diff --git "a/data/IPhO_2025.json" "b/data/IPhO_2025.json" new file mode 100644--- /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