in the gas phase are related by $c_{\\ell} = k_{\\mathrm{H}} P_{\\text{CO}_2}$, known as Henry's law and where $k_{\\mathrm{H}}$ is called Henry's constant.\n\nData: \nSurface tension of champagne $\\sigma = 47 \\times 10^{-3} \\mathrm{J} \\cdot \\mathrm{m}^{-2}$ \nDensity of the liquid $\\rho_{\\ell} = 1.0 \\times 10^{3} \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$ \nHenry's constant at $T_{0} = 20^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(20^{\\circ}\\mathrm{C}) = 3.3 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nHenry's constant at $T_{0} = 6^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(6^{\\circ}\\mathrm{C}) = 5.4 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nAtmospheric pressure $P_{0} = 1 \\mathrm{bar} = 1.0 \\times 10^{5} \\mathrm{Pa}$ \nGases are ideal with an adiabatic coefficient $\\gamma = 1.3$ \n\n[figure1] \nFig. 1. A glass filled with champagne. \n\n[Part B - Acoustic emission of a bursting bubble] \n\nSmall bubbles are nearly spherical as they reach the free surface. Once the liquid film separating the bubble from the air thins out sufficiently, a circular hole of radius $r$ forms in the film and,driven by surface tension, opens very quickly (Fig. 4. left). The hole opens at constant speed $v_{\\mathrm{f}}$ (Fig. 4. right). The film outside the rim remains still, with constant thickness $h$. \n\n[figure4]\nFig. 4. (Left) (a) Bubble at the surface: (1) liquid, (2) air at pressure $P_{0}$ and (3), $\\text{CO}_2$ at pressure $P_{\\mathrm{b}}$, ($\\beta$) and ($\\gamma$) retraction of the liquid film, where the rim is in dark blue, ($\\delta$) bubble collapse. (Right) Retraction of the liquid film at time $t$. Top: sketch of the pierced film seen from above. Bottom: cross-section of the rim and the retracting film. During $\\mathrm{d}t$ the rim accumulates nearby liquid (dotted). \n\nDue to dissipative processes, only half of the difference of the surface energy between $t$ and $t + \\mathrm{d}t$ of the rim and the accumulated liquid is transformed into kinetic energy. We further assume that the variation of the surface of the rim is negligible compared to that of the film. \n\n(B.1) Express $v_{\\mathrm{f}}$ in terms of $\\rho_{\\ell}$, $\\sigma$ and $h$. \n\n[figure5]\nFig. 5. (Left) a Helmholtz resonator. (Right) a bubble as an oscillator. \n\nWhen the film bursts, it releases internal pressure and emits a sound. We model this acoustic emission by a Helmholtz resonator: a cavity open to the atmosphere at $P_{0}$ through a bottleneck aperture of area $S$ (Fig. 5. left). In the neck, a mass $m_{\\mathrm{p}}$ makes small amplitude position oscillations due to the pressure forces it experiences as the gas in the cavity expands or compresses adiabatically. The gravity force on $m_{\\mathrm{p}}$ is negligible compared to pressure forces. Let $V_{0}$ be the volume of gas under the mass $m_{\\mathrm{p}}$ for $P = P_{0}$ as $z = 0$.", + "question": "Express the frequency of oscillation $f_{0}$ of $m_{\\mathrm{p}}$. Hint: for $\\varepsilon \\ll 1$, $(1 + \\varepsilon)^{\\alpha} \\approx 1 + \\alpha \\varepsilon$.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly states the pressure forces on $m_p$ with $P_0$ is $F_z = P(t) S - P_0 S$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the expression of volume $V(t) = V_0 + S z$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly derives the expression of $P(t)$ with adiabatic reversible process for an ideal gas: $P(t) = P_{0}\\left(\\frac{V_{0}}{V_{0} + Sz}\\right)^{\\gamma} = P_{0} \\left(\\frac{1}{1 + Sz/V_{0}}\\right)^{\\gamma}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly approximates the pressure as $P(t) \\approx P_0(1 - \\gamma \\frac{Sz}{V_0})$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly obtains the exact linearized pressure force $F_z = -\\gamma S^2 P_0 \\frac{z}{V_0}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer applies the Newton's second law and derives $m_p \\dot{z} = -\\gamma S^2 P_0 \\frac{z}{V_0} = 0$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly derives the angular frequency of Harmonic oscillator as $\\omega_{0}^{2} = S^{2} \\frac{P_{0} \\gamma}{m_{p} V_{0}}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly derives the expression of $f_0$ as $f_{0} = \\frac{1}{2\\pi} \\sqrt{\\frac{S^{2} P_{0} \\gamma}{m_{p} V_{0}}}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$f_{0} = \\frac{1}{2\\pi} \\sqrt{\\frac{S^{2} P_{0} \\gamma}{m_{p} V_{0}}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.1 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_3_a_1.png", + "image_question/IPhO_2025_3_b_1.png", + "image_question/IPhO_2025_3_b_2.png" + ] + }, + { + "id": "IPhO_2025_3_B_3", + "context": "[Champagne] \n\nChampagne is a French sparkling wine. Fermentation of sugars produces carbon dioxide $(\\text{CO}_2)$ in the bottle. The molar concentration of $(\\text{CO}_2)$ in the liquid phase $c_{\\ell}$ and the partial pressure $P_{\\text{CO}_2}$ in the gas phase are related by $c_{\\ell} = k_{\\mathrm{H}} P_{\\text{CO}_2}$, known as Henry's law and where $k_{\\mathrm{H}}$ is called Henry's constant.\n\nData: \nSurface tension of champagne $\\sigma = 47 \\times 10^{-3} \\mathrm{J} \\cdot \\mathrm{m}^{-2}$ \nDensity of the liquid $\\rho_{\\ell} = 1.0 \\times 10^{3} \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$ \nHenry's constant at $T_{0} = 20^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(20^{\\circ}\\mathrm{C}) = 3.3 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nHenry's constant at $T_{0} = 6^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(6^{\\circ}\\mathrm{C}) = 5.4 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nAtmospheric pressure $P_{0} = 1 \\mathrm{bar} = 1.0 \\times 10^{5} \\mathrm{Pa}$ \nGases are ideal with an adiabatic coefficient $\\gamma = 1.3$ \n\n[figure1] \nFig. 1. A glass filled with champagne. \n\n[Part B - Acoustic emission of a bursting bubble] \n\nSmall bubbles are nearly spherical as they reach the free surface. Once the liquid film separating the bubble from the air thins out sufficiently, a circular hole of radius $r$ forms in the film and,driven by surface tension, opens very quickly (Fig. 4. left). The hole opens at constant speed $v_{\\mathrm{f}}$ (Fig. 4. right). The film outside the rim remains still, with constant thickness $h$. \n\n[figure4]\nFig. 4. (Left) (a) Bubble at the surface: (1) liquid, (2) air at pressure $P_{0}$ and (3), $\\text{CO}_2$ at pressure $P_{\\mathrm{b}}$, ($\\beta$) and ($\\gamma$) retraction of the liquid film, where the rim is in dark blue, ($\\delta$) bubble collapse. (Right) Retraction of the liquid film at time $t$. Top: sketch of the pierced film seen from above. Bottom: cross-section of the rim and the retracting film. During $\\mathrm{d}t$ the rim accumulates nearby liquid (dotted). \n\nDue to dissipative processes, only half of the difference of the surface energy between $t$ and $t + \\mathrm{d}t$ of the rim and the accumulated liquid is transformed into kinetic energy. We further assume that the variation of the surface of the rim is negligible compared to that of the film. \n\n(B.1) Express $v_{\\mathrm{f}}$ in terms of $\\rho_{\\ell}$, $\\sigma$ and $h$. \n\n[figure5]\nFig. 5. (Left) a Helmholtz resonator. (Right) a bubble as an oscillator. \n\nWhen the film bursts, it releases internal pressure and emits a sound. We model this acoustic emission by a Helmholtz resonator: a cavity open to the atmosphere at $P_{0}$ through a bottleneck aperture of area $S$ (Fig. 5. left). In the neck, a mass $m_{\\mathrm{p}}$ makes small amplitude position oscillations due to the pressure forces it experiences as the gas in the cavity expands or compresses adiabatically. The gravity force on $m_{\\mathrm{p}}$ is negligible compared to pressure forces. Let $V_{0}$ be the volume of gas under the mass $m_{\\mathrm{p}}$ for $P = P_{0}$ as $z = 0$. \n\n(B.2) Express the frequency of oscillation $f_{0}$ of $m_{\\mathrm{p}}$. Hint: for $\\varepsilon \\ll 1$, $(1 + \\varepsilon)^{\\alpha} \\approx 1 + \\alpha \\varepsilon$. \n\nThe Helmholtz model may be used for a bubble of radius $a$. $V_{0}$ is the volume of the closed bubble. From litterature,the mass of the equivalent of the piston is $m_{p} = 8 \\rho_{g} r^{3}/3$ where $r$ is the radius of the circular aperture and $\\rho_{g} = 1.8 \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$ is the density of the gas (Fig. 5. right). During the bursting process, $r$ goes from 0 to $r_{\\mathrm{c}}$, given by $r_{\\mathrm{c}} = \\frac{2}{\\sqrt{3}} a^{2} \\sqrt{\\frac{\\rho_{\\ell} g_{0}}{\\sigma}}$. At the same time, the frequency of emitted sound increases until a maximum value of $40 \\mathrm{kHz}$ and the bursting time is $t_{b} = 3 \\times 10^{-2} \\mathrm{ms}$.", + "question": "(1) Find the radius $a$ of the champagne film separating the bubble from the atmosphere. Express your answer in $mm$. \n(2) Find the thickness $h$ of the champagne film separating the bubble from the atmosphere. Express your answer in $\\mu m$.", + "marking": [ + [ + "Award 0.1 pt if the answer states that the maximal value of $f_0$ is $f_0 = 40 \\mathrm{kHz}$ obtained for $r = r_c$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly derives either: (i) the exact expression $f_0 = \\frac{1}{2 \\pi} \\sqrt{\\frac{\\gamma P_{0}}{\\rho_{g}}} \\sqrt{\\frac{3\\sqrt{3}\\pi}{16a} \\sqrt{\\frac{\\rho_{\\ell} g_{0}}{\\sigma}}}$, or (ii) the expression $a = \\frac{3\\sqrt{3}}{64\\pi} \\frac{\\gamma P_{0}}{\\rho_{g} f_{0}^{2}} \\sqrt{\\frac{\\rho_{\\ell} g_{0}}{\\sigma}}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly obtains the exact numerical value of $a$ within the range [0.5 mm, 0.6 mm]. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly derives the relationship between $t_b$, $v_f$ and $r_c$: $v_f = \\frac{r_c}{t_b}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly derives the expression of $h$ as $h = \\frac{2\\sigma}{\\rho_{\\ell} v_{f}^{2}}$ or $h = \\frac{3 t_{b}^{2}}{2a^{4}} \\sqrt{\\frac{\\sigma^{3}}{\\rho_{\\ell}^{3} g_{0}}}$ (both expressions are correct). Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly calculates the numerical value of $h$ as $h = 3.7 \\mathrm{\\mu m}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{[0.5, 0.6]}", + "\\boxed{3.7}" + ], + "answer_type": [ + "Numerical Value", + "Numerical Value" + ], + "unit": [ + "$mm$", + "$\\mu m$" + ], + "points": [ + 0.6, + 0.5 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_3_a_1.png", + "image_question/IPhO_2025_3_b_1.png", + "image_question/IPhO_2025_3_b_2.png" + ] + }, + { + "id": "IPhO_2025_3_C_1", + "context": "[Champagne] \n\nChampagne is a French sparkling wine. Fermentation of sugars produces carbon dioxide $(\\text{CO}_2)$ in the bottle. The molar concentration of $(\\text{CO}_2)$ in the liquid phase $c_{\\ell}$ and the partial pressure $P_{\\text{CO}_2}$ in the gas phase are related by $c_{\\ell} = k_{\\mathrm{H}} P_{\\text{CO}_2}$, known as Henry's law and where $k_{\\mathrm{H}}$ is called Henry's constant.\n\nData: \nSurface tension of champagne $\\sigma = 47 \\times 10^{-3} \\mathrm{J} \\cdot \\mathrm{m}^{-2}$ \nDensity of the liquid $\\rho_{\\ell} = 1.0 \\times 10^{3} \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$ \nHenry's constant at $T_{0} = 20^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(20^{\\circ}\\mathrm{C}) = 3.3 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nHenry's constant at $T_{0} = 6^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(6^{\\circ}\\mathrm{C}) = 5.4 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nAtmospheric pressure $P_{0} = 1 \\mathrm{bar} = 1.0 \\times 10^{5} \\mathrm{Pa}$ \nGases are ideal with an adiabatic coefficient $\\gamma = 1.3$ \n\n[figure1] \nFig. 1. A glass filled with champagne. \n\n[Part C - Popping champagne] \n\nIn a bottle, the total quantity of $\\text{CO}_2$ is $n_{\\mathrm{T}} = 0.2 \\mathrm{mol}$, either dissolved in the volume $V_{\\mathrm{L}} = 750 \\mathrm{mL}$ of liquid champagne, or as a gas in the volume $V_{\\mathrm{G}} = 25 \\mathrm{mL}$ under the cork (Fig. 6. left). $V_{\\mathrm{G}}$ contains only $\\text{CO}_2$. The equilibrium between both $\\text{CO}_2$ phases follows Henry's Law. We suppose that the fast gaseous $\\text{CO}_2$ expansion when the bottle is opened, is adiabatic and reversible. Ambient temperature $T_{0}$ and pressure $P_{0} = 1$ bar are constant. \n\n[figure6]\nFig. 6. Left: traditional bottleneck: (1) surrounding air, (2) cork stopper, (3) headspace, (4) liquid champagne. Right: Two phenomena observed while opening the bottle at two different temperatures (adapted from [2]).", + "question": "(1) Give the numerical value of the pressure $P_{\\mathrm{i}}$ of gaseous $\\text{CO}_2$ in the bottle for $T_{0} = 6^{\\circ}\\mathrm{C}$. Express your answer in units of bar. \n(2) Give the numerical value of the pressure $P_{\\mathrm{i}}$ of gaseous $\\text{CO}_2$ in the bottle for $T_{0} = 20^{\\circ}\\mathrm{C}$. Express your answer in units of bar.", + "marking": [ + [ + "Award 0.1 pt if the answer correctly states the conservation of $\\text{CO}_2$ molecules: $n_{T} = n_{V} + n_{L} = n_{V} + k_{H}(T_{0}) P_{i} V_{L}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly provides the expression of $P_i$: $P_{i} = \\frac{n_{T}}{V_{L} k_{H}(T_{0}) + \\frac{V_{G}}{R T_{0}}} = \\frac{\\frac{n_{T} R T_{0}}{V_{G}}}{1 + R T_{0} k_{H}(T_{0}) \\frac{V_{L}}{V_{G}}}$ (equivalent forms are also correct). Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the value of $P_i$ for $T_0 = 6^{\\circ} C$: $P_i = 4.81 \\mathrm{bar}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly gives the value of $P_i$ for $T_0 = 20^{\\circ} C$: $P_i = 7.76 \\mathrm{bar}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{4.81}", + "\\boxed{7.76}" + ], + "answer_type": [ + "Numerical Value", + "Numerical Value" + ], + "unit": [ + "bar", + "bar" + ], + "points": [ + 0.2, + 0.2 + ], + "modality": "text+variable figure", + "field": "Thermodynamics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_3_a_1.png", + "image_question/IPhO_2025_3_c_1.png" + ] + }, + { + "id": "IPhO_2025_3_C_2", + "context": "[Champagne] \n\nChampagne is a French sparkling wine. Fermentation of sugars produces carbon dioxide $(\\text{CO}_2)$ in the bottle. The molar concentration of $(\\text{CO}_2)$ in the liquid phase $c_{\\ell}$ and the partial pressure $P_{\\text{CO}_2}$ in the gas phase are related by $c_{\\ell} = k_{\\mathrm{H}} P_{\\text{CO}_2}$, known as Henry's law and where $k_{\\mathrm{H}}$ is called Henry's constant.\n\nData: \nSurface tension of champagne $\\sigma = 47 \\times 10^{-3} \\mathrm{J} \\cdot \\mathrm{m}^{-2}$ \nDensity of the liquid $\\rho_{\\ell} = 1.0 \\times 10^{3} \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$ \nHenry's constant at $T_{0} = 20^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(20^{\\circ}\\mathrm{C}) = 3.3 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nHenry's constant at $T_{0} = 6^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(6^{\\circ}\\mathrm{C}) = 5.4 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nAtmospheric pressure $P_{0} = 1 \\mathrm{bar} = 1.0 \\times 10^{5} \\mathrm{Pa}$ \nGases are ideal with an adiabatic coefficient $\\gamma = 1.3$ \n\n[figure1] \nFig. 1. A glass filled with champagne. \n\n[Part C - Popping champagne] \n\nIn a bottle, the total quantity of $\\text{CO}_2$ is $n_{\\mathrm{T}} = 0.2 \\mathrm{mol}$, either dissolved in the volume $V_{\\mathrm{L}} = 750 \\mathrm{mL}$ of liquid champagne, or as a gas in the volume $V_{\\mathrm{G}} = 25 \\mathrm{mL}$ under the cork (Fig. 6. left). $V_{\\mathrm{G}}$ contains only $\\text{CO}_2$. The equilibrium between both $\\text{CO}_2$ phases follows Henry's Law. We suppose that the fast gaseous $\\text{CO}_2$ expansion when the bottle is opened, is adiabatic and reversible. Ambient temperature $T_{0}$ and pressure $P_{0} = 1$ bar are constant. \n\n[figure6]\nFig. 6. Left: traditional bottleneck: (1) surrounding air, (2) cork stopper, (3) headspace, (4) liquid champagne. Right: Two phenomena observed while opening the bottle at two different temperatures (adapted from [2]). \n\n(C.1) Give the numerical value of the pressure $P_{\\mathrm{i}}$ of gaseous $\\text{CO}_2$ in the bottle for $T_{0} = 6^{\\circ}\\mathrm{C}$ and $T_{0} = 20^{\\circ}\\mathrm{C}$. \n\nAnother step of champagne production (not described here) leads to the following values of $P_{i}$ that we will use for the next questions: $P_i = 4.69$ bar at $T_{0} = 6^{\\circ}\\mathrm{C}$ and $P_{i} = 7.45$ bar at $T_{0} = 20^{\\circ}\\mathrm{C}$. \n\nDuring bottle opening, two different phenomena can be observed, depending on $T_{0}$ (Fig. 6. right). \n- either a blue fog appears, due to the formation of solid $\\text{CO}_2$ crystals (but water condensation is inhibited); \n- or a grey-white fog appears, due to water vapor condensation in the air surrounding the bottleneck. In this latter case, there is no formation of $\\text{CO}_2$ solid crystals. \n\nThe saturated vapor pressure $P_{\\text{sat}}^{\\text{CO}_2}$ for the $\\text{CO}_2$ solid/gas transition follows: $\\log_{10}\\left( \\frac{P_{\\text{sat}}^{\\text{CO}_2}}{P_{0}}\\right) = A - \\frac{B}{T + C}$ with $T$ in $\\mathrm{K}$, $A = 6.81$, $B = 1.30 \\times 10^{3} \\mathrm{K}$ and $C = -3.49 \\mathrm{K}$.", + "question": "(1) Give the numerical value $T_{\\mathrm{f}}$ of the $\\text{CO}_2$ gas at the end of the expansion, after opening a bottle, if $T_{0} = 6^{\\circ}\\mathrm{C}$, if no phase transition occured. Express your answer in $K$. \n\n(2) Give the numerical value $T_{\\mathrm{f}}$ of the $\\text{CO}_2$ gas at the end of the expansion, after opening a bottle, if $T_{0} = 20^{\\circ}\\mathrm{C}$, if no phase transition occured. Express your answer in $K$. \n\n(3) Choose which statements are true (several statements possible): \n(A) At $T_{0} = 6^{\\circ}\\mathrm{C}$ a grey-white fog appears while opening the bottle. \n(B) At $T_{0} = 6^{\\circ}\\mathrm{C}$ a blue fog appears while opening the bottle. \n(C) At $T_{0} = 20^{\\circ}\\mathrm{C}$ a grey-white fog appears while opening the bottle. \n(D) At $T_{0} = 20^{\\circ}\\mathrm{C}$ a blue fog appears while opening the bottle.", + "marking": [ + [ + "Award 0.1 pt if the answer states that the adiabatic reversible expansion goes from $P_i$ to $P_0$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer provides the correct literal expression for $T_f$: $T_{f} = T_{0} (\\frac{P_{i}}{P_{0}})^{(1/\\gamma) - 1}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly calculates both the pressure and temperature for $T_0 = 6^{\\circ} C$: $P_i = 4.69 \\mathrm{bar}$ and $T_f = 195.3 K$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly calculates both the pressure and temperature for $T_0 = 20^{\\circ} C$: $P_i = 7.45 \\mathrm{bar}$ and $T_f = 184.3 K$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer states either: (i) the idea of comparison between $P_{\\text{sat}}$ and $P_0$, or (ii) the evaluation of the transition temperature at $P_0$ and comparison with $T_f$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer performs the correct numerical comparisons for both temperature cases (showing $P_{\\text{sat}}^{\\text{CO}_2}$ > $P_0$ or $T_f$ < $T_{\\text{trans}}$). Otherwise, award 0 pt.", + "Award 0.1 pt if and only if the answer identifies both correct statements: A and D. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{195.3}", + "\\boxed{184.3}", + "\\boxed{A, D}" + ], + "answer_type": [ + "Numerical Value", + "Numerical Value", + "Multiple Choice" + ], + "unit": [ + "K", + "K", + null + ], + "points": [ + 0.2, + 0.2, + 0.3 + ], + "modality": "text+variable figure", + "field": "Thermodynamics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_3_a_1.png", + "image_question/IPhO_2025_3_c_1.png" + ] + }, + { + "id": "IPhO_2025_3_C_3", + "context": "[Champagne] \n\nChampagne is a French sparkling wine. Fermentation of sugars produces carbon dioxide $(\\text{CO}_2)$ in the bottle. The molar concentration of $(\\text{CO}_2)$ in the liquid phase $c_{\\ell}$ and the partial pressure $P_{\\text{CO}_2}$ in the gas phase are related by $c_{\\ell} = k_{\\mathrm{H}} P_{\\text{CO}_2}$, known as Henry's law and where $k_{\\mathrm{H}}$ is called Henry's constant.\n\nData: \nSurface tension of champagne $\\sigma = 47 \\times 10^{-3} \\mathrm{J} \\cdot \\mathrm{m}^{-2}$ \nDensity of the liquid $\\rho_{\\ell} = 1.0 \\times 10^{3} \\mathrm{kg} \\cdot \\mathrm{m}^{-3}$ \nHenry's constant at $T_{0} = 20^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(20^{\\circ}\\mathrm{C}) = 3.3 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nHenry's constant at $T_{0} = 6^{\\circ}\\mathrm{C}$, $k_{\\mathrm{H}}(6^{\\circ}\\mathrm{C}) = 5.4 \\times 10^{-4} \\mathrm{mol} \\cdot \\mathrm{m}^{-3} \\cdot \\mathrm{Pa}^{-1}$ \nAtmospheric pressure $P_{0} = 1 \\mathrm{bar} = 1.0 \\times 10^{5} \\mathrm{Pa}$ \nGases are ideal with an adiabatic coefficient $\\gamma = 1.3$ \n\n[figure1] \nFig. 1. A glass filled with champagne. \n\n[Part C - Popping champagne] \n\nIn a bottle, the total quantity of $\\text{CO}_2$ is $n_{\\mathrm{T}} = 0.2 \\mathrm{mol}$, either dissolved in the volume $V_{\\mathrm{L}} = 750 \\mathrm{mL}$ of liquid champagne, or as a gas in the volume $V_{\\mathrm{G}} = 25 \\mathrm{mL}$ under the cork (Fig. 6. left). $V_{\\mathrm{G}}$ contains only $\\text{CO}_2$. The equilibrium between both $\\text{CO}_2$ phases follows Henry's Law. We suppose that the fast gaseous $\\text{CO}_2$ expansion when the bottle is opened, is adiabatic and reversible. Ambient temperature $T_{0}$ and pressure $P_{0} = 1$ bar are constant. \n\n[figure6]\nFig. 6. Left: traditional bottleneck: (1) surrounding air, (2) cork stopper, (3) headspace, (4) liquid champagne. Right: Two phenomena observed while opening the bottle at two different temperatures (adapted from [2]). \n\n(C.1) Give the numerical value of the pressure $P_{\\mathrm{i}}$ of gaseous $\\text{CO}_2$ in the bottle for $T_{0} = 6^{\\circ}\\mathrm{C}$ and $T_{0} = 20^{\\circ}\\mathrm{C}$. \n\nAnother step of champagne production (not described here) leads to the following values of $P_{i}$ that we will use for the next questions: $P_i = 4.69$ bar at $T_{0} = 6^{\\circ}\\mathrm{C}$ and $P_{i} = 7.45$ bar at $T_{0} = 20^{\\circ}\\mathrm{C}$. \n\nDuring bottle opening, two different phenomena can be observed, depending on $T_{0}$ (Fig. 6. right). \n- either a blue fog appears, due to the formation of solid $\\text{CO}_2$ crystals (but water condensation is inhibited); \n- or a grey-white fog appears, due to water vapor condensation in the air surrounding the bottleneck. In this latter case, there is no formation of $\\text{CO}_2$ solid crystals. \n\nThe saturated vapor pressure $P_{\\text{sat}}^{\\text{CO}_2}$ for the $\\text{CO}_2$ solid/gas transition follows: $\\log_{10}\\left( \\frac{P_{\\text{sat}}^{\\text{CO}_2}}{P_{0}}\\right) = A - \\frac{B}{T + C}$ with $T$ in $\\mathrm{K}$, $A = 6.81$, $B = 1.30 \\times 10^{3} \\mathrm{K}$ and $C = -3.49 \\mathrm{K}$. \n\n(C.2) Give the numerical value $T_{\\mathrm{f}}$ of the $\\text{CO}_2$ gas at the end of the expansion, after opening a bottle, if $T_{0} = 6^{\\circ}\\mathrm{C}$ and if $T_{0} = 20^{\\circ}\\mathrm{C}$, if no phase transition occured. Choose which statements are true (several statements possible): \n(A) At $T_{0} = 6^{\\circ}\\mathrm{C}$ a grey-white fog appears while opening the bottle. \n(B) At $T_{0} = 6^{\\circ}\\mathrm{C}$ a blue fog appears while opening the bottle. \n(C) At $T_{0} = 20^{\\circ}\\mathrm{C}$ a grey-white fog appears while opening the bottle. \n(D) At $T_{0} = 20^{\\circ}\\mathrm{C}$ a blue fog appears while opening the bottle. \n\nDuring bottle opening, the cork stopper pops out. We now determine the maximum height $H_{\\mathrm{c}}$ it reaches. Assume that the friction force $F$ due to the bottleneck on the cork stopper is $F = \\alpha A$ where $A$ is the area of contact and $\\alpha$ is a constant to determine. Initially,the pressure force slightly overcomes the friction force. The cork's mass is $m = 10 \\mathrm{g}$, its diameter $d = 1.8 \\mathrm{cm}$ and the length of the cylindrical part initially stuck in the bottleneck is $\\ell_{0} = 2.5 \\mathrm{cm}$. Once the cork has left the bottleneck, you can neglect the net pressure force.", + "question": "Give the numerical value of $H_{\\mathrm{c}}$ if the external temperature is $T_{0} = 6^{\\circ}\\mathrm{C}$. Express your answer in $m$.", + "marking": [ + [ + "Award 0.2 pt if the answer provides the correct expression for $\\alpha$: $\\alpha = (P_i - P_0) \\frac{d}{4\\ell_0}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly expresses the friction work: $W_f = -\\frac{(P_i - P_0)\\pi d^2}{8}\\ell_0$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer states the consequences of adiabatic reversible expansion (either 1st principle with $Q = 0$ or $P V^{\\gamma} = P_i V_G^{\\gamma}$). Otherwise, award 0 pt.", + "Award 0.3 pt if the answer provides the exact expression for work: $W_{CO2\\to cork} = \\frac{P_i V_G}{\\gamma-1} \\left(1 - \\frac{1}{(1 + \\frac{\\pi d^2 \\ell_0}{4V_G})^{\\gamma-1}}\\right)$. If $P$ is considered constant during expansion, award 0 pt.", + "Award 0.1 pt if the answer includes the correct work due to external pressure: $W_e = -P_0 \\cdot \\frac{\\pi d^2}{4} \\ell_0$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer shows correct kinetic energy $E_c$ with all three contributions (even with errors in individual terms): $E_c = -\\delta U_g + W_f + W_e$. If external pressure contribution is missing, award 0 pt.", + "Award 0.1 pt if the answer demonstrates correct energy balance during free flight or uses Newton's second law properly. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer provides the correct numerical value for $H_c = 7.7 \\text{m}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{7.7}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "m" + ], + "points": [ + 1.3 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "IPhO_2025", + "image_question": [ + "image_question/IPhO_2025_3_a_1.png", + "image_question/IPhO_2025_3_c_1.png" + ] + } +] \ No newline at end of file diff --git a/data/NBPhO_2024.json b/data/NBPhO_2024.json new file mode 100644 index 0000000000000000000000000000000000000000..56736473c622f44a95b4b21e01a7744a5435ecaa --- /dev/null +++ b/data/NBPhO_2024.json @@ -0,0 +1,745 @@ +[ + { + "information": "None." + }, + { + "id": "NBPhO_2024_1_1", + "context": "[Four Charges] \n\nFour identical particles are initially in the corners of a square, as shown in the figure. All particles have the same charge $q$, mass $m$, and the same magnitude of initial velocity $v_0$. The directions of the initial velocities are indicated in the figure. You can assume $v \\ll c$ and ignore gravity.\n\n[figure1]", + "question": "After a long time has passed, what is the magnitude of the final velocity $v_f$ of the particles with respect to the center of mass of the system?", + "marking": [ + [ + "Award 0.2 pt if the answer shows the idea of using energy conservation to relate initial and final states. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer recognizes symmetry and equality of particle quantities (e.g., all four particles have same mass and charge). Otherwise, award 0 pt.", + "Award 0.2 pt if the answer expresses total energy as a sum of kinetic and electrostatic energy. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer writes the formula for electrostatic energy with two different distances: $\\frac{k q^2}{L}$ and $\\frac{k q^2}{\\sqrt{2}L}$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer includes the factor $\\frac{1}{2}$ in the electrostatic energy to avoid double counting from pairings. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer correctly states that the final energy is purely kinetic: $E_{\\text{final}} = 4 \\cdot \\frac{1}{2} m v_f^2$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer gives the correct final expression for $v_f$: $v_f = \\sqrt{v_0^2 + \\frac{k q^2}{L m} \\cdot \\frac{4 + \\sqrt{2}}{2}}$. Partial points: award 0.2 pt if only dimensionless factors are missing but the final answer is reasonable. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$v_{f} = \\sqrt{v_0^{2} + \\frac{k{q}^{2}}{Lm} \\frac{4 + \\sqrt{2}}{2}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "point": [ + 2.0 + ], + "modality": "text+variable figure", + "field": "Electromagnetism", + "source": "NBPhO_2024", + "image_question": [ + "image_question/NBPhO_2024_1_1_1.png" + ] + }, + { + "id": "NBPhO_2024_1_2", + "context": "[Four Charges] \n\nFour identical particles are initially in the corners of a square, as shown in the figure. All particles have the same charge $q$, mass $m$, and the same magnitude of initial velocity $v_0$. The directions of the initial velocities are indicated in the figure. You can assume $v \\ll c$ and ignore gravity.\n\n[figure1]\n\n(i) After a long time has passed, what is the magnitude of the final velocity $v_f$ of the particles with respect to the center of mass of the system?\n\nPart (i) is a preliminary question and should not be included in the final answer.", + "question": "What is the angle between the initial velocity $\\vec{v}_0$ and the final velocity $\\vec{v}_f$ of a particle?", + "marking": [ + [ + "Award 1.0 pt if the answer correctly states that the effective repulsive force $F = \\frac{A}{r^2}$ is pointing from the center of the masses (explicit expression or statement is required). Otherwise, award 0 pt.", + "Award 0.5 pt if the answer recognizes the motion as hyperbolic. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer identifies that the central charge $Q_e$ is located at the correct focus of the hyperbola. Otherwise, award 0 pt.", + "Award 1.0 pt if the answer expresses the angle $\\phi$ or $\\alpha$ in terms of geometrical parameters, e.g., $\\cos \\alpha = a/c$, or $\\tan \\alpha = b/a$. Otherwise, award 0 pt.", + "Award 1.0 pt if the answer applies the vis-viva equation or energy conservation to the hyperbolic orbit: $E = \\frac{k q Q_e}{2a} = \\frac{1}{2} m v_0^2 + \\frac{\\sqrt{2} k q Q_e}{L}$. Partial points for angular momentum conservation: award 0.3 pt if the answer mentions that angular momentum is conserved; award 0.3 pt if the answer writes the angular momentum around the center of the masses using $L$ and $v_0$; award 0.4 pt if the answer writes final angular momentum around the center of the masses using $v_f$. Otherwise, award 0 pt.", + "Award 1.0 pt if the answer derives the final answer $\\sin \\phi = \\frac{1}{1 + \\frac{4 L m v_0^2}{k q^2 (4+\\sqrt{2})}}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\sin \\phi = \\frac{1}{1 + \\frac{4 L m v_0^2}{k q^2 (4+\\sqrt{2})}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "point": [ + 5.0 + ], + "modality": "text+variable figure", + "field": "Electromagnetism", + "source": "NBPhO_2024", + "image_question": [ + "image_question/NBPhO_2024_1_1_1.png" + ] + }, + { + "id": "NBPhO_2024_2_1", + "context": "[Oklo Fission Reactor] \n\nBased on the ratio of the uranium isotopes $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$, as well as the abundances of the isotopes produced by nuclear reactors, researchers have established that self-sustaining natural nuclear reactors operated in Oklo ca $T_0 = 1.8 \\times {10}^{9}$ years ago in Gabon, central Africa. For such reactors to exist, two conditions must be met: (a) presence of deposits with high enough concentration of uranium; (b) sufficiently high abundance of $^{235}\\mathrm{U}$ in natural uranium. Rich uranium ores were created by floods: scattered uranium was dissolved in oxygen-rich water and transported by it to underground pools. Significant concentration of oxygen appeared in the atmosphere only around 2.5 billion years ago, so the first condition was not met earlier than that. You will learn below that the abundance of $^{235}\\mathrm{U}$ decreases relatively fast in time, so the second condition ceased to be satisfied soon after the operation of Oklo's reactor. \n\nWhat made the operation of Oklo's reactor possible was a stable influx of ground water that kept the uranium deposits sufficiently wet. Water is the so-called moderator for the fission reactor: it slows down neutrons emerging from fission reactions, dramatically enhancing the chances of a neutron triggering the fission of a next $^{235}\\mathrm{U}$ nucleus. \n\nIn what follows,in addition to $T_0$, you can use the following numerical values. Energy released by the fission of a single $^{235}\\mathrm{U}$ nucleus: $E_0 = 200 \\mathrm{MeV}$. \nHalf-life of $^{235} \\mathrm{U}$: $\\tau_5 \\approx 7 \\times 10^8$ years. \nHalf-life of $^{238}\\mathrm{U}$: $\\tau_8 \\approx 4.5 \\times 10^9$ years. \nLatent heat of evaporation of water: $L = 2260 \\mathrm{kJ} \\mathrm{kg}^{-1}$. \nSpecific heat of water $c = 4200 \\mathrm{J} \\mathrm{kg}^{-1} \\mathrm{K}^{-1}$. \nAbundance of $^{235}\\mathrm{U}$ in natural uranium today: $R = 0.72\\%$. We define abundance as the number of atoms of the isotope, normalized to the number of atoms of the given element.\n\nAverage abundance of $^{235}\\mathrm{U}$ in the uranium from Oklo's uranium ore today: $R_{O} = 0.62\\%$. \nThe total amount of uranium in Oklo's mine today: $M = 5 \\times {10}^{8}\\mathrm{kg}$. \n\nThe duration of the time period over which \nOklo's reactor operated: $T \\approx 1 \\times {10}^{5}$ year. \nElementary charge: $e = 1.6 \\times {10}^{-19}\\mathrm{C}$. \nAtomic mass unit: $u = 1.66 \\times {10}^{-27} \\mathrm{kg}$. \nAvogadro's number: $N_{A} = 6.02 \\times {10}^{23} \\mathrm{mol}^{-1}$. \n\nNote that: (a) the abundance of other isotopes of uranium besides $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$ is negligibly small; (b) $^{235}\\mathrm{U}$ is not among the decay products of $^{238}\\mathrm{U}$; and (c) fission channels other than the fission of $^{235}\\mathrm{U}$ (e.g., synthesis and fission of plutonium) can be neglected.", + "question": "What was the abundance of $^{235}\\mathrm{U}$ in natural uranium when the Oklo's reactor operated? Express your answer as a percentage.", + "marking": [ + [ + "Award 0.3 pt if the answer correctly gives the amount of $^{238}\\mathrm{U}$ during operation as $\\nu_8^{\\prime} = \\nu_8 2^{T_0 / \\tau_8}$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly gives the amount of $^{235}\\mathrm{U}$ during operation as $\\nu_5^{\\prime} = \\nu_5 2^{T_0 / \\tau_5}$. Otherwise, award 0 pt.", + "Award 0.3 pt if the abundance of $^{235}\\mathrm{U}$ is expressed in terms of $\\nu_5$ and $\\nu_8$, where $\\nu$ is the amount of $^{235} \\mathrm{U}$ at some point in time. Otherwise, award 0 pt.", + "Award 0.3 pt if the ratio $\\nu_5 / \\nu_8$ is expressed in terms of $R$ as $\\nu_5 / \\nu_8 = R / (1 - R)$, where $\\nu$ is the amount of $^{235} \\mathrm{U}$ at some point in time. Otherwise, award 0 pt.", + "Award 0.3 pt if the abundance of $^{235}\\mathrm{U}$ is correctly expressed in terms of $R$, and the final numerical answer $\\approx 3.16\\%$ is given. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$3.16 \\%$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "point": [ + 1.5 + ], + "modality": "text-only", + "field": "Modern Physics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_2_2", + "context": "[Oklo Fission Reactor] \n\nBased on the ratio of the uranium isotopes $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$, as well as the abundances of the isotopes produced by nuclear reactors, researchers have established that self-sustaining natural nuclear reactors operated in Oklo ca $T_0 = 1.8 \\times {10}^{9}$ years ago in Gabon, central Africa. For such reactors to exist, two conditions must be met: (a) presence of deposits with high enough concentration of uranium; (b) sufficiently high abundance of $^{235}\\mathrm{U}$ in natural uranium. Rich uranium ores were created by floods: scattered uranium was dissolved in oxygen-rich water and transported by it to underground pools. Significant concentration of oxygen appeared in the atmosphere only around 2.5 billion years ago, so the first condition was not met earlier than that. You will learn below that the abundance of $^{235}\\mathrm{U}$ decreases relatively fast in time, so the second condition ceased to be satisfied soon after the operation of Oklo's reactor. \n\nWhat made the operation of Oklo's reactor possible was a stable influx of ground water that kept the uranium deposits sufficiently wet. Water is the so-called moderator for the fission reactor: it slows down neutrons emerging from fission reactions, dramatically enhancing the chances of a neutron triggering the fission of a next $^{235}\\mathrm{U}$ nucleus. \n\nIn what follows,in addition to $T_0$, you can use the following numerical values. Energy released by the fission of a single $^{235}\\mathrm{U}$ nucleus: $E_0 = 200 \\mathrm{MeV}$. \nHalf-life of $^{235} \\mathrm{U}$: $\\tau_5 \\approx 7 \\times 10^8$ years. \nHalf-life of $^{238}\\mathrm{U}$: $\\tau_8 \\approx 4.5 \\times 10^9$ years. \nLatent heat of evaporation of water: $L = 2260 \\mathrm{kJ} \\mathrm{kg}^{-1}$. \nSpecific heat of water $c = 4200 \\mathrm{J} \\mathrm{kg}^{-1} \\mathrm{K}^{-1}$. \nAbundance of $^{235}\\mathrm{U}$ in natural uranium today: $R = 0.72\\%$. We define abundance as the number of atoms of the isotope, normalized to the number of atoms of the given element.\n\nAverage abundance of $^{235}\\mathrm{U}$ in the uranium from Oklo's uranium ore today: $R_{O} = 0.62\\%$. \nThe total amount of uranium in Oklo's mine today: $M = 5 \\times {10}^{8}\\mathrm{kg}$. \n\nThe duration of the time period over which \nOklo's reactor operated: $T \\approx 1 \\times {10}^{5}$ year. \nElementary charge: $e = 1.6 \\times {10}^{-19}\\mathrm{C}$. \nAtomic mass unit: $u = 1.66 \\times {10}^{-27} \\mathrm{kg}$. \nAvogadro's number: $N_{A} = 6.02 \\times {10}^{23} \\mathrm{mol}^{-1}$. \n\nNote that: (a) the abundance of other isotopes of uranium besides $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$ is negligibly small; (b) $^{235}\\mathrm{U}$ is not among the decay products of $^{238}\\mathrm{U}$; and (c) fission channels other than the fission of $^{235}\\mathrm{U}$ (e.g., synthesis and fission of plutonium) can be neglected. \n\n(i) What was the abundance of $^{235}\\mathrm{U}$ in natural uranium when the Oklo's reactor operated? \n\nPart (i) is a preliminary question and should not be included in the final answer.", + "question": "What was the average power of the Oklo's reactor? Express your answer in $W$.", + "marking": [ + [ + "Award 0.4 pt if the answer gives the final mass of $^{235}\\mathrm{U}$ by the end of operation as $M_5^{\\prime} = 2^{T_0 / \\tau_5} M R_0$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer gives the final mass of $^{238}\\mathrm{U}$ by the end of operation as $M_8^{\\prime} = 2^{T_0 / \\tau_8} M (1 - R)$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer gives the initial mass of $^{235}\\mathrm{U}$ at the beginning of operation as $2^{T_0 / \\tau_8} M (1 - R) \\cdot \\frac{R^{\\prime}}{1 - R^{\\prime}}$. Otherwise, award 0 pt.", + "Award 0.3 pt if the number of atoms that have undergone fission is computed as $N = N_A \\cdot \\Delta M / 0.235$. Otherwise, award 0 pt.", + "Award 0.2 pt if the total energy is given as $E = N E_0$. Otherwise, award 0 pt.", + "Award 0.2 pt if all units (e.g., mass in $kg$, power in $W$) are correctly converted and used. Otherwise, award 0 pt.", + "Award 0.3 pt if the power is calculated correctly using $P = \\nu E_0 / T$ (where $\\nu$ is the number of $^{235}U$ nuclei that reacted), and the numerical value is approximately $7.73 \\times 10^7$ W. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$7.73 \\times 10^7$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "W" + ], + "point": [ + 2.0 + ], + "modality": "text-only", + "field": "Modern Physics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_2_3", + "context": "[Oklo Fission Reactor] \n\nBased on the ratio of the uranium isotopes $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$, as well as the abundances of the isotopes produced by nuclear reactors, researchers have established that self-sustaining natural nuclear reactors operated in Oklo ca $T_0 = 1.8 \\times {10}^{9}$ years ago in Gabon, central Africa. For such reactors to exist, two conditions must be met: (a) presence of deposits with high enough concentration of uranium; (b) sufficiently high abundance of $^{235}\\mathrm{U}$ in natural uranium. Rich uranium ores were created by floods: scattered uranium was dissolved in oxygen-rich water and transported by it to underground pools. Significant concentration of oxygen appeared in the atmosphere only around 2.5 billion years ago, so the first condition was not met earlier than that. You will learn below that the abundance of $^{235}\\mathrm{U}$ decreases relatively fast in time, so the second condition ceased to be satisfied soon after the operation of Oklo's reactor. \n\nWhat made the operation of Oklo's reactor possible was a stable influx of ground water that kept the uranium deposits sufficiently wet. Water is the so-called moderator for the fission reactor: it slows down neutrons emerging from fission reactions, dramatically enhancing the chances of a neutron triggering the fission of a next $^{235}\\mathrm{U}$ nucleus. \n\nIn what follows,in addition to $T_0$, you can use the following numerical values. Energy released by the fission of a single $^{235}\\mathrm{U}$ nucleus: $E_0 = 200 \\mathrm{MeV}$. \nHalf-life of $^{235} \\mathrm{U}$: $\\tau_5 \\approx 7 \\times 10^8$ years. \nHalf-life of $^{238}\\mathrm{U}$: $\\tau_8 \\approx 4.5 \\times 10^9$ years. \nLatent heat of evaporation of water: $L = 2260 \\mathrm{kJ} \\mathrm{kg}^{-1}$. \nSpecific heat of water $c = 4200 \\mathrm{J} \\mathrm{kg}^{-1} \\mathrm{K}^{-1}$. \nAbundance of $^{235}\\mathrm{U}$ in natural uranium today: $R = 0.72\\%$. We define abundance as the number of atoms of the isotope, normalized to the number of atoms of the given element.\n\nAverage abundance of $^{235}\\mathrm{U}$ in the uranium from Oklo's uranium ore today: $R_{O} = 0.62\\%$. \nThe total amount of uranium in Oklo's mine today: $M = 5 \\times {10}^{8}\\mathrm{kg}$. \n\nThe duration of the time period over which \nOklo's reactor operated: $T \\approx 1 \\times {10}^{5}$ year. \nElementary charge: $e = 1.6 \\times {10}^{-19}\\mathrm{C}$. \nAtomic mass unit: $u = 1.66 \\times {10}^{-27} \\mathrm{kg}$. \nAvogadro's number: $N_{A} = 6.02 \\times {10}^{23} \\mathrm{mol}^{-1}$. \n\nNote that: (a) the abundance of other isotopes of uranium besides $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$ is negligibly small; (b) $^{235}\\mathrm{U}$ is not among the decay products of $^{238}\\mathrm{U}$; and (c) fission channels other than the fission of $^{235}\\mathrm{U}$ (e.g., synthesis and fission of plutonium) can be neglected. \n\n(i) What was the abundance of $^{235}\\mathrm{U}$ in natural uranium when the Oklo's reactor operated? \n\n(ii) What was the average power of the Oklo's reactor? \n\nParts (i)–(ii) are preliminary questions and should not be included in the final answer.", + "question": "Qualitatively explain why was Oklo's reactor operating in a stable regime and did not blow up. Water inflow rate varied over time; what happened to the reactor when the water inflow rate increased two times?", + "marking": [ + [ + "Award 0.3 pt if the answer states that neutrons from fission are unlikely to cause further fission unless slowed down (moderation). Otherwise, award 0 pt.", + "Award 0.2 pt if the answer identifies water as a moderator for slowing down neutrons. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer explains that the reactor was self-regulating because increased power would vaporize water, reducing moderation and slowing the reaction. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer states that when water inflow doubles, the reactor power also doubles. Otherwise, award 0 pt." + ] + ], + "answer": [ + "" + ], + "answer_type": [ + "Open-Ended" + ], + "unit": [ + null + ], + "point": [ + 1.5 + ], + "modality": "text-only", + "field": "Modern Physics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_2_4", + "context": "[Oklo Fission Reactor] \n\nBased on the ratio of the uranium isotopes $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$, as well as the abundances of the isotopes produced by nuclear reactors, researchers have established that self-sustaining natural nuclear reactors operated in Oklo ca $T_0 = 1.8 \\times {10}^{9}$ years ago in Gabon, central Africa. For such reactors to exist, two conditions must be met: (a) presence of deposits with high enough concentration of uranium; (b) sufficiently high abundance of $^{235}\\mathrm{U}$ in natural uranium. Rich uranium ores were created by floods: scattered uranium was dissolved in oxygen-rich water and transported by it to underground pools. Significant concentration of oxygen appeared in the atmosphere only around 2.5 billion years ago, so the first condition was not met earlier than that. You will learn below that the abundance of $^{235}\\mathrm{U}$ decreases relatively fast in time, so the second condition ceased to be satisfied soon after the operation of Oklo's reactor. \n\nWhat made the operation of Oklo's reactor possible was a stable influx of ground water that kept the uranium deposits sufficiently wet. Water is the so-called moderator for the fission reactor: it slows down neutrons emerging from fission reactions, dramatically enhancing the chances of a neutron triggering the fission of a next $^{235}\\mathrm{U}$ nucleus. \n\nIn what follows,in addition to $T_0$, you can use the following numerical values. Energy released by the fission of a single $^{235}\\mathrm{U}$ nucleus: $E_0 = 200 \\mathrm{MeV}$. \nHalf-life of $^{235} \\mathrm{U}$: $\\tau_5 \\approx 7 \\times 10^8$ years. \nHalf-life of $^{238}\\mathrm{U}$: $\\tau_8 \\approx 4.5 \\times 10^9$ years. \nLatent heat of evaporation of water: $L = 2260 \\mathrm{kJ} \\mathrm{kg}^{-1}$. \nSpecific heat of water $c = 4200 \\mathrm{J} \\mathrm{kg}^{-1} \\mathrm{K}^{-1}$. \nAbundance of $^{235}\\mathrm{U}$ in natural uranium today: $R = 0.72\\%$. We define abundance as the number of atoms of the isotope, normalized to the number of atoms of the given element.\n\nAverage abundance of $^{235}\\mathrm{U}$ in the uranium from Oklo's uranium ore today: $R_{O} = 0.62\\%$. \nThe total amount of uranium in Oklo's mine today: $M = 5 \\times {10}^{8}\\mathrm{kg}$. \n\nThe duration of the time period over which \nOklo's reactor operated: $T \\approx 1 \\times {10}^{5}$ year. \nElementary charge: $e = 1.6 \\times {10}^{-19}\\mathrm{C}$. \nAtomic mass unit: $u = 1.66 \\times {10}^{-27} \\mathrm{kg}$. \nAvogadro's number: $N_{A} = 6.02 \\times {10}^{23} \\mathrm{mol}^{-1}$. \n\nNote that: (a) the abundance of other isotopes of uranium besides $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$ is negligibly small; (b) $^{235}\\mathrm{U}$ is not among the decay products of $^{238}\\mathrm{U}$; and (c) fission channels other than the fission of $^{235}\\mathrm{U}$ (e.g., synthesis and fission of plutonium) can be neglected. \n\n(i) What was the abundance of $^{235}\\mathrm{U}$ in natural uranium when the Oklo's reactor operated? \n\n(ii) What was the average power of the Oklo's reactor? \n\n(iii) Qualitatively explain why was Oklo's reactor operating in a stable regime and did not blow up. Water inflow rate varied over time; what happened to the reactor when the water inflow rate increased two times? \n\nParts (i)–(iii) are preliminary questions and should not be included in the final answer.", + "question": "Estimate the total mass of water that flowed into the Oklo's reactor during its operation period. Express your answer in $kg$.", + "marking": [ + [ + "Award 0.5 pt if the answer assumes that all energy from the reactor went into heating and vaporizing water. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer approximates $\\Delta T \\approx 100\\ ^\\circ\\mathrm{C}$, e.g., by considering water flows in at $0^\\circ\\mathrm{C}$ and leaves at $100^\\circ\\mathrm{C}$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer considers both heating and vaporization. Otherwise, award 0 pt.", + "Award 0.3 pt if the amount of energy absorbed by 1kg of water is $E_w = 2.68 \\times 10^6 \\mathrm{J}$ or similar results. Otherwise, award 0 pt.", + "Award 0.6 pt if the total mass of water is correctly computed as $\\nu E_0 / E_w = 9.09 \\times 10^{13} \\mathrm{kg}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$9.09 \\times 10^{13}$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "kg" + ], + "point": [ + 2.0 + ], + "modality": "text-only", + "field": "Modern Physics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_3_1", + "context": "", + "question": "[Sticky Ball] \n\nA glass ball of radius $R$ rests on a flat glass plate. A tiny droplet of water (of surface tension $\\sigma$) is injected with a syringe so that water forms a small thin neck between the ball and the plate. Both the ball and the plate are perfectly hydrophilic, i.e. the contact angle of water is $0^{\\circ}$. Find the increase of the normal force ($\\Delta F$) between the plate and the ball caused by the presence of the neck of water.", + "marking": [ + [ + "Award 0.5 pt if the answer states that the meniscus is (roughly) inverse spherical or uses the constant $r$ to characterise the meniscus as inverse spherical. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer states or uses $r \\ll \\rho \\ll R$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer states that surface tension force at the contact is negligible or that $\\Delta F$ comes from pressure difference (either explicitly or implicitly). Otherwise, award 0 pt.", + "Award 0.7 pt if the answer uses the relation $2r \\cdot 2R \\approx \\rho^2$. If this relation is not found, partial points can be earned as below. (1) Award 0.1 pt if the relation $CT \\approx 2r$ is used or mentioned. (2) Award 0.1 pt if the relation $R + r \\approx R$ is used. (3) Award 0.2 pt if $AC \\approx \\rho$ is stated or used. (4) Award 0.3 pt if the answer uses $AC^2 \\approx CT \\cdot CD$ or states the interesecting secants theorems. Otherwise, award 0 pt.", + "Award 1.0 pt if the pressure difference is given as $\\Delta p = \\sigma / r$. If this equation is not found, partial points can be earned as below. (1) Award 0.2 pt for attempting to use a Laplace-Young formula such as $\\Delta p = \\sigma (1/r_1 + 1/r_2)$. (2) Award 0.3 pt if $\\Delta p = \\sigma / (1/\\rho - 1/r)$ is used. Otherwise, award 0 pt.", + "Award 0.5 pt if the vertical cross-sectional area is given as $S = \\pi \\rho^2$. Otherwise, award 0 pt.", + "Award 0.5 pt if the force difference is expressed as $\\Delta F = \\Delta p \\cdot S$. Otherwise, award 0 pt.", + "Award 0.2 pt if the final expression $\\Delta F \\approx 4 \\pi \\sigma R$ is obtained. Partial points: (1) Award 0.1 pt if the answer states that $\\Delta F > 0$ or notes that the contact force increases. (2) Award 0.1 pt if the correct dimensionless factor $4\\pi$ and correct dimensions are used (only if the approach is correct). Otherwise, award 0 pt." + ], + [ + "Award 0.5 pt if the answer states that the meniscus is (roughly) inverse spherical or uses constant $r$ to characterize the meniscus as inverse spherical. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer uses the small-angle approximation $|\\theta| \\ll 1$. Otherwise, award 0 pt.", + "Award 0.2 pt if the neck radius is approximated as $\\rho \\approx R \\theta$. Otherwise, award 0 pt.", + "Award 0.2 pt if the distance $AC \\approx \\rho$ is used. Otherwise, award 0 pt.", + "Award 0.3 pt if the curvature radius is correctly derived as $r \\approx R \\theta^2 / 4$. Otherwise, award 0 pt.", + "Award 0.5 pt if the surface tension force $\\Delta F_\\sigma$ is stated to be negligible or $\\Delta F = \\Delta F_p$ is used. Otherwise, award 0 pt.", + "Award 1.0 pt if the pressure difference is given as $\\Delta p = \\sigma / r$. If this equation is not found, partial points can be earned as below: (1) Award 0.2 pt if a Laplace-like equation is attempted, such as $\\Delta p = \\sigma (1/r_1 + 1/r_2)$. (2) Award 0.3 pt if the pressure difference is given as $\\Delta p = \\sigma (1/\\rho - 1/r)$. Otherwise, award 0 pt.", + "Award 0.5 pt if the correct effective area $S = \\pi \\rho^2$ is used for computing the force. Otherwise, award 0 pt.", + "Award 0.5 pt if the force from pressure is computed using $\\Delta F_p = - S \\Delta p$. Otherwise, award 0 pt.", + "Award 0.2 pt if the final result $\\Delta F \\approx 4 \\pi \\sigma R$ is obtained. Partial points: (1) Award 0.1 pt if the answer states that $\\Delta F > 0$ or notes that the contact force increases. (2) Award 0.1 pt if the correct dimensionless factor $4\\pi$ and correct dimensions are used (only if the approach is correct). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\Delta F \\approx 4 \\pi \\sigma R$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "point": [ + 4.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_4_1", + "context": "[Totality] \n\nTotal solar eclipses are a rare phenomenon which occur when the Moon completely covers the disk of the Sun for some parts of the Earth. This doesn't happen during every solar eclipse because the Moon's apparent size in the sky is sometimes too small to fully cover the Sun, but also because the Moon's shadow usually misses the Earth due its orbital inclination. As a result, total solar eclipses occur on average every 18 months. \n\nLet us consider a total solar eclipse where during the peak, the centre-points of Earth, the Moon and the Sun lie on a line on the same plane as the equator. We measure that right before the total solar eclipse ends at latitude $\\lambda = 28.5^{\\circ}$, the totality lasts for $t_0 = 2 \\mathrm{min}$. Earth's radius is $r_{e} = 6370 \\mathrm{km}$, Moon's radius is $r_{m} = 1740 \\mathrm{km}$, orbital period of the Moon $T_{m} = 27.3 \\mathrm{d}$,orbital radius of the Moon ${R}_{m} = 384000 \\mathrm{km}$. One day on Earth is $T_0 = 24 \\mathrm{hrs}$.", + "question": "For how long is there a place on Earth where the total solar eclipse is observable? Express your answer in hours.", + "marking": [ + [ + "Award 0.5 pt if the answer explains that the Moon's shadow speed on Earth can be approximated by the Moon's position. Otherwise, award 0 pt.", + "Award 0.5 pt if the Moon's speed is correctly calculated using $v_m = \\frac{2 \\pi R_m}{T_m}$ and the value $v_m \\approx 1.02\\ \\mathrm{km/s}$ is obtained. Otherwise, award 0 pt.", + "Award 0.5 pt if the final eclipse duration expression $T_{\\mathrm{ecl}} = \\frac{2 r_e}{v_m} = \\frac{r_e}{\\pi R_m} T_m$ is correctly obtained, where the numerical result of $T_{\\mathrm{ecl}}$ is $3.46 h$. Partial points: Deduct 0.2 pt for a minor mistake in the final expression (e.g., missing constant or slight dimensional inconsistency). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{3.46}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "h" + ], + "point": [ + 1.5 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_4_2", + "context": "[Totality] \n\nTotal solar eclipses are a rare phenomenon which occur when the Moon completely covers the disk of the Sun for some parts of the Earth. This doesn't happen during every solar eclipse because the Moon's apparent size in the sky is sometimes too small to fully cover the Sun, but also because the Moon's shadow usually misses the Earth due its orbital inclination. As a result, total solar eclipses occur on average every 18 months. \n\nLet us consider a total solar eclipse where during the peak, the centre-points of Earth, the Moon and the Sun lie on a line on the same plane as the equator. We measure that right before the total solar eclipse ends at latitude $\\lambda = 28.5^{\\circ}$, the totality lasts for $t_0 = 2 \\mathrm{min}$. Earth's radius is $r_{e} = 6370 \\mathrm{km}$, Moon's radius is $r_{m} = 1740 \\mathrm{km}$, orbital period of the Moon $T_{m} = 27.3 \\mathrm{d}$,orbital radius of the Moon ${R}_{m} = 384000 \\mathrm{km}$. One day on Earth is $T_0 = 24 \\mathrm{hrs}$. \n\n(i) For how long is there a place on Earth where the total solar eclipse is observable? \n\nPart (i) is a preliminary question and should not be included in the final answer.", + "question": "How many degrees in longitude on Earth does the total solar eclipse cover?", + "marking": [ + [ + "Award 0.3 pt if the answer identifies that the eclipse would cover $\\pi$ radians without the Earth's rotation. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly explains that the angle becomes smaller than $\\pi$ due to Earth's rotation. Otherwise, award 0 pt.", + "Award 0.4 pt if the final formula $\\left|180 \\left(1 - \\frac{2 T_{\\mathrm{ecl}}}{T_0} \\right)\\right|$ and answer $\\approx 128$ degrees is given correctly. Partial points: Deduct 0.3 pt if the answer incorrectly assumes Earth’s rotation goes against the Moon’s shadow and obtains an angle greater than $\\pi$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{128}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "degree" + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_4_3", + "context": "[Totality] \n\nTotal solar eclipses are a rare phenomenon which occur when the Moon completely covers the disk of the Sun for some parts of the Earth. This doesn't happen during every solar eclipse because the Moon's apparent size in the sky is sometimes too small to fully cover the Sun, but also because the Moon's shadow usually misses the Earth due its orbital inclination. As a result, total solar eclipses occur on average every 18 months. \n\nLet us consider a total solar eclipse where during the peak, the centre-points of Earth, the Moon and the Sun lie on a line on the same plane as the equator. We measure that right before the total solar eclipse ends at latitude $\\lambda = 28.5^{\\circ}$, the totality lasts for $t_0 = 2 \\mathrm{min}$. Earth's radius is $r_{e} = 6370 \\mathrm{km}$, Moon's radius is $r_{m} = 1740 \\mathrm{km}$, orbital period of the Moon $T_{m} = 27.3 \\mathrm{d}$,orbital radius of the Moon ${R}_{m} = 384000 \\mathrm{km}$. One day on Earth is $T_0 = 24 \\mathrm{hrs}$. \n\n(i) For how long is there a place on Earth where the total solar eclipse is observable? \n\n(ii) How many degrees in longitude on Earth does the total solar eclipse cover? \n\nParts (i)–(ii) are preliminary questions and should not be included in the final answer.", + "question": "What is the width of the path of totality near the equator? Express your answer in $km$.", + "marking": [ + [ + "Award 0.4 pt if the answer realizes that the velocities are perpendicular at the end of the eclipse. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer computes the width of the eclipse at some point on Earth as $w_\\lambda = v_m t_0 = \\frac{2\\pi R_m t_0}{T_m} \\approx 123 \\mathrm{km}$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly explains how to translate the width to the width at the equator using angular diameter $\\alpha = 2r_m / R_m$ and computes $\\alpha r_e \\approx 57.7 \\mathrm{km}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the final expression $w_\\text{eq} = v_m t_0 + \\alpha r_e = 180 \\mathrm{km}$ is correct. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{180}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "km" + ], + "point": [ + 1.5 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_4_4", + "context": "[Totality] \n\nTotal solar eclipses are a rare phenomenon which occur when the Moon completely covers the disk of the Sun for some parts of the Earth. This doesn't happen during every solar eclipse because the Moon's apparent size in the sky is sometimes too small to fully cover the Sun, but also because the Moon's shadow usually misses the Earth due its orbital inclination. As a result, total solar eclipses occur on average every 18 months. \n\nLet us consider a total solar eclipse where during the peak, the centre-points of Earth, the Moon and the Sun lie on a line on the same plane as the equator. We measure that right before the total solar eclipse ends at latitude $\\lambda = 28.5^{\\circ}$, the totality lasts for $t_0 = 2 \\mathrm{min}$. Earth's radius is $r_{e} = 6370 \\mathrm{km}$, Moon's radius is $r_{m} = 1740 \\mathrm{km}$, orbital period of the Moon $T_{m} = 27.3 \\mathrm{d}$,orbital radius of the Moon ${R}_{m} = 384000 \\mathrm{km}$. One day on Earth is $T_0 = 24 \\mathrm{hrs}$. \n\n(i) For how long is there a place on Earth where the total solar eclipse is observable? \n\n(ii) How many degrees in longitude on Earth does the total solar eclipse cover? \n\n(iii) What is the width of the path of totality near the equator? \n\nParts (i)–(iii) are preliminary questions and should not be included in the final answer.", + "question": "What is the longest amount of time the total eclipse is visible for a single location on Earth? Express your answer in minutes.", + "marking": [ + [ + "Award 0.5 pt if the answer shows that the eclipse is observable for the longest time at the equator. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly finds the velocity of the surface of the Earth as $v_e = \\frac{2\\pi r_e}{T_0} = 0.46 \\mathrm{km/s}$ and the relative speed of the Moon's shadow as $v_{\\text{rel}} = \\sqrt{v_m^2 + v_e^2 - 2 v_m v_e \\cos \\lambda} = 0.654 \\mathrm{km/s}$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer derives the formula for the maximum duration of the eclipse: $t_\\text{eq} = \\frac{w_\\text{eq}}{v_\\text{rel}} = 276 \\mathrm{s} = 4.6 \\mathrm{min}$. Partial points: Deduct 0.3 pt if the angle $\\lambda$ between the two velocities is not taken into account in the relative velocity calculation. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{4.6}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "min" + ], + "point": [ + 1.5 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_4_5", + "context": "[Totality] \n\nTotal solar eclipses are a rare phenomenon which occur when the Moon completely covers the disk of the Sun for some parts of the Earth. This doesn't happen during every solar eclipse because the Moon's apparent size in the sky is sometimes too small to fully cover the Sun, but also because the Moon's shadow usually misses the Earth due its orbital inclination. As a result, total solar eclipses occur on average every 18 months. \n\nLet us consider a total solar eclipse where during the peak, the centre-points of Earth, the Moon and the Sun lie on a line on the same plane as the equator. We measure that right before the total solar eclipse ends at latitude $\\lambda = 28.5^{\\circ}$, the totality lasts for $t_0 = 2 \\mathrm{min}$. Earth's radius is $r_{e} = 6370 \\mathrm{km}$, Moon's radius is $r_{m} = 1740 \\mathrm{km}$, orbital period of the Moon $T_{m} = 27.3 \\mathrm{d}$,orbital radius of the Moon ${R}_{m} = 384000 \\mathrm{km}$. One day on Earth is $T_0 = 24 \\mathrm{hrs}$. \n\n(i) For how long is there a place on Earth where the total solar eclipse is observable? \n\n(ii) How many degrees in longitude on Earth does the total solar eclipse cover? \n\n(iii) What is the width of the path of totality near the equator? \n\n(iv) What is the longest amount of time the total eclipse is visible for a single location on Earth? \n\nParts (i)–(iv) are preliminary questions and should not be included in the final answer.", + "question": "For how long is the total eclipse near the location described in (iii), at the distance of $a = 50 km$ from the centreline of the eclipse path? Express your answer in minutes.", + "marking": [ + [ + "Award 0.5 pt if the answer explains that the relative velocity $v_\\text{rel}$ is approximately as $v_{\\text{rel}} = \\sqrt{v_m^2 + v_e^2 - 2 v_m v_e \\cos \\lambda} = 0.654 \\mathrm{km/s}$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly applies the geometry and gives the correct time as $\\frac{1}{v_\\text{rel}}\\sqrt{w_\\text{eq}^2 - 4a^2} = 230 \\mathrm{s} = 3.8 \\mathrm{min}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{3.8}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "min" + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_4_6", + "context": "[Totality] \n\nTotal solar eclipses are a rare phenomenon which occur when the Moon completely covers the disk of the Sun for some parts of the Earth. This doesn't happen during every solar eclipse because the Moon's apparent size in the sky is sometimes too small to fully cover the Sun, but also because the Moon's shadow usually misses the Earth due its orbital inclination. As a result, total solar eclipses occur on average every 18 months. \n\nLet us consider a total solar eclipse where during the peak, the centre-points of Earth, the Moon and the Sun lie on a line on the same plane as the equator. We measure that right before the total solar eclipse ends at latitude $\\lambda = 28.5^{\\circ}$, the totality lasts for $t_0 = 2 \\mathrm{min}$. Earth's radius is $r_{e} = 6370 \\mathrm{km}$, Moon's radius is $r_{m} = 1740 \\mathrm{km}$, orbital period of the Moon $T_{m} = 27.3 \\mathrm{d}$,orbital radius of the Moon ${R}_{m} = 384000 \\mathrm{km}$. One day on Earth is $T_0 = 24 \\mathrm{hrs}$. \n\n(i) For how long is there a place on Earth where the total solar eclipse is observable? \n\n(ii) How many degrees in longitude on Earth does the total solar eclipse cover? \n\n(iii) What is the width of the path of totality near the equator? \n\n(iv) What is the longest amount of time the total eclipse is visible for a single location on Earth? \n\n(v) For how long is the total eclipse near the location described in (iii), at the distance of $a = 50 km$ from the centreline of the eclipse path? \n\nParts (i)–(v) are preliminary questions and should not be included in the final answer.", + "question": "Find the average time interval (in years) between two total solar eclipses for a given location on Earth by making the following simplifying assumptions:\n\n(a) the average width of the full eclipse path is equal to the arithmetic average of its smallest and largest width; \n(b) typical width of a full eclipse path is half of the average width of the eclipse studied above; \n(c) typical length of a full eclipse path is equal to the length of the eclipse path studied above if the Earth were not rotating; \n(d) total solar eclipses occur with equal likelihood anywhere on Earth.", + "marking": [ + [ + "Award 0.5 pt if the answer explains that the probability per eclipse is equal to the area covered by the eclipse divided by the total area of the earth. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly calculates the area covered by one eclipse as $\\pi r_e (w_\\lambda + w_\\text{eq})/4$ using the given assumptions. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly multiplies the inverse probability with the duration between eclipses and finds the correct answer, i.e. $\\frac{16r_e}{w_\\lambda + w_\\text{eq}} \\cdot 18 \\text{months} \\approx 500 \\text{years}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{500}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "years" + ], + "point": [ + 1.5 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_6_1", + "context": "", + "question": "[Cones] \n\nThe photo below shows a self-anamorphic drawing - a red heart in green background. The reflection of the red heart in the conical mirror is a reduced green heart. What is the apex angle of the conical mirror? Express your answer in degrees. You can take measurements from the photo. The distance where the photo was taken was much larger than the diameter of the red heart.\n\n[figure1]", + "marking": [ + [ + "Award 0.5 pt if the answer states that the image is formed by vertical rays because the camera is far away. Otherwise, award 0 pt.", + "Award 1 pt if the answer gives a correct explicit expression for $\\theta$ in terms of $r$ or another measurable quantity. If this is not found, partial points can be earned as below: (1) Award 0.2 pt if the answer provides a correct geometrical figure of the setup. (2) Award 0.5 pt if the answer derives $r = \\tan \\theta / \\tan 2\\theta$ or equivalent or a correct implicit equation for $\\theta$. Otherwise, award 0 pt.", + "Award 0.5 pt if the final numerical result is correct and within $2\\theta \\in [65^\\circ, 76^\\circ]$. Partial points: Deduct 0.2 pt if only $\\theta$ is given and $2\\theta$ is not reported. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{71}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "degrees" + ], + "point": [ + 2.0 + ], + "modality": "text+data figure", + "field": "Optics", + "source": "NBPhO_2024", + "image_question": [ + "image_question/NBPhO_2024_6_1_1.png" + ] + }, + { + "id": "NBPhO_2024_6_2", + "context": "", + "question": "[Cones] \n\nA point-like puck of mass $m$ can slide frictionlessly along the internal surface of a cone of half apex angle $\\theta$. The gravitational acceleration is $g$ and points along the symmetry axis of the cone at the apex. The puck starts sliding from a point $P$ on the surface of the cone with such a horizontal velocity that it will stay moving at the same fixed height while performing uniform circular motion of radius $R$. What is its speed $v$?", + "marking": [ + [ + "Award 0.5 pt if the answer states the correct balance of forces. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer derives the correct expression for $v$ as $v = \\sqrt{Rg \\cot \\theta}$. Partial points: Deduct 0.1 pt if there are mistakes in trigonometry. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$v = \\sqrt{Rg \\cot \\theta}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_6_3", + "context": "[Cones] \n\n(ii) A point-like puck of mass $m$ can slide frictionlessly along the internal surface of a cone of half apex angle $\\theta$. The gravitational acceleration is $g$ and points along the symmetry axis of the cone at the apex. The puck starts sliding from a point $P$ on the surface of the cone with such a horizontal velocity that it will stay moving at the same fixed height while performing uniform circular motion of radius $R$. What is its speed $v$? \n\nPart (ii) is a preliminary question and should not be included in the final answer.", + "question": "Now the puck starts sliding horizontally from the same point $P$ as before, but its initial speed is reduced to $v/2$. What is the smallest distance between the puck and the cone's axis during the subsequent motion?", + "marking": [ + [ + "Award 0.1 pt if the answer mentions using energy conservation. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer mentions using angular momentum conservation. Otherwise, award 0 pt.", + "Award 0.5 pt if the energy conservation equation is correctly written as $E = \\frac{v^2}{8} + g h_f = \\frac{v_f^2}{2}$. Otherwise, award 0 pt.", + "Award 0.5 pt if the angular momentum conservation equation is correctly written as $\\frac{v}{2} R = v_f R_f$. Otherwise, award 0 pt.", + "Award 0.1 pt if the relation between $h_f$ and $R_f$ is correctly given as $h_f = (R - R_f) \\cot \\theta$. Otherwise, award 0 pt.", + "Award 0.5 pt if the correct third degree polynomial in $R_f$ is derived: $8R_f^3 - 9R R_f^2 + R^3 = 0$. Otherwise, award 0 pt.", + "Award 0.5 pt if the physically meaningful root $R_f = \\frac{1 + \\sqrt{33}}{16}R$ is correctly selected. Otherwise, award 0 pt.", + "Award 0.2 pt if the final answer $R_f \\approx 0.42 R$ is correctly given. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$0.42 R$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "point": [ + 2.5 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_6_4", + "context": "[Cones] \n\n(ii) A point-like puck of mass $m$ can slide frictionlessly along the internal surface of a cone of half apex angle $\\theta$. The gravitational acceleration is $g$ and points along the symmetry axis of the cone at the apex. The puck starts sliding from a point $P$ on the surface of the cone with such a horizontal velocity that it will stay moving at the same fixed height while performing uniform circular motion of radius $R$. What is its speed $v$? \n\n(iii) Now the puck starts sliding horizontally from the same point $P$ as before, but its initial speed is reduced to $v/2$. What is the smallest distance between the puck and the cone's axis during the subsequent motion? \n\nParts (ii)–(iii) are preliminary questions and should not be included in the final answer.", + "question": "Now, the cone and the puck are moved to weightlessness. The puck starts again from the point $P$ with the same velocity as in part (ii). By how many degrees will the radius vector drawn from the cone's axis to the puck rotate during the subsequent motion? Assume that the cone is infinitely long. Express your answer in radians.", + "marking": [ + [ + "Award 0.2 pt if the answer mentions the idea of unfolding the cone. Otherwise, award 0 pt.", + "Award 1.0 pt if the answer states that the puck’s trajectory is a straight line in the folded plane. Otherwise, award 0 pt.", + "Award 0.6 pt if the answer correctly describes a $90^{\\circ}$ ($\\pi/2$) rotation in the folded diagram, or provides a correct corresponding figure. Otherwise, award 0 pt.", + "Award 0.7 pt if the answer correctly relates the rotation in the folded plane to the rotation of the radius vector, noting that a $360^{\\circ}$ rotation of the radius vector corresponds to a $2\\phi$ rotation in the drawing, and correctly derives the number of radians of rotation as $\\frac{\\pi}{2 \\sin \\theta}$. Partial points: if the answer incorrectly uses $\\theta$ instead of $2 \\theta$ as the apex angle, award 0.4 pt. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{\\pi}{2 \\sin \\theta}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + "radians" + ], + "point": [ + 2.5 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_7_1", + "context": "The dispersion relation (i.e. the dependence of the circular frequency $\\omega$ on the wave vector $k = \\frac{2\\pi}{\\lambda}$) of capillary-gravity waves is $\\omega^2 = g k^{\\alpha} + \\frac{\\sigma}{\\rho} k^{\\beta}$, where $\\sigma$ denotes the surface tension, $g = 9.81 m/s^2$, and $\\rho = 1000 \\mathrm{kg} m^{-3}$.", + "question": "(1) Determine the value of the exponent $\\alpha$. \n(2) Determine the value of the exponent $\\beta$.", + "marking": [ + [ + "Award 0.2 pt if the answer applies dimensional analysis (or an equivalent technique) to determine the exponents. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly states the units of surface tension $\\sigma$ as $\\mathrm{kg/s^2}$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly finds $\\alpha = 1$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly finds $\\beta = 3$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\alpha = 1$}", + "\\boxed{$\\beta = 3$}" + ], + "answer_type": [ + "Numerical Value", + "Numerical Value" + ], + "unit": [ + null, + null + ], + "point": [ + 0.5, + 0.5 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_7_2", + "context": "The dispersion relation (i.e. the dependence of the circular frequency $\\omega$ on the wave vector $k = \\frac{2\\pi}{\\lambda}$) of capillary-gravity waves is $\\omega^2 = g k^{\\alpha} + \\frac{\\sigma}{\\rho} k^{\\beta}$, where $\\sigma$ denotes the surface tension, $g = 9.81 m/s^2$, and $\\rho = 1000 \\mathrm{kg} m^{-3}$.\n\n[figure1] \n\n(i) Determine the values of the exponents $\\alpha$ and $\\beta$. \n\nPart (i) is a preliminary question and should not be included in the final answer.", + "question": "In the figure, we can see how an object moving with a constant speed $U = 60 \\mathrm{cm} \\mathrm{s}^{-1}$ generates a wake - a set of waves of different wavelengths. Pay attention to the short-wavelength waves whose wave crest extends from the object almost up to the edges of the photo: the presence of a very long wavefront testifies that for these particular waves, the phase and group velocities are equal.\n\n(1) Determine the expression of the surface tension $\\sigma$ of water. \n(2) Estimate the numerical value of $\\sigma$ in $g/s^2$. \nYou can take measurements from the photo. Note that while phase speed is the speed of a constant phase of the wave, the group speed $v_g = \\frac{\\mathrm{d} \\omega}{\\mathrm{d} k}$ is the speed of a wave packet (a train of waves).", + "marking": [ + [ + "Award 0.6 pt if the answer correctly identifies $\\frac{d\\omega}{dk} = \\omega / k$. Otherwise, award 0 pt.", + "Award 0.6 pt if the answer gives the correct angle $\\sin(\\mu) = \\omega / (U k)$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer measures $\\mu$ from the picture. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer provides the measured $\\mu$ within the range $[20^{\\circ}, 30^{\\circ}]$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer correctly derivatives $d\\omega/dk$ from the dispersion relation and obtains $2 \\omega \\frac{d\\omega}{dk} = g + 3 \\frac{\\sigma}{\\rho} k^2$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer correctly derives the expression for $\\sigma$ as $\\sigma = \\frac{\\rho U^4 \\sin^4 \\mu}{4 g}$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly obtains the final value $\\sigma \\approx 60 g s^{-2}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\sigma = \\frac{\\rho U^4 {\\sin}^{4} \\mu}{4g}$}", + "\\boxed{60}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null, + "g/s^2" + ], + "point": [ + 1.5, + 1.5 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [ + "image_question/NBPhO_2024_7_2_1.png" + ] + }, + { + "id": "NBPhO_2024_8_1", + "context": "Two airplanes pass each other while flying at a constant altitudes. While they have identical airspeeds, their ground speeds are $v_1$ and $v_2$, respectively, and the angle between the velocity vectors is $\\alpha$.", + "question": "Based on the above knowledge, what is the minimal possible value of the airspeed of the planes?", + "marking": [ + [ + "Award 0.2 pt if the answer represents the problem using vectors and correctly adds the velocities as $\\vec{v}_1 = \\vec{w} + \\vec{u}_1$ and $\\vec{v}_2 = \\vec{w} + \\vec{u}_2$, where $\\vec{w}$ is the speed of the wind at the airplanes' altitude, $\\vec{u}_1$ and $\\vec{u}_2$ are the planes' respective speeds in absence of wind. Partial points: award 0.1 pt if the answer has the wrong order or sign in vector addition. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer identifies that the wind velocity vector $\\vec{w}$ lies on the perpendicular bisector of segment $(AB)$, where the segment $(OA)$ corresponds to the vector $\\vec{v}_1$ ($\\vec{v}_1 = \\vec{w} + \\vec{u}_1$), the segment $(OB)$ corresponds to the vector $\\vec{v}_2$ ($\\vec{v}_2 = \\vec{w} + \\vec{u}_2$), $\\vec{w}$ is the speed of the wind at the airplanes' altitude, $\\vec{u}_1$ and $\\vec{u}_2$ are the planes' respective speeds in absence of wind. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer states that the minimum airspeed requires $\\vec{w}$ to lie on the intersection of $AB$ and the perpendicular bisector, where the segment $(OA)$ corresponds to the vector $\\vec{v}_1$ ($\\vec{v}_1 = \\vec{w} + \\vec{u}_1$), the segment $(OB)$ corresponds to the vector $\\vec{v}_2$ ($\\vec{v}_2 = \\vec{w} + \\vec{u}_2$), $\\vec{w}$ is the speed of the wind at the airplanes' altitude, $\\vec{u}_1$ and $\\vec{u}_2$ are the planes' respective speeds in absence of wind. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer calculates the correct formula for the minimum airspeed as $|\\vec{u}_1|_{\\min} = \\frac{|\\vec{v}_1 - \\vec{v}_2|}{2} = \\frac{1}{2} \\sqrt{v_1^2 + v_2^2 - 2 v_1 v_2 \\cos \\alpha}$. Partial points: award 0.2 pt if the answer has a small error but is otherwise reasonable with correct units, or if the answer is not expanded when it can be simply expanded. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{1}{2} \\sqrt{v_1^{2} + v_2^{2} - 2 v_1 v_2 \\cos \\alpha}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_8_2", + "context": "Two airplanes pass each other while flying at a constant altitudes. While they have identical airspeeds, their ground speeds are $v_1$ and $v_2$, respectively, and the angle between the velocity vectors is $\\alpha$.", + "question": "Based on the above knowledge, what is the minimal possible value of the wind speed $|\\vec{w}|$ at the altitudes of the planes?", + "marking": [ + [ + "Award 0.5 pt if the answer represents the problem using vectors and correctly adds the velocities as $\\vec{v}_1 = \\vec{w} + \\vec{u}_1$ and $\\vec{v}_2 = \\vec{w} + \\vec{u}_2$, where $\\vec{w}$ is the speed of the wind at the airplanes' altitude, $\\vec{u}_1$ and $\\vec{u}_2$ are the planes' respective speeds in absence of wind. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer states that the wind velocity vector $\\vec{w}$ lies on the perpendicular bisector of segment $(AB)$, where the segment $(OA)$ corresponds to the vector $\\vec{v}_1$ ($\\vec{v}_1 = \\vec{w} + \\vec{u}_1$), the segment $(OB)$ corresponds to the vector $\\vec{v}_2$ ($\\vec{v}_2 = \\vec{w} + \\vec{u}_2$), $\\vec{w}$ is the speed of the wind at the airplanes' altitude, $\\vec{u}_1$ and $\\vec{u}_2$ are the planes' respective speeds in absence of wind. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer states that the wind velocity $\\vec{w}$ is smallest when it is perpendicular to $l$, where $l$ is the perpendicular bisector of the segment $(AB)$, the segment $(OA)$ corresponds to the vector $\\vec{v}_1$ ($\\vec{v}_1 = \\vec{w} + \\vec{u}_1$), the segment $(OB)$ corresponds to the vector $\\vec{v}_2$ ($\\vec{v}_2 = \\vec{w} + \\vec{u}_2$), $\\vec{w}$ is the speed of the wind at the airplanes' altitude, $\\vec{u}_1$ and $\\vec{u}_2$ are the planes' respective speeds in absence of wind. Otherwise, award 0 pt.", + "Award 1.5 pt if the answer correctly calculates the minimal wind speed as $|\\vec{w}|_{\\min} = \\frac{|v_1^2 - v_2^2|}{2\\sqrt{v_1^2 + v_2^2 - 2 v_1 v_2 \\cos \\alpha}}$. Partial points: award 1.0 pt if there is a small error but the answer is otherwise reasonable with correct units, or if the answer is not expanded when it can be simply expanded. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{\\left| v_1^{2} - v_2^{2} \\right|}{2 \\sqrt{v_1^{2} + v_2^{2} - 2 v_1 v_2 \\cos \\alpha}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "point": [ + 3.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_8_3", + "context": "Two airplanes pass each other while flying at a constant altitudes. While they have identical airspeeds, their ground speeds are $v_1$ and $v_2$, respectively, and the angle between the velocity vectors is $\\alpha$.", + "question": "If now $v_1 = v_2 = v$, but it is known that the airspeed of one of the planes is two times bigger than that of the other. What is the minimal possible value of the wind speed at the altitudes of the planes?", + "marking": [ + [ + "Award 0.2 pt if the answer represents the problem using vectors and correctly adds the velocities as $\\vec{v}_1 = \\vec{w} + \\vec{u}_1$ and $\\vec{v}_2 = \\vec{w} + \\vec{u}_2$, where $\\vec{w}$ is the speed of the wind at the airplanes' altitude, $\\vec{u}_1$ and $\\vec{u}_2$ are the planes' respective speeds in absence of wind. Otherwise, award 0 pt.", + "Award 1.3 pt if the answer states that the wind velocity vector $\\vec{w}$ lies on the Apollonius circle (name not required). Partial points: award 0.5 pt if the answer realizes it is a circle but provides an incorrect one. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly calculates the radius $R_A$ of the Apollonius circle as $R_A = \\frac{2|\\vec{v}_1 - \\vec{v}_2|}{3}$, where $R_A$ is the distance from the circle's center to its intersection point with the line $AB$. Otherwise, award 0 pt.", + "Award 1 pt if the answer correctly calculates the minimal wind speed as $|\\vec{w}|_{\\min} = \\left| \\frac{4}{3}\\vec{v}_2 - \\frac{1}{3}\\vec{v}_1 \\right| - \\frac{2}{3}|\\vec{v}_2 - \\vec{v}_1| = \\frac{\\sqrt{17 - 8 \\cos \\alpha} - 4 \\sin(\\frac{\\alpha}{2})}{3} v$. Partial points: award 0.5 pt if there is a small error but the answer is otherwise reasonable with correct units, or if the answer is not expanded when it can be simply expanded. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{\\sqrt{17 - 8 \\cos \\alpha} - 4 \\sin(\\frac{\\alpha}{2})}{3} v$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "point": [ + 3.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [] + }, + { + "id": "NBPhO_2024_9_1", + "context": "Three identical small iron balls were initially arranged in an equilateral triangle formation, connected by massless nonstretchable threads. Upon being thrown into the air, the system experienced the following conditions: (a) all threads were taut initially; (b) all balls possessed different velocities; (c) all velocities were confined to the plane of the triangle as the system underwent free fall within Earth's gravitational field. At a certain moment $t = 0$, two threads ruptured, leaving two balls tethered together while the third ball separated from the rest of the system. The accompanying diagram depicts the positions of all three balls and the remaining thread within the plane of their initial arrangement at a later moment of time $t = T$ when all the balls were still continuing their free fall. To answer the questions below, you can take measurements from the figure.\n\n[figure1]", + "question": "By how many degrees did the remaining thread rotate during the time period from $t = 0$ to $t = T$? Express your answer in radians.", + "marking": [ + [ + "Award 0.5 pt if the answer shows understanding of the situation, i.e., recognizes that the motion is rotating in a plane perpendicular to the ground. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly determines the center of mass $G$ and explains that $|GA| = 2|GM|$, where $A$ is the position of the detached ball and $M$ is the midpoint of the remaining string. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer draws or clearly describes the trajectory after separation and does so correctly, i.e., the lines of motion of $A$ and $M$ are parallel but not colinear, representing the motion of the center of mass of each part. Otherwise, award 0 pt.", + "Award 1.0 pt if the answer correctly shows that $\\omega_1 = \\omega_2$ based on the triangle configuration and the two connected balls, where $\\omega_1$ is the angular velocity of the detached ball and $\\omega_2$ is the angular velocity of the two balls that remain connected. Otherwise, award 0 pt.", + "Award 0.6 pt for the answer correctly writing each of the following formulas (0.2 pt each): $v_1 = \\omega_1 r$, $s_1 = v_1 T$, and $\\alpha = \\omega_2 T$, where $v_1$ is the linear speed of the detached ball, $\\omega_1$ is the angular velocity of the detached ball, $r$ is the radius of its circular trajectory, $s_1$ is the arc length traveled by the detached ball after separation, $T$ is the travel time after separation, $\\alpha$ is the rotation angle of the two connected balls, and $\\omega_2$ is the angular velocity of the two connected balls. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer obtains the correct final expression for $\\alpha$ as $\\alpha = \\frac{s_1}{r}$, where $\\alpha$ is the rotation angle of the two connected balls, $s_1$ is the arc length traveled by the detached ball after separation, and $r$ is the radius of its circular trajectory. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer obtains the correct numerical final value $\\alpha \\approx 5.6 \\text{rad}$ (allowing for some tolerance for small measurement errors). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{5.6}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "radian" + ], + "point": [ + 4.0 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [ + "image_question/NBPhO_2024_9_1_1.png" + ] + }, + { + "id": "NBPhO_2024_9_2", + "context": "Three identical small iron balls were initially arranged in an equilateral triangle formation, connected by massless nonstretchable threads. Upon being thrown into the air, the system experienced the following conditions: (a) all threads were taut initially; (b) all balls possessed different velocities; (c) all velocities were confined to the plane of the triangle as the system underwent free fall within Earth's gravitational field. At a certain moment $t = 0$, two threads ruptured, leaving two balls tethered together while the third ball separated from the rest of the system. The accompanying diagram depicts the positions of all three balls and the remaining thread within the plane of their initial arrangement at a later moment of time $t = T$ when all the balls were still continuing their free fall. To answer the questions below, you can take measurements from the figure.\n\n[figure1] \n\n(i) By how many degrees did the remaining thread rotate during the time period from $t = 0$ to $t = T$? \n\nPart (i) is a preliminary question and should not be included in the final answer.", + "question": "Was the rotation clockwise or counterclockwise?\n\n(A) Clockwise \n(B) Counterclockwise", + "marking": [ + [ + "Award 1.0 pt if the answer correctly determines that the rotation is counterclockwise or selects option B as the final answer. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "point": [ + 1.0 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "NBPhO_2024", + "image_question": [ + "image_question/NBPhO_2024_9_1_1.png" + ] + } +] \ No newline at end of file diff --git a/data/NBPhO_2025.json b/data/NBPhO_2025.json new file mode 100644 index 0000000000000000000000000000000000000000..c121a8111e3c3255c3bd45a1947a99ea48b705c7 --- /dev/null +++ b/data/NBPhO_2025.json @@ -0,0 +1,686 @@ +[ + { + "information": "None." + }, + { + "id": "NBPhO_2025_1_1", + "context": "[Flying Dumbbell] \n\nIn this problem, we shall study the dynamics of a dumbbell consisting of two steel balls, each with radius $r = 1 \\mathrm{cm}$, connected by a steel rod with diameter $d = 1 \\mathrm{mm}$ and length $l = 10 \\mathrm{cm}$, in the absence of gravity. Unless instructed otherwise, assume steel is perfectly elastic. You may simplify your calculations by assuming $l \\gg r$.", + "question": "Given that the Young's modulus of steel is $Y = 2 \\times 10^{11} \\mathrm{Pa}$ and the density of steel is $\\rho = 7800 \\mathrm{kg}\\mathrm{m}^{-3}$, determine the period $T$ of free longitudinal oscillations of the dumbbell. (Do not consider oscillations with standing waves in the rod where the balls remain almost at rest.) Young's modulus is the ratio of a material's stress (force per unit area) to its strain (relative deformation). Express your answer in $\\mathrm{ms}$.", + "marking": [ + [ + "Award 0.5 pt if the answer explains that the oscillation is symmetric around the centre of the rod (or invokes Newton's third law). Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly expresses the stiffness of the half-rod as $k = Y \\frac{\\pi}{2} d^2 / l$. Partial points: award 0.3 pt if there is a minor mistake in the stiffness expression. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly gives the mass of the ball as $m = \\frac{4}{3} \\pi r^3 \\rho$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer realises that the system can be treated as a spring. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly obtains the oscillation period formula as $T = 2 \\pi \\sqrt{m/k}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly obtains the final answer for the oscillation period as $T \\approx 0.64 \\mathrm{ms}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{0.64}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "ms" + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2025", + "image_question": [] + }, + { + "id": "NBPhO_2025_1_2", + "context": "[Flying Dumbbell] \n\nIn this problem, we shall study the dynamics of a dumbbell consisting of two steel balls, each with radius $r = 1 \\mathrm{cm}$, connected by a steel rod with diameter $d = 1 \\mathrm{mm}$ and length $l = 10 \\mathrm{cm}$, in the absence of gravity. Unless instructed otherwise, assume steel is perfectly elastic. You may simplify your calculations by assuming $l \\gg r$. \n\n(i) Given that the Young's modulus of steel is $Y = 2 \\times 10^{11} \\mathrm{Pa}$ and the density of steel is $\\rho = 7800 \\mathrm{kg}\\mathrm{m}^{-3}$, determine the period $T$ of free longitudinal oscillations of the dumbbell. (Do not consider oscillations with standing waves in the rod where the balls remain almost at rest.) Young's modulus is the ratio of a material's stress (force per unit area) to its strain (relative deformation). \n\nPart (i) is a preliminary question and should not be included in the final answer.", + "question": "Estimate the impact time $\\tau$ when a steel ball bounces off a steel wall. Express your answer in $\\mathrm{\\mu s}$.", + "marking": [ + [ + "Award 0.5 pt if the answer realises that the compressed ball is essentially a compression wave. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly gives the formula for the speed of sound as $c_s = \\sqrt{Y / \\rho}$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly gives the relation between time, radius and speed as $\\tau \\approx 2 r / c_s = 2r \\sqrt{\\rho / Y}$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly obtains the final answer $\\tau \\approx 4 \\mathrm{\\mu s}$. Otherwise, award 0 pt." + ], + [ + "Award 0.5 pt if the answer realises that the ball can be thought of as a spring. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly approximates the ball as a spring of stiffness $\\kappa \\sim Y r$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly gives the relation between the spring constant and the time of frequency as $\\tau \\approx 2 \\pi \\sqrt{m / \\kappa}$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly obtains the final answer $\\tau \\approx 4 \\mathrm{\\mu s}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{4}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "$\\mu s$" + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2025", + "image_question": [] + }, + { + "id": "NBPhO_2025_1_4", + "context": "[Flying Dumbbell] \n\nIn this problem, we shall study the dynamics of a dumbbell consisting of two steel balls, each with radius $r = 1 \\mathrm{cm}$, connected by a steel rod with diameter $d = 1 \\mathrm{mm}$ and length $l = 10 \\mathrm{cm}$, in the absence of gravity. Unless instructed otherwise, assume steel is perfectly elastic. You may simplify your calculations by assuming $l \\gg r$. \n\n(i) Given that the Young's modulus of steel is $Y = 2 \\times 10^{11} \\mathrm{Pa}$ and the density of steel is $\\rho = 7800 \\mathrm{kg}\\mathrm{m}^{-3}$, determine the period $T$ of free longitudinal oscillations of the dumbbell. (Do not consider oscillations with standing waves in the rod where the balls remain almost at rest.) Young's modulus is the ratio of a material's stress (force per unit area) to its strain (relative deformation).\n\n(ii) Estimate the impact time $\\tau$ when a steel ball bounces off a steel wall. \n\n(iii) The dumbbell moves toward a steel wall with velocity $\\vec{v} = -v \\hat{x}$, with its axis perpendicular to the wall, and bounces back. Here, $\\hat{x}$ denotes a unit vector along the axis perpendicular to the wall. Sketch how the $x$-component $v_x$ of the front ball's velocity (the ball that collides with the wall) depends on time. \n\nParts (i)–(iii) are preliminary questions and should not be included in the final answer.", + "question": "Now, the dumbbell moves toward a steel wall with velocity $\\vec{v} = -v \\hat{x}$ as before, but its axis forms an angle $\\alpha$ with the $x$-axis. The interaction of the front ball with the wall depends qualitatively on the angle $\\alpha$, with a transition from one type of interaction to another occurring at $\\alpha = \\alpha_{0}$. Determine the value of $\\alpha_{0}$. Express your answer in degrees. Hint: $\\min \\left(\\frac{\\sin x}{x}\\right) \\approx -0.217$.", + "marking": [ + [ + "Award 0.5 pt if the answer realises it behaves as in the previous question (balls at velocity $-v$ and $v$, center of mass at rest), but it now also rotates and oscillates around the centre of mass. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly gives the expression for the angular speed of rotation as $\\Omega = 2 v \\sin \\alpha / l$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly gives the expression for the oscillation amplitude as $a = v \\cos \\alpha \\sqrt{m/k} = v \\cos \\alpha / \\omega$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer realises that the difference in interaction is whether the first ball bounces once or twice. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly gives the formula for the distance of the front ball to the wall over time as $\\frac{l}{2} \\cos \\alpha - \\left[ \\frac{l}{2} - a \\sin(\\omega t) \\right] \\cos(\\alpha + \\Omega t)$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer realises that if the distance of the front ball to the wall over time is over 0 for all $t > 0$, the first ball does not hit the wall twice. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly finds the critical angle $\\alpha_0 \\approx 25^{\\circ}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{25^{\\circ}}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "degrees" + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2025", + "image_question": [] + }, + { + "id": "NBPhO_2025_1_5", + "context": "[Flying Dumbbell] \n\nIn this problem, we shall study the dynamics of a dumbbell consisting of two steel balls, each with radius $r = 1 \\mathrm{cm}$, connected by a steel rod with diameter $d = 1 \\mathrm{mm}$ and length $l = 10 \\mathrm{cm}$, in the absence of gravity. Unless instructed otherwise, assume steel is perfectly elastic. You may simplify your calculations by assuming $l \\gg r$. \n\n(i) Given that the Young's modulus of steel is $Y = 2 \\times 10^{11} \\mathrm{Pa}$ and the density of steel is $\\rho = 7800 \\mathrm{kg}\\mathrm{m}^{-3}$, determine the period $T$ of free longitudinal oscillations of the dumbbell. (Do not consider oscillations with standing waves in the rod where the balls remain almost at rest.) Young's modulus is the ratio of a material's stress (force per unit area) to its strain (relative deformation).\n\n(ii) Estimate the impact time $\\tau$ when a steel ball bounces off a steel wall. \n\n(iii) The dumbbell moves toward a steel wall with velocity $\\vec{v} = -v \\hat{x}$, with its axis perpendicular to the wall, and bounces back. Here, $\\hat{x}$ denotes a unit vector along the axis perpendicular to the wall. Sketch how the $x$-component $v_x$ of the front ball's velocity (the ball that collides with the wall) depends on time. \n\n(iv) Now, the dumbbell moves toward a steel wall with velocity $\\vec{v} = -v \\hat{x}$ as before, but its axis forms an angle $\\alpha$ with the $x$-axis. The interaction of the front ball with the wall depends qualitatively on the angle $\\alpha$, with a transition from one type of interaction to another occurring at $\\alpha = \\alpha_{0}$. Determine the value of $\\alpha_{0}$. Hint: $\\min \\left(\\frac{\\sin x}{x}\\right) \\approx -0.217$. \n\nParts (i)–(iv) are preliminary questions and should not be included in the final answer.", + "question": "Under the assumptions of the previous task, let $\\alpha > \\alpha_{0}$. Additionally, assume that while steel is highly elastic, it is not infinitely so: any oscillations excited in the rod will decay by the time the rear ball collides with the wall. Determine the speed with which the centre of mass of the dumbbell departs from the wall.", + "marking": [ + [ + "Award 0.2 pt if the answer realises that the dumbbell rotates around its centre of mass after the first collision. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer realises that the longitudinal oscillations have decayed by the time of the second collision. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly gives the expression for the velocity of the ball at the moment of the second collision as $v \\sin \\alpha$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly gives the expression for the component of the ball's velocity in the direction of the surface normal at the moment of the second collision as $v \\sin^2 \\alpha$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer realises that the component of velocity of the second ball in the direction of the surface normal is also $v \\sin^2 \\alpha$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer realises that the speed of the centre of mass is $v \\sin^2 \\alpha$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$v \\sin^{2} \\alpha$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2025", + "image_question": [] + }, + { + "id": "NBPhO_2025_2_1", + "context": "[Evaporation] \n\nFor the subsequent tasks, the graph shows how the density of saturated water vapour in $\\mathrm{g} \\mathrm{m}^{-3}$ depends on the temperature in $^\\circ C$.\n\n[figure1]\n\nYou may also use the following characteristics of water. Specific heat capacity $c = 4200 \\mathrm{J} \\mathrm{kg}^{-1} \\mathrm{K}^{-1}$; latent heat of vaporization $L = 2260 \\mathrm{kJ} \\mathrm{kg}^{-1}$; density $\\rho = 1000 \\mathrm{kg} \\mathrm{m}^{-3}$; molar mass of water $\\mu = 18 \\mathrm{g} \\mathrm{mol}^{-1}$. You may also assume water vapour to behave as an ideal gas. The universal gas constant is $R = 8.31 \\mathrm{J} \\mathrm{mol}^{-1} \\mathrm{K}^{-1}$.", + "question": "A cylinder contains a certain amount of water at temperature $T_{0} = 90^{\\circ} \\mathrm{C}$, as shown in the figure. The cross-sectional area of the piston is $S = 1 \\mathrm{dm}^{2}$. What is the minimum pulling force required to move the piston? Express your answer in $N$. The pressure of the surrounding air is $p_{0} = 100 \\mathrm{kPa}$.\n\n[figure2]", + "marking": [ + [ + "Award 0.8 pt if the answer realises that the pressure inside the cylinder equals the saturated vapour pressure of water at temperature $T_0$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer reads the density $\\rho$ from the graph in the range $[400, 440] \\mathrm{g m^{-3}}$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer uses the ideal gas law to obtain an expression for the pressure $p_1$ at temperature $T_0$ as $p_1 = \\rho R T / \\mu$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer correctly gives the expression for the force needed to pull the piston as $S (p_0 - p_1)$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly obtains the numerical value of the minimum pulling force required to move the piston as $F \\approx 300 \\mathrm{N}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{300}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "N" + ], + "points": [ + 2.0 + ], + "modality": "text+data figure", + "field": "Thermodynamics", + "source": "NBPhO_2025", + "image_question": [ + "image_question/NBPhO_2025_2_1_1.png", + "image_question/NBPhO_2025_2_1_2.png" + ] + }, + { + "id": "NBPhO_2025_2_2", + "context": "[Evaporation] \n\nFor the subsequent tasks, the graph shows how the density of saturated water vapour in $\\mathrm{g} \\mathrm{m}^{-3}$ depends on the temperature in $^\\circ C$.\n\n[figure1]\n\nYou may also use the following characteristics of water. Specific heat capacity $c = 4200 \\mathrm{J} \\mathrm{kg}^{-1} \\mathrm{K}^{-1}$; latent heat of vaporization $L = 2260 \\mathrm{kJ} \\mathrm{kg}^{-1}$; density $\\rho = 1000 \\mathrm{kg} \\mathrm{m}^{-3}$; molar mass of water $\\mu = 18 \\mathrm{g} \\mathrm{mol}^{-1}$. You may also assume water vapour to behave as an ideal gas. The universal gas constant is $R = 8.31 \\mathrm{J} \\mathrm{mol}^{-1} \\mathrm{K}^{-1}$. \n\n(i) A cylinder contains a certain amount of water at temperature $T_{0} = 90^{\\circ} \\mathrm{C}$, as shown in the figure. The cross-sectional area of the piston is $S = 1 \\mathrm{dm}^{2}$. What is the minimum pulling force required to move the piston? The pressure of the surrounding air is $p_{0} = 100 \\mathrm{kPa}$.\n\n[figure2]\n\nPart (i) is a preliminary question and should not be included in the final answer.", + "question": "If the piston is pulled so that it shifts by $a = 3 \\mathrm{dm}$, the water cools to a temperature of $T_{1} = 89^{\\circ} \\mathrm{C}$; what is the mass of the water under the piston? Express your answer in $g$.", + "marking": [ + [ + "Award 0.2 pt if the answer reads the vapour density $\\rho_1$ from the graph in the range $[390, 420] \\mathrm{g m^{-3}}$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly gives the expression for the mass of water vapour as $m_v = S a \\rho_1$, where $S$ is the cross-sectional area of the piston, and $\\rho_1$ is the vapour density. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly gives the expression for the latent heat as $m_v L$, where $m_v$ is the mass of vapour. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly gives the expression for the heat lost by water as $(m - m_v) \\cdot c \\cdot (T_0 - T_1)$, where $m$ is the mass of the water, $m_v$ is the mass of vapour, and the water cools from $T_0$ to $T_1$. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer correctly applies the energy conservation equation $m_v L = (m - m_v) \\cdot c \\cdot (T_0 - T_1)$, where $m$ is the mass of the water, $m_v$ is the mass of vapour, and the water cools from $T_0$ to $T_1$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly obtains the expression for the mass of water $m = \\frac{\\rho_1 S a L}{c (T_0 - T_1)}$, where $\\rho_1$ is the vapour density, and the water cools from $T_0$ to $T_1$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly obtains the numerical value of the mass of water $m \\in [630, 680] \\mathrm{g}$ with the correct dimension. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{[630, 680]}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "g" + ], + "points": [ + 2.0 + ], + "modality": "text+data figure", + "field": "Thermodynamics", + "source": "NBPhO_2025", + "image_question": [ + "image_question/NBPhO_2025_2_1_1.png", + "image_question/NBPhO_2025_2_1_2.png" + ] + }, + { + "id": "NBPhO_2025_2_3", + "context": "[Evaporation] \n\nFor the subsequent tasks, the graph shows how the density of saturated water vapour in $\\mathrm{g} \\mathrm{m}^{-3}$ depends on the temperature in $^\\circ C$.\n\n[figure1]\n\nYou may also use the following characteristics of water. Specific heat capacity $c = 4200 \\mathrm{J} \\mathrm{kg}^{-1} \\mathrm{K}^{-1}$; latent heat of vaporization $L = 2260 \\mathrm{kJ} \\mathrm{kg}^{-1}$; density $\\rho = 1000 \\mathrm{kg} \\mathrm{m}^{-3}$; molar mass of water $\\mu = 18 \\mathrm{g} \\mathrm{mol}^{-1}$. You may also assume water vapour to behave as an ideal gas. The universal gas constant is $R = 8.31 \\mathrm{J} \\mathrm{mol}^{-1} \\mathrm{K}^{-1}$. \n\n(i) A cylinder contains a certain amount of water at temperature $T_{0} = 90^{\\circ} \\mathrm{C}$, as shown in the figure. The cross-sectional area of the piston is $S = 1 \\mathrm{dm}^{2}$. What is the minimum pulling force required to move the piston? The pressure of the surrounding air is $p_{0} = 100 \\mathrm{kPa}$.\n\n[figure2]\n\n(ii) If the piston is pulled so that it shifts by $a = 3 \\mathrm{dm}$, the water cools to a temperature of $T_{1} = 89^{\\circ} \\mathrm{C}$; what is the mass of the water under the piston? \n\nWater evaporation has a cooling effect the intensity of which depends on the relative humidity and air convection intensity. It appears, however, that once a dynamical thermal equilibrium is reached, the equilibrium temperature of a wet surface depends only on the relative humidity and the temperature of air and does not depend on the convection speed (as long as the convection is not too weak). This is so because the two competing processes determining the equilibrium state both depend on the thickness of the laminar (non-turbulent) surface layer exactly in the same way. In what follows we shall use the following assumptions. (a) Atop a wet surface (such as a sweating bare skin), there is a layer with a laminar flow of a certain thickness $d$. (b) Atop the laminar layer, the surrounding turbulent flow keeps a constant temperature $T$ and relative humidity $r$, both equal to the respective values in the bulk of the surrounding air. (c) Heat flux from beneath the wet surface (e.g. through the skin) can be neglected. (d) The heat conductivity of air $\\kappa = 30 \\mathrm{mW} \\mathrm{m}^{-1} \\mathrm{K}^{-1}$ at $T = 70^{\\circ} \\mathrm{C}$ (neglect the temperature dependence), and the diffusivity of water molecules in air $D = 26 \\mathrm{mm}^{2} \\mathrm{s}^{-1}$. Neglect the dependence of $D$ on the temperature. Note that the particle flux (net number of molecules passing a cross-section in y-z-plane per second and per cross-sectional area) can be found as $J = D \\frac{\\mathrm{d} n}{\\mathrm{d} x}$, where $n$ denotes the number density (number of molecules per volume). \n\nParts (i)–(ii) are preliminary questions and should not be included in the final answer.", + "question": "Determine the temperature of sweating human skin in a sauna if the air temperature $T = 110^{\\circ}\\mathrm{C}$ and $r = 3\\%$. Express your answer in $^{\\circ}\\mathrm{C}$.", + "marking": [ + [ + "Award 0.4 pt if the answer states that at equilibrium the heat going away from the skin (up) is equal to the heat going to the skin (down) due to evaporation. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly gives the expression for the heat flux down as $\\kappa \\frac{dT}{dx}$, where $\\kappa$ is the heat conductivity of air and $T$ is the temperature. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly gives the expression for the heat flux up as $\\frac{D L \\mu}{R} \\frac{d}{dx} \\frac{P}{T}$, where $D$ is the diffusivity of water molecules in air, $L$ is the latent heat of vaporization, $\\mu$ is the molar mass of water, $R$ is the universal gas constant, $P$ is the vapour pressure, and $T$ is the temperature. Partial points: award 0.3 pt if the answer only correctly gives the magnitude of the heat flux up as $L J m$, where $J$ is the particle flux and $m$ is the mass of one molecule; award 0.1 pt if the answer correctly gives the expression for the mass of one molecule as $m = \\mu / N_A$, where $N_A$ is Avogadro's number; award 0.1 pt if the answer correctly gives the expression $n = P / (T k_B)$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer deduces that the direction of the heat flow is opposite to $\\frac{dn}{dx}$, explicitly mentioning or implying the existence of the minus sign in the equations. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly states the relation between the pressure of the water $P$ and the relative humidity $r$ as $P = r p$, where $p$ is the saturation pressure of vapour. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer correctly integrates to obtain $\\kappa (T - T_s) = \\frac{D L \\mu}{R} \\left[ \\frac{p(T_s)}{T_s} - \\frac{r p(T)}{T} \\right]$, where the index $s$ denotes quantities evaluated at the skin surface and $r$ is the relative humidity. Alternatively, if a change from $d$ to $\\Delta$ in the derivatives is made, it must be properly justified: for heat conductivity, no explicit explanation is needed, but for Fick's law, the answer must state that $J$ is constant because the number of particles is conserved. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly substitutes $\\rho = p \\mu / (R T)$ for the density. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer reads the density $\\rho$ correctly from the graph as $\\rho_1 \\in [800, 815] \\mathrm{g m^{-3}}$. Otherwise, award 0 pt.", + "Award 0.8 pt if the answer correctly applies the graphical method to find that $\\rho(T_s) = r p(T) + \\frac{\\kappa}{D L} (T - T_s)$ defines a straight line in $(T, \\rho)$ graph, where the index $s$ denotes quantities evaluated at the skin surface. Alternatively, any other valid numerical method that is explained is accepted. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly gives the numerical final result of the temperature of sweating human skin in a sauna as $T_s \\in [36, 47] ^\\circ \\mathrm{C}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{[36, 47]}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "$^{\\circ}\\mathrm{C}$" + ], + "points": [ + 3.0 + ], + "modality": "text+data figure", + "field": "Thermodynamics", + "source": "NBPhO_2025", + "image_question": [ + "image_question/NBPhO_2025_2_1_1.png", + "image_question/NBPhO_2025_2_1_2.png" + ] + }, + { + "id": "NBPhO_2025_3_1", + "context": "[Nuclear Reactors] \n\nIn order to maintain a chain reaction in a modern thermal-neutron nuclear reactor one needs three things: 1. nuclear fuel (e.g. $\\mathrm{U}^{235}$), 2. moderator (e.g. water) and 3. coolant. In most cases the moderator acts as the coolant as well. Neutrons released from a thermal fission of $\\mathrm{U}^{235}$ have a mean kinetic energy of approximately $E_{0} = 2 \\mathrm{MeV}$. However, neutrons which are that fast are inefficient in triggering fission of $\\mathrm{U}^{235}$: neutrons need to be slowed down to an average kinetic energy of $E_{f} = 0.025 \\mathrm{eV}$. In what follows, justify why non-relativistic approximations can be used unless explicitly instructed otherwise.", + "question": "The rest energy of neutrons $m_{\\mathrm{n}} c^{2} = 940 \\mathrm{MeV}$, the Boltzmann constant $k_{\\mathrm{B}} = 1.38 \\times 10^{-23} \\mathrm{J} \\mathrm{K}^{-1}$, and the elementary charge $e = 1.602 \\times 10^{-19} \\mathrm{C}$. \n\n(1) What is the required speed of neutrons, i.e. the speed $v_f$ of neutrons with kinetic energy $E_{f}$? Express your answer in $m/s$. \n(2) What is the temperature $T_{f}$ of a neutron gas where the average kinetic energy of neutrons is $E_{f}$? Express your answer in $K$. \n(3) What is the initial speed of neutrons, i.e. the speed $v_0$ of neutrons with energy $E_{0}$? Express your answer in $m/s$.", + "marking": [ + [ + "Award 0.3 pt if the answer correctly expresses the required velocity of the neutrons with the kinetic energy $E_f$ as $v_f = \\sqrt{2 E_f / m}$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly calculates both $v_f = 2.2 \\times 10^3 \\mathrm{m/s}$ ($v_f$ is the velocity with $E_f$) and $v_0 = 2.0 \\times 10^7 \\mathrm{m/s}$ ($v_0$ is the velocity with $E_0$) for the given cases. Partial points: award 0.3 pt if only one numerical value is correct. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly uses $E_f = \\frac{3}{2} k_B T$, where $k_B$ is the Boltzmann constant and $T$ is the temperature. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer uses $E_f = \\frac{3}{2} k_B T$ to correctly calculate the temperature of a neutron gas with $E_f$ as $T_f = 193 \\mathrm{K}$. Otherwise, award 0 pt.", + "Award 0.6 pt if the answer justifies the validity of the classical (non-relativistic) approach for both cases, for example by showing that the speed is much less than the speed of light or that the kinetic energy is significantly less than the rest energy $E_f \\ll m c^2$. Partial points: award 0.3 pt if the answer uses $E_f = k_B T$ without justification and finds $T = 290 \\mathrm{K}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$2.2 \\times 10^3$}", + "\\boxed{193}", + "\\boxed{$2.0 \\times 10^7$}" + ], + "answer_type": [ + "Numerical Value", + "Numerical Value", + "Numerical Value" + ], + "unit": [ + "m/s", + "K", + "m/s" + ], + "points": [ + 0.5, + 0.5, + 1.0 + ], + "modality": "text-only", + "field": "Thermodynamics", + "source": "NBPhO_2025", + "image_question": [] + }, + { + "id": "NBPhO_2025_3_3", + "context": "[Nuclear Reactors] \n\nIn order to maintain a chain reaction in a modern thermal-neutron nuclear reactor one needs three things: 1. nuclear fuel (e.g. $\\mathrm{U}^{235}$), 2. moderator (e.g. water) and 3. coolant. In most cases the moderator acts as the coolant as well. Neutrons released from a thermal fission of $\\mathrm{U}^{235}$ have a mean kinetic energy of approximately $E_{0} = 2 \\mathrm{MeV}$. However, neutrons which are that fast are inefficient in triggering fission of $\\mathrm{U}^{235}$: neutrons need to be slowed down to an average kinetic energy of $E_{f} = 0.025 \\mathrm{eV}$. In what follows, justify why non-relativistic approximations can be used unless explicitly instructed otherwise.\n\n(i) The rest energy of neutrons $m_{\\mathrm{n}} c^{2} = 940 \\mathrm{MeV}$, the Boltzmann constant $k_{\\mathrm{B}} = 1.38 \\times 10^{-23} \\mathrm{J} \\mathrm{K}^{-1}$, and the elementary charge $e = 1.602 \\times 10^{-19} \\mathrm{C}$. What is the required speed of neutrons, i.e. the speed $v_f$ of neutrons with kinetic energy $E_{f}$? What is the temperature $T_{f}$ of a neutron gas where the average kinetic energy of neutrons is $E_{f}$? \n\n(ii) What is the initial speed of neutrons, i.e. the speed $v_0$ of neutrons with energy $E_{0}$? \n\nParts (i)–(ii) are preliminary questions and should not be included in the final answer.", + "question": "(1) From a completely nonrelativistic point of view, what should be the mass $M$ of the moderator's atoms to slow down the fast neutrons as efficiently as possible? Express $M$ in terms of $m_{\\mathrm{n}}$. \n(2) If the mass of the moderator's atoms were to be $M = 135 m_{\\mathrm{n}}$, how many collisions with such atoms at a temperature much lower than $T_{f}$ would a fast neutron need to experience to slow down from $E_{0}$ to $E_{f}$? Assume that all collisions are elastic and central.", + "marking": [ + [ + "Award 0.3 pt if the answer states that $T \\ll T_f$, so the moderator atoms are essentially at rest. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer justifies that the maximum momentum transfer occurs when $m_n = M$, where $m_n$ is the neutron mass and $M$ is the mass of the moderator atom. Otherwise, award 0 pt.", + "Award 0.6 pt if the answer applies both momentum conservation $m_1 (v_{1,f} - v_{1,i}) = m_2 (v_{2,i} - v_{2,f})$ and kinetic energy conservation $m_1 (v_{1,f}^2 - v_{1,i}^2) = m_2 (v_{2,i}^2 - v_{2,f}^2)$, where $m_1$ and $m_2$ are the particle masses, $v_{1,i}$ and $v_{1,f}$ are the initial and final velocities of particle 1, and $v_{2,i}$ and $v_{2,f}$ are the initial and final velocities of particle 2. Partial points: award 0.3 pt if the answer applies only momentum conservation or only energy conservation. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer expresses $u = v \\frac{m_1 - m_2}{m_1 + m_2}$, where $u$ is the speed of the neutron after collision and $v$ is the speed of the neutron before collision. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer expresses $v_f = v_0 \\left( \\frac{m_1 - m_2}{m_1 + m_2} \\right)^N$, where $v_f$ is the final velocity after $N$ collisions, $v_0$ is the initial velocity of the neutron, $m_1$ and $m_2$ are the masses, and $N$ is the number of collisions. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly calculates the number of needed collisions from $E_0$ to $E_f$ as $N = 614$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$M = m_n$}", + "\\boxed{614}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 0.7, + 1.8 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2025", + "image_question": [] + }, + { + "id": "NBPhO_2025_3_4", + "context": "[Nuclear Reactors] \n\nIn order to maintain a chain reaction in a modern thermal-neutron nuclear reactor one needs three things: 1. nuclear fuel (e.g. $\\mathrm{U}^{235}$), 2. moderator (e.g. water) and 3. coolant. In most cases the moderator acts as the coolant as well. Neutrons released from a thermal fission of $\\mathrm{U}^{235}$ have a mean kinetic energy of approximately $E_{0} = 2 \\mathrm{MeV}$. However, neutrons which are that fast are inefficient in triggering fission of $\\mathrm{U}^{235}$: neutrons need to be slowed down to an average kinetic energy of $E_{f} = 0.025 \\mathrm{eV}$. In what follows, justify why non-relativistic approximations can be used unless explicitly instructed otherwise.\n\n(i) The rest energy of neutrons $m_{\\mathrm{n}} c^{2} = 940 \\mathrm{MeV}$, the Boltzmann constant $k_{\\mathrm{B}} = 1.38 \\times 10^{-23} \\mathrm{J} \\mathrm{K}^{-1}$, and the elementary charge $e = 1.602 \\times 10^{-19} \\mathrm{C}$. What is the required speed of neutrons, i.e. the speed $v_f$ of neutrons with kinetic energy $E_{f}$? What is the temperature $T_{f}$ of a neutron gas where the average kinetic energy of neutrons is $E_{f}$? \n\n(ii) What is the initial speed of neutrons, i.e. the speed $v_0$ of neutrons with energy $E_{0}$? \n\n(iii) From a completely nonrelativistic point of view, what should be the mass of the moderator's atoms to slow down the fast neutrons as efficiently as possible? If the mass of the moderator's atoms were to be $M = 135 m_{\\mathrm{n}}$, how many collisions with such atoms at a temperature much lower than $T_{f}$ would a fast neutron need to experience to slow down from $E_{0}$ to $E_{f}$? Assume that all collisions are elastic and central. \n\nParts (i)–(iii) are preliminary questions and should not be included in the final answer.", + "question": "Nuclear fuel, i.e. $\\mathrm{U}^{235}$, is placed inside metal rods and pressurized with helium gas to $p_{0} = 2.5 \\mathrm{MPa}$. During operation, as $\\mathrm{U}^{235}$ keeps on fissioning inside the fuel rods, there is a build up of gas inside the rods. With a non-invasive ultrasound measurement we can measure that the gas pressure inside the rod after it is finally picked out from the core is $p = 6.5 \\mathrm{MPa}$. \n\n(1) Assuming that the gas released inside the rods is completely made of xenon isotope ${}_{54}^{135}\\mathrm{Xe}$ and that the initial gas volume drops from $V_{0} = 18 \\mathrm{cm}^{3}$ to $V = 9 \\mathrm{cm}^{3}$ due to the swelling of the fuel pellets, how many moles of xenon are released from fission? \n(2) What is the ratio of helium to xenon inside the rod? The measurements are done at $T_{0} = 20^{\\circ}\\mathrm{C}$; the universal gas constant $R = 8.31 \\mathrm{J} \\mathrm{mol}^{-1} \\mathrm{K}^{-1}$.", + "marking": [ + [ + "Award 0.3 pt if the answer correctly applies Boyle's law. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer correctly applies Dalton's law. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer expresses $n_{\\mathrm{Xe}} = p_{\\mathrm{Xe}} V / (R T_0)$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly calculates $n_{\\mathrm{Xe}} = 5.5 \\times 10^{-3} \\mathrm{mol}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly expresses $n_{\\mathrm{He}} = p_{\\mathrm{He}} V_0 / (R T_0)$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly calculates $\\frac{n_{\\mathrm{He}}}{n_{\\mathrm{Xe}}} = 3.3$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$5.5 \\times 10^{-3}$}", + "\\boxed{3.3}" + ], + "answer_type": [ + "Numerical Value", + "Numerical Value" + ], + "unit": [ + "mol", + null + ], + "points": [ + 1.1, + 0.4 + ], + "modality": "text-only", + "field": "Thermodynamics", + "source": "NBPhO_2025", + "image_question": [] + }, + { + "id": "NBPhO_2025_5_1", + "context": "", + "question": "[Throwing] \n\nA drone starts from the origin at rest and accelerates horizontally with an acceleration $g$ to the $+x$ direction. Simultaneously, a ball is thrown from the point with coordinates $(x, y) = (-h, -h)$. What is the minimum initial speed $v_0$ the ball needs to hit the drone? The free fall acceleration $g$ is antiparallel to the $y$-axis.", + "marking": [ + [ + "Award 1.0 pt if the answer identifies the idea of switching to the coaccelerating frame, where the drone is at rest and an effective gravitational field is present. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer states that the ball gains a horizontal acceleration $g$, where $g$ is the free-fall acceleration magnitude. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly gives the effective gravitational field magnitude as $g \\sqrt{2}$, pointing at a $45^{\\circ}$ angle from the drone to the throwing point. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer applies the energy conservation equation $\\frac{1}{2} m v_0^2 = m (g \\sqrt{2}) (h \\sqrt{2})$, where $m$ is the ball mass, $v_0$ is the initial speed, and $h$ is the given distance parameter. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly obtains the minimum initial speed $v_0 = 2 \\sqrt{g h}$. Otherwise, award 0 pt." + ], + [ + "Award 0.6 pt if the answer correctly writes the three kinematic equations: (1) In time $t$, the drone travels a distance $s = \\frac{1}{2} gt^2$; (2) For a collision to occur at time $t$, the ball must travel a vertical distance $h$, giving $h = v t \\sin \\alpha - \\frac{1}{2} g t^2$; (3) The horizontal distance $h + s$ gives $h + s = v t \\cos \\alpha$, where $\\alpha$ is the launch angle, $v$ is the initial speed, and $t$ is the flight time. Partial points: award 0.2 pt for each of the three equations. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer derives the condition $\\sin \\alpha = \\cos \\alpha$ from the kinematic equations. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer explicitly states $\\alpha = 45^{\\circ}$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer uses $v_x = v_y$ to argue that the highest point of the trajectory must be as low as possible to minimize the initial speed. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer applies energy conservation or the respective kinematical equation. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly obtains the minimum initial speed $v_0 = 2 \\sqrt{g h}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$v_{0} = 2 \\sqrt{gh}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2025", + "image_question": [] + }, + { + "id": "NBPhO_2025_5_2", + "context": "", + "question": "[Throwing] \n\nA stone is thrown from point $S$ (shown in the figure below) with an initial speed $v$. A boy at point $B$ wishes to hit the stone in midair by throwing a ball simultaneously with the stone's release. He wants to use the minimum possible speed $u$ that will still allow the ball to hit the stone in midair. After calculating the stone's trajectory, he determines the optimal trajectory for the ball and throws it according to his calculations. The collision point $C$ is shown in the figure. Using the scale provided and necessary measurements from the figure: \n\n(1) Find the initial speed $v$ of the stone. Express your answer in $m/s$. \n(2) Find the initial speed $u$ of the ball. Express your answer in $m/s$. \nThe free fall acceleration is $g = 9.8 m s^{-2}$.\n\n[figure1]", + "marking": [ + [ + "Award 0.5 pt if the answer identifies the idea of switching to the free-falling frame. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer states explicitly or implicitly that the stone and the ball travel in straight lines in the free-falling frame. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly writes $SC' = v t$ and $BC' = u t$, where $v$ is the initial speed of the stone, $u$ is the initial speed of the ball, $t$ is the collision time, and $C'$ is the point obtained in the free-falling frame by shifting the collision point $C$ vertically upward by $h = \\frac{1}{2} g t^2$. Partial points: award 0.1 pt for each correct equation. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer notices that we need to minimize $\\frac{|SC'|}{|BC'|}$ (or maximizing its reciprocal), where $C'$ is defined as the point obtained by shifting $C$ vertically upward by $h = \\frac{1}{2} g t^2$ in the free-falling frame. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer correctly applies the sine theorem to minimize $\\frac{|SC'|}{|BC'|}$, where $C'$ is defined as the point obtained by shifting $C$ vertically upward by $h = \\frac{1}{2} g t^2$ in the free-falling frame. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer states that the maximum of $\\frac{|SC'|}{|BC'|}$ occurs when $\\angle C'BS = 90^{\\circ}$, where $C'$ is defined as the point obtained by shifting $C$ vertically upward by $h = \\frac{1}{2} g t^2$ in the free-falling frame. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer states that in the free-falling frame the collision point $C'$ is shifted upwards with respect to $S$, $B$, and $C$ by a distance $h = \\frac{1}{2} g t^2$, where $g$ is the gravitational acceleration. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly expresses $|CC'| = \\frac{1}{2} g t^2$, where $C'$ is defined as the point obtained by shifting $C$ vertically upward by $h = \\frac{1}{2} g t^2$ in the free-falling frame. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer provides a well-drawn and correct geometrical construction showing $S$, $B$, $C$, and $C'$ (where $C'$ is defined as the point obtained by shifting $C$ vertically upward by $h = \\frac{1}{2} g t^2$ in the free-falling frame). Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly gives $v = \\sqrt{g} |SC'| / \\sqrt{2 |CC'|}$ and $u = \\sqrt{g} |BC'| / \\sqrt{2 |CC'|}$, where $|SC'|$ and $|BC'|$ are distances from $S$ and $B$ to $C'$ in the free-falling frame, and $C'$ is defined as the point obtained by shifting $C$ vertically upward by $h = \\frac{1}{2} g t^2$. Partial points: award 0.1 pt for each correct formula. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly calculates $v \\in [11.8, 12.7] \\mathrm{m/s}$ and $u \\in [10.5, 11.4] \\mathrm{m/s}$. Partial points: award 0.1 pt for each correct value (only if the method is correct). Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{[11.8, 12.7]}", + "\\boxed{[10.5, 11.4]}" + ], + "answer_type": [ + "Numerical Value", + "Numerical Value" + ], + "unit": [ + "m/s", + "m/s" + ], + "points": [ + 2.0, + 2.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "NBPhO_2025", + "image_question": [ + "image_question/NBPhO_2025_5_2_1.png" + ] + }, + { + "id": "NBPhO_2025_6_1", + "context": "", + "question": "[Birds] \n\nA long and thin homogenous beam with uniform thickness and square cross-section floats horizontally in water with its top surface parallel to the water surface. A bird lands on one end of the beam, and as a result, the beam sinks so that the edge of the upper face on the bird's side is exactly at the same height as the water surface, while at the other end of the beam the lower face does not rise above the water. What is the maximum number of such birds that this beam can hold above water?", + "marking": [ + [ + "Award 0.6 pt if the answer uses any correct torque balance to solve the problem. Partial points: award 0.3 pt if the answer uses any correct force balance with the bird present. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly identifies the moment arm for the force of the bird as $\\frac{1}{2} L$, where $L$ is the beam length. Otherwise, award 0 pt.", + "Award 0.6 pt if the answer explicitly states or derives that the centre of mass of a triangle is located at the intersection of its medians, a distance $\\frac{2}{3} L$ away from the bird. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly gives the moment arm for the buoyancy force as $\\frac{1}{6} L$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer justifies the numerical result for the maximum number of birds is 4. Otherwise, award 0 pt.", + "Award 2.0 pts if the answer explicitly states that the final result for the maximum number of birds does not depend on fixing any unknown parameters. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{4}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 4.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2025", + "image_question": [] + }, + { + "id": "NBPhO_2025_7_1", + "context": "[Charged Rod] \n\nA rod of mass $m$ carries a charge $q$; both the charge and the mass are homogeneously distributed over its entire length $l$. The system is in homogeneous magnetic field of strength $B$, parallel to the $z$-axis whereas the rod is in the x-y-plane. Neglect any forces except for the Lorentz force. One end of the rod is painted red, and the other - blue.", + "question": "Consider the case when the rod rotates around its centre of mass. What should be the angular speed $\\omega$ for the mechanical tension force at the centre of the rod to be zero?", + "marking": [ + [ + "Award 0.4 pt if the answer considers forces on an infinitesimal part of the rod. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer equates the Lorentz force and the centrifugal force with justification, i.e., writes $\\mathrm{d}q v B = \\mathrm{d}m \\omega^2 r$, where $\\mathrm{d}q$ is the infinitesimal charge, $v$ is the tangential velocity, $B$ is the magnetic field strength, $\\mathrm{d}m$ is the infinitesimal mass, $\\omega$ is the angular speed, and $r$ is the distance from the axis of rotation. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer uses the relation $\\omega = \\frac{v}{r}$, where $v$ is the tangential velocity and $r$ is the distance from the axis of rotation. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer uses the ratio $\\frac{\\mathrm{d}q}{\\mathrm{d}m} = \\frac{q}{m}$, where $q$ is the total charge of the rod and $m$ is the total mass of the rod. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer expresses the angular speed as $\\omega = \\frac{qB}{m}$, where $q$ is the total charge, $B$ is the magnetic field strength, and $m$ is the total mass of the rod. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\omega = \\frac{Bq}{m}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Electromagnetism", + "source": "NBPhO_2025", + "image_question": [] + }, + { + "id": "NBPhO_2025_7_2", + "context": "[Charged Rod] \n\nA rod of mass $m$ carries a charge $q$; both the charge and the mass are homogeneously distributed over its entire length $l$. The system is in homogeneous magnetic field of strength $B$, parallel to the $z$-axis whereas the rod is in the x-y-plane. Neglect any forces except for the Lorentz force. One end of the rod is painted red, and the other - blue. \n\n(i) Consider the case when the rod rotates around its centre of mass. What should be the angular speed $\\omega$ for the mechanical tension force at the centre of the rod to be zero? \n\nPart (i) is a preliminary question and should not be included in the final answer.", + "question": "Consider now a case when initially the blue end of the rod is at the origin $(x = y = 0)$, and the red end at $x = l$. The blue end's initial speed is zero while the red end's speed is $v$, parallel to the y-axis. It turns out that after a certain time $t$, the red end passes through the origin. \n\n(1) Find the smallest possible value for $t$. \n(2) Express the corresponding value of $v$ in terms of $m$, $q$ and $l$.", + "marking": [ + [ + "Award 0.5 pt if the answer deduces, with justification, that the net force on the rod is $\\vec{F} = q \\vec{v}_C \\times \\vec{B}$, where $q$ is the total charge, $\\vec{v}_C$ is the velocity of the center of mass, and $\\vec{B}$ is the magnetic field. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer obtains that the velocity of the center of mass is $v_C = v/2$, where $v$ is the initial velocity of the red end. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer justifies that the center of mass moves on a circular path. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer expresses the radius of the circular path of the center of mass as $R = \\frac{m v}{2 q B}$, where $m$ is the total mass, $q$ is the total charge, and $B$ is the magnetic field strength. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer concludes that the angular velocity of the center of mass is $\\omega = \\frac{q B}{m}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer expresses the angular velocity of the rotation of the rod around the center of mass as $\\Omega = v/l$, where $l$ is the length of the rod. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer justifies that $\\Omega$ is conserved, where $\\Omega$ is the angular velocity of the rotation of the rod around the center of mass. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer argues that $t < 2\\pi/\\omega$ is possible only if $R = l/2$. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer justifies that in this case, the red end will never end up at the origin. Otherwise, award 0 pt.", + "Award 0.4 pt if the answer justifies that the condition for the red end to reach the origin after time $T$ is $\\Omega T = \\pi + 2\\pi k$ with $k \\in \\mathbb{Z}_{\\geq 0}$. Otherwise, award 0 pt.", + "Award 0.6 pt if the answer expresses the final result as $v = \\frac{q B l}{m} \\left( \\frac{1}{2} + k \\right)$, where $k$ is a non-negative integer. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\frac{2 \\pi}{\\omega}$}", + "\\boxed{$v = \\frac{lBq}{2m}$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 2.0, + 2.0 + ], + "modality": "text-only", + "field": "Electromagnetism", + "source": "NBPhO_2025", + "image_question": [] + }, + { + "id": "NBPhO_2025_8_1", + "context": "[Phase Spiral] \n\nHere we shall study the motion of Milky Way stars in the Solar neighbourhood in the direction of the $z$-axis, i.e. perpendicular to the galactic plane. For our purposes, we can model the galactic gravity field as being created by a continuous mass density $\\rho$ (that accounts for the masses of stars, dark matter, gas, interstellar dust, etc), and assume that this mass forms an infinite mirror-symmetric plate, i.e. $\\rho \\equiv \\rho(z)$ and $\\rho(z) = \\rho(-z)$ is independent of $x$ and $y$. Throughout the problem, you may assume that each star's total energy is conserved over the entire considered time period. Gravitational constant $G = 6.67 \\times 10^{-11} \\mathrm{m}^{3} \\mathrm{kg}^{-1} \\mathrm{s}^{-2} = 4.30 \\times 10^{-3} \\mathrm{pc} \\mathrm{M}_{\\odot}^{-1} (\\mathrm{km}/\\mathrm{s})^{2}$.", + "question": "Assuming that the mass density is constant over the plate's thickness. i.e. $\\rho(z) = \\rho_{0}$, what is the acceleration $a_{z}$ of a star at a distance $z$ from the mid-plane?", + "marking": [ + [ + "Award 0.3 pt if the answer identifies that the gravitational acceleration obeys Gauss' law, i.e., the number of field lines passing through a closed surface is proportional to the enclosed mass. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer correctly writes the formula relating the mass inside with gravitational flus, e.g., $\\iint g \\cdot \\mathrm{d}A = -4\\pi G M$, where $g$ is the gravitational field (acceleration), $G$ is the gravitational constant, and $M$ is a point mass. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer applies Gauss' law to a cuboid of area $A$ and half-thickness $z$, obtaining $-2 a_z A = -4\\pi G (2A z \\rho_0)$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the final result for the acceleration at distance $z$ from the mid-plane as $a_z = -4\\pi G \\rho_0 z$. Otherwise, award 0 pt." + ], + [ + "Award 0.5 pt if the answer finds the acceleration of a thin disk by integrating the surface contribution over the plate. Partial points: award 0.3 pt if the answer writes the correct integral; award 0.2 pt if the answer gives the correct evaluation, including finding that the acceleration is independent of the displacement from the surface. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer infers that only the layers within $-a < z < a$ contribute to the final acceleration. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives the final result for the acceleration at distance $z$ from the mid-plane as $a_z = -4\\pi G \\rho_0 z$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$a_{z} = -4 \\pi G \\rho_{0} z$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2025", + "image_question": [] + }, + { + "id": "NBPhO_2025_8_2", + "context": "[Phase Spiral] \n\nHere we shall study the motion of Milky Way stars in the Solar neighbourhood in the direction of the $z$-axis, i.e. perpendicular to the galactic plane. For our purposes, we can model the galactic gravity field as being created by a continuous mass density $\\rho$ (that accounts for the masses of stars, dark matter, gas, interstellar dust, etc), and assume that this mass forms an infinite mirror-symmetric plate, i.e. $\\rho \\equiv \\rho(z)$ and $\\rho(z) = \\rho(-z)$ is independent of $x$ and $y$. Throughout the problem, you may assume that each star's total energy is conserved over the entire considered time period. Gravitational constant $G = 6.67 \\times 10^{-11} \\mathrm{m}^{3} \\mathrm{kg}^{-1} \\mathrm{s}^{-2} = 4.30 \\times 10^{-3} \\mathrm{pc} \\mathrm{M}_{\\odot}^{-1} (\\mathrm{km}/\\mathrm{s})^{2}$. \n\n(i) Assuming that the mass density is constant over the plate's thickness. i.e. $\\rho(z) = \\rho_{0}$, what is the acceleration $a_{z}$ of a star at a distance $z$ from the mid-plane? \n\nPart (i) is a preliminary question and should not be included in the final answer.", + "question": "Consider a star that starts with zero velocity at a distance of $z = a$ from the mid-plane. With what period does it start oscillating around the mid-plane?", + "marking": [ + [ + "Award 0.3 pt if the answer notices that the movement is that of a harmonic oscillator. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer correctly gives the oscillation period as $T = \\sqrt{\\frac{\\pi}{G \\rho_0}}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$\\sqrt{\\frac{\\pi}{G \\rho_0}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.5 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "NBPhO_2025", + "image_question": [] + }, + { + "id": "NBPhO_2025_8_4", + "context": "[Phase Spiral] \n\nHere we shall study the motion of Milky Way stars in the Solar neighbourhood in the direction of the $z$-axis, i.e. perpendicular to the galactic plane. For our purposes, we can model the galactic gravity field as being created by a continuous mass density $\\rho$ (that accounts for the masses of stars, dark matter, gas, interstellar dust, etc), and assume that this mass forms an infinite mirror-symmetric plate, i.e. $\\rho \\equiv \\rho(z)$ and $\\rho(z) = \\rho(-z)$ is independent of $x$ and $y$. Throughout the problem, you may assume that each star's total energy is conserved over the entire considered time period. Gravitational constant $G = 6.67 \\times 10^{-11} \\mathrm{m}^{3} \\mathrm{kg}^{-1} \\mathrm{s}^{-2} = 4.30 \\times 10^{-3} \\mathrm{pc} \\mathrm{M}_{\\odot}^{-1} (\\mathrm{km}/\\mathrm{s})^{2}$. \n\n(i) Assuming that the mass density is constant over the plate's thickness. i.e. $\\rho(z) = \\rho_{0}$, what is the acceleration $a_{z}$ of a star at a distance $z$ from the mid-plane? \n\n(ii) Consider a star that starts with zero velocity at a distance of $z = a$ from the mid-plane. With what period does it start oscillating around the mid-plane? \n\nIn reality, density decreases with growing $|z|$. Measuring density has been a great challenge because of contributions from dark and other difficult-to-see matter. Here, we consider a breakthrough method of doing it. Consider the distributions of the stars in our neighbourhood on the $z$-$v_{z}$ phase plane, where each star is a dot with coordinates $(v_{z}, z)$; $v_{z}$ denotes the $z$-component the star's velocity, and $z$ - the vertical coordinate. Initially, these dots were distributed nearly homogeneously, but some time ago, the Milky Way was perturbed externally, probably by a passing-by dwarf galaxy; this shuffled the positions and velocities of stars, creating a bar-shaped overdensity region. When moving within that bar-shaped region from the centre to the periphery, the total energy per mass of stars increased monotonously. Over time, this overdensity region started \"winding up\", due to the oscillation periods of stars in the vertical plane depending on their oscillation amplitude $z_{\\mathrm{m}}$, and evolved into a spiral pictured below (Antoja et al. 2018, Nature 561, 360). An observation that you need to exploit below is that the ordering of stars by energies along the spiral today remains the same as it was at the time of perturbation. \n\nThe oscillation period of stars depends on the amplitude $z_{\\mathrm{m}}$ because the gravitational potential (the potential energy per mass) $\\Phi(z)$ is not parabolic. In such a case, the period can be approximately found by substituting the real $\\Phi(z)$ with a $k z^{2}$ matching $\\Phi(z)$ at $z = z_{\\mathrm{m}}$, i. e. with $k = \\Phi(z_{\\mathrm{m}}) / z_{\\mathrm{m}}^{2}$. \n\n[figure1] \n\n(iii) At the intersection points of the spiral with $v_z = 0$, calculate $\\Phi(z)$ by interpolating data linearly where appropriate; plot your results (this follows the analysis of Guo et al. 2024, ApJ, 960, 133) \n\nParts (i)–(iii) are preliminary questions and should not be included in the final answer.", + "question": "Assuming that the mass density is almost constant for $|z| \\leq 0.3 \\mathrm{kpc}$, what is the mass density near $z = 0$? Express your answer in $\\mathrm{kg} / \\mathrm{m}^3$.", + "marking": [ + [ + "Award 0.8 pt if the answer connects the first value of $\\Phi(z_1)$ with $\\rho_0$ by assuming constant mass density. Partial points: award 0.6 pt if the answer obtains correct values of $\\Phi(z)$ but does not use the first data point. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly obtains the final expression for $\\rho_0$ as $\\rho_0 = \\frac{\\Phi(z_1)}{2\\pi G z_1^2}$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer correctly obtains the numerical value of $\\rho_0$ within the range $[5.5 \\times 10^{-21}, 6.7 \\times 10^{-21}]$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{[5.5 \\times 10^{-21}, 6.7 \\times 10^{-21}]}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "$\\mathrm{kg} / \\mathrm{m}^3$." + ], + "points": [ + 1.0 + ], + "modality": "text+data figure", + "field": "Modern Physics", + "source": "NBPhO_2025", + "image_question": [ + "image_question/NBPhO_2025_8_3_1.png" + ] + }, + { + "id": "NBPhO_2025_8_5", + "context": "[Phase Spiral] \n\nHere we shall study the motion of Milky Way stars in the Solar neighbourhood in the direction of the $z$-axis, i.e. perpendicular to the galactic plane. For our purposes, we can model the galactic gravity field as being created by a continuous mass density $\\rho$ (that accounts for the masses of stars, dark matter, gas, interstellar dust, etc), and assume that this mass forms an infinite mirror-symmetric plate, i.e. $\\rho \\equiv \\rho(z)$ and $\\rho(z) = \\rho(-z)$ is independent of $x$ and $y$. Throughout the problem, you may assume that each star's total energy is conserved over the entire considered time period. Gravitational constant $G = 6.67 \\times 10^{-11} \\mathrm{m}^{3} \\mathrm{kg}^{-1} \\mathrm{s}^{-2} = 4.30 \\times 10^{-3} \\mathrm{pc} \\mathrm{M}_{\\odot}^{-1} (\\mathrm{km}/\\mathrm{s})^{2}$. \n\n(i) Assuming that the mass density is constant over the plate's thickness. i.e. $\\rho(z) = \\rho_{0}$, what is the acceleration $a_{z}$ of a star at a distance $z$ from the mid-plane? \n\n(ii) Consider a star that starts with zero velocity at a distance of $z = a$ from the mid-plane. With what period does it start oscillating around the mid-plane? \n\nIn reality, density decreases with growing $|z|$. Measuring density has been a great challenge because of contributions from dark and other difficult-to-see matter. Here, we consider a breakthrough method of doing it. Consider the distributions of the stars in our neighbourhood on the $z$-$v_{z}$ phase plane, where each star is a dot with coordinates $(v_{z}, z)$; $v_{z}$ denotes the $z$-component the star's velocity, and $z$ - the vertical coordinate. Initially, these dots were distributed nearly homogeneously, but some time ago, the Milky Way was perturbed externally, probably by a passing-by dwarf galaxy; this shuffled the positions and velocities of stars, creating a bar-shaped overdensity region. When moving within that bar-shaped region from the centre to the periphery, the total energy per mass of stars increased monotonously. Over time, this overdensity region started \"winding up\", due to the oscillation periods of stars in the vertical plane depending on their oscillation amplitude $z_{\\mathrm{m}}$, and evolved into a spiral pictured below (Antoja et al. 2018, Nature 561, 360). An observation that you need to exploit below is that the ordering of stars by energies along the spiral today remains the same as it was at the time of perturbation. \n\nThe oscillation period of stars depends on the amplitude $z_{\\mathrm{m}}$ because the gravitational potential (the potential energy per mass) $\\Phi(z)$ is not parabolic. In such a case, the period can be approximately found by substituting the real $\\Phi(z)$ with a $k z^{2}$ matching $\\Phi(z)$ at $z = z_{\\mathrm{m}}$, i. e. with $k = \\Phi(z_{\\mathrm{m}}) / z_{\\mathrm{m}}^{2}$. \n\n[figure1] \n\n(iii) At the intersection points of the spiral with $v_z = 0$, calculate $\\Phi(z)$ by interpolating data linearly where appropriate; plot your results (this follows the analysis of Guo et al. 2024, ApJ, 960, 133) \n\n(iv) Assuming that the mass density is almost constant for $|z| \\leq 0.3 \\mathrm{kpc}$, what is the mass density near $z = 0$? \n\nParts (i)–(iv) are preliminary questions and should not be included in the final answer.", + "question": "Dark matter is an \"invisible\" form of matter that only interacts by gravity. In general, it is found that dark matter forms halos that extend significantly farther than visible matter structures. By assuming that the dark matter density doesn't vary significantly within the volume of interest and that it starts dominating far away from the galactic plane, from around $z = 0.7 \\mathrm{kpc}$, estimate the local dark matter density $\\rho_{\\mathrm{DM}}$. Express your answer in $\\mathrm{kg} / \\mathrm{m}^3$.", + "marking": [ + [ + "Award 0.7 pt if the answer obtains an expression for the total mass per unit area (surface density) enclosed within height $z$ using the constant density approximation, e.g., $\\Sigma(z) = \\rho_0 z = \\frac{\\Phi(z)}{2\\pi G z}$, where $\\Sigma(z)$ is the surface density, $\\rho_0$ is the assumed constant mass density over the plate's thickness, $\\Phi(z)$ is the gravitational potential per unit mass, and $G$ is the gravitational constant. Otherwise, award 0 pt.", + "Award 0.9 pt if the answer takes the difference between the total surface densities at two heights $z_6$ and $z_5$ where dark matter dominates at $z_6$ (and not at $z_5$), and sets $\\Sigma(z_6) - \\Sigma(z_5) = \\rho_{\\mathrm{DM}} (z_6 - z_5)$ to isolate the dark matter contribution, where $\\rho_{\\mathrm{DM}}$ is the local dark matter density. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer expresses $\\rho_{\\mathrm{DM}}$ in terms of $\\Phi(z)$ at the two heights as $\\rho_{\\mathrm{DM}} \\approx \\frac{1}{2\\pi G (z_6 - z_5)} \\left(\\frac{\\Phi(z_6)}{z_6} - \\frac{\\Phi(z_5)}{z_5}\\right)$. Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives a numerical value of the local dark matter density $\\rho_{\\mathrm{DM}}$ within the range $[6.9 \\times 10^{-22} \\mathrm{kg/m^3}, 8.5 \\times 10^{-22} \\mathrm{kg/m^3}]$. Otherwise, award 0 pt." + ], + [ + "Award 1.6 pt if the answer uses the previous relation between density and potential to express $(z_6 - z_5) \\rho_{\\mathrm{DM}} = z_6 \\rho(z_6) - z_5 \\rho(z_5)$, where the difference between the total surface densities at two heights $z_6$ and $z_5$. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer presents a final explicit expression for $\\rho_{\\mathrm{DM}}$ based on $z_6$ and $z_5$, e.g., $\\rho_{\\mathrm{DM}} = \\frac{1}{2\\pi G (z_6 - z_5)} \\left(\\frac{\\Phi(z_6)}{z_6} - \\frac{\\Phi(z_5)}{z_5}\\right).$ Otherwise, award 0 pt.", + "Award 0.1 pt if the answer gives a numerical value of the local dark matter density $\\rho_{\\mathrm{DM}}$ within the range $[6.9 \\times 10^{-22} \\mathrm{kg/m^3}, 8.5 \\times 10^{-22} \\mathrm{kg/m^3}]$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{[6.9 \\times 10^{-22}, 8.5 \\times 10^{-22}]}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "$\\mathrm{kg} / \\mathrm{m}^3$" + ], + "points": [ + 2.0 + ], + "modality": "text+data figure", + "field": "Modern Physics", + "source": "NBPhO_2025", + "image_question": [ + "image_question/NBPhO_2025_8_3_1.png" + ] + }, + { + "id": "NBPhO_2025_8_6", + "context": "[Phase Spiral] \n\nHere we shall study the motion of Milky Way stars in the Solar neighbourhood in the direction of the $z$-axis, i.e. perpendicular to the galactic plane. For our purposes, we can model the galactic gravity field as being created by a continuous mass density $\\rho$ (that accounts for the masses of stars, dark matter, gas, interstellar dust, etc), and assume that this mass forms an infinite mirror-symmetric plate, i.e. $\\rho \\equiv \\rho(z)$ and $\\rho(z) = \\rho(-z)$ is independent of $x$ and $y$. Throughout the problem, you may assume that each star's total energy is conserved over the entire considered time period. Gravitational constant $G = 6.67 \\times 10^{-11} \\mathrm{m}^{3} \\mathrm{kg}^{-1} \\mathrm{s}^{-2} = 4.30 \\times 10^{-3} \\mathrm{pc} \\mathrm{M}_{\\odot}^{-1} (\\mathrm{km}/\\mathrm{s})^{2}$. \n\n(i) Assuming that the mass density is constant over the plate's thickness. i.e. $\\rho(z) = \\rho_{0}$, what is the acceleration $a_{z}$ of a star at a distance $z$ from the mid-plane? \n\n(ii) Consider a star that starts with zero velocity at a distance of $z = a$ from the mid-plane. With what period does it start oscillating around the mid-plane? \n\nIn reality, density decreases with growing $|z|$. Measuring density has been a great challenge because of contributions from dark and other difficult-to-see matter. Here, we consider a breakthrough method of doing it. Consider the distributions of the stars in our neighbourhood on the $z$-$v_{z}$ phase plane, where each star is a dot with coordinates $(v_{z}, z)$; $v_{z}$ denotes the $z$-component the star's velocity, and $z$ - the vertical coordinate. Initially, these dots were distributed nearly homogeneously, but some time ago, the Milky Way was perturbed externally, probably by a passing-by dwarf galaxy; this shuffled the positions and velocities of stars, creating a bar-shaped overdensity region. When moving within that bar-shaped region from the centre to the periphery, the total energy per mass of stars increased monotonously. Over time, this overdensity region started \"winding up\", due to the oscillation periods of stars in the vertical plane depending on their oscillation amplitude $z_{\\mathrm{m}}$, and evolved into a spiral pictured below (Antoja et al. 2018, Nature 561, 360). An observation that you need to exploit below is that the ordering of stars by energies along the spiral today remains the same as it was at the time of perturbation. \n\nThe oscillation period of stars depends on the amplitude $z_{\\mathrm{m}}$ because the gravitational potential (the potential energy per mass) $\\Phi(z)$ is not parabolic. In such a case, the period can be approximately found by substituting the real $\\Phi(z)$ with a $k z^{2}$ matching $\\Phi(z)$ at $z = z_{\\mathrm{m}}$, i. e. with $k = \\Phi(z_{\\mathrm{m}}) / z_{\\mathrm{m}}^{2}$. \n\n[figure1] \n\n(iii) At the intersection points of the spiral with $v_z = 0$, calculate $\\Phi(z)$ by interpolating data linearly where appropriate; plot your results (this follows the analysis of Guo et al. 2024, ApJ, 960, 133) \n\n(iv) Assuming that the mass density is almost constant for $|z| \\leq 0.3 \\mathrm{kpc}$, what is the mass density near $z = 0$? \n\n(v) Dark matter is an \"invisible\" form of matter that only interacts by gravity. In general, it is found that dark matter forms halos that extend significantly farther than visible matter structures. By assuming that the dark matter density doesn't vary significantly within the volume of interest and that it starts dominating far away from the galactic plane, from around $z = 0.7 \\mathrm{kpc}$, estimate the local dark matter density $\\rho_{\\mathrm{DM}}$. \n\nParts (i)–(v) are preliminary questions and should not be included in the final answer.", + "question": "How long ago did the perturbation occur? Express your answer in $s$.", + "marking": [ + [ + "Award 1.0 pt if the answer includes the idea of using differences in the winding rate between two points on the spiral. Otherwise, award 0 pt.", + "Award 0.5 pt if the answer expresses the angular frequency $\\omega$ in terms of $\\Phi(z)$ by assuming a harmonic oscillator, i.e., $\\omega(z) = \\sqrt{\\frac{2\\Phi(z)}{z^2}}$, where $G$ is the gravitational constant, $\\rho_0$ is the constant mass density, $\\Phi(z)$ is the gravitational potential per unit mass at height $z$, and $z$ is the distance from the mid-plane. Otherwise, award 0 pt.", + "Award 0.3 pt if the answer picks two labeled points on the spiral (e.g., $z_1$ and $z_6$) and connects the age of the spiral, $\\omega$ and the winding amount via $T_0 = 2.5 \\frac{2\\pi}{\\omega(z_6) - \\omega(z_1)}$. Otherwise, award 0 pt.", + "Award 0.2 pt if the answer gives a numerical value for the time of the perturbation that falls within the range $[1.7 \\times 10^{16} \\mathrm{s}, 2.1 \\times 10^{16} \\mathrm{s}$. Otherwise, award 0 pt." + ] + ], + "answer": [ + "\\boxed{$[1.7 \\times 10^{16} \\mathrm{s}, 2.1 \\times 10^{16} \\mathrm{s}$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "s" + ], + "points": [ + 2.0 + ], + "modality": "text+data figure", + "field": "Modern Physics", + "source": "NBPhO_2025", + "image_question": [ + "image_question/NBPhO_2025_8_3_1.png" + ] + } +] \ No newline at end of file diff --git a/data/PanMechanics_2024.json b/data/PanMechanics_2024.json new file mode 100644 index 0000000000000000000000000000000000000000..4bf4f3a363c4adf6b15fd51655c94af65fb97b08 --- /dev/null +++ b/data/PanMechanics_2024.json @@ -0,0 +1,704 @@ +[ + { + "information": "若有需要，取重力加速度 $g = 10 m/s^2$ 及 重力常数 $G = 6.67 \\times 10^{-11} N m^2/\\mathrm{kg}^2$（若没有特别注明，取所有摩擦力为零）。" + }, + { + "id": "PanMechanics_2024_1", + "context": "", + "question": "由三根质量为 $M$ 、长度为 $L$ 的相同均匀杆组成一个三角形。它通过顶部的枢轴铰接在垂直平面上，如图所示。这个物理摆的小振荡周期是多少？杆子通过其质心的转动惯量为 $I_{\\mathrm{CM}} = \\frac{1}{12} ML^2$。\n\n[figure1]\n\n(A) $\\sqrt{\\frac{3L}{2g}}$ \n(B) $2\\pi \\sqrt{\\frac{3L}{g}}$ \n(C) $\\pi \\sqrt{\\frac{3L}{g}}$ \n(D) $\\pi \\sqrt{\\frac{3ML}{g}}$ \n(E) $\\pi \\sqrt{\\frac{2\\sqrt{3}L}{g}}$", + "marking": [], + "answer": [ + "\\boxed{E}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_1_1.png" + ] + }, + { + "id": "PanMechanics_2024_2", + "context": "", + "question": "一辆以 $36 m/s$ 速度行驶的卡车经过一辆以 $45 m/s$ 速度朝相反方向行驶的警车。如果警笛相对于警车的频率为 $500 Hz$,那么当警车接近卡车时,卡车内的观察者听到的频率是多少？(空气中的声速为 $343 m/s$) \n\n(A) $396 \\mathrm{Hz}$ \n(B) $636 \\mathrm{Hz}$ \n(C) $361 \\mathrm{Hz}$ \n(D) $393 \\mathrm{Hz}$ \n(E) $617 \\mathrm{Hz}$", + "marking": [], + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [] + }, + { + "id": "PanMechanics_2024_3", + "context": "", + "question": "一颗卫星绕 X 行星做圆形轨道运行，且轨道距离行星表面非常近。要估计行星 X 的密度，我们只需测量：\n\n(A) 卫星的周期 \n(B) 轨道半径 \n(C) 卫星的速度 \n(D) 行星 X 的质量 \n(E) 卫星的质量", + "marking": [], + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [] + }, + { + "id": "PanMechanics_2024_4", + "context": "质量为 $M$ 的三角楔子置于水平无摩擦的地面上。将质量为 $m$ 的木块放在楔子上，如图所示。木块和楔子之间没有摩擦力。系统从静止状态释放。给定 $M = 3m$ 和 $\\alpha = 45^{\\circ}$。", + "question": "求三角楔子加速度的大小。\n\n(A) $g/6$ \n(B) $g/7$ \n(C) $g/4$ \n(D) $g$ \n(E) 0", + "marking": [], + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_4_1.png" + ] + }, + { + "id": "PanMechanics_2024_5", + "context": "质量为 $M$ 的三角楔子置于水平无摩擦的地面上。将质量为 $m$ 的木块放在楔子上，如图所示。木块和楔子之间没有摩擦力。系统从静止状态释放。给定 $M = 3m$ 和 $\\alpha = 45^{\\circ}$。", + "question": "若当木块滑到地面时，楔子相对地面的速度为 $1 m/s$。求木块在楔子上离地面的初始高度（假设木块为无体积的重点）。 \n\n(A) $0.60 m$ \n(B) $0.82 m$ \n(C) $1.00 m$ \n(D) $1.05 m$ \n(E) $1.40 m$", + "marking": [], + "answer": [ + "\\boxed{E}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_4_1.png" + ] + }, + { + "id": "PanMechanics_2024_6", + "context": "", + "question": "在一维运动中，力 $F = -(m/b) v^2$ 作用在质量为 $m$ 的粒子上，其中 $v$ 是粒子的速度，$b$ 是常数。在 $t = 0 s$ 时，该粒子位于 $x = 0 m$。哪一个是粒子随时间变化的可能位置？\n\n(A) $x(t) = b \\ln (\\frac{t}{1 \\mathrm{s}})$ \n(B) $x(t) = b \\ln (\\frac{t}{1 \\mathrm{s}} + 1)$ \n(C) $x(t) = b \\frac{t / 1\\mathrm{s}}{2 + ( t/1\\mathrm{s})^2}$ \n(D) $x(t) = \\frac{b}{t/1\\mathrm{s}}$ \n(E) $x(t) = b \\sin (t/1\\mathrm{s})$", + "marking": [], + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [] + }, + { + "id": "PanMechanics_2024_7", + "context": "一个体重 $60 \\mathrm{kg}$ 的人以 $5 m/s$ 的初始速度沿着半径为 $3 m$、质量为 $100 \\mathrm{kg}$ 的固定均匀圆形平台的切线跑步，如图所示。平台本来静止，当人跳上及静止在平台上后，平台绕中心的垂直轴旋转。圆形平台通过其质心的转动惯量为 $I_{\\mathrm{CM}} = \\frac{1}{2} M R^2$。", + "question": "求该人跳上平台后系统的角速度。\n\n(A) 0.500 rad/s \n(B) 0.250 rad/s \n(C) 1.33 rad/s \n(D) 0.909 rad/s \n(E) 1.705 rad/s", + "marking": [], + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_7_1.png" + ] + }, + { + "id": "PanMechanics_2024_8", + "context": "一个体重 $60 \\mathrm{kg}$ 的人以 $5 m/s$ 的初始速度沿着半径为 $3 m$、质量为 $100 \\mathrm{kg}$ 的固定均匀圆形平台的切线跑步，如图所示。平台本来静止，当人跳上及静止在平台上后，平台绕中心的垂直轴旋转。圆形平台通过其质心的转动惯量为 $I_{\\mathrm{CM}} = \\frac{1}{2} M R^2$。", + "question": "找出总机械能的损失。\n\n(A) $150 J$ \n(B) $341 J$ \n(C) $257 J$ \n(D) $457 J$ \n(E) $0 J$", + "marking": [], + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_7_1.png" + ] + }, + { + "id": "PanMechanics_2024_9", + "context": "两个 $1.0 \\mathrm{kg}$ 的粒子以 $(40.0 m/s) \\hat{l}$ 和 $(-20.0 m/s) \\hat{l}$ 的速度沿直线相互移动并發生碰撞。碰撞后，其中一个粒子以 $30.0 m/s$ 的速度离开。在碰撞过程中，两颗粒子共损失了 $100 \\mathrm{J}$ 的能量。", + "question": "求碰撞后另一个粒子的速度。\n\n(A) $33.2 m/s$ \n(B) $36.1 m/s$ \n(C) $17.3 m/s$ \n(D) $26.8 m/s$ \n(E) $30.0 m/s$", + "marking": [], + "answer": [ + "\\boxed{E}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [] + }, + { + "id": "PanMechanics_2024_10", + "context": "两个 $1.0 \\mathrm{kg}$ 的粒子以 $(40.0 m/s) \\hat{l}$ 和 $(-20.0 m/s) \\hat{l}$ 的速度沿直线相互移动并發生碰撞。碰撞后，其中一个粒子以 $30.0 m/s$ 的速度离开。在碰撞过程中，两颗粒子共损失了 $100 \\mathrm{J}$ 的能量。", + "question": "求碰撞后粒子速度之间的夹角。\n\n(A) $141^{\\circ}$ \n(B) $105^{\\circ}$ \n(C) $70.5^{\\circ}$ \n(D) $96.4^{\\circ}$ \n(E) $48.2^{\\circ}$", + "marking": [], + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [] + }, + { + "id": "PanMechanics_2024_11", + "context": "", + "question": "如图所示，粒子在 $x = a$ 点从静止状态释放，并根据图中所示的势能函数 $U(x)$ 沿 $x$ 轴移动。图中 $U(a) = U(e)$。粒子其后的运动为：\n\n(A) 移动到 $x = e$ 左侧的点，停止并保持静止。\n(B) 在 $x = a$ 及 $x = e$ 之间来回移动。\n(C) 以不同的速度移动到无穷大 $(x \\rightarrow \\infty)$。\n(D) 移动到 $x = b$，并保持静止状态。\n(E) 移动到 $x = e$，然后移动到 $x = d$，并保持静止状态。", + "marking": [], + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_11_1.png" + ] + }, + { + "id": "PanMechanics_2024_12", + "context": "", + "question": "逃离木星引力的最低速度为 60 公里/秒。假设木星的半径为 70,000 公里，那么 80 公斤重的宇航员在木星上的重量是多少？\n\n(A) $1029 N$ \n(B) $1371 N$ \n(C) $2057 N$ \n(D) $2742 N$ \n(E) $4114 N$", + "marking": [], + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [] + }, + { + "id": "PanMechanics_2024_13", + "context": "", + "question": "下列哪一个人必须是非惯性观察者？将地面视为惯性系。应考虑空气摩擦力。\nI. 一个人的位置被另一个观察者描述为 $y(t) = -\\frac{g}{2} t^2$。\nII. 坐在固定在地面上旋转的旋转木马边缘的人。\nIII. 一个人垂直向上跳跃。而此刻，当人处于最高位置时。\nIV. 一个人垂直向上跳跃。而此刻，人还在上升的时候。\nV. 一个戴着打开的降落伞进行跳伞的人。\n\n(A) 只有 I, IV 和 V \n(B) 只有 I 和 II \n(C) 只有 I,II,IV 和 V \n(D) 只有 II, III 和 IV \n(E) I, II, III, IV 和 V", + "marking": [], + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [] + }, + { + "id": "PanMechanics_2024_14", + "context": "", + "question": "如图所示一个 3.0 kg 的三角体，求推动三角体的力 $F$，使在三角块上的 1.0 kg 方形块不会沿斜面移动。假设所有表面都是无摩擦的。\n\n(A) $15 N$ \n(B) $20 N$ \n(C) $25 N$ \n(D) $40 N$ \n(E) $45 N$", + "marking": [], + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_14_1.png" + ] + }, + { + "id": "PanMechanics_2024_15", + "context": "", + "question": "一个边长为 $L$ 的正方体平稳地漂浮在容器内静止的水中。此时有一半的立方体位于水面以下。再将密度为水四分之一的液体添加到容器中，使立方体完全浸没在液体的表面下，而液体和水不混合，液体留在水上面。添加液体后，立方体从原來的水面上升了多少？\n\n(A) $L/6$ \n(B) $L/3$ \n(C) $L/2$ \n(D) $L/4$ \n(E) $L/5$", + "marking": [], + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [] + }, + { + "id": "PanMechanics_2024_16", + "context": "", + "question": "一个盒子由两根具有相同线性质量密度的绳子悬挂在天花板上，如图所示。求弦 1 的基频 $f_1$ 与弦 2 的基频 $f_2$ 之比，$f_1 / f_2$。\n\n(A) $\\sqrt{3 \\sqrt{3}}$ \n(B) 3 \n(C) $3 \\sqrt{3}$ \n(D) $\\sqrt{6}$ \n(E) $\\sqrt{3}/4$", + "marking": [], + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_16_1.png" + ] + }, + { + "id": "PanMechanics_2024_17_1", + "context": "质量为 $m$ 的质点附着在力常数为 $k$ 的弹簧上，在粗糙表面上沿 X 轴移动。以原点为弹簧自然长度时的位置。", + "question": "当 $t = 0$ 时，粒子在 $x_0 \\neq 0$ 及静止。假设弹簧力足够大，使得粒子在恒定的摩擦力 $f$ 下移动。在时间 $0 \\leq t \\leq \\tau$ 内，求 $x(t)$，其中 $\\tau$ 是 $t = 0$ 后粒子第一次停止的时间。用 $k$、$m$、$f$ 和 $x_0$ 表示 $x(t)$。设静摩擦系数为 0.03，动摩擦系数为 0.01，$m = 1 \\mathrm{kg}$，$k = 10 \\mathrm{N}/\\mathrm{m}$，重力加速度 $g = 10 m/s^2$。", + "marking": [], + "answer": [ + "\\boxed{$x(t) = (x_0 - \\frac{f}{k}) \\cos(\\sqrt{\\frac{k}{m}}t) + \\frac{f}{k}$}", + "\\boxed{$x(t) = (x_0 + \\frac{f}{k}) \\cos(\\sqrt{\\frac{k}{m}}t) - \\frac{f}{k}$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null + ], + "points": [ + 8.0, + 8.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [] + }, + { + "id": "PanMechanics_2024_17_2", + "context": "质量为 $m$ 的质点附着在力常数为 $k$ 的弹簧上，在粗糙表面上沿 X 轴移动。以原点为弹簧自然长度时的位置。\n(a) 当 $t = 0$ 时，粒子在 $x_0 \\neq 0$ 及静止。假设弹簧力足够大，使得粒子在恒定的摩擦力 $f$ 下移动。在时间 $0 \\leq t \\leq \\tau$ 内，求 $x(t)$，其中 $\\tau$ 是 $t = 0$ 后粒子第一次停止的时间。用 $k$、$m$、$f$ 和 $x_0$ 表示 $x(t)$。设静摩擦系数为 0.03，动摩擦系数为 0.01，$m = 1 \\mathrm{kg}$，$k = 10 \\mathrm{N}/\\mathrm{m}$，重力加速度 $g = 10 m/s^2$。\n注意：（a）是前置问题，请不要写入最终答案中。", + "question": "设 $x_0 = 1 m$。使用 (a) 或其他方式，找到粒子永久停止时的最终位置。（单位用 $m$ 表示）", + "marking": [], + "answer": [ + "\\boxed{-0.02}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "m" + ], + "points": [ + 8.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [] + }, + { + "id": "PanMechanics_2024_17_3", + "context": "质量为 $m$ 的质点附着在力常数为 $k$ 的弹簧上，在粗糙表面上沿 X 轴移动。以原点为弹簧自然长度时的位置。\n(a) 当 $t = 0$ 时，粒子在 $x_0 \\neq 0$ 及静止。假设弹簧力足够大，使得粒子在恒定的摩擦力 $f$ 下移动。在时间 $0 \\leq t \\leq \\tau$ 内，求 $x(t)$，其中 $\\tau$ 是 $t = 0$ 后粒子第一次停止的时间。用 $k$、$m$、$f$ 和 $x_0$ 表示 $x(t)$。设静摩擦系数为 0.03，动摩擦系数为 0.01，$m = 1 \\mathrm{kg}$，$k = 10 \\mathrm{N}/\\mathrm{m}$，重力加速度 $g = 10 m/s^2$。\n(b) 设 $x_0 = 1 m$。使用 (a) 或其他方式，找到粒子永久停止时的最终位置。（单位用 $m$ 表示）。\n注意：(a) 和 (b) 都是前置问题，请不要写入最终答案中。", + "question": "求出粒子的总移动距离。（单位用 $m$ 表示）", + "marking": [], + "answer": [ + "\\boxed{49.98}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "m" + ], + "points": [ + 6.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [] + }, + { + "id": "PanMechanics_2024_17_4", + "context": "质量为 $m$ 的质点附着在力常数为 $k$ 的弹簧上，在粗糙表面上沿 X 轴移动。以原点为弹簧自然长度时的位置。\n(a) 当 $t = 0$ 时，粒子在 $x_0 \\neq 0$ 及静止。假设弹簧力足够大，使得粒子在恒定的摩擦力 $f$ 下移动。在时间 $0 \\leq t \\leq \\tau$ 内，求 $x(t)$，其中 $\\tau$ 是 $t = 0$ 后粒子第一次停止的时间。用 $k$、$m$、$f$ 和 $x_0$ 表示 $x(t)$。设静摩擦系数为 0.03，动摩擦系数为 0.01，$m = 1 \\mathrm{kg}$，$k = 10 \\mathrm{N}/\\mathrm{m}$，重力加速度 $g = 10 m/s^2$。\n(b) 设 $x_0 = 1 m$。使用 (a) 或其他方式，找到粒子永久停止时的最终位置。（单位用 $m$ 表示）。\n(c) 求出粒子的总移动距离。（单位用 $m$ 表示）\n注意：(a)、(b) 和 (c) 都是前置问题，请不要写入最终答案中。", + "question": "求粒子永久停止之前所经过的总时间。（单位用 $s$ 表示）", + "marking": [], + "answer": [ + "\\boxed{48.68}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "s" + ], + "points": [ + 3.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [] + }, + { + "id": "PanMechanics_2024_18_1", + "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。", + "question": "求 $k_N$ 。答案以 $K$ 表示。", + "marking": [], + "answer": [ + "\\boxed{$k_N = (N-1) K$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_18_1.png", + "image_question/PanMechanics_2024_18_2.png" + ] + }, + { + "id": "PanMechanics_2024_18_2", + "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。", + "question": "求释放前处于平衡状态的弹簧的总长度 $L_0$。答案以 $M$，$g$ 和 $K$ 表示。", + "marking": [], + "answer": [ + "\\boxed{$L_0 = \\frac{Mg}{2K}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_18_1.png", + "image_question/PanMechanics_2024_18_2.png" + ] + }, + { + "id": "PanMechanics_2024_18_3", + "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。", + "question": "应用牛顿第二定律，写出顶部 $x_N$、底部 $x_1$ 和第 $n$ 个质量 $x_n$ 的运动方程，而 $1 < n < N$，答案以 $m_N$、$k_N$、$g$ 以及其他质量的坐标 $x_2, x_3, \\ldots$（如果需要）表示。", + "marking": [], + "answer": [ + "\\boxed{$m_N \\ddot{x}_1 = -k_N (x_1 - x_2) + m_N g$}", + "\\boxed{$m_N \\ddot{x}_n = k_N (x_{n+1} - 2x_n + x_{n-1}) + m_N g$}", + "\\boxed{$m_N \\ddot{x}_N = k_N (x_{N-1} - x_N) + m_N g$}" + ], + "answer_type": [ + "Equation", + "Equation", + "Equation" + ], + "unit": [ + null, + null, + null + ], + "points": [ + 2.0, + 2.0, + 2.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_18_1.png", + "image_question/PanMechanics_2024_18_2.png" + ] + }, + { + "id": "PanMechanics_2024_18_4", + "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 \\mathrm{s}$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 \\mathrm{s}$ 时，$x_N = 0 \\mathrm{m}$。现在考虑 $N = 2$、$m_N = 1 \\mathrm{kg}$ 且 $k_N = 1 \\mathrm{N}/\\mathrm{m}$ 的情况（$g = 10 m/s^2$）。", + "question": "求系统质心的加速度。（向下为正，单位用 $m/s^2$ 表示）", + "marking": [], + "answer": [ + "\\boxed{10}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "m/s^2" + ], + "points": [ + 2.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_18_1.png", + "image_question/PanMechanics_2024_18_2.png" + ] + }, + { + "id": "PanMechanics_2024_18_5", + "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。现在考虑 $N = 2$、$m_N = 1 \\mathrm{kg}$ 且 $k_N = 1 \\mathrm{N}/\\mathrm{m}$ 的情况（$g = 10 m/s^2$）。", + "question": "求出两个质量随时间变化的距离函数：$d(t) = x_1(t) - x_2(t)$。(表达式中重力加速度用 $g$ 表示)", + "marking": [], + "answer": [ + "\\boxed{$d(t) = g \\cos(\\sqrt{2} t)$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 5.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_18_1.png", + "image_question/PanMechanics_2024_18_2.png" + ] + }, + { + "id": "PanMechanics_2024_18_6", + "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。现在考虑 $N = 2$、$m_N = 1 \\mathrm{kg}$ 且 $k_N = 1 \\mathrm{N}/\\mathrm{m}$ 的情况（$g = 10 m/s^2$）。", + "question": "当两个质量碰撞时（设碰撞时间为 $\\tau$），设底部质量从 $t = 0 \\mathrm{s}$ 的下降距离为 $D_2 = x_1(\\tau) - x_1(0) = \\gamma L_0$。求 $\\gamma$ 的数值。", + "marking": [], + "answer": [ + "\\boxed{0.117}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 6.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_18_1.png", + "image_question/PanMechanics_2024_18_2.png" + ] + }, + { + "id": "PanMechanics_2024_18_7", + "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。现在考虑 $N = 3$ 的情况。", + "question": "为了使弹簧的总质量和总弹簧常数 $K$ 与 $N = 2$、$m_N = 1 kg$ 且 $k_N = 1 N/m$ 的情况相同，（1）求出对应的 $m_N$（单位用 $kg$ 表示），（2）求出对应的 $k_N$（单位用 $N/m$ 表示）。", + "marking": [], + "answer": [ + "\\boxed{2/3}", + "\\boxed{2}" + ], + "answer_type": [ + "Numerical Value", + "Numerical Value" + ], + "unit": [ + "kg", + "N/m" + ], + "points": [ + 1.0, + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_18_1.png", + "image_question/PanMechanics_2024_18_2.png" + ] + }, + { + "id": "PanMechanics_2024_18_8", + "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。现在考虑 $N = 3$ 的情况。", + "question": "求解底部质量随时间变化的位置：$x_1(t)$。提示：尝试先找出质心的运动方程，$x_1$ 和 $x_3$ 之间的差的运动方程，及另一个由 $x_1, x_2$ 及 $x_3$ 的线性组合组成的量的运动方程。", + "marking": [], + "answer": [ + "\\boxed{$x_1(t) = \\frac{5}{9}g + \\frac{1}{2}gt^2 + \\frac{g}{2} \\cos(\\sqrt{3}t) - \\frac{g}{18} \\cos(3t)$}" + ], + "answer_type": [ + "Equation" + ], + "unit": [ + null + ], + "points": [ + 8.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_18_1.png", + "image_question/PanMechanics_2024_18_2.png" + ] + }, + { + "id": "PanMechanics_2024_18_9", + "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。\n\n(f) 考虑 $N = 2$、$m_N = 1 \\mathrm{kg}$ 且 $k_N = 1 \\mathrm{N}/\\mathrm{m}$ 的情况（$g = 10 m/s^2$）。当两个质量碰撞时（设碰撞时间为 $\\tau$），设底部质量从 $t = 0 \\mathrm{s}$ 的下降距离为 $D_2 = x_1(\\tau) - x_1(0) = \\gamma L_0$。求 $\\gamma$ 的数值。\n注意：(f) 是前置问题，请不要写入最终答案中。\n\n现在考虑 $N = 3$ 的情况。", + "question": "底部质量经过 (f) 部分中的 $\\tau$ 时间后：\n\n（1）求下降的距离 $D_3 = x_1(\\tau) - x_1(0)$（用 $L_0$ 表示）。\n（2）比较 $D_3$ 及在 (f) 部分所得距离 $D_2$，看看哪一个比较小，在答案中写上 $D_3$ 或 $D_2$。", + "marking": [], + "answer": [ + "\\boxed{$D_3 = 0.054 L_0$}", + "\\boxed{$D_3$}" + ], + "answer_type": [ + "Expression", + "Open-Ended" + ], + "unit": [ + null + ], + "points": [ + 1.0, + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2024", + "image_question": [ + "image_question/PanMechanics_2024_18_1.png", + "image_question/PanMechanics_2024_18_2.png" + ] + } +] \ No newline at end of file diff --git a/data/PanMechanics_2025.json b/data/PanMechanics_2025.json new file mode 100644 index 0000000000000000000000000000000000000000..9def754b947b7a591e322c665b3a5249ac921ca5 --- /dev/null +++ b/data/PanMechanics_2025.json @@ -0,0 +1,548 @@ +[ + { + "information": "若有需要，取重力加速度 $g = 9.80 m/s^2$ 及 重力常数 $G = 6.67 \\times 10^{-11} N m^2/\\mathrm{kg}^2$（若没有特别注明，取所有摩擦力为零）。" + }, + { + "id": "PanMechanics_2025_1", + "context": "", + "question": "一辆重量为 800 公斤的赛车沿着半径为 $R = 100$ 米的圆形赛道行驶。赛道倾斜角度为 $\\beta = 30.0^{\\circ}$，如图所示。轮胎与赛道之间的静摩擦系数和动摩擦系数分别为 $\\mu_s = 0.300$ 和 $\\mu_k = 0.200$。保留 3 位有效数字，在什么速度范围内，轮胎不会相对于赛道打滑？您可以忽略空气阻力和滚动摩擦，这样轮胎和赛道之间在赛道切线方向上就没有摩擦力。\n\n[figure1]\n\n(A) $v \\leq 32.2 m/s$. \n(B) $15.2 m/s \\leq v$. \n(C) $v \\leq 38.4 m/s$. \n(D) $15.2 m/s \\leq v \\leq 32.2 m/s$. \n(E) $20.7 m/s \\leq v \\leq 38.4 m/s$.", + "marking": [], + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [ + "image_question/PanMechanics_2025_1_1.png" + ] + }, + { + "id": "PanMechanics_2025_2", + "context": "", + "question": "一辆重量为 800 公斤的赛车沿着半径为 $R = 100$ 米的圆形赛道行驶。赛道倾斜角度为 $\\beta = 30.0^{\\circ}$，如图所示。轮胎与赛道之间的静摩擦系数和动摩擦系数分别为 $\\mu_s = 0.600$ 和 $\\mu_k = 0.400$。保留 3 位有效数字，在什么速度范围内，轮胎不会相对于赛道打滑？您可以忽略空气阻力和滚动摩擦，这样轮胎和赛道之间在赛道切线方向上就没有摩擦力。\n\n[figure1]\n\n(A) $10.4 m/s \\leq v$. \n(B) $v \\leq 35.1 m/s$. \n(C) $v \\leq 42.0 m/s$. \n(D) $10.4 m/s \\leq v \\leq 42.0 m/s$. \n(E) $18.5 m/s \\leq v \\leq 35.1 m/s$.", + "marking": [], + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [ + "image_question/PanMechanics_2025_1_1.png" + ] + }, + { + "id": "PanMechanics_2025_3", + "context": "", + "question": "一个装有弹簧的玩具静止地放在水平、无摩擦的表面上。当弹簧松开时，玩具会分裂成三块，A、B 和 C，且各自沿着表面滑动。A、B 和 C 的质量分别为 $m_A$、$m_B$ 和 $m_C$。若已知 A 和 B 的速度之间的角度为 $120^{\\circ}$，问必须满足以下哪个条件才能确保 C 的速率为三者中最快？\n\n(A) $m_C < m_A + m_B$. \n(B) $m_C < \\frac{m_A + m_B}{2}$. \n(C) $m_C < \\frac{\\sqrt{3}}{2} (m_A + m_B)$. \n(D) $m_C < \\frac{\\min{m_A, m_B}}{2}$. \n(E) $m_C < \\frac{\\sqrt{3}}{2} \\min{m_A, m_B}$.", + "marking": [], + "answer": [ + "\\boxed{E}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [] + }, + { + "id": "PanMechanics_2025_4", + "context": "", + "question": "质量为 $4.00 \\times 10^{30}$ 千克的恒星 A 正朝某个方向移动，质量为 $2.00 \\times 10^{30}$ 千克的恒星 B 位于其前方并正朝同一方向移动。恒星 A 和恒星 B 的初速度分别为 10 公里/秒和 40 公里/秒，它们的初始距离为 $2.00 \\times 10^8$ 公里。求它们的最大距离。\n\n(A) $2.58 \\times 10^8$ 公里. \n(B) $3.33 \\times 10^8$ 公里. \n(C) $4.75 \\times 10^8$ 公里. \n(D) $5.29 \\times 10^8$ 公里. \n(E) $6.14 \\times 10^8$ 公里.", + "marking": [], + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [] + }, + { + "id": "PanMechanics_2025_5", + "context": "", + "question": "在一个半径为 $R$ 的均匀球体上挖出一个直径为 $R$ 的洞，使原球体的中心 $O$ 位于洞的表面，如图所示。洞的中心和 $O$ 都位于 $y$ 轴上。求该物体绕通过 $O$ 并与洞表面相切的轴（图中的 $x$ 轴）的转动惯量，以 $R$ 和物体的质量 $M$ 表达。\n\n[figure1]\n\n(A) $\\frac{1}{3} MR^2$. \n(B) $\\frac{57}{140} MR^2$. \n(C) $\\frac{463}{1120} MR^2$. \n(D) $\\frac{31}{70} MR^2$. \n(E) $\\frac{249}{560} MR^2$.", + "marking": [], + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [ + "image_question/PanMechanics_2025_5_1.png" + ] + }, + { + "id": "PanMechanics_2025_6", + "context": "", + "question": "一个均匀的矩形板，其一边固定在地面上，且板面与垂直方向成角度 $\\theta_0$，其中 $0^{\\circ} \\leq \\theta_0 \\leq 90^{\\circ}$。如果 $\\theta_0$ 大于某个角度，则在放开板体使其在重力作用下绕固定边缘旋转的那一刻，板体的另一端的加速度的垂直向下分量大于 $g$。求该角度。\n\n(A) $\\sin^{-1} \\frac{1}{3}$. \n(B) $\\sin^{-1} \\sqrt{\\frac{1}{3}}$. \n(C) $\\sin^{-1} \\frac{2}{3}$. \n(D) $45^{\\circ}$. \n(E) $\\sin^{-1} \\sqrt{\\frac{2}{3}}$.", + "marking": [], + "answer": [ + "\\boxed{E}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [] + }, + { + "id": "PanMechanics_2025_7", + "context": "", + "question": "一个均匀的矩形板，其一边固定在地面上，且板面与垂直方向成角度 $\\theta_0$，其中 $0^{\\circ} \\leq \\theta_0 \\leq 90^{\\circ}$。如果 $\\theta_0$ 大于某个角度，则在放开板体使其在重力作用下绕固定边缘旋转的那一刻，板体的另一端的加速度的垂直向下分量大于 $g$。当板体以上述角度被放开的瞬间，地面所施加的反作用力的大小和垂直分力分别是多少？\n\n(A) $\\sqrt{\\frac{3}{8}} Mg$, $\\frac{1}{2} Mg$ 向上. \n(B) $\\sqrt{\\frac{3}{8}} Mg$, $\\frac{1}{2} Mg$ 向下. \n(C) $\\frac{\\sqrt{3}}{2} Mg$, $\\frac{1}{2} Mg$ 向上. \n(D) $\\frac{\\sqrt{3}}{2} Mg$, $\\frac{1}{2} Mg$ 向下. \n(E) 此瞬间没有任何反作用力.", + "marking": [], + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [] + }, + { + "id": "PanMechanics_2025_8", + "context": "", + "question": "一个均匀的矩形板，其一边固定在地面上，且板面与垂直方向成角度 $\\theta_0$，其中 $0^{\\circ} \\leq \\theta_0 \\leq 90^{\\circ}$。如果 $\\theta_0$ 大于某个角度，则在放开板体使其在重力作用下绕固定边缘旋转的那一刻，板体的另一端的加速度的垂直向下分量大于 $g$。如果 $\\theta_0$ 小于上述临界角，则当板旋转到某个角度时，板体的另一端的加速度的垂直向下分量大于 $g$。如果 $\\theta_0 = 0^{\\circ}$，求该角度。\n\n(A) $\\sin^{-1} \\frac{\\sqrt{2}-1}{3}$. \n(B) $\\cos^{-1} \\frac{1+\\sqrt{3}}{3}$. \n(C) $\\cos^{-1} \\frac{1+\\sqrt{2}}{3}$. \n(D) $\\sin^{-1} \\frac{1+\\sqrt{2}}{3}$. \n(E) $\\cos^{-1} \\frac{\\sqrt{2}-1}{3}$.", + "marking": [], + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [] + }, + { + "id": "PanMechanics_2025_9", + "context": "", + "question": "一根均匀的细杆，质量为 $M$，长度为 $a$，最初静止在光滑的地板上。它的一端被一个点质量 $m$ 撞击，该点质量的速率为 $v$，垂直于杆。碰撞后，点质量嵌入杆中，如图所示。取 $M = 6.00$ 千克，$m = 2.00$ 千克，$a = 10.0$ 米，$v = 20.0$ 米/秒。求刚碰撞后瞬间杆另一端的速度。\n\n[figure1]\n\n(A) 0 米/秒. \n(B) 向右 6.67 米/秒. \n(C) 向右 10.2 米/秒. \n(D) 向左 3.45 米/秒. \n(E) 向左 5.71 米/秒.", + "marking": [], + "answer": [ + "\\boxed{E}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [ + "image_question/PanMechanics_2025_9_1.png" + ] + }, + { + "id": "PanMechanics_2025_10", + "context": "", + "question": "大炮 A 和大炮 B 可以以相同的速率 $u$ 发射炮弹。大炮 B 位于高为 $h$ 的悬崖顶。大炮 A 位于悬崖左方地面上，与悬崖的距离为 $d$。在某一时刻，大炮 B 向左水平发射一枚炮弹。大炮 A 同时以一定的仰角 $\\theta$ 发射一枚炮弹，如图所示。求出使得两枚炮弹可能相撞的 $\\theta$。\n\n[figure1]\n\n(A) $\\tan^{-1} \\frac{h}{d}$. \n(B) $\\tan^{-1} \\frac{d}{h}$. \n(C) $2 \\tan^{-1} \\frac{h}{d}$. \n(D) $2 \\tan^{-1} \\frac{d}{h}$. \n(E) $90^{\\circ}$.", + "marking": [], + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [ + "image_question/PanMechanics_2025_10_1.png" + ] + }, + { + "id": "PanMechanics_2025_11", + "context": "", + "question": "大炮 A 和大炮 B 可以以相同的速率 $u$ 发射炮弹。大炮 B 位于高为 $h$ 的悬崖顶。大炮 A 位于悬崖左方地面上，与悬崖的距离为 $d$。在某一时刻，大炮 B 向左水平发射一枚炮弹。大炮 A 同时以一定的仰角 $\\theta$ 发射一枚炮弹，如图所示。当速度 $u$ 低于多少时，两颗炮弹不可能相撞？\n\n[figure1]\n\n(A) $\\frac{(h^2+d^2)}{hd} \\sqrt{gh}$. \n(B) $\\frac{1}{2} \\frac{(h^2+d^2)}{hd} \\sqrt{gh}$. \n(C) $\\frac{1}{\\sqrt{2}} \\frac{(h^2+d^2)}{hd} \\sqrt{gh}$. \n(D) $\\frac{1}{2\\sqrt{2}} \\frac{(h^2+d^2)}{hd} \\sqrt{gh}$. \n(E) $\\frac{1}{4} \\frac{(h^2+d^2)}{hd} \\sqrt{gh}$.", + "marking": [], + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [ + "image_question/PanMechanics_2025_10_1.png" + ] + }, + { + "id": "PanMechanics_2025_12", + "context": "", + "question": "如图所示，一根刚性均匀杆，其右端受到向右 $2F$ 的力，左端受到向右 $F$ 的力。杆中点处的张力是多少？\n\n[figure1]\n\n(A) 0. \n(B) $F/2$. \n(C) $F$. \n(D) $3F/2$. \n(E) $2F$.", + "marking": [], + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [ + "image_question/PanMechanics_2025_12_1.png" + ] + }, + { + "id": "PanMechanics_2025_13", + "context": "", + "question": "考虑一个牛顿摆，其中只有两个球 A 和 B，它们的质量分别为 $m_A$ 和 $m_B$，且这两个质量可能不同。每个球都用一根绳子悬挂在一水平杆上固定的点。B 最初处于静止状态，A 最初在一定高度开始向下摆动并以速率 $u$ 撞击 B，如图所示。此时，两根绳子都是垂直的。碰撞是理想的弹性碰撞。碰撞后，两个球向上摆动，然后再次向下摆动，并在相同的最低位置再次碰撞。求 $m_B / m_A$。\n\n[figure1]\n\n(A) 1/3. \n(B) 1/2. \n(C) 1. \n(D) 2. \n(E) 3.", + "marking": [], + "answer": [ + "\\boxed{E}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [ + "image_question/PanMechanics_2025_13_1.png" + ] + }, + { + "id": "PanMechanics_2025_14", + "context": "", + "question": "考虑一个牛顿摆，其中只有两个球 A 和 B，它们的质量分别为 $m_A$ 和 $m_B$，且这两个质量可能不同。每个球都用一根绳子悬挂在一水平杆上固定的点。B 最初处于静止状态，A 最初在一定高度开始向下摆动并以速率 $u$ 撞击 B，如图所示。此时，两根绳子都是垂直的。碰撞是理想的弹性碰撞。碰撞后，两个球向上摆动，然后再次向下摆动，并在相同的最低位置再次碰撞。第二次碰撞后 A 和 B 的速度分别是多少？取向右的速度为正。\n\n[figure1]\n\n(A) A: $-u$, B: 0. \n(B) A: $u$, B: 0. \n(C) A: $\\frac{m_A-m_B}{m_A+m_B} u$, B: $\\frac{2m_A}{m_A+m_B} u$. \n(D) A: 0, B: $\\sqrt{\\frac{m_A}{m_B}} u$. \n(E) A: 0, B: $\\frac{m_A}{m_B} u$.", + "marking": [], + "answer": [ + "\\boxed{A}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [ + "image_question/PanMechanics_2025_13_1.png" + ] + }, + { + "id": "PanMechanics_2025_15", + "context": "", + "question": "考虑如图所示的装置。4 个金属球完全相同，质量均为 $m$。磁铁的质量为 $M = 5m$。整个装置放置在气垫轨道上，这样可以忽略所有摩擦力。最初，球 A 从左侧很远的地方以初速率 $u$ 向右移动。球 A 撞到磁铁后，球 D 被射向右侧，并在很远的地方达到最终速率 $v = 9u$。碰撞后，磁铁与球 A、B 和 C 粘在一起。由于没有摩擦，球在轨道上滑动而不旋转。分别用 $K_i$ 和 $K_f$ 表示整个系统的初动能和终动能。求 $K_f / K_i$。\n\n[figure1]\n\n(A) 81. \n(B) 89. \n(C) 105. \n(D) 121. \n(E) 137.", + "marking": [], + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [ + "image_question/PanMechanics_2025_15_1.png" + ] + }, + { + "id": "PanMechanics_2025_16", + "context": "", + "question": "一根 $2.00$ 米长的均匀细绳的一端连接到振荡器上，另一端固定。当振荡器设置为进行振幅为 $1.00$ 毫米、频率为 $10.0$ 赫兹的简谐运动时，会产生横向驻波。沿弦传播的波速为 $41.0$ 米/秒。由于振荡幅度很小，可以忽略由于振荡引起的弦长变化，故振荡器到固定端的距离可以取为 $2.00$ 米。求产生的驻波的最大振幅。\n\n(A) 1.00 毫米. \n(B) 6.53 毫米. \n(C) 9.35 毫米. \n(D) 13.1 毫米. \n(E) 18.7 毫米.", + "marking": [], + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [] + }, + { + "id": "PanMechanics_2025_17_1", + "context": "", + "question": "如图所示，两个均匀薄圆盘的质量分别为 $m$ 和 $4m$，半径分别为 $a$ 和 $2a$，由一根穿过它们中心的无质量的刚性杆牢固地固定住，该杆长度为 $l = \\sqrt{24} a$。该组件放在坚固平坦的表面上，并使其在表面上滚动而不滑动。绕杆轴的角速率为 $\\omega$。\n\n[figure1]\n\n求杆与水平面的夹角 $\\theta$。（单位用 $\\mathrm{rad}$ 表示）", + "marking": [], + "answer": [ + "\\boxed{0.201}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "$\\mathrm{rad}$" + ], + "points": [ + 4.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [ + "image_question/PanMechanics_2025_17_1.png" + ] + }, + { + "id": "PanMechanics_2025_17_2", + "context": "", + "question": "如图所示，两个均匀薄圆盘的质量分别为 $m$ 和 $4m$，半径分别为 $a$ 和 $2a$，由一根穿过它们中心的无质量的刚性杆牢固地固定住，该杆长度为 $l = \\sqrt{24} a$。该组件放在坚固平坦的表面上，并使其在表面上滚动而不滑动。绕杆轴的角速率为 $\\omega$。\n\n[figure1]\n\n求组件质心绕 $z$ 轴的角速率。", + "marking": [], + "answer": [ + "\\boxed{$\\frac{\\omega}{5}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 7.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [ + "image_question/PanMechanics_2025_17_1.png" + ] + }, + { + "id": "PanMechanics_2025_17_3", + "context": "", + "question": "如图所示，两个均匀薄圆盘的质量分别为 $m$ 和 $4m$，半径分别为 $a$ 和 $2a$，由一根穿过它们中心的无质量的刚性杆牢固地固定住，该杆长度为 $l = \\sqrt{24} a$。该组件放在坚固平坦的表面上，并使其在表面上滚动而不滑动。绕杆轴的角速率为 $\\omega$。\n\n[figure1]\n\n求绕点 $O$ 的轨道角动量的大小，即忽略在质心参考系中观察到的由于自转而产生的自旋角动量而只计算由于组件质心运动而产生的角动量。", + "marking": [], + "answer": [ + "\\boxed{$\\frac{1944 \\sqrt{24}}{125} m \\omega a^2$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 11.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [ + "image_question/PanMechanics_2025_17_1.png" + ] + }, + { + "id": "PanMechanics_2025_18_1", + "context": "", + "question": "考虑两个质量分别为 $m$ 和 $M$ 的物体在光滑水平面上作一维运动运动并发生弹性碰撞。$m$ 和 $M$ 的初速度分别为 $u$ 和 $U$，碰撞后的速度分别为 $v$ 和 $V$。\n\n(1) 求 $v$ 的表达式，用 $m$, $M$, $u$ 和 $U$ 表示。\n(2) 求 $V$ 的表达式，用 $m$, $M$, $u$ 和 $U$ 表示。", + "marking": [], + "answer": [ + "\\boxed{$v = \\frac{m-M}{m+M} u + \\frac{2M}{m+M} U$}", + "\\boxed{$V = \\frac{2m}{m+M} u - \\frac{m-M}{m+M} U$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 2.0, + 2.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [] + }, + { + "id": "PanMechanics_2025_18_2", + "context": "考虑两个质量分别为 $m$ 和 $M$ 的物体在光滑水平面上作一维运动运动并发生弹性碰撞。$m$ 和 $M$ 的初速度分别为 $u$ 和 $U$，碰撞后的速度分别为 $v$ 和 $V$。\n\n已知 $m < M$ 且 $m$ 最初处于静止状态，而 $M$ 在右侧以速度 $U$ 向 $m$ 移动。将向右的方向视为正方向，因此 $U < 0$。初次碰撞后，$m$ 向左移动并在一定距离处撞到一墙壁，然后反弹回来再次向右移动。然后它可能再次撞到 $M$ 并再反弹回来撞到墙壁。将 $m$ 和 $M$ 之间的初次碰撞称为第 0 次碰撞。我们假设所有碰撞都是弹性碰撞。设 $m$ 和 $M$ 在第 $n$ 次碰撞之前的速度分别为 $v_n$ 和 $V_n$，因此 $V_0 = U$ 和 $v_0 = 0$。\n\n[figure1]", + "question": "用相空间中的一个点 $(\\sqrt{M}V, \\sqrt{m}v)$ 表示两个物体的运动状态。因此，在第一次碰撞之前，状态为 $(\\sqrt{M}V_0, \\sqrt{m}v_0) = (\\sqrt{M}U, 0)$。在第二次碰撞之前，状态为 $(\\sqrt{M}V_1, \\sqrt{m}v_1)$。用 $m$ 和 $M$ 表示如图所示的角度 $\\theta$。", + "marking": [], + "answer": [ + "\\boxed{$\\theta = \\tan^{-1} \\frac{2 \\sqrt{Mm}}{M-m}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 6.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [ + "image_question/PanMechanics_2025_18_1.png" + ] + }, + { + "id": "PanMechanics_2025_18_3", + "context": "考虑两个质量分别为 $m$ 和 $M$ 的物体在光滑水平面上作一维运动运动并发生弹性碰撞。$m$ 和 $M$ 的初速度分别为 $u$ 和 $U$，碰撞后的速度分别为 $v$ 和 $V$。\n\n已知 $m < M$ 且 $m$ 最初处于静止状态，而 $M$ 在右侧以速度 $U$ 向 $m$ 移动。将向右的方向视为正方向，因此 $U < 0$。初次碰撞后，$m$ 向左移动并在一定距离处撞到一墙壁，然后反弹回来再次向右移动。然后它可能再次撞到 $M$ 并再反弹回来撞到墙壁。将 $m$ 和 $M$ 之间的初次碰撞称为第 0 次碰撞。我们假设所有碰撞都是弹性碰撞。设 $m$ 和 $M$ 在第 $n$ 次碰撞之前的速度分别为 $v_n$ 和 $V_n$，因此 $V_0 = U$ 和 $v_0 = 0$。\n\n[figure1]\n\n用相空间中的一个点 $(\\sqrt{M}V, \\sqrt{m}v)$ 表示两个物体的运动状态。因此，在第一次碰撞之前，状态为 $(\\sqrt{M}V_0, \\sqrt{m}v_0) = (\\sqrt{M}U, 0)$。在第二次碰撞之前，状态为 $(\\sqrt{M}V_1, \\sqrt{m}v_1)$。", + "question": "如上所述，或以其他方法：(1) 求 $v_n$ 的表达式，用 $m$, $M$, 和 $U$ 表示。\n(2) 求 $V_n$ 的表达式，用 $m$, $M$, 和 $U$ 表示。", + "marking": [], + "answer": [ + "\\boxed{$v_n = -\\sqrt{\\frac{M}{m}} U \\sin\\left( n \\tan^{-1} \\frac{2\\sqrt{Mm}}{M-m} \\right)$}", + "\\boxed{$V_n = U \\cos\\left( n \\tan^{-1} \\frac{2\\sqrt{Mm}}{M-m} \\right)$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null + ], + "points": [ + 6.5, + 6.5 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [ + "image_question/PanMechanics_2025_18_1.png" + ] + }, + { + "id": "PanMechanics_2025_18_4", + "context": "考虑两个质量分别为 $m$ 和 $M$ 的物体在光滑水平面上作一维运动运动并发生弹性碰撞。$m$ 和 $M$ 的初速度分别为 $u$ 和 $U$，碰撞后的速度分别为 $v$ 和 $V$。\n\n已知 $m < M$ 且 $m$ 最初处于静止状态，而 $M$ 在右侧以速度 $U$ 向 $m$ 移动。将向右的方向视为正方向，因此 $U < 0$。初次碰撞后，$m$ 向左移动并在一定距离处撞到一墙壁，然后反弹回来再次向右移动。然后它可能再次撞到 $M$ 并再反弹回来撞到墙壁。将 $m$ 和 $M$ 之间的初次碰撞称为第 0 次碰撞。我们假设所有碰撞都是弹性碰撞。设 $m$ 和 $M$ 在第 $n$ 次碰撞之前的速度分别为 $v_n$ 和 $V_n$，因此 $V_0 = U$ 和 $v_0 = 0$。上述过程重复一定次数，直到系统达到不再发生碰撞的状态。", + "question": "假设两物体之间以及 $m$ 与墙壁之间的碰撞总次数为 $N$。如果画出 $N$ 与 $\\sqrt{M/m}$ 的关系图，那么当$M \\gg m$ 时，这些点渐近位于一条直线上。求这条直线的斜率是多少？", + "marking": [], + "answer": [ + "\\boxed{$\\pi$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 7.0 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanMechanics_2025", + "image_question": [] + } +] \ No newline at end of file diff --git a/data/PanPhO_2024.json b/data/PanPhO_2024.json new file mode 100644 index 0000000000000000000000000000000000000000..2814f7376c6c49881e90713bddd13910bc094c32 --- /dev/null +++ b/data/PanPhO_2024.json @@ -0,0 +1,850 @@ +[ + { + "information": "None." + }, + { + "id": "PanPhO_2024_1_1", + "context": "We consider an inhomogeneous cylinder whose have are mode of two materials of different densities. The cylinder has radius $r$ and length $L$ and its cross section is shown in the Figure 1. The bottom half made of a material whose density is $1 kg / m^3$ while the upper half is mode of a material whose density is $c kg / m^3$, where $c$ is a parameter $0 < c < 1$. In the problem, we can use the fact that for a half-cylinder of radius $r$, the center of mass (CM) is located at a distance of $\\frac{4 r}{3 \\pi}$ from the axis of the half-cylinder, as shown in the Figure 2.\n\n[figure1]\n\n[figure2]", + "question": "(1) Compute the total mass $M$ of the entire inhomogeneous cylinder. Express the answer in terms of $r, L, c$. \n(2) Compute the distance $d$ between the geometrical center and the center of mass of the entire cylinder. Express the answer in terms of $r, L, c$.", + "marking": [], + "answer": [ + "\\boxed{$M = \\frac{\\pi r^2 L}{2}(1+c)$}", + "\\boxed{$d = \\frac{4 r}{3 \\pi} \\left(\\frac{1-c}{1+c}\\right)$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 1.0, + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_1_1_1.png", + "image_question/PanPhO_2024_1_1_2.png" + ] + }, + { + "id": "PanPhO_2024_1_2", + "context": "We consider an inhomogeneous cylinder whose have are mode of two materials of different densities. The cylinder has radius $r$ and length $L$ and its cross section is shown in the Figure 1. The bottom half made of a material whose density is $1 kg / m^3$ while the upper half is mode of a material whose density is $c kg / m^3$, where $c$ is a parameter $0 < c < 1$. In the problem, we can use the fact that for a half-cylinder of radius $r$, the center of mass (CM) is located at a distance of $\\frac{4 r}{3 \\pi}$ from the axis of the half-cylinder, as shown in the Figure 2.\n\n[figure1]\n\n[figure2]", + "question": "Compute the moment of inertia $I$ of the entire inhomogeneous cylinder with respect to its geometrical axis. Express the answers in terms of $r, L, c$.", + "marking": [], + "answer": [ + "\\boxed{$I = \\frac{\\pi r^4 L}{4} (1+c)$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_1_1_1.png", + "image_question/PanPhO_2024_1_1_2.png" + ] + }, + { + "id": "PanPhO_2024_1_3", + "context": "We consider an inhomogeneous cylinder whose have are mode of two materials of different densities. The cylinder has radius $r$ and length $L$ and its cross section is shown in the Figure 1. The bottom half made of a material whose density is $1 kg / m^3$ while the upper half is mode of a material whose density is $c kg / m^3$, where $c$ is a parameter $0 < c < 1$. In the problem, we can use the fact that for a half-cylinder of radius $r$, the center of mass (CM) is located at a distance of $\\frac{4 r}{3 \\pi}$ from the axis of the half-cylinder, as shown in the Figure 2.\n\n[figure1]\n\n[figure2]\n\nDenote $M$ as the total mass of the entire inhomogeneous cylinder, $d$ as the distance between the geometrical center and the center of mass of the entire cylinder, $I$ as the moment of inertia of the entire inhomogeneous cylinder with respect to its geometrical axis.", + "question": "The geometrical axis of the cylinder is fixed in a horizontal position, but the cylinder is free to rotate without any friction around the axis. If the cylinder oscillates around its stable equilibrium position with small amplitude, calculate the period $T$ of oscillation of the cylinder. Express the answer in terms of $M, I, r, d$ and the gravitational acceleration $g$.", + "marking": [], + "answer": [ + "\\boxed{$T = 2 \\pi \\sqrt{\\frac{I}{M g d}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_1_1_1.png", + "image_question/PanPhO_2024_1_1_2.png" + ] + }, + { + "id": "PanPhO_2024_1_4", + "context": "We consider an inhomogeneous cylinder whose have are mode of two materials of different densities. The cylinder has radius $r$ and length $L$ and its cross section is shown in the Figure 1. The bottom half made of a material whose density is $1 kg / m^3$ while the upper half is mode of a material whose density is $c kg / m^3$, where $c$ is a parameter $0 < c < 1$. In the problem, we can use the fact that for a half-cylinder of radius $r$, the center of mass (CM) is located at a distance of $\\frac{4 r}{3 \\pi}$ from the axis of the half-cylinder, as shown in the Figure 2.\n\n[figure1]\n\n[figure2]\n\nDenote $M$ as the total mass of the entire inhomogeneous cylinder, $d$ as the distance between the geometrical center and the center of mass of the entire cylinder, $I$ as the moment of inertia of the entire inhomogeneous cylinder with respect to its geometrical axis.\n\nNow we assume that the cylinder is completely free to move on a horizontal table under the gravity. We assume that the coefficient of static friction between the cylinder and the table is infinite, such that the cylinder cannot slide. Suppose that at time $t=0$, the cylinder is in its equilibrium position with an initial angular velocity $\\omega_0$.", + "question": "If $\\omega_0$ is sufficiently small, the cylinder will undergo a period motion around its stable equilibrium. What is the period $T$ of oscillation if the amplitude of the oscillation is small? Express the answer in terms of $M, l, r, d$ and the gravitational acceleration $g$.", + "marking": [], + "answer": [ + "\\boxed{$T = 2 \\pi \\sqrt{\\frac{I + M(r^2 - 2 r d)}{M g d}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_1_1_1.png", + "image_question/PanPhO_2024_1_1_2.png" + ] + }, + { + "id": "PanPhO_2024_1_5", + "context": "We consider an inhomogeneous cylinder whose have are mode of two materials of different densities. The cylinder has radius $r$ and length $L$ and its cross section is shown in the Figure 1. The bottom half made of a material whose density is $1 kg / m^3$ while the upper half is mode of a material whose density is $c kg / m^3$, where $c$ is a parameter $0 < c < 1$. In the problem, we can use the fact that for a half-cylinder of radius $r$, the center of mass (CM) is located at a distance of $\\frac{4 r}{3 \\pi}$ from the axis of the half-cylinder, as shown in the Figure 2.\n\n[figure1]\n\n[figure2]\n\nDenote $M$ as the total mass of the entire inhomogeneous cylinder, $d$ as the distance between the geometrical center and the center of mass of the entire cylinder, $I$ as the moment of inertia of the entire inhomogeneous cylinder with respect to its geometrical axis.\n\nNow we assume that the cylinder is completely free to move on a horizontal table under the gravity. We assume that the coefficient of static friction between the cylinder and the table is infinite, such that the cylinder cannot slide. Suppose that at time $t=0$, the cylinder is in its equilibrium position with an initial angular velocity $\\omega_0$. If $\\omega_0$ is sufficiently small, the cylinder will undergo a period motion around its stable equilibrium.", + "question": "What is the minimum value of $\\omega_0$ that allows the cylinder to roll forever in the same direction. Express the answer in terms of $M, l, r, d$.", + "marking": [], + "answer": [ + "\\boxed{$\\omega_0 = \\sqrt{\\frac{4 M g d}{M (r-d)^2 + (I - M d^2)}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_1_1_1.png", + "image_question/PanPhO_2024_1_1_2.png" + ] + }, + { + "id": "PanPhO_2024_2_1", + "context": "A closed container is divided into three compartments, A, B, and C, by two partitions, $D_1$ and $D_2$, as shown in the figure. Each compartment is filled with the same monoatomic ideal gas with pressure $P$, volume $V$, and absolute temperature $T$ as shown in the figure. The mass of partition $D_1$ is $m$, which can slide freely without friction, while partition $D_2$ is fixed and has a small valve on it. Now, the valve on partition $D_2$ is opened, allowing the gases in compartments B and C to mix and the entire system to reach equilibrium while maintaining a constant temperature $T_0$.\n\n[figure1]", + "question": "After the entire system reaches equilibrium: \n(1) What is the pressure $P_A$ of the gas in compartment A? Express your answer in terms of $P_0$. \n(2) What is the pressure $P_B$ of the gas in compartment B? Express your answer in terms of $P_0$. \n(3) What is the pressure $P_C$ of the gas in compartment C? Express your answer in terms of $P_0$. \n(4) What is the volume $V_A$ of the gas in compartment A? Express your answer in terms of $V_0$. \n(5) What is the volume $V_B$ of the gas in compartment B? Express your answer in terms of $V_0$. \n(6) What is the volume $V_C$ of the gas in compartment C? Express your answer in terms of $V_0$. \nNote: The definitions of $P_0$ and $V_0$ are given in the figure.", + "marking": [], + "answer": [ + "\\boxed{$P_A = \\frac{4}{3} P_0$}", + "\\boxed{$P_B = \\frac{4}{3} P_0$}", + "\\boxed{$P_C = \\frac{4}{3} P_0$}", + "\\boxed{$V_A = \\frac{3}{4} V_0$}", + "\\boxed{$V_B = \\frac{5}{4} V_0$}", + "\\boxed{$V_C = V_0$}" + ], + "answer_type": [ + "Expression", + "Expression", + "Expression", + "Expression", + "Expression", + "Expression" + ], + "unit": [ + null, + null, + null, + null, + null, + null + ], + "points": [ + 0.5, + 0.5, + 0.5, + 0.5, + 0.5, + 0.5 + ], + "modality": "text+variable figure", + "field": "Thermodynamics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_2_1_1.png" + ] + }, + { + "id": "PanPhO_2024_2_2", + "context": "A closed container is divided into three compartments, A, B, and C, by two partitions, $D_1$ and $D_2$, as shown in the figure. Each compartment is filled with the same monoatomic ideal gas with pressure $P$, volume $V$, and absolute temperature $T$ as shown in the figure. The mass of partition $D_1$ is $m$, which can slide freely without friction, while partition $D_2$ is fixed and has a small valve on it. Now, the valve on partition $D_2$ is opened, allowing the gases in compartments B and C to mix and the entire system to reach equilibrium while maintaining a constant temperature $T_0$.\n\n[figure1]", + "question": "How much total heat is absorbed by the gases in compartments B and C during the process of the entire system reaching equilibrium? Express your answer in terms of $P_0$ and $V_0$. \nNote: The definitions of $P_0$ and $V_0$ are given in the figure.", + "marking": [], + "answer": [ + "\\boxed{$0.288 P_0 V_0$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+variable figure", + "field": "Thermodynamics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_2_1_1.png" + ] + }, + { + "id": "PanPhO_2024_2_3", + "context": "A closed container is divided into three compartments, A, B, and C, by two partitions, $D_1$ and $D_2$, as shown in the figure. Each compartment is filled with the same monoatomic ideal gas with pressure $P$, volume $V$, and absolute temperature $T$ as shown in the figure. The mass of partition $D_1$ is $m$, which can slide freely without friction, while partition $D_2$ is fixed and has a small valve on it. Now, the valve on partition $D_2$ is opened, allowing the gases in compartments B and C to mix and the entire system to reach equilibrium while maintaining a constant temperature $T_0$.\n\n[figure1]", + "question": "Calculate the change in entropy $\\Delta S$ of the entire system during the process of reaching equilibrium. Express your answer in terms of $P_0$, $V_0$, and $T_0$. \nNote: The definitions of $P_0$, $V_0$, and $T_0$ are given in the figure.", + "marking": [], + "answer": [ + "\\boxed{$\\Delta S \\approx 0.236 \\frac{P_0 V_0}{T_0}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 4.0 + ], + "modality": "text+variable figure", + "field": "Thermodynamics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_2_1_1.png" + ] + }, + { + "id": "PanPhO_2024_3_2", + "context": "[Trapped Ball] \n\nAs shown in the figure, a ball (modelled as a point charge of magnitude $q > 0$) of mass $m$ is trapped in a spherical cavity of radius $R$ carved from an infinite grounded conductor. The charge is at a distance $z$ from the center. In the problem, the gravity can be ignored.\n\n[figure1]", + "question": "Find the magnitude of the electric force $F(z)$ acting on the ball in terms of $q$, $z$, $R$ and the vacuum permittivity $\\varepsilon_0$.", + "marking": [], + "answer": [ + "\\boxed{$\\frac{1}{4 \\pi \\varepsilon_0} \\frac{q^2 R z}{(R^2 - z^2)^2}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 4.0 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_3_1_1.png" + ] + }, + { + "id": "PanPhO_2024_3_3", + "context": "[Trapped Ball] \n\nAs shown in the figure, a ball (modelled as a point charge of magnitude $q > 0$) of mass $m$ is trapped in a spherical cavity of radius $R$ carved from an infinite grounded conductor. The charge is at a distance $z$ from the center. In the problem, the gravity can be ignored.\n\n[figure1]", + "question": "If the ball is released at the center with a very small speed, find the speed of the ball $v$ when it is at a distance $R / 2$ from the center of the conductor. Express the answer in terms of $q, m, R$ and the vacuum permittivity $\\varepsilon_0$.", + "marking": [], + "answer": [ + "\\boxed{$v = \\sqrt{\\frac{1}{12 \\pi \\varepsilon_0} \\frac{q^2}{m R}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 4.0 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_3_1_1.png" + ] + }, + { + "id": "PanPhO_2024_4_1", + "context": "[In this question, all answers cannot be expressed in terms of any trigonometrical functions.] \n\nAn ice hemisphere with radius $R$ and refractive index $n$ lies on a warm flat table and melts slowly. The rate of heat transfer from the table to the ice is proportional to the area of contact between them. It is known that the ice hemisphere completely melts in time $T_{0}$. Throughout the process, a laser beam incident on the ice from above. The beam is vertically incident at a distance of $R / 2$ from the axis of symmetry (see figure).\n\nAssume that the temperature of the ice and the surrounding atmosphere are $0^{\\circ} \\mathrm{C}$ and remains constant during the melting process. The laser beam does not transfer energy to the ice. The melting water immediately flows off the table, and the ice does not move along the table.\n\n[figure1]", + "question": "What is the position of the point on the table, $x_{0} = x(t=0)$, where the beam hit at time $t = 0$? Express the answer in terms of $n$ and $R$.", + "marking": [], + "answer": [ + "\\boxed{$x_0 = \\frac{2 R}{\\sqrt{12 n^2 - 3} + 1}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 4.0 + ], + "modality": "text+variable figure", + "field": "Optics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_4_1_1.png" + ] + }, + { + "id": "PanPhO_2024_4_2", + "context": "[In this question, all answers cannot be expressed in terms of any trigonometrical functions.] \n\nAn ice hemisphere with radius $R$ and refractive index $n$ lies on a warm flat table and melts slowly. The rate of heat transfer from the table to the ice is proportional to the area of contact between them. It is known that the ice hemisphere completely melts in time $T_{0}$. Throughout the process, a laser beam incident on the ice from above. The beam is vertically incident at a distance of $R / 2$ from the axis of symmetry (see figure).\n\nAssume that the temperature of the ice and the surrounding atmosphere are $0^{\\circ} \\mathrm{C}$ and remains constant during the melting process. The laser beam does not transfer energy to the ice. The melting water immediately flows off the table, and the ice does not move along the table.\n\n[figure1]", + "question": "What is the height of the ice $z(t)$ as a function of time $t$? Express the answer in terms of $R$ and $T_{0}$.", + "marking": [], + "answer": [ + "\\boxed{$z(t) = R (1 - \\frac{t}{T_0})$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+variable figure", + "field": "Thermodynamics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_4_1_1.png" + ] + }, + { + "id": "PanPhO_2024_4_3", + "context": "[In this question, all answers cannot be expressed in terms of any trigonometrical functions.] \n\nAn ice hemisphere with radius $R$ and refractive index $n$ lies on a warm flat table and melts slowly. The rate of heat transfer from the table to the ice is proportional to the area of contact between them. It is known that the ice hemisphere completely melts in time $T_{0}$. Throughout the process, a laser beam incident on the ice from above. The beam is vertically incident at a distance of $R / 2$ from the axis of symmetry (see figure).\n\nAssume that the temperature of the ice and the surrounding atmosphere are $0^{\\circ} \\mathrm{C}$ and remains constant during the melting process. The laser beam does not transfer energy to the ice. The melting water immediately flows off the table, and the ice does not move along the table.\n\n[figure1]", + "question": "What is the position of the point on the table, $x(t)$, where the beam hit for $t \\geq 0$? Express the answer in terms of $n, R, T_{0}$ and $t$.", + "marking": [], + "answer": [ + "\\boxed{$x(t) = \\frac{R}{2}$ for $t \\geq \\frac{\\sqrt{3}}{2} T_0$}", + "\\boxed{$x(t) = \\frac{R}{2} - \\frac{R}{2} \\left(1 - \\frac{4}{\\sqrt{12 n^2 - 3} + 1}\\right) \\left(1 - \\frac{2}{\\sqrt{3}} \\frac{t}{T_0}\\right)$ for $t < \\frac{\\sqrt{3}}{2} T_0$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 1.5, + 1.5 + ], + "modality": "text+variable figure", + "field": "Optics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_4_1_1.png" + ] + }, + { + "id": "PanPhO_2024_5_1", + "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part A: Gravitational fluctuations on the orbit of the satellite]", + "question": "Here we only consider gravity from the earth and consider the earth as a homogeneous ideal ball. Give the periodicity $T$ of a satellite rotating around the earth. Please use second as the unit and give three significant figures.", + "marking": [], + "answer": [ + "\\boxed{$3.15 \\times 10^{5}$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "s" + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_5_1_1.png" + ] + }, + { + "id": "PanPhO_2024_5_2", + "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part A: Gravitational fluctuations on the orbit of the satellite]\n\n[figure2]", + "question": "Since the shape and density of the earth is inhomogeneous, the satellite will feel an additional acceleration $\\delta a$ in addition to the uniform circular motion. To simplify the calculation, let us model the inhomogeneity of the earth as follows: Consider an ideal ball with mass $M - 2m$. Two additional point masses (each has mass $m$) are put to diametrically opposite points on the equator of the earth. Assume that the satellite orbit and the earth are in the same plane, with angle $\\theta$ between them. Give the precise formula to calculate $\\delta a$: \n(1) the $x$-component of the acceleration $\\delta a_{x}$; \n(2) the $y$-component of the acceleration $\\delta a_{y}$. \nPlease express your answer in terms of $G$, $M$, $m$, $R$, $r$, and $\\theta$.", + "marking": [], + "answer": [ + "\\boxed{$\\delta a_{x} = -\\frac{G(M - 2m)}{R^2} - \\frac{G m (R - r \\cos \\theta)}{(R^2 + r^2 - 2 R r \\cos \\theta)^{3/2}} - \\frac{G m (R + r \\cos \\theta)}{(R^2 + r^2 + 2 R r \\cos \\theta)^{3/2}}$}", + "\\boxed{$\\delta a_{y} = \\frac{G m r \\sin \\theta}{(R^2 + r^2 - 2 R r \\cos \\theta)^{3/2}} - \\frac{G m r \\sin \\theta}{(R^2 + r^2 + 2 R r \\cos \\theta)^{3/2}}$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 1.0, + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_5_1_1.png", + "image_question/PanPhO_2024_5_2_1.png" + ] + }, + { + "id": "PanPhO_2024_5_3", + "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part A: Gravitational fluctuations on the orbit of the satellite]\n\n[figure2]\n\nSince the shape and density of the earth is inhomogeneous, the satellite will feel an additional acceleration $\\delta a$ in addition to the uniform circular motion. To simplify the calculation, let us model the inhomogeneity of the earth as follows: Consider an ideal ball with mass $M - 2m$. Two additional point masses (each has mass $m$) are put to diametrically opposite points on the equator of the earth. Assume that the satellite orbit and the earth are in the same plane, with angle $\\theta$ between them.", + "question": "Calculate all the possible periodicities for $\\delta a$. Please use second as the unit and give three significant figures.", + "marking": [], + "answer": [ + "\\boxed{$3.39 \\times 10^{4}$}", + "\\boxed{$5.95 \\times 10^{4}$}" + ], + "answer_type": [ + "Numerical Value", + "Numerical Value" + ], + "unit": [ + "s", + "s" + ], + "points": [ + 1.0, + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_5_1_1.png", + "image_question/PanPhO_2024_5_2_1.png" + ] + }, + { + "id": "PanPhO_2024_5_4", + "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part A: Gravitational fluctuations on the orbit of the satellite]\n\n[figure2]\n\nSince the shape and density of the earth is inhomogeneous, the satellite will feel an additional acceleration $\\delta a$ in addition to the uniform circular motion. To simplify the calculation, let us model the inhomogeneity of the earth as follows: Consider an ideal ball with mass $M - 2m$. Two additional point masses (each has mass $m$) are put to diametrically opposite points on the equator of the earth. Assume that the satellite orbit and the earth are in the same plane, with angle $\\theta$ between them.", + "question": "In the $R \\gg r$ limit, give the leading order expression (the lowest nonzero order in the Taylor expansion of $r / R$) for $\\delta a$: \n(1) the $x$-component of the acceleration $\\delta a_{x}$; \n(2) the $y$-component of the acceleration $\\delta a_{y}$. \nPlease express your answer in terms of $G$, $m$, $r$, $R$ and $\\theta$.", + "marking": [], + "answer": [ + "\\boxed{$\\delta a_{x} \\simeq \\frac{3 G m r^{2} (1 - 3 \\cos^2 \\theta)}{R^4}$}", + "\\boxed{$\\delta a_{y} \\simeq \\frac{3 G m r^{2} \\sin 2 \\theta}{R^4}$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 1.0, + 1.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_5_1_1.png", + "image_question/PanPhO_2024_5_2_1.png" + ] + }, + { + "id": "PanPhO_2024_5_5", + "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part A: Gravitational fluctuations on the orbit of the satellite]\n\n[figure2]\n\nSince the shape and density of the earth is inhomogeneous, the satellite will feel an additional acceleration $\\delta a$ in addition to the uniform circular motion. To simplify the calculation, let us model the inhomogeneity of the earth as follows: Consider an ideal ball with mass $M - 2m$. Two additional point masses (each has mass $m$) are put to diametrically opposite points on the equator of the earth. Assume that the satellite orbit and the earth are in the same plane, with angle $\\theta$ between them.", + "question": "Estimate the typical value of $\\delta a$ with the unit of $m/s^2$ (an error within two orders of magnitude will be considered as correct).", + "marking": [], + "answer": [ + "\\boxed{$[10^{-11}, 10^{-7}]$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "m/s^2" + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_5_1_1.png", + "image_question/PanPhO_2024_5_2_1.png" + ] + }, + { + "id": "PanPhO_2024_5_6", + "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part A: Gravitational fluctuations on the orbit of the satellite]\n\n[figure2]\n\nSince the shape and density of the earth is inhomogeneous, the satellite will feel an additional acceleration $\\delta a$ in addition to the uniform circular motion. To simplify the calculation, let us model the inhomogeneity of the earth as follows: Consider an ideal ball with mass $M - 2m$. Two additional point masses (each has mass $m$) are put to diametrically opposite points on the equator of the earth. Assume that the satellite orbit and the earth are in the same plane, with angle $\\theta$ between them.", + "question": "In satellite experiments, we are interested in the gravitational waves with a particular periodicity (such as periods between 1-1000 seconds). Thus, if the periodicity of the gravitational fluctuation is too long, it will not interfere the gravitational wave measurement. Assume the satellite is co-rotating in the same direction with the spinning direction of the earth. In the Taylor expansion of $\\delta a$, calculate the component with period closest to 1000s. Denote this component as $\\delta a_{1000}$. Estimate the value of $\\delta a_{1000} / \\delta a$ for $\\theta = \\pi / 3$. (an error within two orders of magnitude will be considered as correct)", + "marking": [], + "answer": [ + "\\boxed{$[10^{-143}, 10^{-139}]$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_5_1_1.png", + "image_question/PanPhO_2024_5_2_1.png" + ] + }, + { + "id": "PanPhO_2024_5_7", + "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part B: Free electrons from the solar wind]\n\nConsider the laser signal between the satellites. Although the space between satellites is close to the vacuum, but it is not the absolute vacuum. In particular, solar wind will introduce free electrons. Let the number density of the free electrons be $N_{e}$, the electric charge of an electron be $e$, the electron mass be $m_{e}$. And we ignore other media apart from these electrons.", + "question": "Assume that the electrons move freely in the electric field produced by the laser. Calculate the acceleration of the electron $d \\mathbf{v}_{e} / d t$ as a function of the electric field $\\mathbf{E}$ produced by the laser.", + "marking": [], + "answer": [ + "\\boxed{$\\frac{d \\mathbf{v}_{e}}{d t} = -e \\frac{\\mathbf{E}}{m_{e}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_5_1_1.png" + ] + }, + { + "id": "PanPhO_2024_5_8", + "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part B: Free electrons from the solar wind]\n\nConsider the laser signal between the satellites. Although the space between satellites is close to the vacuum, but it is not the absolute vacuum. In particular, solar wind will introduce free electrons. Let the number density of the free electrons be $N_{e}$, the electric charge of an electron be $e$, the electron mass be $m_{e}$. And we ignore other media apart from these electrons.\n\nAssume that the electrons move freely in the electric field produced by the laser. Denote the velocity of the electron as $\\mathbf{v}_e$, and the electric field produced by the laser as $\\mathbf{E}$.", + "question": "Calculate the time dependence of the electric current, $\\frac{d \\mathbf{J}}{d t}$, from the free electrons.", + "marking": [], + "answer": [ + "\\boxed{$\\frac{d \\mathbf{J}}{d t} = \\frac{N_{e} e^{2} \\mathbf{E}}{m_{e}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_5_1_1.png" + ] + }, + { + "id": "PanPhO_2024_5_9", + "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part B: Free electrons from the solar wind]\n\nConsider the laser signal between the satellites. Although the space between satellites is close to the vacuum, but it is not the absolute vacuum. In particular, solar wind will introduce free electrons. Let the number density of the free electrons be $N_{e}$, the electric charge of an electron be $e$, the electron mass be $m_{e}$. And we ignore other media apart from these electrons.\n\nAssume that the electrons move freely in the electric field produced by the laser. Denote the velocity of the electron as $\\mathbf{v}_e$, and the electric field produced by the laser as $\\mathbf{E}$.", + "question": "Calculate the phase speed of the laser $v_{p}$ in the environment of the free electrons (since $v_{p}$ is very close to the speed of light, the higher order difference between $v_{p}$ and the speed of light can be ignored). \n\nHint: from the Maxwell equations, one can derive that $\\frac{\\partial^{2} \\mathbf{E}}{\\partial t^{2}} - c^{2} \\nabla^{2} \\mathbf{E} + c^{2} \\mu_{0} \\frac{d \\mathbf{J}}{d t} = 0$, where $c$ is the speed of light in vacuum.", + "marking": [], + "answer": [ + "\\boxed{$v_p \\simeq c \\left(1 + \\frac{\\mu_{0} N_{e} e^{2} \\lambda^{2}}{8 \\pi^{2} m_{e}}\\right)$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_5_1_1.png" + ] + }, + { + "id": "PanPhO_2024_5_10", + "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part B: Free electrons from the solar wind]\n\nConsider the laser signal between the satellites. Although the space between satellites is close to the vacuum, but it is not the absolute vacuum. In particular, solar wind will introduce free electrons. Let the number density of the free electrons be $N_{e}$, the electric charge of an electron be $e$, the electron mass be $m_{e}$. And we ignore other media apart from these electrons.", + "question": "Let $N_{e} = 10 \\mathrm{cm}^{-3}$. Calculate the phase error of the laser between two satellites. In other words, if there were no free electrons, the laser waveform arrived at a satellite is $\\cos \\theta$. Now with free electrons, the same wave form at the same moment changes into $\\cos (\\theta + \\delta \\theta)$. Calculate the value of $\\delta \\theta$. Express your answer in radians, and give the numerical value with three significant figures.", + "marking": [], + "answer": [ + "\\boxed{$5.19 \\times 10^{-6}$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Optics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_5_1_1.png" + ] + }, + { + "id": "PanPhO_2024_5_11", + "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part C: Shot noise]\n\nAny precision measurements are limited by the uncertainty principle of quantum mechanics. Assume that every photon's arrival time at the detector can be considered as independent stochastic processes. Also, in actual experiments, phase error of the laser is more important. But here for simplicity, here we only estimate photon number errors.", + "question": "During a certain period of time, the average photon number in the laser is $N$. In this case, the error in the photon number measurement in the laser is $\\Delta N = N^{\\alpha}$. Find $\\alpha$.", + "marking": [], + "answer": [ + "\\boxed{$1/2$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Modern Physics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_5_1_1.png" + ] + }, + { + "id": "PanPhO_2024_5_12", + "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part C: Shot noise]\n\nAny precision measurements are limited by the uncertainty principle of quantum mechanics. Assume that every photon's arrival time at the detector can be considered as independent stochastic processes. Also, in actual experiments, phase error of the laser is more important. But here for simplicity, here we only estimate photon number errors.", + "question": "If we request that in one second, the relative error of photon number measurement is $\\frac{\\Delta N}{N} < 3 \\times 10^{-6}$. Calculate the minimal power of laser $P_{\\text{rec}}$ that the satellite should receive. Express your answer in the unit of $W$.", + "marking": [], + "answer": [ + "\\boxed{$2 \\times 10^{-8}$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "W" + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Modern Physics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_5_1_1.png" + ] + }, + { + "id": "PanPhO_2024_5_13", + "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part C: Shot noise]\n\nAny precision measurements are limited by the uncertainty principle of quantum mechanics. Assume that every photon's arrival time at the detector can be considered as independent stochastic processes. Also, in actual experiments, phase error of the laser is more important. But here for simplicity, here we only estimate photon number errors.", + "question": "Assume the laser arrived at a satellite is emitted from the other satellite from the three-satellite system. Estimate: in an ideal case, what is the minimal emission power of laser $P_{\\text{emit}}$ from the other satellite (can be considered to be correct if the order-of-magnitude is correct). Express your answer in the unit of $W$.", + "marking": [], + "answer": [ + "\\boxed{$[1, 9]$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "W" + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Modern Physics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_5_1_1.png" + ] + }, + { + "id": "PanPhO_2024_6_1", + "context": "[Metric-modified geodesic and heat conduction] \n\nSolving physics, such as wave propagation, geodesics, and thermal conduction, on a curved surface in 3D requires a thorough understanding of metrics and differential geometry. However, there can be significant simplifications for systems with spatial symmetry or by adopting coordinate transformation. In this question, we will go through two problems for physics on a curved surface. The first one is light propagating on a curved surface. Figure 1 (a) shows a circular cone with height 5 mm and a base diameter of $2 \\rho_{0} = 10 \\mathrm{mm}$, joining to a flat surface. The flat surface has a circular hole of the same diameter so that as a whole, there is only one single surface with the cone part indicating the 'curved space.' The entire surface, including the flat surface and the cone, have a very thin surface so that light can be effectively confined on such a surface. We have assumed the cone is joint smoothly to the flat surface. \n\n[figure1]\n\nFigure 1(a) depicts a curved surface created by connecting a circular cone to a flat surface with a hole of the same size as the cone's base. In Figure 1(b), we present a top view of this surface. Light confined to such a surface originates on the flat surface at the bottom, undergoes bending due to the cone, and exits in a different direction. \n\n[Part A: Geodesic on A Rotational Symmetric Curved Surface]\n\nIn mechanics, we are aware that when a system exhibits rotational symmetry, we can simplify the derivation of dynamics by applying the conservation of angular momentum. For instance, we can employ the conservation of angular momentum to derive Kepler's laws. In the current scenario, we consider a normalized angular momentum, denoted as L, which is defined as: \n$L = \\hat{z} \\cdot \\mathbf{\\rho} \\times \\frac{d \\mathbf{\\rho}}{d s} = \\rho^{2} \\frac{d \\phi}{d s}$ \nHere, $\\mathbf{\\rho}$ represents the projected position vector on the two-dimensional $x-y$ plane, given by $\\mathbf{\\rho} = x \\hat{x} + y \\hat{y} = \\hat{x} \\rho \\cos \\phi + \\hat{y} \\rho \\sin \\phi$, with the projected cone center as the origin. $\\rho$ is the magnitude of vector $\\mathbf{\\rho}$ and $s$ is the arc length along the path of light on the surface.", + "question": "Given that the infinitesimal arc length on the cone satisfies $d s^{2} = d x^{2} + d y^{2} + d z^{2}$ and $z = z(\\rho)$ is the height at that point, find the equation that the geodesic on the cone satisfies, in terms of $\\rho^{\\prime}(\\phi), \\rho, L$, and $\\theta$. Instead of using arc length $s$ to parametrize the geodesic, we have used $\\phi$ for parametrization.", + "marking": [], + "answer": [ + "\\boxed{$\\rho^{\\prime}(\\phi)^{2} = \\frac{\\sin^{2} \\theta}{L^{2}} \\rho^{2} (\\rho^{2} - L^{2})$}" + ], + "answer_type": [ + "Equation" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_6_1_1.png" + ] + }, + { + "id": "PanPhO_2024_6_2", + "context": "[Metric-modified geodesic and heat conduction] \n\nSolving physics, such as wave propagation, geodesics, and thermal conduction, on a curved surface in 3D requires a thorough understanding of metrics and differential geometry. However, there can be significant simplifications for systems with spatial symmetry or by adopting coordinate transformation. In this question, we will go through two problems for physics on a curved surface. The first one is light propagating on a curved surface. Figure 1 (a) shows a circular cone with height 5 mm and a base diameter of $2 \\rho_{0} = 10 \\mathrm{mm}$, joining to a flat surface. The flat surface has a circular hole of the same diameter so that as a whole, there is only one single surface with the cone part indicating the 'curved space.' The entire surface, including the flat surface and the cone, have a very thin surface so that light can be effectively confined on such a surface. We have assumed the cone is joint smoothly to the flat surface. \n\n[figure1]\n\nFigure 1(a) depicts a curved surface created by connecting a circular cone to a flat surface with a hole of the same size as the cone's base. In Figure 1(b), we present a top view of this surface. Light confined to such a surface originates on the flat surface at the bottom, undergoes bending due to the cone, and exits in a different direction. \n\n[Part A: Geodesic on A Rotational Symmetric Curved Surface]\n\nIn mechanics, we are aware that when a system exhibits rotational symmetry, we can simplify the derivation of dynamics by applying the conservation of angular momentum. For instance, we can employ the conservation of angular momentum to derive Kepler's laws. In the current scenario, we consider a normalized angular momentum, denoted as L, which is defined as: \n$L = \\hat{z} \\cdot \\mathbf{\\rho} \\times \\frac{d \\mathbf{\\rho}}{d s} = \\rho^{2} \\frac{d \\phi}{d s}$ \nHere, $\\mathbf{\\rho}$ represents the projected position vector on the two-dimensional $x-y$ plane, given by $\\mathbf{\\rho} = x \\hat{x} + y \\hat{y} = \\hat{x} \\rho \\cos \\phi + \\hat{y} \\rho \\sin \\phi$, with the projected cone center as the origin. $\\rho$ is the magnitude of vector $\\mathbf{\\rho}$ and $s$ is the arc length along the path of light on the surface.", + "question": "For a light starting on the flat surface with a perpendicular distance of $\\rho_{0} / 2$ to the origin, what will be minimal $\\rho$ the light can go.", + "marking": [], + "answer": [ + "\\boxed{$\\frac{\\rho_{0}}{2}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_6_1_1.png" + ] + }, + { + "id": "PanPhO_2024_6_3", + "context": "[Metric-modified geodesic and heat conduction] \n\nSolving physics, such as wave propagation, geodesics, and thermal conduction, on a curved surface in 3D requires a thorough understanding of metrics and differential geometry. However, there can be significant simplifications for systems with spatial symmetry or by adopting coordinate transformation. In this question, we will go through two problems for physics on a curved surface. The first one is light propagating on a curved surface. Figure 1 (a) shows a circular cone with height 5 mm and a base diameter of $2 \\rho_{0} = 10 \\mathrm{mm}$, joining to a flat surface. The flat surface has a circular hole of the same diameter so that as a whole, there is only one single surface with the cone part indicating the 'curved space.' The entire surface, including the flat surface and the cone, have a very thin surface so that light can be effectively confined on such a surface. We have assumed the cone is joint smoothly to the flat surface. \n\n[figure1]\n\nFigure 1(a) depicts a curved surface created by connecting a circular cone to a flat surface with a hole of the same size as the cone's base. In Figure 1(b), we present a top view of this surface. Light confined to such a surface originates on the flat surface at the bottom, undergoes bending due to the cone, and exits in a different direction. \n\n[Part A: Geodesic on A Rotational Symmetric Curved Surface]\n\nIn mechanics, we are aware that when a system exhibits rotational symmetry, we can simplify the derivation of dynamics by applying the conservation of angular momentum. For instance, we can employ the conservation of angular momentum to derive Kepler's laws. In the current scenario, we consider a normalized angular momentum, denoted as L, which is defined as: \n$L = \\hat{z} \\cdot \\mathbf{\\rho} \\times \\frac{d \\mathbf{\\rho}}{d s} = \\rho^{2} \\frac{d \\phi}{d s}$ \nHere, $\\mathbf{\\rho}$ represents the projected position vector on the two-dimensional $x-y$ plane, given by $\\mathbf{\\rho} = x \\hat{x} + y \\hat{y} = \\hat{x} \\rho \\cos \\phi + \\hat{y} \\rho \\sin \\phi$, with the projected cone center as the origin. $\\rho$ is the magnitude of vector $\\mathbf{\\rho}$ and $s$ is the arc length along the path of light on the surface.", + "question": "What is the deflection angle $\\gamma$ by comparing the entering and exit rays on the flat surface? Express the numerical answer in degrees. \nHint: you may need to use $\\int \\frac{d x}{x \\sqrt{x^{2}-1}} = \\tan^{-1} \\sqrt{x^{2}-1} + c$.", + "marking": [], + "answer": [ + "\\boxed{49.7}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "degree" + ], + "points": [ + 7.0 + ], + "modality": "text+variable figure", + "field": "Optics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_6_1_1.png" + ] + }, + { + "id": "PanPhO_2024_6_4", + "context": "[Metric-modified geodesic and heat conduction] \n\nSolving physics, such as wave propagation, geodesics, and thermal conduction, on a curved surface in 3D requires a thorough understanding of metrics and differential geometry. However, there can be significant simplifications for systems with spatial symmetry or by adopting coordinate transformation. In this question, we will go through two problems for physics on a curved surface. The first one is light propagating on a curved surface. Figure 1 (a) shows a circular cone with height 5 mm and a base diameter of $2 \\rho_{0} = 10 \\mathrm{mm}$, joining to a flat surface. The flat surface has a circular hole of the same diameter so that as a whole, there is only one single surface with the cone part indicating the 'curved space.' The entire surface, including the flat surface and the cone, have a very thin surface so that light can be effectively confined on such a surface. We have assumed the cone is joint smoothly to the flat surface. \n\n[figure1]\n\nFigure 1(a) depicts a curved surface created by connecting a circular cone to a flat surface with a hole of the same size as the cone's base. In Figure 1(b), we present a top view of this surface. Light confined to such a surface originates on the flat surface at the bottom, undergoes bending due to the cone, and exits in a different direction.\n\n[Part B: Heat Conduction on A Spherical Surface]\n\nA usual trick is to search for a coordinate transform from the curved surface (represented by the Cartesian coordinates $(x, y, z)$) to a 2-dimensional coordinates system $X-Y$ plane so that the physics on the $(X, Y)$ just looks like a flat plane. For a unit spherical surface, such a map is the stereographic projection \n$(X, Y) = \\left(\\frac{x}{z+1}, \\frac{y}{z+1}\\right) = (\\rho \\cos \\phi, \\rho \\sin \\phi)$ \nSuppose now we consider heat conduction problem on such a spherical surface, i.e. a very thin shell of spherical surface. The steady-state heat conduction has the temperature profile satisfying the Laplace equation \n$\\nabla^{2} T(\\theta, \\phi) = 0$ \nwhile temperature profile is independent of radial distance $r$ in spherical coordinate $(r, \\theta, \\phi)$. The spherical surface is at radius $r=1$.", + "question": "Write down the Laplace equation that $T$ satisfies on the $(X, Y)$ coordinate and prove it. \nHint: For convenience, we are given the Laplacian in spherical and cylindrical coordinates as \nIn spherical coordinate $(r, \\theta, \\phi)$: \n$\\nabla^{2} f = \\frac{1}{r^{2}} \\partial_{r}\\left(r^{2} \\partial_{r} f\\right) + \\frac{1}{r^{2} \\sin \\theta} \\partial_{\\theta} \\left(\\sin \\theta \\partial_{\\theta} f\\right) + \\frac{1}{r^{2} \\sin^{2} \\theta} \\partial_{\\phi}^{2} f$ \ncylindrical coordinate $(\\rho, \\phi, z)$: \n$\\nabla^{2} f = \\frac{1}{\\rho} \\partial_{\\rho} \\left(\\rho \\partial_{\\rho} f\\right) + \\frac{1}{\\rho^{2}} \\partial_{\\phi}^{2} f + \\partial_{z}^{2} f$.", + "marking": [], + "answer": [ + "\\boxed{$\\frac{1}{\\rho} \\partial_{\\rho} \\left(\\rho \\partial_{\\rho} T\\right) + \\frac{1}{\\rho^{2}} \\partial_{\\phi}^{2} T = 0$}" + ], + "answer_type": [ + "Equation" + ], + "unit": [ + null + ], + "points": [ + 4.0 + ], + "modality": "text+variable figure", + "field": "Thermodynamics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_6_1_1.png" + ] + }, + { + "id": "PanPhO_2024_6_5", + "context": "[Metric-modified geodesic and heat conduction] \n\nSolving physics, such as wave propagation, geodesics, and thermal conduction, on a curved surface in 3D requires a thorough understanding of metrics and differential geometry. However, there can be significant simplifications for systems with spatial symmetry or by adopting coordinate transformation. In this question, we will go through two problems for physics on a curved surface. The first one is light propagating on a curved surface. Figure 1 (a) shows a circular cone with height 5 mm and a base diameter of $2 \\rho_{0} = 10 \\mathrm{mm}$, joining to a flat surface. The flat surface has a circular hole of the same diameter so that as a whole, there is only one single surface with the cone part indicating the 'curved space.' The entire surface, including the flat surface and the cone, have a very thin surface so that light can be effectively confined on such a surface. We have assumed the cone is joint smoothly to the flat surface. \n\n[figure1]\n\nFigure 1(a) depicts a curved surface created by connecting a circular cone to a flat surface with a hole of the same size as the cone's base. In Figure 1(b), we present a top view of this surface. Light confined to such a surface originates on the flat surface at the bottom, undergoes bending due to the cone, and exits in a different direction.\n\n[Part B: Heat Conduction on A Spherical Surface]\n\nA usual trick is to search for a coordinate transform from the curved surface (represented by the Cartesian coordinates $(x, y, z)$) to a 2-dimensional coordinates system $X-Y$ plane so that the physics on the $(X, Y)$ just looks like a flat plane. For a unit spherical surface, such a map is the stereographic projection \n$(X, Y) = \\left(\\frac{x}{z+1}, \\frac{y}{z+1}\\right) = (\\rho \\cos \\phi, \\rho \\sin \\phi)$ \nSuppose now we consider heat conduction problem on such a spherical surface, i.e. a very thin shell of spherical surface. The steady-state heat conduction has the temperature profile satisfying the Laplace equation \n$\\nabla^{2} T(\\theta, \\phi) = 0$ \nwhile temperature profile is independent of radial distance $r$ in spherical coordinate $(r, \\theta, \\phi)$. The spherical surface is at radius $r=1$.\n\n[figure2]\n\nNow, the figure 2 gives the thin shell in the shape of lamp shade, which is in a hemi-spherical surface with a circular opening at the top. The whole shape still has a rotational symmetry about the vertical z-axis. The bottom of the lamp shade is kept at temperature $T_{b}$ (at $\\theta = \\frac{\\pi}{2}$ for spherical polar coordinate) and the top is kept at temperature $T_{t}$ (at $\\theta = \\theta_{t}$).", + "question": "Solve the temperature profile $T(\\theta)$, as a function of $\\theta$, with such rotational symmetry.", + "marking": [], + "answer": [ + "\\boxed{$T(\\theta) = (T_{t}-T_{b}) \\frac{\\ln \\left(\\tan \\frac{\\theta}{2}\\right)}{\\ln \\left(\\tan \\frac{\\theta_t}{2}\\right)} + T_{b}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 4.0 + ], + "modality": "text+variable figure", + "field": "Thermodynamics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_6_1_1.png", + "image_question/PanPhO_2024_6_5_1.png" + ] + }, + { + "id": "PanPhO_2024_6_6", + "context": "[Metric-modified geodesic and heat conduction] \n\nSolving physics, such as wave propagation, geodesics, and thermal conduction, on a curved surface in 3D requires a thorough understanding of metrics and differential geometry. However, there can be significant simplifications for systems with spatial symmetry or by adopting coordinate transformation. In this question, we will go through two problems for physics on a curved surface. The first one is light propagating on a curved surface. Figure 1 (a) shows a circular cone with height 5 mm and a base diameter of $2 \\rho_{0} = 10 \\mathrm{mm}$, joining to a flat surface. The flat surface has a circular hole of the same diameter so that as a whole, there is only one single surface with the cone part indicating the 'curved space.' The entire surface, including the flat surface and the cone, have a very thin surface so that light can be effectively confined on such a surface. We have assumed the cone is joint smoothly to the flat surface. \n\n[figure1]\n\nFigure 1(a) depicts a curved surface created by connecting a circular cone to a flat surface with a hole of the same size as the cone's base. In Figure 1(b), we present a top view of this surface. Light confined to such a surface originates on the flat surface at the bottom, undergoes bending due to the cone, and exits in a different direction.\n\n[Part C: Heat Conduction on A Spherical Surface]\n\n[figure2]\n\nNow, we consider the top opening is tilted about the $y$-axis in breaking rotational symmetry. Suppose the top opening is still a circle on the spherical surface passing through $(x, y, z) = (0, 0, 1)$ and $(\\sin \\alpha, 0, \\cos \\alpha)$ as diameter and its normal is on the $x-z$ plane.", + "question": "Determine where the top opening is mapped on $(X, Y)$ plane through the stereographic projection map. Write down the equation of the circle that $X$ and $Y$ satisfy. \nHint: the answer is still a circle in $X$ and $Y$ coordinates.", + "marking": [], + "answer": [ + "\\boxed{$\\left(X - \\frac{1}{2} \\tan \\frac{\\alpha}{2}\\right)^{2} + Y^{2} = \\frac{1}{4} \\tan^{2} \\frac{\\alpha}{2}$}" + ], + "answer_type": [ + "Equation" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+variable figure", + "field": "Thermodynamics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_6_1_1.png", + "image_question/PanPhO_2024_6_6_1.png" + ] + }, + { + "id": "PanPhO_2024_6_7", + "context": "[Metric-modified geodesic and heat conduction] \n\nSolving physics, such as wave propagation, geodesics, and thermal conduction, on a curved surface in 3D requires a thorough understanding of metrics and differential geometry. However, there can be significant simplifications for systems with spatial symmetry or by adopting coordinate transformation. In this question, we will go through two problems for physics on a curved surface. The first one is light propagating on a curved surface. Figure 1 (a) shows a circular cone with height 5 mm and a base diameter of $2 \\rho_{0} = 10 \\mathrm{mm}$, joining to a flat surface. The flat surface has a circular hole of the same diameter so that as a whole, there is only one single surface with the cone part indicating the 'curved space.' The entire surface, including the flat surface and the cone, have a very thin surface so that light can be effectively confined on such a surface. We have assumed the cone is joint smoothly to the flat surface. \n\n[figure1]\n\nFigure 1(a) depicts a curved surface created by connecting a circular cone to a flat surface with a hole of the same size as the cone's base. In Figure 1(b), we present a top view of this surface. Light confined to such a surface originates on the flat surface at the bottom, undergoes bending due to the cone, and exits in a different direction.\n\n[Part C: Heat Conduction on A Spherical Surface]\n\n[figure2]\n\nNow, we consider the top opening is tilted about the $y$-axis in breaking rotational symmetry. Suppose the top opening is still a circle on the spherical surface passing through $(x, y, z) = (0, 0, 1)$ and $(\\sin \\alpha, 0, \\cos \\alpha)$ as diameter and its normal is on the $x-z$ plane.\n\nThe top opening of the surface is mapped onto the (X, Y) plane through the stereographic projection, resulting in a circle. The coordinates $X$ and $Y$ satisfy the equation of this circle.", + "question": "Solve the temperature profile $T(X, Y)$ when the bottom opening is kept at temperature $T_{b}$ and the top is kept at temperature $T_{t}$. You can leave your answer in terms of $X$ and $Y$ coordinates. \nHint: In the stereographic projected domain X-Y plane, Laplace equation is satisfied and you can further use general method of image, like the one used in solving electrostatic problem by putting two point charges on the $X$-axis with undetermined charges.", + "marking": [], + "answer": [ + "\\boxed{$T = \\frac{T_{t}-T_{b}}{2 \\ln \\mu} \\ln \\left(\\frac{(X-\\mu)^{2} + Y^{2}}{\\left(X-\\mu^{-1}\\right)^{2} + Y^{2}}\\right) + 2 T_{b} - T_{t}$ or $T = \\frac{T_t - T_b}{2 \\ln \\mu} \\ln \\left( \\frac{\\tan^2(\\theta/2) - 2\\mu \\tan(\\theta/2) \\cos \\phi + \\mu^2}{\\tan^2(\\theta/2) - 2\\mu^{-1} \\tan(\\theta/2) \\cos \\phi + \\mu^{-2}} \\right) + 2T_b - T_t$, where $\\mu = \\frac{1+\\sqrt{1-4a^2}}{2a}$ and $a = \\frac{1}{2} \\tan \\frac{\\alpha}{2}$.}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 8.0 + ], + "modality": "text+variable figure", + "field": "Thermodynamics", + "source": "PanPhO_2024", + "image_question": [ + "image_question/PanPhO_2024_6_1_1.png", + "image_question/PanPhO_2024_6_6_1.png" + ] + } +] \ No newline at end of file diff --git a/data/PanPhO_2025.json b/data/PanPhO_2025.json new file mode 100644 index 0000000000000000000000000000000000000000..24e05f316072cd05f7d913c6ffac1de554711558 --- /dev/null +++ b/data/PanPhO_2025.json @@ -0,0 +1,1201 @@ +[ + { + "information": "None." + }, + { + "id": "PanPhO_2025_1_1", + "context": "As shown in the figure, consider a lollipop made of a solid sphere of mass $m$ and radius $r$ that is radially pierced by a massless stick. The free end of the stick is pivoted on the ground. The sphere rolls on the ground without slipping, with its centre moving in a circle of radius $R$ with angular velocity $\\Omega$ (along $-\\hat{z}$ direction). The moment of inertia of a solid sphere along the symmetric axis with mass $m$ and radius $r$ is $I = \\frac{2}{5} m r^{2}$.\n\n[figure1]", + "question": "What is the (instantaneous) total angular velocity $\\vec{\\omega}$ of the lollipop? Express your answer in terms of $r$, $R$, $\\Omega$, $\\hat{x}$ and $\\hat{z}$.", + "marking": [], + "answer": [ + "\\boxed{$\\vec{\\omega} = \\frac{R \\Omega}{r} \\hat{x}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_1_1_1.png" + ] + }, + { + "id": "PanPhO_2025_1_2", + "context": "As shown in the figure, consider a lollipop made of a solid sphere of mass $m$ and radius $r$ that is radially pierced by a massless stick. The free end of the stick is pivoted on the ground. The sphere rolls on the ground without slipping, with its centre moving in a circle of radius $R$ with angular velocity $\\Omega$ (along $-\\hat{z}$ direction). The moment of inertia of a solid sphere along the symmetric axis with mass $m$ and radius $r$ is $I = \\frac{2}{5} m r^{2}$.\n\n[figure1]", + "question": "What is the angular momentum $\\vec{L}$ of the lollipop? Express your answer in terms of $m, r, R, \\Omega, \\hat{x}$ and $\\hat{z}$.", + "marking": [], + "answer": [ + "\\boxed{$\\vec{L} = \\frac{7}{5} m r R \\Omega \\hat{x} - m R^{2} \\Omega \\hat{z}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 4.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_1_1_1.png" + ] + }, + { + "id": "PanPhO_2025_1_3", + "context": "As shown in the figure, consider a lollipop made of a solid sphere of mass $m$ and radius $r$ that is radially pierced by a massless stick. The free end of the stick is pivoted on the ground. The sphere rolls on the ground without slipping, with its centre moving in a circle of radius $R$ with angular velocity $\\Omega$ (along $-\\hat{z}$ direction). The moment of inertia of a solid sphere along the symmetric axis with mass $m$ and radius $r$ is $I = \\frac{2}{5} m r^{2}$.\n\n[figure1]", + "question": "What is the normal force $\\vec{N}$ between the ground and the sphere?", + "marking": [], + "answer": [ + "\\boxed{$\\vec{N} = \\frac{7}{5} m r \\Omega^{2} + m g$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_1_1_1.png" + ] + }, + { + "id": "PanPhO_2025_2_1", + "context": "In 2018, the Nobel Prize in physics was awarded to Arthur Ashkin, for the creation of the \"laser tweezer\", a device that allows one to hold and move transparent microscopic objects with the help of light. In this device, a parallel beam of light from a laser passes through a converging lens $L$ and hits a microparticle $M$, which can also be considered as a converging lens. Point $F$ is the common focus of $L$ and $M$. The light intensity in the beam is $I = 1.00 \\mu \\mathrm{W} / \\mathrm{cm}^{2}$, the beam radius is $R = 1.00 \\mathrm{cm}$, and the focal length of the lens $L$ is $F = 10.0 \\mathrm{cm}$. Ignore the absorption and reflection of light. Speed of light in vacuum is $c = 3 \\times 10^{8} \\mathrm{m} / \\mathrm{s}$.\n\n[figure1]", + "question": "Find the expression of the force due to the light acting on the lens $L$, in the setup shown in Fig.2a. Express the answer in terms of $I$, $R$, $F$, $c$, and $\\hat{y}$.", + "marking": [], + "answer": [ + "\\boxed{$-\\hat{y} \\frac{\\pi I R^{4}}{4 c F^{2}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Optics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_2_1_1.png" + ] + }, + { + "id": "PanPhO_2025_2_2", + "context": "In 2018, the Nobel Prize in physics was awarded to Arthur Ashkin, for the creation of the \"laser tweezer\", a device that allows one to hold and move transparent microscopic objects with the help of light. In this device, a parallel beam of light from a laser passes through a converging lens $L$ and hits a microparticle $M$, which can also be considered as a converging lens. Point $F$ is the common focus of $L$ and $M$. The light intensity in the beam is $I = 1.00 \\mu \\mathrm{W} / \\mathrm{cm}^{2}$, the beam radius is $R = 1.00 \\mathrm{cm}$, and the focal length of the lens $L$ is $F = 10.0 \\mathrm{cm}$. Ignore the absorption and reflection of light. Speed of light in vacuum is $c = 3 \\times 10^{8} \\mathrm{m} / \\mathrm{s}$.\n\n[figure1]", + "question": "Calculate the (numerical value) magnitude of the force (expressed in $N$) acting on the microparticle $M$, in the setup shown in Fig.2a.", + "marking": [], + "answer": [ + "\\boxed{$[2.60 \\times 10^{-17}, 2.64 \\times 10^{-17}]$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "N" + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Optics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_2_1_1.png" + ] + }, + { + "id": "PanPhO_2025_2_3", + "context": "In 2018, the Nobel Prize in physics was awarded to Arthur Ashkin, for the creation of the \"laser tweezer\", a device that allows one to hold and move transparent microscopic objects with the help of light. In this device, a parallel beam of light from a laser passes through a converging lens $L$ and hits a microparticle $M$, which can also be considered as a converging lens. Point $F$ is the common focus of $L$ and $M$. The light intensity in the beam is $I = 1.00 \\mu \\mathrm{W} / \\mathrm{cm}^{2}$, the beam radius is $R = 1.00 \\mathrm{cm}$, and the focal length of the lens $L$ is $F = 10.0 \\mathrm{cm}$. Ignore the absorption and reflection of light. Speed of light in vacuum is $c = 3 \\times 10^{8} \\mathrm{m} / \\mathrm{s}$.\n\nNext, the left half of the lens $L$ is covered by a diaphragm, as shown in Fig.2b.\n\n[figure1]", + "question": "Find the expression of the force $\\vec{F}$ acting on the microparticle in this case. Express the answer in terms of $I$, $R$, $F$, $c$, $\\hat{i}$, and $\\hat{j}$.", + "marking": [], + "answer": [ + "\\boxed{$\\vec{F} = -\\frac{2 I R^{3}}{3 c F} \\hat{i} + \\frac{\\pi I R^{4}}{8 c F^{2}} \\hat{j}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Optics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_2_1_1.png" + ] + }, + { + "id": "PanPhO_2025_2_4", + "context": "In 2018, the Nobel Prize in physics was awarded to Arthur Ashkin, for the creation of the \"laser tweezer\", a device that allows one to hold and move transparent microscopic objects with the help of light. In this device, a parallel beam of light from a laser passes through a converging lens $L$ and hits a microparticle $M$, which can also be considered as a converging lens. Point $F$ is the common focus of $L$ and $M$. The light intensity in the beam is $I = 1.00 \\mu \\mathrm{W} / \\mathrm{cm}^{2}$, the beam radius is $R = 1.00 \\mathrm{cm}$, and the focal length of the lens $L$ is $F = 10.0 \\mathrm{cm}$. Ignore the absorption and reflection of light. Speed of light in vacuum is $c = 3 \\times 10^{8} \\mathrm{m} / \\mathrm{s}$.\n\nNext, the left half of the lens $L$ is covered by a diaphragm, as shown in Fig.2b.\n\n[figure1]", + "question": "Calculate the (numerical value) magnitude of the force in this case. Express the force vector $\\vec{F}$ in component form with numerical values, following the format: $\\vec{F} = (a \\hat{i} + b \\hat{j}) N$.", + "marking": [], + "answer": [ + "\\boxed{$\\vec{F} = -2.24 \\times 10^{-16} \\hat{i} + 1.32 \\times 10^{-17} \\hat{j}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + "N" + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Optics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_2_1_1.png" + ] + }, + { + "id": "PanPhO_2025_3_1", + "context": "The toroidal cavity is designed to confine charged particles for nuclear fusion. This geometry enables charged particles to follow helical magnetic field lines, allowing them to remain suspended within the cavity without coming into contact with the walls. As depicted in the figure, consider a toroidal cavity with an outer radius $R_{0}$ and a circular cross-section of radius $r_{0}$, where $r_{0} \\ll R_{0}$. In the figure, $O$ represents the origin.\n\n[figure1]", + "question": "If a wire is uniformly and tightly wound around the toroidal cavity for $N$ turns, and the magnetic field inside the cavity is given as: $\\vec{B}_{1}(r) = f(r)(\\sin \\phi \\hat{x} - \\cos \\phi \\hat{y})$, find $f(r)$ in terms of $N$, $I$, $R_{o}$, $r_{0}$ and relevant physical constants. Here $r$, $\\phi$ represent the radial and angular coordinates in a polar coordinate system on the plane.", + "marking": [], + "answer": [ + "\\boxed{$f(r) = \\frac{\\mu_{0} N I}{2 \\pi r}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.5 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_3_1_1.png" + ] + }, + { + "id": "PanPhO_2025_3_2", + "context": "The toroidal cavity is designed to confine charged particles for nuclear fusion. This geometry enables charged particles to follow helical magnetic field lines, allowing them to remain suspended within the cavity without coming into contact with the walls. As depicted in the figure, consider a toroidal cavity with an outer radius $R_{0}$ and a circular cross-section of radius $r_{0}$, where $r_{0} \\ll R_{0}$. In the figure, $O$ represents the origin.\n\n[figure1]\n\nHowever, the diamagnetic drifty caused by the gradient of the toroidal magnetic field tends to push the particles outward, leading to a loss of confinement.", + "question": "To address this, a uniform magnetic field $\\vec{B}_{2} = B_{0} \\hat{z}$ is applied along the $z$-direction. In this uniform magnetic field $\\vec{B}_{2}$, a particle of mass $m$ and charge $q$ moves in uniform circular motion with a radius $R_{o}$, find the angular frequency $\\omega_{0}$ of this circular motion. Express the answer in terms of $q$, $m$, $B_{0}$.", + "marking": [], + "answer": [ + "\\boxed{$\\omega_0 = \\frac{q B_{0}}{m}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.5 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_3_1_1.png" + ] + }, + { + "id": "PanPhO_2025_3_3", + "context": "The toroidal cavity is designed to confine charged particles for nuclear fusion. This geometry enables charged particles to follow helical magnetic field lines, allowing them to remain suspended within the cavity without coming into contact with the walls. As depicted in the figure, consider a toroidal cavity with an outer radius $R_{0}$ and a circular cross-section of radius $r_{0}$, where $r_{0} \\ll R_{0}$. In the figure, $O$ represents the origin.\n\n[figure1]\n\nHowever, the diamagnetic drifty caused by the gradient of the toroidal magnetic field tends to push the particles outward, leading to a loss of confinement.\n\nTo address this, a uniform magnetic field $\\vec{B}_{2} = B_{0} \\hat{z}$ is applied along the $z$-direction. In this uniform magnetic field $\\vec{B}_{2}$, a particle of mass $m$ and charge $q$ moves in uniform circular motion with a radius $R_{o}$, the angular frequency of this circular motion is denoted by $\\omega_{0}$.", + "question": "For a charged particle with mass $m$ and charge $q$, write down the equations of motion in the $r$, $\\phi$, $z$ directions for its motion inside the toroidal cavity with $\\vec{B}_{1}$ and $\\vec{B}_{2}$. Express three equations in term of the dimensionless parameter $\\alpha = \\frac{\\mu_{0} N I}{2 \\pi R_{0} B_{0}}$, $\\dot{r}$, $\\dot{\\phi}$, $\\dot{z}$, $\\ddot{r}$, $\\ddot{\\phi}$, and $\\ddot{z}$.", + "marking": [], + "answer": [ + "\\boxed{$m \\left(\\ddot{r} - r \\dot{\\phi}^{2}\\right) + q B_{0} \\left(-\\frac{\\alpha R_{0} \\dot{z}}{r} - r \\dot{\\phi} \\right) = 0$}", + "\\boxed{$m (2 \\dot{r} \\dot{\\phi} + r \\ddot{\\phi}) + q B_{0} \\dot{r} = 0$}", + "\\boxed{$m \\ddot{z} + q B_{0} \\left( \\frac{\\alpha R_{0} \\dot{r}}{r} \\right) = 0$}" + ], + "answer_type": [ + "Equation", + "Equation", + "Equation" + ], + "unit": [ + null, + null, + null + ], + "points": [ + 0.67, + 0.67, + 0.66 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_3_1_1.png" + ] + }, + { + "id": "PanPhO_2025_3_4", + "context": "The toroidal cavity is designed to confine charged particles for nuclear fusion. This geometry enables charged particles to follow helical magnetic field lines, allowing them to remain suspended within the cavity without coming into contact with the walls. As depicted in the figure, consider a toroidal cavity with an outer radius $R_{0}$ and a circular cross-section of radius $r_{0}$, where $r_{0} \\ll R_{0}$. In the figure, $O$ represents the origin.\n\n[figure1]\n\nHowever, the diamagnetic drifty caused by the gradient of the toroidal magnetic field tends to push the particles outward, leading to a loss of confinement.\n\nPart (B): To address this, a uniform magnetic field $\\vec{B}_{2} = B_{0} \\hat{z}$ is applied along the $z$-direction. In this uniform magnetic field $\\vec{B}_{2}$, a particle of mass $m$ and charge $q$ moves in uniform circular motion with a radius $R_{o}$, the angular frequency of this circular motion is denoted by $\\omega_{0}$. \n\nPart (C): For a charged particle with mass $m$ and charge $q$, write down the equations of motion in the $r$, $\\phi$, $z$ directions for its motion inside the toroidal cavity with $\\vec{B}_{1}$ and $\\vec{B}_{2}$. Express three equations in term of the dimensionless parameter $\\alpha = \\frac{\\mu_{0} N I}{2 \\pi R_{0} B_{0}}$, $\\dot{r}$, $\\dot{\\phi}$, $\\dot{z}$, $\\ddot{r}$, $\\ddot{\\phi}$, and $\\ddot{z}$. (Parts (B) and (C) are preliminary questions and do not include in the final answer)", + "question": "Based on the equations of motion derived in part (C), consider the case where the charged particle is slightly perturbed from the circular motion described in part (B). Let the perturbed motion be expressed as: $r(t) = R_{0} + \\delta r(t)$, $\\phi(t) = -\\omega_{0} t + \\delta \\phi(t)$, $z(t) = \\delta z(t)$. Here, we assume $\\delta r$, $\\delta z \\ll R_{0}$ and $\\delta \\phi \\ll 1$. Derive three equations of motion that are expanded to the first order in $\\delta r(t)$, $\\delta \\phi(t)$, $\\delta z(t)$.", + "marking": [], + "answer": [ + "\\boxed{$\\delta \\ddot{r} + \\omega_{0} R_{0} \\delta \\dot{\\phi} - \\alpha \\omega_{0} \\delta \\dot{z} = 0$}", + "\\boxed{$R_{0} \\delta \\ddot{\\phi} - \\omega_{0} \\delta \\dot{r} = 0$}", + "\\boxed{$\\delta \\ddot{z} + \\alpha \\omega_{0} \\delta \\dot{r} = 0$}" + ], + "answer_type": [ + "Equation", + "Equation", + "Equation" + ], + "unit": [ + null, + null, + null + ], + "points": [ + 0.67, + 0.67, + 0.66 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_3_1_1.png" + ] + }, + { + "id": "PanPhO_2025_3_5", + "context": "The toroidal cavity is designed to confine charged particles for nuclear fusion. This geometry enables charged particles to follow helical magnetic field lines, allowing them to remain suspended within the cavity without coming into contact with the walls. As depicted in the figure, consider a toroidal cavity with an outer radius $R_{0}$ and a circular cross-section of radius $r_{0}$, where $r_{0} \\ll R_{0}$. In the figure, $O$ represents the origin.\n\n[figure1]\n\nHowever, the diamagnetic drifty caused by the gradient of the toroidal magnetic field tends to push the particles outward, leading to a loss of confinement.\n\nPart (B): To address this, a uniform magnetic field $\\vec{B}_{2} = B_{0} \\hat{z}$ is applied along the $z$-direction. In this uniform magnetic field $\\vec{B}_{2}$, a particle of mass $m$ and charge $q$ moves in uniform circular motion with a radius $R_{o}$, the angular frequency of this circular motion is denoted by $\\omega_{0}$. \n\nPart (C): For a charged particle with mass $m$ and charge $q$, write down the equations of motion in the $r$, $\\phi$, $z$ directions for its motion inside the toroidal cavity with $\\vec{B}_{1}$ and $\\vec{B}_{2}$. Express three equations in term of the dimensionless parameter $\\alpha = \\frac{\\mu_{0} N I}{2 \\pi R_{0} B_{0}}$, $\\dot{r}$, $\\dot{\\phi}$, $\\dot{z}$, $\\ddot{r}$, $\\ddot{\\phi}$, and $\\ddot{z}$.\n\nPart (D): Based on the equations of motion derived in part (C), consider the case where the charged particle is slightly perturbed from the circular motion described in part (B). Let the perturbed motion be expressed as: $r(t) = R_{0} + \\delta r(t)$, $\\phi(t) = -\\omega_{0} t + \\delta \\phi(t)$, $z(t) = \\delta z(t)$. Here, we assume $\\delta r$, $\\delta z \\ll R_{0}$ and $\\delta \\phi \\ll 1$. Derive three equations of motion that are expanded to the first order in $\\delta r(t)$, $\\delta \\phi(t)$, $\\delta z(t)$. (Parts (B), (C), and (D) are preliminary questions and do not include in the final answer)", + "question": "Given initial conditions: $\\delta r(0) = 0$, $\\delta \\dot{r}(0) = v_{0}$, $\\delta \\phi(0) = \\delta \\dot{\\phi}(0) = \\delta z(0) = \\delta \\dot{z}(0) = 0$. Define $\\Omega_0$ as $\\sqrt{1+\\alpha^{2}} \\omega_{0}$. \n\n(1) Find the expression of $\\delta r(t)$. \n(2) Find the expression of $\\delta z(t)$.", + "marking": [], + "answer": [ + "\\boxed{$\\delta r(t) = \\frac{v_{0} \\sin \\left(\\Omega_{0} t\\right)}{\\Omega_{0}}$}", + "\\boxed{$\\delta z(t) = -\\frac{\\alpha v_{0}\\left(1 - \\cos \\Omega_{0} t\\right)}{\\Omega_0 \\sqrt{1+\\alpha^{2}}}$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 1.0, + 1.0 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_3_1_1.png" + ] + }, + { + "id": "PanPhO_2025_3_6", + "context": "The toroidal cavity is designed to confine charged particles for nuclear fusion. This geometry enables charged particles to follow helical magnetic field lines, allowing them to remain suspended within the cavity without coming into contact with the walls. As depicted in the figure, consider a toroidal cavity with an outer radius $R_{0}$ and a circular cross-section of radius $r_{0}$, where $r_{0} \\ll R_{0}$. In the figure, $O$ represents the origin.\n\n[figure1]\n\nHowever, the diamagnetic drifty caused by the gradient of the toroidal magnetic field tends to push the particles outward, leading to a loss of confinement.\n\nPart (B): To address this, a uniform magnetic field $\\vec{B}_{2} = B_{0} \\hat{z}$ is applied along the $z$-direction. In this uniform magnetic field $\\vec{B}_{2}$, a particle of mass $m$ and charge $q$ moves in uniform circular motion with a radius $R_{o}$, the angular frequency of this circular motion is denoted by $\\omega_{0}$. \n\nPart (C): For a charged particle with mass $m$ and charge $q$, write down the equations of motion in the $r$, $\\phi$, $z$ directions for its motion inside the toroidal cavity with $\\vec{B}_{1}$ and $\\vec{B}_{2}$. Express three equations in term of the dimensionless parameter $\\alpha = \\frac{\\mu_{0} N I}{2 \\pi R_{0} B_{0}}$, $\\dot{r}$, $\\dot{\\phi}$, $\\dot{z}$, $\\ddot{r}$, $\\ddot{\\phi}$, and $\\ddot{z}$.\n\nPart (D): Based on the equations of motion derived in part (C), consider the case where the charged particle is slightly perturbed from the circular motion described in part (B). Let the perturbed motion be expressed as: $r(t) = R_{0} + \\delta r(t)$, $\\phi(t) = -\\omega_{0} t + \\delta \\phi(t)$, $z(t) = \\delta z(t)$. Here, we assume $\\delta r$, $\\delta z \\ll R_{0}$ and $\\delta \\phi \\ll 1$. Derive three equations of motion that are expanded to the first order in $\\delta r(t)$, $\\delta \\phi(t)$, $\\delta z(t)$. (Parts (B), (C), and (D) are preliminary questions and do not include in the final answer)", + "question": "To prevent the charged particle from colliding with the walls of the toroidal cavity after being perturbed: \n\n(1) If $\\alpha > 1$, what conditions must $\\frac{v_{0}}{r_{0} \\omega_{0}}$ satisfy? \n(2) If $\\alpha < 1$, what conditions must $\\frac{v_{0}}{r_{0} \\omega_{0}}$ satisfy?", + "marking": [], + "answer": [ + "\\boxed{$\\frac{v_{0}}{r_{0} \\omega_{0}} < \\frac{1+\\alpha^2}{2\\alpha}$}", + "\\boxed{$\\frac{v_{0}}{r_{0} \\omega_{0}} < 1$}" + ], + "answer_type": [ + "Inequality", + "Inequality" + ], + "unit": [ + null, + null + ], + "points": [ + 1.5, + 1.5 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_3_1_1.png" + ] + }, + { + "id": "PanPhO_2025_4_1", + "context": "[Harrison and the Longitude Problem] \n\nAccurate measurement of the longitude was a long-standing problem in sea navigation. The earliest solution to this problem was to compare the local time at a location with that of a meridian. However, there was not a clock accurate enough to preserve the absolute time of another location during a journey, until the inventions by the $18^{\\text{th}}$ century English clockmaker John Harrison.\n\nLet us start our discussion on the simplest type of clock, which is maintained by a vertically hung simple pendulum. It is made of a heavy bob of mass $m$, hung by a rod of length $L$ and negligible mass, to a hinge. The timekeeping is evidently sensitive to temperature fluctuations.", + "question": "For a simple pendulum clock, would thermal expansion make it go faster or slower?\n\n(A) Faster. \n(B) Slower.", + "marking": [], + "answer": [ + "\\boxed{B}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 0.5 + ], + "modality": "text-only", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [] + }, + { + "id": "PanPhO_2025_4_2", + "context": "[Harrison and the Longitude Problem] \n\nAccurate measurement of the longitude was a long-standing problem in sea navigation. The earliest solution to this problem was to compare the local time at a location with that of a meridian. However, there was not a clock accurate enough to preserve the absolute time of another location during a journey, until the inventions by the $18^{\\text{th}}$ century English clockmaker John Harrison.\n\nLet us start our discussion on the simplest type of clock, which is maintained by a vertically hung simple pendulum. It is made of a heavy bob of mass $m$, hung by a rod of length $L$ and negligible mass, to a hinge. The timekeeping is evidently sensitive to temperature fluctuations.\n\n[figure1]", + "question": "In adjustment, Harrison proposed a modification to the rod of a pendulum, now consisting of two types of metals (of thermal expansion coefficients $\\alpha_{1}$ and $\\alpha_{2}$ respectively) by installing a central piece of metal 2 of length $l^{\\prime}$.\n\n(1) Which of the 3 proposals in Fig. 4a can suppress the temperature fluctuations? \n(2) How long should the middle piece $l^{\\prime}$ of metal 2 be such that the total length of the rod $L$ is independent of temperature fluctuations in this proposal? Express your answer in terms of $L$, $\\alpha_{1}$ and $\\alpha_{2}$.", + "marking": [], + "answer": [ + "\\boxed{B}", + "\\boxed{$l^{\\prime} = \\frac{\\alpha_{1}}{\\alpha_{2}-\\alpha_{1}} L$}" + ], + "answer_type": [ + "Multiple Choice", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 0.75, + 0.75 + ], + "modality": "text+data figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_4_2_1.png" + ] + }, + { + "id": "PanPhO_2025_4_3", + "context": "[Harrison and the Longitude Problem] \n\nAccurate measurement of the longitude was a long-standing problem in sea navigation. The earliest solution to this problem was to compare the local time at a location with that of a meridian. However, there was not a clock accurate enough to preserve the absolute time of another location during a journey, until the inventions by the $18^{\\text{th}}$ century English clockmaker John Harrison.\n\nLet us start our discussion on the simplest type of clock, which is maintained by a vertically hung simple pendulum. It is made of a heavy bob of mass $m$, hung by a rod of length $L$ and negligible mass, to a hinge.\n\n[figure1]", + "question": "One problem is that simple pendulums were affected by the motion of the ship it was on. To evaluate this effect, consider a ship's journey between two cities separated by $D = 96 \\mathrm{km}$ shown in the figure. Starting from rest at noon, the ship first accelerated forward at a constant rate, then maintained a constant velocity $v_{0} = 27 \\mathrm{km} \\mathrm{h}^{-1}$ until it decelerated at the same rate to arrive at the destination. The two cities are in the same time zone, and upon arrival, the local time was recorded as 16:00:00. Assuming that the pendulum clock was perfectly calibrated with real time upon departure, calculate the numerical value of the time difference between the pendulum clock and the real time at the end of the journey? (expressed in $s$)", + "marking": [], + "answer": [ + "\\boxed{$1.8 \\times 10^{-4}$}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + "s" + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_4_3_1.png" + ] + }, + { + "id": "PanPhO_2025_4_4", + "context": "[Harrison and the Longitude Problem] \n\nAccurate measurement of the longitude was a long-standing problem in sea navigation. The earliest solution to this problem was to compare the local time at a location with that of a meridian. However, there was not a clock accurate enough to preserve the absolute time of another location during a journey, until the inventions by the $18^{\\text{th}}$ century English clockmaker John Harrison.\n\nLet us start our discussion on the simplest type of clock, which is maintained by a vertically hung simple pendulum. It is made of a heavy bob of mass $m$, hung by a rod of length $L$ and negligible mass, to a hinge. The timekeeping is evidently sensitive to temperature fluctuations. Another problem is that simple pendulums were affected by the motion of the ship it was on.\n\n To design a clock that can mitigate the effects discussed in the previous part, Harrison built the following clock, known as the \"H1\". Let us consider a simplified model of the H1 clock, shown in the figure.\n\nTwo massive dumbbells, each of length $\\ell$ joining two metal balls of mass $M$, are connected by two identical springs of stiffness $k$. The middle contacts ensure that the rotations of the dumbbells are always in antiphase. The oscillations of the dumbbells are used to determine time.\n\n[figure1]", + "question": "Just from the simplified design, one can see that its accuracy is not affected by translational motion of the ship. Which of the following correctly summarizes the reason behind it? Select only one option.\n\n(A) Spring can be made exceedingly stiff, such that elastic forces dominate by orders of magnitude over gravitational forces or any net force due to acceleration of the ship. \n(B) Stabilization of the clock is actualized with the springs in the H1 clock design. Therefore, the dumbbells stay level during ship motion, and their oscillations are always perpendicular to gravity. \n(C) The H1 clock is effectively equivalent to a gyroscope, which removes any bias due to translations of the boat owing to the mechanism of precession. \n(D) In the non-inertial frame of the ship, the metal balls on each dumbbell experience equal and opposite torques due to translational effects, so they cancel out and do not affect oscillations.", + "marking": [], + "answer": [ + "\\boxed{D}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 0.5 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_4_4_1.png" + ] + }, + { + "id": "PanPhO_2025_4_5", + "context": "[Harrison and the Longitude Problem] \n\nAccurate measurement of the longitude was a long-standing problem in sea navigation. The earliest solution to this problem was to compare the local time at a location with that of a meridian. However, there was not a clock accurate enough to preserve the absolute time of another location during a journey, until the inventions by the $18^{\\text{th}}$ century English clockmaker John Harrison.\n\nLet us start our discussion on the simplest type of clock, which is maintained by a vertically hung simple pendulum. It is made of a heavy bob of mass $m$, hung by a rod of length $L$ and negligible mass, to a hinge. The timekeeping is evidently sensitive to temperature fluctuations. Another problem is that simple pendulums were affected by the motion of the ship it was on.\n\n To design a clock that can mitigate the effects discussed in the previous part, Harrison built the following clock, known as the \"H1\". Let us consider a simplified model of the H1 clock, shown in the figure.\n\nTwo massive dumbbells, each of length $\\ell$ joining two metal balls of mass $M$, are connected by two identical springs of stiffness $k$. The middle contacts ensure that the rotations of the dumbbells are always in antiphase. The oscillations of the dumbbells are used to determine time.\n\n[figure1]", + "question": "Determine the period $T$ of small oscillations of the H1 clock design in terms of $M$, $k$ and $l$.", + "marking": [], + "answer": [ + "\\boxed{$T = 2 \\pi \\sqrt{\\frac{M}{2 k}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.5 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_4_4_1.png" + ] + }, + { + "id": "PanPhO_2025_4_6", + "context": "[Harrison and the Longitude Problem] \n\nAccurate measurement of the longitude was a long-standing problem in sea navigation. The earliest solution to this problem was to compare the local time at a location with that of a meridian. However, there was not a clock accurate enough to preserve the absolute time of another location during a journey, until the inventions by the $18^{\\text{th}}$ century English clockmaker John Harrison.\n\nLet us start our discussion on the simplest type of clock, which is maintained by a vertically hung simple pendulum. It is made of a heavy bob of mass $m$, hung by a rod of length $L$ and negligible mass, to a hinge. The timekeeping is evidently sensitive to temperature fluctuations. Another problem is that simple pendulums were affected by the motion of the ship it was on.\n\n To design a clock that can mitigate the effects discussed in the previous part, Harrison built the following clock, known as the \"H1\". Let us consider a simplified model of the H1 clock, shown in the figure (on the left).\n\nTwo massive dumbbells, each of length $\\ell$ joining two metal balls of mass $M$, are connected by two identical springs of stiffness $k$. The middle contacts ensure that the rotations of the dumbbells are always in antiphase. The oscillations of the dumbbells are used to determine time.\n\n[figure1]", + "question": "However, Harrison's H1 design still has many problems. For instance, let us consider the rocking of the ship, under the influence of waves. Let us consider a small model toy boat with the clock on board that moves on a sinusoidal landscape of the form $z = A_{z} \\sin (2 \\pi x / \\lambda)$ at a constant speed $v_{s}$, as shown in the figure (on the right).\n\nWhat is the time-averaged new oscillation period of the clock $$, considering that $v_{s} T^{\\prime} \\ll A_{z} \\ll \\lambda$, and that the dimensions of the boat/clock are much less than $A_{z}$? Express your answer as the ratio $\\frac{-T}{T}$.\n\nHint: the radius of curvature $R$ of a curve $z(x)$ is given by $\\frac{1}{R} \\approx \\frac{\\mathrm{d}^{2} z}{\\mathrm{d} x^{2}}$.", + "marking": [], + "answer": [ + "\\boxed{$\\frac{-T}{T} = -\\frac{2 \\pi^{4} A_{z}^{2} M v_{s}^{2}}{k \\lambda^{4}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+illustration figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_4_6_1.png" + ] + }, + { + "id": "PanPhO_2025_5_1", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART I: No friction between the ball and the wall. \n\nFirstly, we begin with a case where the wall and the ball are frictionless. Suppose the force exerting on the ball by the wall at the contact point $P$ is $\\vec{F} = -N \\hat{e}*{p}$ which is only the normal force.", + "question": "Write down the position vector $\\vec{r}$ of point $C$ in terms of $m$, $g$, $R$, $r$, $\\phi$, $z$, their derivatives and basis vectors in cylindrical coordinates.", + "marking": [], + "answer": [ + "\\boxed{$\\vec{r} = (R-r) \\hat{e}_{\\rho} + z \\hat{e}_{z}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png" + ] + }, + { + "id": "PanPhO_2025_5_2", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART I: No friction between the ball and the wall. \n\nFirstly, we begin with a case where the wall and the ball are frictionless. Suppose the force exerting on the ball by the wall at the contact point $P$ is $\\vec{F} = -N \\hat{e}*{p}$ which is only the normal force.", + "question": "Given the initial condition: at time $t=0$, $z(0)=H$, $\\phi(0)=0$, $\\dot{z}(0)=0$ and $\\dot{\\phi}(0)=\\Omega>0$. \n\n(1) Find $\\phi(t)$ as a function of $t$. \n(2) Find $z(t)$ as a function of $t$. \n(3) Find $N(t)$ as a function of $t$.", + "marking": [], + "answer": [ + "\\boxed{$\\phi(t) = \\Omega t$}", + "\\boxed{$z(t) = H - \\frac{1}{2} g t^2$}", + "\\boxed{$N(t) = m(R - r)\\Omega^2$}" + ], + "answer_type": [ + "Expression", + "Expression", + "Expression" + ], + "unit": [ + null + ], + "points": [ + 0.67, + 0.67, + 0.66 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png" + ] + }, + { + "id": "PanPhO_2025_5_3", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART II. Rolling without slipping on the wall. \n\nAs we see in PART I, without friction, obviously, the ball will only accelerate downward. Now, we consider the ball is rolling without slipping on the wall surface. Suppose the forces exerting on the ball by the wall at the contact point $P$ include the normal force and the static friction as \n$\\vec{F} = -N \\hat{e}_{\\rho} + F_{\\phi} \\hat{e}_{\\phi} + F_{z} \\hat{e}_{z}$\nUsing cylindrical coordinates, let the position of the center of mass (CM) of the ball as \n$\\vec{r} = \\rho \\hat{e}_{\\rho} + z \\hat{e}_{z}$ \nand the angular velocity vector of the ball with respect to the CM as \n$\\vec{\\omega} = \\omega_{\\rho} \\hat{e}_{\\rho} + \\omega_{\\phi} \\hat{e}_{\\phi} + \\omega_{z} \\hat{e}_{z}$\n\nAnswer the following questions in terms of $m, g, R, r, \\phi, z, N, I, F_{\\phi}, F_{z}, \\omega_{\\rho}, \\omega_{\\phi}, \\omega_{z}$ and their derivatives.", + "question": "Find three equations of motion describing the center of mass (CM) of the ball.", + "marking": [], + "answer": [ + "\\boxed{$m(R-r) \\dot{\\phi}^{2} = N$}", + "\\boxed{$m(R-r) \\ddot{\\phi} = F_{\\phi}$}", + "\\boxed{$m \\ddot{z} = F_{z} - m g$}" + ], + "answer_type": [ + "Equation", + "Equation", + "Equation" + ], + "unit": [ + null + ], + "points": [ + 1.0, + 1.0, + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png" + ] + }, + { + "id": "PanPhO_2025_5_4", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART II. Rolling without slipping on the wall. \n\nAs we see in PART I, without friction, obviously, the ball will only accelerate downward. Now, we consider the ball is rolling without slipping on the wall surface. Suppose the forces exerting on the ball by the wall at the contact point $P$ include the normal force and the static friction as \n$\\vec{F} = -N \\hat{e}_{\\rho} + F_{\\phi} \\hat{e}_{\\phi} + F_{z} \\hat{e}_{z}$\nUsing cylindrical coordinates, let the position of the center of mass (CM) of the ball as \n$\\vec{r} = \\rho \\hat{e}_{\\rho} + z \\hat{e}_{z}$ \nand the angular velocity vector of the ball with respect to the CM as \n$\\vec{\\omega} = \\omega_{\\rho} \\hat{e}_{\\rho} + \\omega_{\\phi} \\hat{e}_{\\phi} + \\omega_{z} \\hat{e}_{z}$\n\nAnswer the following questions in terms of $m, g, R, r, \\phi, z, N, I, F_{\\phi}, F_{z}, \\omega_{\\rho}, \\omega_{\\phi}, \\omega_{z}$ and their derivatives.", + "question": "Write down a set of differential equations for $\\omega_{\\rho}$, $\\omega_{\\phi}$ and $\\omega_{z}$.", + "marking": [], + "answer": [ + "\\boxed{$I \\dot{\\omega}_{\\rho} - I \\dot{\\phi} \\omega_{\\phi} = 0$}", + "\\boxed{$I \\dot{\\omega}_{\\phi} + I \\dot{\\phi} \\omega_{\\rho} = -r F_{z}$}", + "\\boxed{$I \\dot{\\omega}_{z} = r F_{\\phi}$}" + ], + "answer_type": [ + "Equation", + "Equation", + "Equation" + ], + "unit": [ + null + ], + "points": [ + 1.0, + 1.0, + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png" + ] + }, + { + "id": "PanPhO_2025_5_5", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART II. Rolling without slipping on the wall. \n\nAs we see in PART I, without friction, obviously, the ball will only accelerate downward. Now, we consider the ball is rolling without slipping on the wall surface. Suppose the forces exerting on the ball by the wall at the contact point $P$ include the normal force and the static friction as \n$\\vec{F} = -N \\hat{e}_{\\rho} + F_{\\phi} \\hat{e}_{\\phi} + F_{z} \\hat{e}_{z}$\nUsing cylindrical coordinates, let the position of the center of mass (CM) of the ball as \n$\\vec{r} = \\rho \\hat{e}_{\\rho} + z \\hat{e}_{z}$ \nand the angular velocity vector of the ball with respect to the CM as \n$\\vec{\\omega} = \\omega_{\\rho} \\hat{e}_{\\rho} + \\omega_{\\phi} \\hat{e}_{\\phi} + \\omega_{z} \\hat{e}_{z}$\n\nAnswer the following questions in terms of $m, g, R, r, \\phi, z, N, I, F_{\\phi}, F_{z}, \\omega_{\\rho}, \\omega_{\\phi}, \\omega_{z}$ and their derivatives.", + "question": "Write down equations of the rolling without slipping condition.", + "marking": [], + "answer": [ + "\\boxed{$(R-r) \\dot{\\phi} + r \\omega_{z} = 0$}", + "\\boxed{$\\dot{z} - r \\omega_{\\phi} = 0$}" + ], + "answer_type": [ + "Equation", + "Equation" + ], + "unit": [ + null + ], + "points": [ + 1.0, + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png" + ] + }, + { + "id": "PanPhO_2025_5_6", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART II. Rolling without slipping on the wall. \n\nAs we see in PART I, without friction, obviously, the ball will only accelerate downward. Now, we consider the ball is rolling without slipping on the wall surface. Suppose the forces exerting on the ball by the wall at the contact point $P$ include the normal force and the static friction as \n$\\vec{F} = -N \\hat{e}_{\\rho} + F_{\\phi} \\hat{e}_{\\phi} + F_{z} \\hat{e}_{z}$\nUsing cylindrical coordinates, let the position of the center of mass (CM) of the ball as \n$\\vec{r} = \\rho \\hat{e}_{\\rho} + z \\hat{e}_{z}$ \nand the angular velocity vector of the ball with respect to the CM as \n$\\vec{\\omega} = \\omega_{\\rho} \\hat{e}_{\\rho} + \\omega_{\\phi} \\hat{e}_{\\phi} + \\omega_{z} \\hat{e}_{z}$\n\nAnswer the following questions in terms of $m, g, R, r, \\phi, z, N, I, F_{\\phi}, F_{z}, \\omega_{\\rho}, \\omega_{\\phi}, \\omega_{z}$ and their derivatives. \n\n Previous questions: \n(1) Find three equations of motion describing the center of mass (CM) of the ball. \n(2)Write down a set of differential equations for $\\omega_{\\rho}$, $\\omega_{\\phi}$ and $\\omega_{z}$. \n(3) Write down equations of the rolling without slipping condition. (This is a preliminary question and do not include in the final answer)", + "question": "Find, from the equations of motion, that the rate of change of the $z$-component (vertical) of the total angular momentum of the ball with respective to the origin $O$.", + "marking": [], + "answer": [ + "\\boxed{$m(R-r)^{2} \\dot{\\phi} + I \\omega_{z}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png" + ] + }, + { + "id": "PanPhO_2025_5_7", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART II. Rolling without slipping on the wall. \n\nAs we see in PART I, without friction, obviously, the ball will only accelerate downward. Now, we consider the ball is rolling without slipping on the wall surface. Suppose the forces exerting on the ball by the wall at the contact point $P$ include the normal force and the static friction as \n$\\vec{F} = -N \\hat{e}_{\\rho} + F_{\\phi} \\hat{e}_{\\phi} + F_{z} \\hat{e}_{z}$\nUsing cylindrical coordinates, let the position of the center of mass (CM) of the ball as \n$\\vec{r} = \\rho \\hat{e}_{\\rho} + z \\hat{e}_{z}$ \nand the angular velocity vector of the ball with respect to the CM as \n$\\vec{\\omega} = \\omega_{\\rho} \\hat{e}_{\\rho} + \\omega_{\\phi} \\hat{e}_{\\phi} + \\omega_{z} \\hat{e}_{z}$\n\nAnswer the following questions in terms of $m, g, R, r, \\phi, z, N, I, F_{\\phi}, F_{z}, \\omega_{\\rho}, \\omega_{\\phi}, \\omega_{z}$ and their derivatives. \n\n Previous questions: \n(1) Find three equations of motion describing the center of mass (CM) of the ball. \n(2)Write down a set of differential equations for $\\omega_{\\rho}$, $\\omega_{\\phi}$ and $\\omega_{z}$. \n(3) Write down equations of the rolling without slipping condition. (This is a preliminary question and do not include in the final answer)", + "question": "Consider the initial condition that at the top of the hole as at time $t = 0$, $z(0) = H$, $\\phi(0) = 0$, $\\dot{z}(0) = 0$ and $\\dot{\\phi}(0) = \\Omega > 0$. \n\n Show that the vertical motion of the ball is a simple harmonic motion. Write down the equation of the vertical motion, i.e., express $\\ddot{z}$ in terms of $z, \\Omega, K, H, g$.", + "marking": [], + "answer": [ + "\\boxed{$\\ddot{z} = -\\frac{K}{K+1} \\Omega^{2}\\left(z - H + \\frac{g}{\\Omega^{2} K}\\right)$}" + ], + "answer_type": [ + "Equation" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png" + ] + }, + { + "id": "PanPhO_2025_5_8", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART II. Rolling without slipping on the wall. \n\nAs we see in PART I, without friction, obviously, the ball will only accelerate downward. Now, we consider the ball is rolling without slipping on the wall surface. Suppose the forces exerting on the ball by the wall at the contact point $P$ include the normal force and the static friction as \n$\\vec{F} = -N \\hat{e}_{\\rho} + F_{\\phi} \\hat{e}_{\\phi} + F_{z} \\hat{e}_{z}$\nUsing cylindrical coordinates, let the position of the center of mass (CM) of the ball as \n$\\vec{r} = \\rho \\hat{e}_{\\rho} + z \\hat{e}_{z}$ \nand the angular velocity vector of the ball with respect to the CM as \n$\\vec{\\omega} = \\omega_{\\rho} \\hat{e}_{\\rho} + \\omega_{\\phi} \\hat{e}_{\\phi} + \\omega_{z} \\hat{e}_{z}$\n\nAnswer the following questions in terms of $m, g, R, r, \\phi, z, N, I, F_{\\phi}, F_{z}, \\omega_{\\rho}, \\omega_{\\phi}, \\omega_{z}$ and their derivatives. \n\nConsider the initial condition that at the top of the hole as at time $t = 0$, $z(0) = H$, $\\phi(0) = 0$, $\\dot{z}(0) = 0$ and $\\dot{\\phi}(0) = \\Omega > 0$. \n\nSince the vertical motion is a simple harmonic motion, if the golf ball has a tiny non-zero initial downward motion at the top of the hole, the ball will first roll down and, at the time that the golfer is happy about finishing the hole, the ball will come back up and out again. That is the golfer's nightmare! \n\nConsider the ball is a uniform solid sphere, i.e. $K = \\frac{2}{5}$.", + "question": "Find the angular frequency of the vertical simple harmonic motion, $\\Omega_{z}$, in terms of the initial angular velocity of the center of mass (CM) of the ball around the hole, $\\Omega$.", + "marking": [], + "answer": [ + "\\boxed{$\\Omega_{z} = \\sqrt{\\frac{K}{K+1}} \\Omega$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png" + ] + }, + { + "id": "PanPhO_2025_5_9", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART II. Rolling without slipping on the wall. \n\nAs we see in PART I, without friction, obviously, the ball will only accelerate downward. Now, we consider the ball is rolling without slipping on the wall surface. Suppose the forces exerting on the ball by the wall at the contact point $P$ include the normal force and the static friction as \n$\\vec{F} = -N \\hat{e}_{\\rho} + F_{\\phi} \\hat{e}_{\\phi} + F_{z} \\hat{e}_{z}$\nUsing cylindrical coordinates, let the position of the center of mass (CM) of the ball as \n$\\vec{r} = \\rho \\hat{e}_{\\rho} + z \\hat{e}_{z}$ \nand the angular velocity vector of the ball with respect to the CM as \n$\\vec{\\omega} = \\omega_{\\rho} \\hat{e}_{\\rho} + \\omega_{\\phi} \\hat{e}_{\\phi} + \\omega_{z} \\hat{e}_{z}$\n\nAnswer the following questions in terms of $m, g, R, r, \\phi, z, N, I, F_{\\phi}, F_{z}, \\omega_{\\rho}, \\omega_{\\phi}, \\omega_{z}$ and their derivatives. \n\nConsider the initial condition that at the top of the hole as at time $t = 0$, $z(0) = H$, $\\phi(0) = 0$, $\\dot{z}(0) = 0$ and $\\dot{\\phi}(0) = \\Omega > 0$. \n\nSince the vertical motion is a simple harmonic motion, if the golf ball has a tiny non-zero initial downward motion at the top of the hole, the ball will first roll down and, at the time that the golfer is happy about finishing the hole, the ball will come back up and out again. That is the golfer's nightmare! \n\nConsider the ball is a uniform solid sphere, i.e. $K = \\frac{2}{5}$.", + "question": "Find minimum depth of the hole $H_{m}$ such that the ball will not touch the bottom of the hole in terms of $g$ and $\\Omega$.", + "marking": [], + "answer": [ + "\\boxed{$\\frac{2 g}{\\Omega^{2} K}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png" + ] + }, + { + "id": "PanPhO_2025_5_10", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART II. Rolling without slipping on the wall. \n\nAs we see in PART I, without friction, obviously, the ball will only accelerate downward. Now, we consider the ball is rolling without slipping on the wall surface. Suppose the forces exerting on the ball by the wall at the contact point $P$ include the normal force and the static friction as \n$\\vec{F} = -N \\hat{e}_{\\rho} + F_{\\phi} \\hat{e}_{\\phi} + F_{z} \\hat{e}_{z}$\nUsing cylindrical coordinates, let the position of the center of mass (CM) of the ball as \n$\\vec{r} = \\rho \\hat{e}_{\\rho} + z \\hat{e}_{z}$ \nand the angular velocity vector of the ball with respect to the CM as \n$\\vec{\\omega} = \\omega_{\\rho} \\hat{e}_{\\rho} + \\omega_{\\phi} \\hat{e}_{\\phi} + \\omega_{z} \\hat{e}_{z}$\n\nAnswer the following questions in terms of $m, g, R, r, \\phi, z, N, I, F_{\\phi}, F_{z}, \\omega_{\\rho}, \\omega_{\\phi}, \\omega_{z}$ and their derivatives. \n\nConsider the initial condition that at the top of the hole as at time $t = 0$, $z(0) = H$, $\\phi(0) = 0$, $\\dot{z}(0) = 0$ and $\\dot{\\phi}(0) = \\Omega > 0$. \n\nSince the vertical motion is a simple harmonic motion, if the golf ball has a tiny non-zero initial downward motion at the top of the hole, the ball will first roll down and, at the time that the golfer is happy about finishing the hole, the ball will come back up and out again. That is the golfer's nightmare! \n\nConsider the ball is a uniform solid sphere, i.e. $K = \\frac{2}{5}$.\n\n[Where does the energy go?]", + "question": "Using energy conservation: \n(1) Find the magnitude of angular velocity component $\\omega_{z}$ of the ball at the lowest point in its vertical motion. \n(2) Find the magnitude of angular velocity component $\\omega_{\\phi}$ of the ball at the lowest point in its vertical motion. \n(3) Find the magnitude of angular velocity component $\\omega_{\\rho}$ of the ball at the lowest point in its vertical motion, in terms of $g$, $r$, and $\\Omega$.", + "marking": [], + "answer": [ + "\\boxed{$\\omega_{z} = -\\frac{R-r}{r} \\Omega$}", + "\\boxed{$\\omega_{\\phi} = 0$}", + "\\boxed{$\\omega_{\\rho} = \\frac{5 g}{\\Omega r}$}" + ], + "answer_type": [ + "Expression", + "Expression", + "Expression" + ], + "unit": [ + null, + null, + null + ], + "points": [ + 0.67, + 0.67, + 0.66 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png" + ] + }, + { + "id": "PanPhO_2025_5_11", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART II. Rolling without slipping on the wall. \n\nAs we see in PART I, without friction, obviously, the ball will only accelerate downward. Now, we consider the ball is rolling without slipping on the wall surface. Suppose the forces exerting on the ball by the wall at the contact point $P$ include the normal force and the static friction as \n$\\vec{F} = -N \\hat{e}_{\\rho} + F_{\\phi} \\hat{e}_{\\phi} + F_{z} \\hat{e}_{z}$\nUsing cylindrical coordinates, let the position of the center of mass (CM) of the ball as \n$\\vec{r} = \\rho \\hat{e}_{\\rho} + z \\hat{e}_{z}$ \nand the angular velocity vector of the ball with respect to the CM as \n$\\vec{\\omega} = \\omega_{\\rho} \\hat{e}_{\\rho} + \\omega_{\\phi} \\hat{e}_{\\phi} + \\omega_{z} \\hat{e}_{z}$\n\nAnswer the following questions in terms of $m, g, R, r, \\phi, z, N, I, F_{\\phi}, F_{z}, \\omega_{\\rho}, \\omega_{\\phi}, \\omega_{z}$ and their derivatives. \n\nConsider the initial condition that at the top of the hole as at time $t = 0$, $z(0) = H$, $\\phi(0) = 0$, $\\dot{z}(0) = 0$ and $\\dot{\\phi}(0) = \\Omega > 0$. \n\nSince the vertical motion is a simple harmonic motion, if the golf ball has a tiny non-zero initial downward motion at the top of the hole, the ball will first roll down and, at the time that the golfer is happy about finishing the hole, the ball will come back up and out again. That is the golfer's nightmare! \n\nConsider the ball is a uniform solid sphere, i.e. $K = \\frac{2}{5}$. \n\nTo help explaining this phenomenon, one can investigate the motion of the ball in a reference frame $S^{\\prime}(x^{\\prime}, y^{\\prime}, z^{\\prime})$ that rotates about the $z$-axis with the same angular velocity $\\Omega \\hat{e}_{z}$ as the motion of the ball around the hole. In this $S^{\\prime}$ frame, the $\\phi^{\\prime}$ value of the position of the ball is fixed at $\\phi^{\\prime} = 0$.\n\nIn this rotating frame, there are three fictitious forces including the centrifugal force, the Euler force $\\vec{F}_{E} = -m \\dot{\\vec{\\Omega}} \\times \\vec{r}^{\\prime}$ and the Coriolis force $\\vec{F}_{C} = -2 m \\vec{\\Omega} \\times \\frac{d}{d t} \\vec{r}^{\\prime}$, where $\\vec{r}^{\\prime}$ is the position vector in $S^{\\prime}$ frame.\n\n[figure2] \nFigure 2. Top view of the axes of the rotating frame $S^{\\prime}$. The fixed and rotating frames share the same origin and vertical axis: $z$ and $z^{\\prime}$.", + "question": "Find the $z$-component of the centrifugal force of the entire ball.", + "marking": [], + "answer": [ + "\\boxed{0}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png", + "image_question/PanPhO_2025_5_11_1.png" + ] + }, + { + "id": "PanPhO_2025_5_12", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART II. Rolling without slipping on the wall. \n\nAs we see in PART I, without friction, obviously, the ball will only accelerate downward. Now, we consider the ball is rolling without slipping on the wall surface. Suppose the forces exerting on the ball by the wall at the contact point $P$ include the normal force and the static friction as \n$\\vec{F} = -N \\hat{e}_{\\rho} + F_{\\phi} \\hat{e}_{\\phi} + F_{z} \\hat{e}_{z}$\nUsing cylindrical coordinates, let the position of the center of mass (CM) of the ball as \n$\\vec{r} = \\rho \\hat{e}_{\\rho} + z \\hat{e}_{z}$ \nand the angular velocity vector of the ball with respect to the CM as \n$\\vec{\\omega} = \\omega_{\\rho} \\hat{e}_{\\rho} + \\omega_{\\phi} \\hat{e}_{\\phi} + \\omega_{z} \\hat{e}_{z}$\n\nAnswer the following questions in terms of $m, g, R, r, \\phi, z, N, I, F_{\\phi}, F_{z}, \\omega_{\\rho}, \\omega_{\\phi}, \\omega_{z}$ and their derivatives. \n\nConsider the initial condition that at the top of the hole as at time $t = 0$, $z(0) = H$, $\\phi(0) = 0$, $\\dot{z}(0) = 0$ and $\\dot{\\phi}(0) = \\Omega > 0$. \n\nSince the vertical motion is a simple harmonic motion, if the golf ball has a tiny non-zero initial downward motion at the top of the hole, the ball will first roll down and, at the time that the golfer is happy about finishing the hole, the ball will come back up and out again. That is the golfer's nightmare! \n\nConsider the ball is a uniform solid sphere, i.e. $K = \\frac{2}{5}$. \n\nTo help explaining this phenomenon, one can investigate the motion of the ball in a reference frame $S^{\\prime}(x^{\\prime}, y^{\\prime}, z^{\\prime})$ that rotates about the $z$-axis with the same angular velocity $\\Omega \\hat{e}_{z}$ as the motion of the ball around the hole. In this $S^{\\prime}$ frame, the $\\phi^{\\prime}$ value of the position of the ball is fixed at $\\phi^{\\prime} = 0$.\n\nIn this rotating frame, there are three fictitious forces including the centrifugal force, the Euler force $\\vec{F}_{E} = -m \\dot{\\vec{\\Omega}} \\times \\vec{r}^{\\prime}$ and the Coriolis force $\\vec{F}_{C} = -2 m \\vec{\\Omega} \\times \\frac{d}{d t} \\vec{r}^{\\prime}$, where $\\vec{r}^{\\prime}$ is the position vector in $S^{\\prime}$ frame.\n\n[figure2] \nFigure 2. Top view of the axes of the rotating frame $S^{\\prime}$. The fixed and rotating frames share the same origin and vertical axis: $z$ and $z^{\\prime}$.", + "question": "Find the $z$-component of the Euler force of the entire ball.", + "marking": [], + "answer": [ + "\\boxed{0}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png", + "image_question/PanPhO_2025_5_11_1.png" + ] + }, + { + "id": "PanPhO_2025_5_13", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART II. Rolling without slipping on the wall. \n\nAs we see in PART I, without friction, obviously, the ball will only accelerate downward. Now, we consider the ball is rolling without slipping on the wall surface. Suppose the forces exerting on the ball by the wall at the contact point $P$ include the normal force and the static friction as \n$\\vec{F} = -N \\hat{e}_{\\rho} + F_{\\phi} \\hat{e}_{\\phi} + F_{z} \\hat{e}_{z}$\nUsing cylindrical coordinates, let the position of the center of mass (CM) of the ball as \n$\\vec{r} = \\rho \\hat{e}_{\\rho} + z \\hat{e}_{z}$ \nand the angular velocity vector of the ball with respect to the CM as \n$\\vec{\\omega} = \\omega_{\\rho} \\hat{e}_{\\rho} + \\omega_{\\phi} \\hat{e}_{\\phi} + \\omega_{z} \\hat{e}_{z}$\n\nAnswer the following questions in terms of $m, g, R, r, \\phi, z, N, I, F_{\\phi}, F_{z}, \\omega_{\\rho}, \\omega_{\\phi}, \\omega_{z}$ and their derivatives. \n\nConsider the initial condition that at the top of the hole as at time $t = 0$, $z(0) = H$, $\\phi(0) = 0$, $\\dot{z}(0) = 0$ and $\\dot{\\phi}(0) = \\Omega > 0$. \n\nSince the vertical motion is a simple harmonic motion, if the golf ball has a tiny non-zero initial downward motion at the top of the hole, the ball will first roll down and, at the time that the golfer is happy about finishing the hole, the ball will come back up and out again. That is the golfer's nightmare! \n\nConsider the ball is a uniform solid sphere, i.e. $K = \\frac{2}{5}$. \n\nTo help explaining this phenomenon, one can investigate the motion of the ball in a reference frame $S^{\\prime}(x^{\\prime}, y^{\\prime}, z^{\\prime})$ that rotates about the $z$-axis with the same angular velocity $\\Omega \\hat{e}_{z}$ as the motion of the ball around the hole. In this $S^{\\prime}$ frame, the $\\phi^{\\prime}$ value of the position of the ball is fixed at $\\phi^{\\prime} = 0$.\n\nIn this rotating frame, there are three fictitious forces including the centrifugal force, the Euler force $\\vec{F}_{E} = -m \\dot{\\vec{\\Omega}} \\times \\vec{r}^{\\prime}$ and the Coriolis force $\\vec{F}_{C} = -2 m \\vec{\\Omega} \\times \\frac{d}{d t} \\vec{r}^{\\prime}$, where $\\vec{r}^{\\prime}$ is the position vector in $S^{\\prime}$ frame.\n\n[figure2] \nFigure 2. Top view of the axes of the rotating frame $S^{\\prime}$. The fixed and rotating frames share the same origin and vertical axis: $z$ and $z^{\\prime}$.", + "question": "Find the $z$-component of the Coriolis force of the entire ball.", + "marking": [], + "answer": [ + "\\boxed{0}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png", + "image_question/PanPhO_2025_5_11_1.png" + ] + }, + { + "id": "PanPhO_2025_5_14", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART II. Rolling without slipping on the wall. \n\nAs we see in PART I, without friction, obviously, the ball will only accelerate downward. Now, we consider the ball is rolling without slipping on the wall surface. Suppose the forces exerting on the ball by the wall at the contact point $P$ include the normal force and the static friction as \n$\\vec{F} = -N \\hat{e}_{\\rho} + F_{\\phi} \\hat{e}_{\\phi} + F_{z} \\hat{e}_{z}$\nUsing cylindrical coordinates, let the position of the center of mass (CM) of the ball as \n$\\vec{r} = \\rho \\hat{e}_{\\rho} + z \\hat{e}_{z}$ \nand the angular velocity vector of the ball with respect to the CM as \n$\\vec{\\omega} = \\omega_{\\rho} \\hat{e}_{\\rho} + \\omega_{\\phi} \\hat{e}_{\\phi} + \\omega_{z} \\hat{e}_{z}$\n\nAnswer the following questions in terms of $m, g, R, r, \\phi, z, N, I, F_{\\phi}, F_{z}, \\omega_{\\rho}, \\omega_{\\phi}, \\omega_{z}$ and their derivatives. \n\nConsider the initial condition that at the top of the hole as at time $t = 0$, $z(0) = H$, $\\phi(0) = 0$, $\\dot{z}(0) = 0$ and $\\dot{\\phi}(0) = \\Omega > 0$. \n\nSince the vertical motion is a simple harmonic motion, if the golf ball has a tiny non-zero initial downward motion at the top of the hole, the ball will first roll down and, at the time that the golfer is happy about finishing the hole, the ball will come back up and out again. That is the golfer's nightmare! \n\nConsider the ball is a uniform solid sphere, i.e. $K = \\frac{2}{5}$. \n\nTo help explaining this phenomenon, one can investigate the motion of the ball in a reference frame $S^{\\prime}(x^{\\prime}, y^{\\prime}, z^{\\prime})$ that rotates about the $z$-axis with the same angular velocity $\\Omega \\hat{e}_{z}$ as the motion of the ball around the hole. In this $S^{\\prime}$ frame, the $\\phi^{\\prime}$ value of the position of the ball is fixed at $\\phi^{\\prime} = 0$.\n\nIn this rotating frame, there are three fictitious forces including the centrifugal force, the Euler force $\\vec{F}_{E} = -m \\dot{\\vec{\\Omega}} \\times \\vec{r}^{\\prime}$ and the Coriolis force $\\vec{F}_{C} = -2 m \\vec{\\Omega} \\times \\frac{d}{d t} \\vec{r}^{\\prime}$, where $\\vec{r}^{\\prime}$ is the position vector in $S^{\\prime}$ frame.\n\n[figure2] \nFigure 2. Top view of the axes of the rotating frame $S^{\\prime}$. The fixed and rotating frames share the same origin and vertical axis: $z$ and $z^{\\prime}$. \n\nThe rotation of the ball about the $y^{\\prime}$-axis (or the $\\phi$-componet in cylindrical coordinates) is coupled with the vertical motion of the ball when it is rolling without slipping. Therefore, knowing the torque of this rotation will give us insight into why the ball can move up and down under gravity. Let the angular velocity of the ball in this rotating frame as $\\vec{\\omega}^{\\prime}$.", + "question": "Find the torque with respect to the center of mass (CM) of the ball due to centrifugal force.", + "marking": [], + "answer": [ + "\\boxed{0}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 1.5 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png", + "image_question/PanPhO_2025_5_11_1.png" + ] + }, + { + "id": "PanPhO_2025_5_15", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART II. Rolling without slipping on the wall. \n\nAs we see in PART I, without friction, obviously, the ball will only accelerate downward. Now, we consider the ball is rolling without slipping on the wall surface. Suppose the forces exerting on the ball by the wall at the contact point $P$ include the normal force and the static friction as \n$\\vec{F} = -N \\hat{e}_{\\rho} + F_{\\phi} \\hat{e}_{\\phi} + F_{z} \\hat{e}_{z}$\nUsing cylindrical coordinates, let the position of the center of mass (CM) of the ball as \n$\\vec{r} = \\rho \\hat{e}_{\\rho} + z \\hat{e}_{z}$ \nand the angular velocity vector of the ball with respect to the CM as \n$\\vec{\\omega} = \\omega_{\\rho} \\hat{e}_{\\rho} + \\omega_{\\phi} \\hat{e}_{\\phi} + \\omega_{z} \\hat{e}_{z}$\n\nAnswer the following questions in terms of $m, g, R, r, \\phi, z, N, I, F_{\\phi}, F_{z}, \\omega_{\\rho}, \\omega_{\\phi}, \\omega_{z}$ and their derivatives. \n\nConsider the initial condition that at the top of the hole as at time $t = 0$, $z(0) = H$, $\\phi(0) = 0$, $\\dot{z}(0) = 0$ and $\\dot{\\phi}(0) = \\Omega > 0$. \n\nSince the vertical motion is a simple harmonic motion, if the golf ball has a tiny non-zero initial downward motion at the top of the hole, the ball will first roll down and, at the time that the golfer is happy about finishing the hole, the ball will come back up and out again. That is the golfer's nightmare! \n\nConsider the ball is a uniform solid sphere, i.e. $K = \\frac{2}{5}$. \n\nTo help explaining this phenomenon, one can investigate the motion of the ball in a reference frame $S^{\\prime}(x^{\\prime}, y^{\\prime}, z^{\\prime})$ that rotates about the $z$-axis with the same angular velocity $\\Omega \\hat{e}_{z}$ as the motion of the ball around the hole. In this $S^{\\prime}$ frame, the $\\phi^{\\prime}$ value of the position of the ball is fixed at $\\phi^{\\prime} = 0$.\n\nIn this rotating frame, there are three fictitious forces including the centrifugal force, the Euler force $\\vec{F}_{E} = -m \\dot{\\vec{\\Omega}} \\times \\vec{r}^{\\prime}$ and the Coriolis force $\\vec{F}_{C} = -2 m \\vec{\\Omega} \\times \\frac{d}{d t} \\vec{r}^{\\prime}$, where $\\vec{r}^{\\prime}$ is the position vector in $S^{\\prime}$ frame.\n\n[figure2] \nFigure 2. Top view of the axes of the rotating frame $S^{\\prime}$. The fixed and rotating frames share the same origin and vertical axis: $z$ and $z^{\\prime}$. \n\nThe rotation of the ball about the $y^{\\prime}$-axis (or the $\\phi$-componet in cylindrical coordinates) is coupled with the vertical motion of the ball when it is rolling without slipping. Therefore, knowing the torque of this rotation will give us insight into why the ball can move up and down under gravity. Let the angular velocity of the ball in this rotating frame as $\\vec{\\omega}^{\\prime}$.", + "question": "Find the torque with respect to the center of mass (CM) of the ball due to Euler force.", + "marking": [], + "answer": [ + "\\boxed{0}" + ], + "answer_type": [ + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 1.5 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png", + "image_question/PanPhO_2025_5_11_1.png" + ] + }, + { + "id": "PanPhO_2025_5_16", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART II. Rolling without slipping on the wall. \n\nAs we see in PART I, without friction, obviously, the ball will only accelerate downward. Now, we consider the ball is rolling without slipping on the wall surface. Suppose the forces exerting on the ball by the wall at the contact point $P$ include the normal force and the static friction as \n$\\vec{F} = -N \\hat{e}_{\\rho} + F_{\\phi} \\hat{e}_{\\phi} + F_{z} \\hat{e}_{z}$\nUsing cylindrical coordinates, let the position of the center of mass (CM) of the ball as \n$\\vec{r} = \\rho \\hat{e}_{\\rho} + z \\hat{e}_{z}$ \nand the angular velocity vector of the ball with respect to the CM as \n$\\vec{\\omega} = \\omega_{\\rho} \\hat{e}_{\\rho} + \\omega_{\\phi} \\hat{e}_{\\phi} + \\omega_{z} \\hat{e}_{z}$\n\nAnswer the following questions in terms of $m, g, R, r, \\phi, z, N, I, F_{\\phi}, F_{z}, \\omega_{\\rho}, \\omega_{\\phi}, \\omega_{z}$ and their derivatives. \n\nConsider the initial condition that at the top of the hole as at time $t = 0$, $z(0) = H$, $\\phi(0) = 0$, $\\dot{z}(0) = 0$ and $\\dot{\\phi}(0) = \\Omega > 0$. \n\nSince the vertical motion is a simple harmonic motion, if the golf ball has a tiny non-zero initial downward motion at the top of the hole, the ball will first roll down and, at the time that the golfer is happy about finishing the hole, the ball will come back up and out again. That is the golfer's nightmare! \n\nConsider the ball is a uniform solid sphere, i.e. $K = \\frac{2}{5}$. \n\nTo help explaining this phenomenon, one can investigate the motion of the ball in a reference frame $S^{\\prime}(x^{\\prime}, y^{\\prime}, z^{\\prime})$ that rotates about the $z$-axis with the same angular velocity $\\Omega \\hat{e}_{z}$ as the motion of the ball around the hole. In this $S^{\\prime}$ frame, the $\\phi^{\\prime}$ value of the position of the ball is fixed at $\\phi^{\\prime} = 0$.\n\nIn this rotating frame, there are three fictitious forces including the centrifugal force, the Euler force $\\vec{F}_{E} = -m \\dot{\\vec{\\Omega}} \\times \\vec{r}^{\\prime}$ and the Coriolis force $\\vec{F}_{C} = -2 m \\vec{\\Omega} \\times \\frac{d}{d t} \\vec{r}^{\\prime}$, where $\\vec{r}^{\\prime}$ is the position vector in $S^{\\prime}$ frame.\n\n[figure2] \nFigure 2. Top view of the axes of the rotating frame $S^{\\prime}$. The fixed and rotating frames share the same origin and vertical axis: $z$ and $z^{\\prime}$. \n\nThe rotation of the ball about the $y^{\\prime}$-axis (or the $\\phi$-componet in cylindrical coordinates) is coupled with the vertical motion of the ball when it is rolling without slipping. Therefore, knowing the torque of this rotation will give us insight into why the ball can move up and down under gravity. Let the angular velocity of the ball in this rotating frame as $\\vec{\\omega}^{\\prime}$.", + "question": "Find the torque $\\vec{\\tau}$ with respect to the center of mass (CM) of the ball due to Coriolis force.", + "marking": [], + "answer": [ + "\\boxed{$\\vec{\\tau} = I (\\vec{\\omega}^{\\prime} \\times \\vec{\\Omega})$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.5 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png", + "image_question/PanPhO_2025_5_11_1.png" + ] + }, + { + "id": "PanPhO_2025_5_17", + "context": "[Golfer's Nightmare] \n\nWhat happens when a golf ball rolls along the inner vertical wall of the cylindrical hole under gravity? Normally, one thinks the ball will go in and never come back up. However, it is often observed that the ball first rolls down along the wall and but then it rolls back up without touching the bottom of the hole. \n\nSome mathematics Identities may be useful in this problem: \n\n$\\vec{A} \\cdot (\\vec{B} \\times \\vec{C}) = \\vec{B} \\cdot (\\vec{C} \\times \\vec{A}) = \\vec{C} \\cdot(\\vec{A} \\times \\vec{B})$ \n$\\vec{A} \\times (\\vec{B} \\times \\vec{C}) = (\\vec{A} \\cdot \\vec{C}) \\vec{B} - (\\vec{A} \\cdot \\vec{B}) \\vec{C})$ \n\nTo understand this phenomenon, we consider a ball rolls without slipping along the vertical wall of a cylindrical hole at all time. The hole has a depth $H$ and radius $R$. The ball is spherically symmetrical, has a mass $m$, a radius $r$ and a moment of inertia $I = K m r^{2}$, where $K$ is a numerical constant that depends on the mass distribution inside the ball, such as $K = 2 / 5$ for a uniform sphere and $K = 2 / 3$ for a spherical thin shell, etc. Let \n$\\vec{r}$ be the position vector of the center of the ball;\n$\\vec{\\omega}$ be the angular velocity vector of the rotation of the ball; \n$\\vec{g}$ be the constant acceleration due to gravity pointing towards $-z$ direction, i.e. $\\vec{g} = -g \\hat{e}_{z}$\nWe ignore air friction in this problem. \n\nThe situation is shown in Figure 1 in cylindrical coordinates. Point $C$ is the center of the ball. Point $P$ is the contact point between the ball and the wall. The basis vectors $\\hat{e}_{\\rho}$ and $\\hat{e}_{\\phi}$ are the basis vectors in cylindrical coordinates $(\\rho, \\phi, z)$, where $\\rho$ is the perpendicular distance from $C$ to the $z$-axis, $\\phi$ is the azimuthal angle measured from $x$-axis and $z$ is the vertical distance of $C$ from the $xy$-plane containing the origin $O$. Given the basis vectors in cylindrical coordinates as \n\n$\\hat{e}_{\\rho} = \\cos \\phi \\hat{i} + \\sin \\phi \\hat{j}$, $\\hat{e}_{\\phi} = -\\sin \\phi \\hat{i} + \\cos \\phi \\hat{j}$, $\\hat{e}_{z} = \\hat{k}$, \n\nwhere $\\hat{i}, \\hat{j}$ and $\\hat{k}$ are the unit vectors of the Cartesian coordinates along $x$, $y$ and $z$ axis, respectively.\n\n[figure1]\nFigure 1. (a) The ball and the cylinder. (b) Top view of the situation. \n\nPART II. Rolling without slipping on the wall. \n\nAs we see in PART I, without friction, obviously, the ball will only accelerate downward. Now, we consider the ball is rolling without slipping on the wall surface. Suppose the forces exerting on the ball by the wall at the contact point $P$ include the normal force and the static friction as \n$\\vec{F} = -N \\hat{e}_{\\rho} + F_{\\phi} \\hat{e}_{\\phi} + F_{z} \\hat{e}_{z}$\nUsing cylindrical coordinates, let the position of the center of mass (CM) of the ball as \n$\\vec{r} = \\rho \\hat{e}_{\\rho} + z \\hat{e}_{z}$ \nand the angular velocity vector of the ball with respect to the CM as \n$\\vec{\\omega} = \\omega_{\\rho} \\hat{e}_{\\rho} + \\omega_{\\phi} \\hat{e}_{\\phi} + \\omega_{z} \\hat{e}_{z}$\n\nAnswer the following questions in terms of $m, g, R, r, \\phi, z, N, I, F_{\\phi}, F_{z}, \\omega_{\\rho}, \\omega_{\\phi}, \\omega_{z}$ and their derivatives. \n\nConsider the initial condition that at the top of the hole as at time $t = 0$, $z(0) = H$, $\\phi(0) = 0$, $\\dot{z}(0) = 0$ and $\\dot{\\phi}(0) = \\Omega > 0$. \n\nSince the vertical motion is a simple harmonic motion, if the golf ball has a tiny non-zero initial downward motion at the top of the hole, the ball will first roll down and, at the time that the golfer is happy about finishing the hole, the ball will come back up and out again. That is the golfer's nightmare! \n\nConsider the ball is a uniform solid sphere, i.e. $K = \\frac{2}{5}$. \n\nTo help explaining this phenomenon, one can investigate the motion of the ball in a reference frame $S^{\\prime}(x^{\\prime}, y^{\\prime}, z^{\\prime})$ that rotates about the $z$-axis with the same angular velocity $\\Omega \\hat{e}_{z}$ as the motion of the ball around the hole. In this $S^{\\prime}$ frame, the $\\phi^{\\prime}$ value of the position of the ball is fixed at $\\phi^{\\prime} = 0$.\n\nIn this rotating frame, there are three fictitious forces including the centrifugal force, the Euler force $\\vec{F}_{E} = -m \\dot{\\vec{\\Omega}} \\times \\vec{r}^{\\prime}$ and the Coriolis force $\\vec{F}_{C} = -2 m \\vec{\\Omega} \\times \\frac{d}{d t} \\vec{r}^{\\prime}$, where $\\vec{r}^{\\prime}$ is the position vector in $S^{\\prime}$ frame.\n\n[figure2] \nFigure 2. Top view of the axes of the rotating frame $S^{\\prime}$. The fixed and rotating frames share the same origin and vertical axis: $z$ and $z^{\\prime}$. \n\nThe rotation of the ball about the $y^{\\prime}$-axis (or the $\\phi$-componet in cylindrical coordinates) is coupled with the vertical motion of the ball when it is rolling without slipping. Therefore, knowing the torque of this rotation will give us insight into why the ball can move up and down under gravity. Let the angular velocity of the ball in this rotating frame as $\\vec{\\omega}^{\\prime}$.", + "question": "Which fictitious force can yield a torque which corresponds to the ball rolling up the wall? \n\n(A) Centrifugal force. \n(B) Euler force. \n(C) Coriolis force.", + "marking": [], + "answer": [ + "\\boxed{C}" + ], + "answer_type": [ + "Multiple Choice" + ], + "unit": [ + null + ], + "points": [ + 1.5 + ], + "modality": "text+variable figure", + "field": "Mechanics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_5_1_1.png", + "image_question/PanPhO_2025_5_11_1.png" + ] + }, + { + "id": "PanPhO_2025_6_1", + "context": "[Generation of ultrashort electromagnetic pulse] \n\nThe Nobel Prize in Physics 2018 \\& 2023 were awarded to pioneers who contributed to \"Method of generating high-intensity, ultra-short optical pulses\" and \"Generation of attosecond pulses of light for the study of electron dynamics in matter\". Attosecond pulse refers to electromagnetic field with a duration on the order of $10^{-18}$ second. The advent of attosecond technique has made possible the study of ultrafast dynamics in physical, chemical and biological systems at a record high temporal resolution. Thus far, the most widely used method to generate attosecond pulse (Nobel Prize in Physics 2023) is to rely on the interaction of gas molecules and intensive femtosecond laser pulse (Nobel Prize in Physics 2018). In this question, we will explore some important aspects of the short pulse generation.\n\nThe following identity may be useful:\n$\\int_{-\\infty}^{\\infty} e^{-a \\omega^{2}} e^{-i \\omega t} d \\omega = \\sqrt{\\frac{\\pi}{a}} \\exp \\left(-\\frac{t^{2}}{4 a}\\right)$ \n\nPhysical constants: \nElectric charge: $e = 1.60 \\times 10^{-19} \\mathrm{C}$ \nElectron mass: $m_{e} = 9.11 \\times 10^{-31} \\mathrm{kg}$ \nSpeed of light in vacuum: $c = 3.00 \\times 10^{8} \\mathrm{m} / \\mathrm{s}$ \nPlanck constant: $h = 6.63 \\times 10^{-34} \\mathrm{J} \\mathrm{s}$ \n\nPart A: Ultrashort laser pulse \n\nThe simplest form of pulsed electromagnetic field is a sinusoidal wave (center frequency $\\omega_{0}$) dressed in a Gaussian profile with peak $E_{0}$ and a standard deviation of $\\tau_{p} / \\sqrt{2}$, as shown in the figure.\n\n[figure1]", + "question": "(1) Please find the complex expression $a(t)$ of a Gaussian signal in the time domain. \n(2) Calculate the value of the average electric field of a Gaussian laser pulse over time.", + "marking": [], + "answer": [ + "\\boxed{$a(t) = E_{0} e^{-\\frac{t^{2}}{\\tau_{p}^{2}}} e^{i(\\omega_{0} t + \\varphi_{0})}$}", + "\\boxed{0}" + ], + "answer_type": [ + "Expression", + "Numerical Value" + ], + "unit": [ + null + ], + "points": [ + 1.5, + 1.5 + ], + "modality": "text+illustration figure", + "field": "Optics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_6_1_1.png" + ] + }, + { + "id": "PanPhO_2025_6_2", + "context": "[Generation of ultrashort electromagnetic pulse] \n\nThe Nobel Prize in Physics 2018 \\& 2023 were awarded to pioneers who contributed to \"Method of generating high-intensity, ultra-short optical pulses\" and \"Generation of attosecond pulses of light for the study of electron dynamics in matter\". Attosecond pulse refers to electromagnetic field with a duration on the order of $10^{-18}$ second. The advent of attosecond technique has made possible the study of ultrafast dynamics in physical, chemical and biological systems at a record high temporal resolution. Thus far, the most widely used method to generate attosecond pulse (Nobel Prize in Physics 2023) is to rely on the interaction of gas molecules and intensive femtosecond laser pulse (Nobel Prize in Physics 2018). In this question, we will explore some important aspects of the short pulse generation.\n\nThe following identity may be useful:\n$\\int_{-\\infty}^{\\infty} e^{-a \\omega^{2}} e^{-i \\omega t} d \\omega = \\sqrt{\\frac{\\pi}{a}} \\exp \\left(-\\frac{t^{2}}{4 a}\\right)$ \n\nPhysical constants: \nElectric charge: $e = 1.60 \\times 10^{-19} \\mathrm{C}$ \nElectron mass: $m_{e} = 9.11 \\times 10^{-31} \\mathrm{kg}$ \nSpeed of light in vacuum: $c = 3.00 \\times 10^{8} \\mathrm{m} / \\mathrm{s}$ \nPlanck constant: $h = 6.63 \\times 10^{-34} \\mathrm{J} \\mathrm{s}$ \n\nPart A: Ultrashort laser pulse \n\nThe simplest form of pulsed electromagnetic field is a sinusoidal wave (center frequency $\\omega_{0}$) dressed in a Gaussian profile with peak $E_{0}$ and a standard deviation of $\\tau_{p} / \\sqrt{2}$, as shown in the figure.\n\n[figure1]", + "question": "The most commonly used laser for generation femtosecond pulses employs titanium-doped sapphire (Ti:sapphire) crystal as its gain medium. It emits light with photon energy around 1.55 eV. Please find the expression $a(t)$ of Gaussian pulse (Full-width at half maximum 35 fs in duration) out of such laser cavity.", + "marking": [], + "answer": [ + "\\boxed{$a(t) = E_{0} e^{-\\frac{t^{2}}{(4.2 \\times 10^{-12} s)^{2}}} \\cos \\left(2 \\pi \\times 375 \\times 10^{12} \\mathrm{Hz} \\times t + \\varphi_{0}\\right)$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text+illustration figure", + "field": "Optics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_6_1_1.png" + ] + }, + { + "id": "PanPhO_2025_6_3", + "context": "[Generation of ultrashort electromagnetic pulse] \n\nThe Nobel Prize in Physics 2018 \\& 2023 were awarded to pioneers who contributed to \"Method of generating high-intensity, ultra-short optical pulses\" and \"Generation of attosecond pulses of light for the study of electron dynamics in matter\". Attosecond pulse refers to electromagnetic field with a duration on the order of $10^{-18}$ second. The advent of attosecond technique has made possible the study of ultrafast dynamics in physical, chemical and biological systems at a record high temporal resolution. Thus far, the most widely used method to generate attosecond pulse (Nobel Prize in Physics 2023) is to rely on the interaction of gas molecules and intensive femtosecond laser pulse (Nobel Prize in Physics 2018). In this question, we will explore some important aspects of the short pulse generation.\n\nThe following identity may be useful:\n$\\int_{-\\infty}^{\\infty} e^{-a \\omega^{2}} e^{-i \\omega t} d \\omega = \\sqrt{\\frac{\\pi}{a}} \\exp \\left(-\\frac{t^{2}}{4 a}\\right)$ \n\nPhysical constants: \nElectric charge: $e = 1.60 \\times 10^{-19} \\mathrm{C}$ \nElectron mass: $m_{e} = 9.11 \\times 10^{-31} \\mathrm{kg}$ \nSpeed of light in vacuum: $c = 3.00 \\times 10^{8} \\mathrm{m} / \\mathrm{s}$ \nPlanck constant: $h = 6.63 \\times 10^{-34} \\mathrm{J} \\mathrm{s}$ \n\nPart A: Ultrashort laser pulse \n\nThe simplest form of pulsed electromagnetic field is a sinusoidal wave (center frequency $\\omega_{0}$) dressed in a Gaussian profile with peak $E_{0}$ and a standard deviation of $\\tau_{p} / \\sqrt{2}$, as shown in the figure.\n\n[figure1]\n\nA Gaussian pulse that is only dependent on time cannot propagate in real space. Consider a Gaussian pulse propagating in one dimensional vacuum along the $z$-axis.", + "question": "Please find the expression the electric field $E(z, t)$ of a Gaussian pulse and show that your answer can indeed propagate. Hint: you may use the complex form to describe the propagating wave.", + "marking": [], + "answer": [ + "\\boxed{$E(z, t) = E_{0} \\int_{-\\infty}^{\\infty} \\exp \\left(-\\frac{\\tau_{p}^{2}(\\omega - \\omega_{0})^{2}}{4}\\right) e^{i(\\omega t - \\frac{\\omega}{c} z)} d \\omega$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_6_1_1.png" + ] + }, + { + "id": "PanPhO_2025_6_4", + "context": "[Generation of ultrashort electromagnetic pulse] \n\nThe Nobel Prize in Physics 2018 \\& 2023 were awarded to pioneers who contributed to \"Method of generating high-intensity, ultra-short optical pulses\" and \"Generation of attosecond pulses of light for the study of electron dynamics in matter\". Attosecond pulse refers to electromagnetic field with a duration on the order of $10^{-18}$ second. The advent of attosecond technique has made possible the study of ultrafast dynamics in physical, chemical and biological systems at a record high temporal resolution. Thus far, the most widely used method to generate attosecond pulse (Nobel Prize in Physics 2023) is to rely on the interaction of gas molecules and intensive femtosecond laser pulse (Nobel Prize in Physics 2018). In this question, we will explore some important aspects of the short pulse generation.\n\nThe following identity may be useful:\n$\\int_{-\\infty}^{\\infty} e^{-a \\omega^{2}} e^{-i \\omega t} d \\omega = \\sqrt{\\frac{\\pi}{a}} \\exp \\left(-\\frac{t^{2}}{4 a}\\right)$ \n\nPhysical constants: \nElectric charge: $e = 1.60 \\times 10^{-19} \\mathrm{C}$ \nElectron mass: $m_{e} = 9.11 \\times 10^{-31} \\mathrm{kg}$ \nSpeed of light in vacuum: $c = 3.00 \\times 10^{8} \\mathrm{m} / \\mathrm{s}$ \nPlanck constant: $h = 6.63 \\times 10^{-34} \\mathrm{J} \\mathrm{s}$ \n\nPart A: Ultrashort laser pulse \n\nThe simplest form of pulsed electromagnetic field is a sinusoidal wave (center frequency $\\omega_{0}$) dressed in a Gaussian profile with peak $E_{0}$ and a standard deviation of $\\tau_{p} / \\sqrt{2}$, as shown in the figure.\n\n[figure1]\n\nA Gaussian pulse that is only dependent on time cannot propagate in real space. Consider a Gaussian pulse propagating in one dimensional vacuum along the $z$-axis.", + "question": "Please find the expression of the electric field $E(t)$ if it is a Gaussian pulse after propagating at a distance $L$ in medium of the frequency dependent refractive index $n(\\omega)$.", + "marking": [], + "answer": [ + "\\boxed{$E(t) = E_{0} \\int_{-\\infty}^{\\infty} \\exp \\left(-\\frac{\\tau_{p}^{2}(\\omega-\\omega_{0})^{2}}{4}\\right) e^{i(\\omega t - n \\frac{\\omega}{c} L)} d \\omega$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 1.0 + ], + "modality": "text+illustration figure", + "field": "Optics", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_6_1_1.png" + ] + }, + { + "id": "PanPhO_2025_6_5", + "context": "[Generation of ultrashort electromagnetic pulse] \n\nThe Nobel Prize in Physics 2018 \\& 2023 were awarded to pioneers who contributed to \"Method of generating high-intensity, ultra-short optical pulses\" and \"Generation of attosecond pulses of light for the study of electron dynamics in matter\". Attosecond pulse refers to electromagnetic field with a duration on the order of $10^{-18}$ second. The advent of attosecond technique has made possible the study of ultrafast dynamics in physical, chemical and biological systems at a record high temporal resolution. Thus far, the most widely used method to generate attosecond pulse (Nobel Prize in Physics 2023) is to rely on the interaction of gas molecules and intensive femtosecond laser pulse (Nobel Prize in Physics 2018). In this question, we will explore some important aspects of the short pulse generation.\n\nThe following identity may be useful:\n$\\int_{-\\infty}^{\\infty} e^{-a \\omega^{2}} e^{-i \\omega t} d \\omega = \\sqrt{\\frac{\\pi}{a}} \\exp \\left(-\\frac{t^{2}}{4 a}\\right)$ \n\nPhysical constants: \nElectric charge: $e = 1.60 \\times 10^{-19} \\mathrm{C}$ \nElectron mass: $m_{e} = 9.11 \\times 10^{-31} \\mathrm{kg}$ \nSpeed of light in vacuum: $c = 3.00 \\times 10^{8} \\mathrm{m} / \\mathrm{s}$ \nPlanck constant: $h = 6.63 \\times 10^{-34} \\mathrm{J} \\mathrm{s}$ \n\nPart B: Dispersion \n\nBefore entering the attosecond $(10^{-18} \\mathrm{s})$ regime, it historically took numerous effort of researchers to just generate femtosecond pulses ($1 \\mathrm{fs} = 10^{-15} \\mathrm{s}$), which now can be readily obtained from a standard Ti:sapphire laser and serve as the starting point to generate the even shorter attosecond pulse. One challenge at the time was to devise a laser cavity that can fight against the strong dispersion arise from traversing the Ti:sapphire crystal, an indispensable element in which amplification takes place. The dispersion here means the frequency dependent refractive index in the Ti:sapphire crystal, which has a strong absorption at 2.5 eV.\n\n[figure1]", + "question": "Assuming the transition responsible for the absorption can be described with a classical model for an oscillating bound electron (mass $m$) oscillating at a characteristic frequency $(\\Omega_{0})$ about a nucleus. In the presence of an external AC E-field of amplitude $E_{0}$ oscillating at single frequency $\\omega$, please find the largest possible displacement $X_{0}$ of the electron. The damping force of the oscillator can be described by $f_{d} = -m \\gamma v$ where $v$ is the velocity of the oscillator and $\\gamma$ is a single parameter to describe the total effect from energy loss of all kinds.", + "marking": [], + "answer": [ + "\\boxed{$X_{0} = \\frac{1}{m(\\omega^{2} - \\Omega_{0}^{2}) - i \\omega m \\gamma} e E_{0}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 4.0 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_6_5_1.png" + ] + }, + { + "id": "PanPhO_2025_6_6", + "context": "[Generation of ultrashort electromagnetic pulse] \n\nThe Nobel Prize in Physics 2018 \\& 2023 were awarded to pioneers who contributed to \"Method of generating high-intensity, ultra-short optical pulses\" and \"Generation of attosecond pulses of light for the study of electron dynamics in matter\". Attosecond pulse refers to electromagnetic field with a duration on the order of $10^{-18}$ second. The advent of attosecond technique has made possible the study of ultrafast dynamics in physical, chemical and biological systems at a record high temporal resolution. Thus far, the most widely used method to generate attosecond pulse (Nobel Prize in Physics 2023) is to rely on the interaction of gas molecules and intensive femtosecond laser pulse (Nobel Prize in Physics 2018). In this question, we will explore some important aspects of the short pulse generation.\n\nThe following identity may be useful:\n$\\int_{-\\infty}^{\\infty} e^{-a \\omega^{2}} e^{-i \\omega t} d \\omega = \\sqrt{\\frac{\\pi}{a}} \\exp \\left(-\\frac{t^{2}}{4 a}\\right)$ \n\nPhysical constants: \nElectric charge: $e = 1.60 \\times 10^{-19} \\mathrm{C}$ \nElectron mass: $m_{e} = 9.11 \\times 10^{-31} \\mathrm{kg}$ \nSpeed of light in vacuum: $c = 3.00 \\times 10^{8} \\mathrm{m} / \\mathrm{s}$ \nPlanck constant: $h = 6.63 \\times 10^{-34} \\mathrm{J} \\mathrm{s}$ \n\nPart B: Dispersion \n\nBefore entering the attosecond $(10^{-18} \\mathrm{s})$ regime, it historically took numerous effort of researchers to just generate femtosecond pulses ($1 \\mathrm{fs} = 10^{-15} \\mathrm{s}$), which now can be readily obtained from a standard Ti:sapphire laser and serve as the starting point to generate the even shorter attosecond pulse. One challenge at the time was to devise a laser cavity that can fight against the strong dispersion arise from traversing the Ti:sapphire crystal, an indispensable element in which amplification takes place. The dispersion here means the frequency dependent refractive index in the Ti:sapphire crystal, which has a strong absorption at 2.5 eV.\n\n[figure1]", + "question": "Suppose the particle density of absorptive $\\mathrm{Ti}^{3+}$ centers is $N$ and $P = \\varepsilon_{0} \\left(\\varepsilon_{r} - 1\\right) E$. \n\n(1) Find the polarization density $\\vec{P}$. \n(2) Find the corresponding dielectric function $\\varepsilon(\\omega)$.", + "marking": [], + "answer": [ + "\\boxed{$\\vec{P} = \\left(\\frac{N e^{2}}{m}\\right) \\frac{1}{(\\Omega_{0}^{2} - \\omega^{2}) + i \\gamma \\omega} \\vec{E}(t)$}", + "\\boxed{$\\varepsilon(\\omega) = 1 + \\left(\\frac{N e^{2}}{\\epsilon_{0} m}\\right) \\frac{1}{\\left(\\Omega_{0}^{2}-\\omega^{2}\\right) + i \\gamma \\omega}$}" + ], + "answer_type": [ + "Expression", + "Expression" + ], + "unit": [ + null, + null + ], + "points": [ + 2.0, + 2.0 + ], + "modality": "text+illustration figure", + "field": "Electromagnetism", + "source": "PanPhO_2025", + "image_question": [ + "image_question/PanPhO_2025_6_5_1.png" + ] + }, + { + "id": "PanPhO_2025_6_7", + "context": "[Generation of ultrashort electromagnetic pulse] \n\nThe Nobel Prize in Physics 2018 \\& 2023 were awarded to pioneers who contributed to \"Method of generating high-intensity, ultra-short optical pulses\" and \"Generation of attosecond pulses of light for the study of electron dynamics in matter\". Attosecond pulse refers to electromagnetic field with a duration on the order of $10^{-18}$ second. The advent of attosecond technique has made possible the study of ultrafast dynamics in physical, chemical and biological systems at a record high temporal resolution. Thus far, the most widely used method to generate attosecond pulse (Nobel Prize in Physics 2023) is to rely on the interaction of gas molecules and intensive femtosecond laser pulse (Nobel Prize in Physics 2018). In this question, we will explore some important aspects of the short pulse generation.\n\nThe following identity may be useful:\n$\\int_{-\\infty}^{\\infty} e^{-a \\omega^{2}} e^{-i \\omega t} d \\omega = \\sqrt{\\frac{\\pi}{a}} \\exp \\left(-\\frac{t^{2}}{4 a}\\right)$ \n\nPhysical constants: \nElectric charge: $e = 1.60 \\times 10^{-19} \\mathrm{C}$ \nElectron mass: $m_{e} = 9.11 \\times 10^{-31} \\mathrm{kg}$ \nSpeed of light in vacuum: $c = 3.00 \\times 10^{8} \\mathrm{m} / \\mathrm{s}$ \nPlanck constant: $h = 6.63 \\times 10^{-34} \\mathrm{J} \\mathrm{s}$ \n\nPart C: Pulse broadening effect \n\nA Gaussian pulse, of which the transient frequency is constant in the time domain, is referred to as a Fourier-transform-limited pulse $(\\tau_{p})$ and has the shortest possible duration at a given bandwidth. After propagating through a dispersive medium of distance $L$, a transform-limited pulse will acquire a broadened pulse. It happens because the lasing frequency is close to the strong absorption line of the gain medium at 2.5 eV, and the high-order dispersion (frequency dependent part in refractive index $n(\\omega)$) will start to take effect. (Hint: The dissipative part during propagation can be ignored.) \n\nIn this part, it is useful to define the propagating function \n$\\beta(\\omega) \\equiv n(\\omega) \\frac{\\omega}{c} \\approx \\beta_{0} + \\beta_{1}(\\omega-\\omega_{0}) + \\beta_{2}(\\omega-\\omega_{0})^{2} + \\cdots$ \nwhich can be expanded around the center frequency $\\omega_{0}$ of the Gaussian pulse.", + "question": "Find the expression of the duration $\\tau$ of the transform-limited pulse after the propagation distance of $L$, in terms of $\\beta_2, \\tau_p$, and $L$, where $\\beta_{2}$ is the second-order propagation factor from expansion about the center frequency $\\omega_{0}$. (You may ignore effect from third-order and above)", + "marking": [], + "answer": [ + "\\boxed{$\\tau = \\tau_{p} \\sqrt{1 + \\frac{16 \\beta_{2}^{2} L^{2}}{\\tau_{p}^{4}}}$}" + ], + "answer_type": [ + "Expression" + ], + "unit": [ + null + ], + "points": [ + 4.0 + ], + "modality": "text-only", + "field": "Optics", + "source": "PanPhO_2025", + "image_question": [] + }, + { + "id": "PanPhO_2025_6_8", + "context": "[Generation of ultrashort electromagnetic pulse] \n\nThe Nobel Prize in Physics 2018 \\& 2023 were awarded to pioneers who contributed to \"Method of generating high-intensity, ultra-short optical pulses\" and \"Generation of attosecond pulses of light for the study of electron dynamics in matter\". Attosecond pulse refers to electromagnetic field with a duration on the order of $10^{-18}$ second. The advent of attosecond technique has made possible the study of ultrafast dynamics in physical, chemical and biological systems at a record high temporal resolution. Thus far, the most widely used method to generate attosecond pulse (Nobel Prize in Physics 2023) is to rely on the interaction of gas molecules and intensive femtosecond laser pulse (Nobel Prize in Physics 2018). In this question, we will explore some important aspects of the short pulse generation.\n\nThe following identity may be useful:\n$\\int_{-\\infty}^{\\infty} e^{-a \\omega^{2}} e^{-i \\omega t} d \\omega = \\sqrt{\\frac{\\pi}{a}} \\exp \\left(-\\frac{t^{2}}{4 a}\\right)$ \n\nPhysical constants: \nElectric charge: $e = 1.60 \\times 10^{-19} \\mathrm{C}$ \nElectron mass: $m_{e} = 9.11 \\times 10^{-31} \\mathrm{kg}$ \nSpeed of light in vacuum: $c = 3.00 \\times 10^{8} \\mathrm{m} / \\mathrm{s}$ \nPlanck constant: $h = 6.63 \\times 10^{-34} \\mathrm{J} \\mathrm{s}$ \n\nPart C: Pulse broadening effect \n\nA Gaussian pulse, of which the transient frequency is constant in the time domain, is referred to as a Fourier-transform-limited pulse $(\\tau_{p})$ and has the shortest possible duration at a given bandwidth. After propagating through a dispersive medium of distance $L$, a transform-limited pulse will acquire a broadened pulse. It happens because the lasing frequency is close to the strong absorption line of the gain medium at 2.5 eV, and the high-order dispersion (frequency dependent part in refractive index $n(\\omega)$) will start to take effect. (Hint: The dissipative part during propagation can be ignored.) \n\nIn this part, it is useful to define the propagating function \n$\\beta(\\omega) \\equiv n(\\omega) \\frac{\\omega}{c} \\approx \\beta_{0} + \\beta_{1}(\\omega-\\omega_{0}) + \\beta_{2}(\\omega-\\omega_{0})^{2} + \\cdots$ \nwhich can be expanded around the center frequency $\\omega_{0}$ of the Gaussian pulse.", + "question": "For an ultrafast femtosecond laser to oscillate in its stationary state, please come up with a design to compensate for such pulse broadening effect.", + "marking": [ + "Award 3.0 pts if the answer correctly identifies either a prism compressor (lower panel), a grating compressor (upper panel), or an equivalent scheme. Otherwise, award 0.0 pts." + ], + "answer": [ + "" + ], + "answer_type": [ + "Open-Ended" + ], + "unit": [ + null + ], + "points": [ + 3.0 + ], + "modality": "text-only", + "field": "Optics", + "source": "PanPhO_2025", + "image_question": [] + }, + { + "id": "PanPhO_2025_6_9", + "context": "[Generation of ultrashort electromagnetic pulse] \n\nThe Nobel Prize in Physics 2018 \\& 2023 were awarded to pioneers who contributed to \"Method of generating high-intensity, ultra-short optical pulses\" and \"Generation of attosecond pulses of light for the study of electron dynamics in matter\". Attosecond pulse refers to electromagnetic field with a duration on the order of $10^{-18}$ second. The advent of attosecond technique has made possible the study of ultrafast dynamics in physical, chemical and biological systems at a record high temporal resolution. Thus far, the most widely used method to generate attosecond pulse (Nobel Prize in Physics 2023) is to rely on the interaction of gas molecules and intensive femtosecond laser pulse (Nobel Prize in Physics 2018). In this question, we will explore some important aspects of the short pulse generation.\n\nThe following identity may be useful:\n$\\int_{-\\infty}^{\\infty} e^{-a \\omega^{2}} e^{-i \\omega t} d \\omega = \\sqrt{\\frac{\\pi}{a}} \\exp \\left(-\\frac{t^{2}}{4 a}\\right)$ \n\nPhysical constants: \nElectric charge: $e = 1.60 \\times 10^{-19} \\mathrm{C}$ \nElectron mass: $m_{e} = 9.11 \\times 10^{-31} \\mathrm{kg}$ \nSpeed of light in vacuum: $c = 3.00 \\times 10^{8} \\mathrm{m} / \\mathrm{s}$ \nPlanck constant: $h = 6.63 \\times 10^{-34} \\mathrm{J} \\mathrm{s}$\n\nPart D: High-harmonic generation of attosecond pulse \n\nTake a short femtosecond pulse from Ti:sapphire laser (center frequency $\\omega_{0}$) and focus it into a gas medium. One could generate light at integer multiples of the driving frequency, often referred to as high-harmonic generation (HHG).", + "question": "One critical step for high-harmonic generation (HHG) is to drive the bound charge of molecules in the non-perturbative regime to initiate the ionization dynamics. If we take hydrogen molecules for HHG, please estimate the peak electric field strength required to initiate the process (expressed in $E > ? \\frac{GV}{m}$).", + "marking": [], + "answer": [ + "\\boxed{$E > 30 \\frac{GV}{m}$}" + ], + "answer_type": [ + "Inequality" + ], + "unit": [ + "$\\frac{GV}{m}$" + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Modern Physics", + "source": "PanPhO_2025", + "image_question": [] + }, + { + "id": "PanPhO_2025_6_10", + "context": "[Generation of ultrashort electromagnetic pulse] \n\nThe Nobel Prize in Physics 2018 \\& 2023 were awarded to pioneers who contributed to \"Method of generating high-intensity, ultra-short optical pulses\" and \"Generation of attosecond pulses of light for the study of electron dynamics in matter\". Attosecond pulse refers to electromagnetic field with a duration on the order of $10^{-18}$ second. The advent of attosecond technique has made possible the study of ultrafast dynamics in physical, chemical and biological systems at a record high temporal resolution. Thus far, the most widely used method to generate attosecond pulse (Nobel Prize in Physics 2023) is to rely on the interaction of gas molecules and intensive femtosecond laser pulse (Nobel Prize in Physics 2018). In this question, we will explore some important aspects of the short pulse generation.\n\nThe following identity may be useful:\n$\\int_{-\\infty}^{\\infty} e^{-a \\omega^{2}} e^{-i \\omega t} d \\omega = \\sqrt{\\frac{\\pi}{a}} \\exp \\left(-\\frac{t^{2}}{4 a}\\right)$ \n\nPhysical constants: \nElectric charge: $e = 1.60 \\times 10^{-19} \\mathrm{C}$ \nElectron mass: $m_{e} = 9.11 \\times 10^{-31} \\mathrm{kg}$ \nSpeed of light in vacuum: $c = 3.00 \\times 10^{8} \\mathrm{m} / \\mathrm{s}$ \nPlanck constant: $h = 6.63 \\times 10^{-34} \\mathrm{J} \\mathrm{s}$ \n\nPart B: Dispersion \n\nBefore entering the attosecond $(10^{-18} \\mathrm{s})$ regime, it historically took numerous effort of researchers to just generate femtosecond pulses ($1 \\mathrm{fs} = 10^{-15} \\mathrm{s}$), which now can be readily obtained from a standard Ti:sapphire laser and serve as the starting point to generate the even shorter attosecond pulse. One challenge at the time was to devise a laser cavity that can fight against the strong dispersion arise from traversing the Ti:sapphire crystal, an indispensable element in which amplification takes place. The dispersion here means the frequency dependent refractive index in the Ti:sapphire crystal, which has a strong absorption at 2.5 eV.\n\n[figure1]\n\nB1: Assuming the transition responsible for the absorption can be described with a classical model for an oscillating bound electron (mass $m$) oscillating at a characteristic frequency $(\\Omega_{0})$ about a nucleus. In the presence of an external AC E-field of amplitude $E_{0}$ oscillating at single frequency $\\omega$, $X_{0}$ is the largest possible displacement of the electron. The damping force of the oscillator can be described by $f_{d} = -m \\gamma v$ where $v$ is the velocity of the oscillator and $\\gamma$ is a single parameter to describe the total effect from energy loss of all kinds. \n\nPart D: High-harmonic generation of attosecond pulse \n\nTake a short femtosecond pulse from Ti:sapphire laser (center frequency $\\omega_{0}$) and focus it into a gas medium. One could generate light at integer multiples of the driving frequency, often referred to as high-harmonic generation (HHG).", + "question": "To allow radiation at new frequencies, please modify the oscillator model correspondingly and show your modification is viable.", + "test": "In the previous analysis (see B1–B2), the motion of a bound electron under an external AC electric field was modeled as a damped driven linear harmonic oscillator. Specifically, for an electron of mass m, natural frequency Ω₀, and damping coefficient γ, subjected to an electric field E(t) = E₀ e^{-iωt}, the equation of motion was:\n\nm ẍ + mγ ẋ + mΩ₀² x = -e E₀ e^{-iωt}.\n\nThis model describes linear response near resonance and explains absorption and dispersion. To account for high-harmonic radiation, the oscillator model must be extended to include anharmonic terms in the potential, leading to nonlinear response.", + "marking": [ + "Award 1.0 pt if the answer writes down or correctly explains the modified equation of motion", + "Award 1.5 pts if the answer shows how the first-order solution leads to $x_1 \\propto E_0 e^{i \\omega_0 t}$", + "Award 1.5 pts if the answer explains that higher-order solutions generate new frequencies, e.g., $x_n \\propto E_0^n e^{i n \\omega_0 t}$." + ], + "answer": [ + "" + ], + "answer_type": [ + "Open-Ended" + ], + "unit": [ + null + ], + "points": [ + 4.0 + ], + "modality": "text-only", + "field": "Optics", + "source": "PanPhO_2025", + "image_question": [] + }, + { + "id": "PanPhO_2025_6_11", + "context": "[Generation of ultrashort electromagnetic pulse] \n\nThe Nobel Prize in Physics 2018 \\& 2023 were awarded to pioneers who contributed to \"Method of generating high-intensity, ultra-short optical pulses\" and \"Generation of attosecond pulses of light for the study of electron dynamics in matter\". Attosecond pulse refers to electromagnetic field with a duration on the order of $10^{-18}$ second. The advent of attosecond technique has made possible the study of ultrafast dynamics in physical, chemical and biological systems at a record high temporal resolution. Thus far, the most widely used method to generate attosecond pulse (Nobel Prize in Physics 2023) is to rely on the interaction of gas molecules and intensive femtosecond laser pulse (Nobel Prize in Physics 2018). In this question, we will explore some important aspects of the short pulse generation.\n\nThe following identity may be useful:\n$\\int_{-\\infty}^{\\infty} e^{-a \\omega^{2}} e^{-i \\omega t} d \\omega = \\sqrt{\\frac{\\pi}{a}} \\exp \\left(-\\frac{t^{2}}{4 a}\\right)$ \n\nPhysical constants: \nElectric charge: $e = 1.60 \\times 10^{-19} \\mathrm{C}$ \nElectron mass: $m_{e} = 9.11 \\times 10^{-31} \\mathrm{kg}$ \nSpeed of light in vacuum: $c = 3.00 \\times 10^{8} \\mathrm{m} / \\mathrm{s}$ \nPlanck constant: $h = 6.63 \\times 10^{-34} \\mathrm{J} \\mathrm{s}$\n\nPart D: High-harmonic generation of attosecond pulse \n\nTake a short femtosecond pulse from Ti:sapphire laser (center frequency $\\omega_{0}$) and focus it into a gas medium. One could generate light at integer multiples of the driving frequency, often referred to as high-harmonic generation (HHG).", + "question": "Show how one could leverage the high-harmonic field to generate attosecond pulse.", + "marking": [ + "Award 2.0 pts if the answer correctly shows that if oscillator is anharmonic, the n-th order dipole is proportional to $E_0^n e^{i n \\omega_0 t}$, thus indicating that the envelope of the high-harmonic field becomes narrower and can be used to generate attosecond pulses. Otherwise, award 0.0 pts." + ], + "answer": [ + "" + ], + "answer_type": [ + "Open-Ended" + ], + "unit": [ + null + ], + "points": [ + 2.0 + ], + "modality": "text-only", + "field": "Optics", + "source": "PanPhO_2025", + "image_question": [] + } +] \ No newline at end of file diff --git a/data/image_question/APhO_2025_1_a_1.png b/data/image_question/APhO_2025_1_a_1.png new file mode 100644 index 0000000000000000000000000000000000000000..6512437c5e0eef0d94a5e186b50b3dfde12e5d00 --- /dev/null +++ b/data/image_question/APhO_2025_1_a_1.png @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:532d2eaa6fd00da0bc63027c196fe9bca48c331f478c3ee52595deaf91177ff5 +size 104197 diff --git a/data/image_question/APhO_2025_1_b_1.png b/data/image_question/APhO_2025_1_b_1.png new file mode 100644 index 0000000000000000000000000000000000000000..1435cf6a7a9114fc3a9cde89905523ac5b1e174f --- /dev/null +++ b/data/image_question/APhO_2025_1_b_1.png @@ -0,0 +1,3 @@ +version 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