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README.md

@@ -5,7 +5,7 @@
 <p align="center">
 <a href="https://huggingface.co/datasets/SciYu/HiPhO">[📊 Dataset]</a>
 <a href="https://github.com/SciYu/HiPhO">[✨ GitHub]</a>
-<a href="https://arxiv.org/abs/2509.07894">[📄 Paper]</a>
+<a href="https://huggingface.co/papers/2509.07894">[📄 Paper]</a>
 </p>
 
 [![License: MIT](https://img.shields.io/badge/License-MIT-blue.svg)](https://opensource.org/license/mit)
@@ -17,7 +17,7 @@
 **HiPhO** (High School Physics Olympiad Benchmark) is the **first benchmark** specifically designed to evaluate the physical reasoning abilities of (M)LLMs on **real-world Physics Olympiads from 2024–2025**.
 
 <div align="center">
-  <img src="img/HiPhO_overview.png" alt="hipho overview five rings" width="600"/>
+  <img src="intro/HiPhO_overview.png" alt="hipho overview five rings" width="600"/>
 </div>
 
 ### ✨ Key Features
@@ -31,7 +31,7 @@
 ## 🏆 IPhO 2025 (Theory) Results
 
 <div align="center">
-  <img src="img/HiPhO_IPhO2025.png" alt="ipho2025 results" width="800"/>
+  <img src="intro/HiPhO_IPhO2025.png" alt="ipho2025 results" width="800"/>
 </div>
 
 - **Top-1 Human Score**: 29.2 / 30.0  
@@ -47,7 +47,7 @@
 ## 📊 Dataset Overview
 
 <div align="center">
-  <img src="img/HiPhO_statistics.png" alt="framework and stats" width="700"/>
+  <img src="intro/HiPhO_statistics.png" alt="framework and stats" width="700"/>
 </div>
 
 HiPhO contains:
@@ -68,7 +68,7 @@ Evaluation is conducted using:
 ## 🖼️ Modality Categorization
 
 <div align="center">
-  <img src="img/HiPhO_modality.png" alt="modality examples" width="700"/>
+  <img src="intro/HiPhO_modality.png" alt="modality examples" width="700"/>
 </div>
 
 - 📝 **Text-Only (TO)**: Pure text, no figures  
@@ -83,7 +83,7 @@ Evaluation is conducted using:
 ## 📈 Main Results
 
 <div align="center">
-  <img src="img/HiPhO_main_results.png" alt="main results medal table" width="700"/>
+  <img src="intro/HiPhO_main_results.png" alt="main results medal table" width="700"/>
 </div>
 
 - **Closed-source reasoning MLLMs** lead the benchmark, earning **6–12 gold medals** (Top 5: Gemini-2.5-Pro, Gemini-2.5-Flash, GPT-5, o3, Grok-4)

data/EuPhO_2024.json

@@ -0,0 +1,302 @@
+[
+    {
+        "information": "None."
+    },
+    {
+        "id": "EuPhO_2024_1_1",
+        "context": "As shown in the figure, a puck (a small disc) with radius $r$ and uniform density is moving on a horizontal plane with the velocity $v_{0}$ without rotation. The puck meets a fixed half-circular wall with a radius $R \\gg r$ and starts to move along the wall. The coefficient of friction with the wall is $\\mu$, and friction with the horizontal plane is negligible.",
+        "question": "Find the velocity of the puck $v_{e}$ when it leaves the wall. Consider different possible cases.",
+        "marking": [
+            [
+                "Award 0.3 pt if the answer realizes that puck is sliding initially. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer realizes that puck may roll without sliding. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer states that sliding ends when roll condition $v = {r\\omega }$ is met. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly equates the normal force with $m v^2 / R$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer uses $$F_f = \\mu N$$ for the friction force. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer gives the correct equation for translational motion: $m \\frac{d v}{d t} = -\\mu m \\frac{v^2}{R}$. Partial points: deduct 0.2 pt for wrong sign. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer gives the integral expression for translational motion with correct initial conditions: $\\int_{v_0}^v \\frac{d v}{v^2} = - \\frac{\\mu}{R} \\int_0^t dt$, where $$t=0$$ is the time at which the puck meets the semicircular wall and has the initial velocity $$v_0$$. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer gives the expression for $$v$$ as a function of time or angle as in $v(t) = \\frac{v_0}{1 + t/\\tau}$ or $v = v_0 \\exp(-\\mu \\varphi)$. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer gives the equation of motion for rotation: $$r \\frac{d \\omega}{d t} = \\frac{\\mu m r^2}{R I} v^2 = \\frac{2\\mu}{R} \\cdot \\frac{v_0^2}{(1 + t/\\tau)^2}$$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer uses $$I = \\frac{1}{2} m r^2$$ as the moment of inertia. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer gives the integral expression for rotational motion with correct initial conditions: $r \\int_0^{\\omega} d \\omega = \\frac{2 v_0}{\\tau} \\int_0^t \\frac{d t}{(1 + t/\\tau)^2}$. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer gives the expression for $$r \\omega$$ as a function of time or angle as in $r \\omega = v_0 \\frac{2t / \\tau}{1 + t/\\tau}$ or $r \\omega = 2v_0 (1 - \\exp(-\\mu \\varphi))$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly finds the time $$\\frac{R}{2 v_0 \\mu}$$ or angle $$\\frac{\\ln(3/2)}{\\mu}$$ for the transition to rolling without sliding. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer obtains the critical coefficient of friction: $$\\mu_c = \\frac{\\ln(3/2)}{\\pi}$$. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer finds the final velocity $$v_e = \\frac{2 v_0}{3}$$ for rolling without sliding. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer finds the velocity $$v_e = v_0 \\exp(-\\pi \\mu)$$ if the puck slides the whole time. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$v_e = \\frac{2 v_0}{3}$}",
+            "\\boxed{$v_e = v_0 \\exp(-\\pi \\mu)$}"
+        ],
+        "answer_type": [
+            "Expression",
+            "Expression"
+        ],
+        "unit": [
+            null,
+            null
+        ],
+        "points": [
+            4.0,
+            4.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "EuPhO_2024",
+        "image_question": [
+            "images_question/EuPhO_2024_1_1_1.png"
+        ]
+    },
+    {
+        "id": "EuPhO_2024_2_1",
+        "context": "Alice and Bob are twin astronauts on a long space mission. After many years, they are finally approaching each other to reunite. Alice's spaceship is moving towards Bob's spaceship at a speed of $u = \\frac{3}{5}c$ ,where $c$ is the speed of light.\n\nDuring their approach, both Alice and Bob send gifts to each other. Alice sends gifts to Bob at regular time intervals $\\Delta t_{0}$ in her own frame of reference, with each gift travelling at a velocity $v = \\frac{4}{5}c$ (again, in her frame of reference). Similarly, Bob sends gifts to Alice at the same regular time intervals $\\Delta t_{0}$ in his own frame of reference, with each gift also travelling at a velocity $v = \\frac{4}{5} c$ in his frame of reference. Assume that the distance $L$ between Alice and Bob is so large that there are many gifts in transit at any given moment.",
+        "question": "In Bob's reference frame: \n\n(1) Find the distance $l_B$ between two successive gifts sent by Alice. \n(2) Find the time interval $\\Delta t_{1}$ at which these gifts from Alice arrive at Bob's spaceship.",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer writes the correct formula for relativistic addition of velocities. Partial points: deduct 0.3 pt for one mistake. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer gives the relative velocity $v_B$ of frames B and G as $v_B = \\frac{35}{37} c$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer finds $l_{A} = \\frac{4}{5} c \\Delta t_0$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer gives correct $\\gamma$ formula $\\gamma_v = \\frac{1}{\\sqrt{1 - v^2/c^2}}$. Partial points: deduct 0.2 pt for one mistake. Otherwise, award 0 pt.",
+                "Award 0.7 pt if the answer states that $l_{1} = l_{2} / \\gamma$ is only true in rest frame. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer uses $l_{A} = l_{G} / \\gamma_v$, where $G$ is the rest frame of the gifts. Partial points: deduct 0.1 pt for each mistake. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer uses $l_{B} = l_{G} / \\gamma_{v_B}$, where $G$ is the rest frame of the gifts. Partial points: deduct 0.1 pt for each mistake. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer gives the correct expression for $l_{B}$: $l_B = v \\Delta t_0 \\frac{\\gamma_v}{\\gamma_{v_B}} = \\frac{16}{37} \\Delta t_0 c$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer gives the correct numerical result $16/37 = 0.\\overline{432}$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer uses $\\Delta t_{1} = l_{B} / v_{B}$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer gives the correct numerical result $16/35 \\approx 0.457$. Otherwise, award 0 pt."
+            ],
+            [
+                "Award 0.5 pt if the answer writes the correct formula for relativistic addition of velocities. Partial points: deduct 0.3 pt for one mistake. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer gives the relative velocity $v_B$ of frames B and G as $v_B = \\frac{35}{37} c$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer gives correct $\\gamma$ formula $\\gamma_v = \\frac{1}{\\sqrt{1 - v^2/c^2}}$. Partial points: deduct 0.2 pt for one mistake. Otherwise, award 0 pt.",
+                "Award 0.7 pt if the answer realizes that two subsequent gifts are sent from the same place in Alice's frame. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer finds $\\Delta {t}_{0,B} = \\gamma_{u} \\Delta t_{0}$. Partial points: deduct 0.1 pt for each mistake. Otherwise, award 0 pt.",
+                "Award 0.7 pt if the answer finds that in Bob's frame,second gift at position $u \\Delta t_{0,B}$ while first gift at $v_{B} \\Delta t_{0,B}$. Partial points: deduct 0.2 pt for each mistake. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer gives the correct expression for $l_{B}$: $l_B = (v_B - u) \\Delta t_0 \\gamma_u = \\frac{16}{37} \\Delta t_0 c$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer gives the correct numerical result $16/37 = 0.\\overline{432}$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer uses $\\Delta t_{1} = l_{B} / v_{B}$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer gives the correct numerical result $16/35 \\approx 0.457$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$l_B = \\frac{16}{37} c \\Delta t_{0}$}",
+            "\\boxed{$\\Delta t_{1} = \\frac{16}{35} \\Delta t_0$}"
+        ],
+        "answer_type": [
+            "Expression",
+            "Expression"
+        ],
+        "unit": [
+            null,
+            null
+        ],
+        "points": [
+            4.0,
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Modern Physics",
+        "source": "EuPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "EuPhO_2024_2_2",
+        "context": "Alice and Bob are twin astronauts on a long space mission. After many years, they are finally approaching each other to reunite. Alice's spaceship is moving towards Bob's spaceship at a speed of $u = \\frac{3}{5}c$ ,where $c$ is the speed of light.\n\nDuring their approach, both Alice and Bob send gifts to each other. Alice sends gifts to Bob at regular time intervals $\\Delta t_{0}$ in her own frame of reference, with each gift travelling at a velocity $v = \\frac{4}{5}c$ (again, in her frame of reference). Similarly, Bob sends gifts to Alice at the same regular time intervals $\\Delta t_{0}$ in his own frame of reference, with each gift also travelling at a velocity $v = \\frac{4}{5} c$ in his frame of reference. Assume that the distance $L$ between Alice and Bob is so large that there are many gifts in transit at any given moment.",
+        "question": "At a given instant, Alice can see a number of gifts moving away from her and a number of gifts moving towards her. What is the ratio between these two numbers?",
+        "marking": [
+            [
+                "Award 0.3 pt if the answer identifies the distance to Bob as $d_B$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer gives the light travel time to Bob as $t_l = d_B / c$, where $d_B$ is the distance to Bob. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer recognizes the need to correct for light travel time in Alice's frame. Otherwise, award 0 pt.",
+                "Award 0.9 pt if the answer gives $d_{AG} = d_B + t_l v$. Partial points: deduct 0.3 pt for each mistake. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer computes the number of gifts sent by Alice to Bob $N_{a \\rightarrow b} = d_{AG} / L_A$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer recognizes the need to correct for light travel time in the incoming direction. Otherwise, award 0 pt.",
+                "Award 0.9 pt if the answer gives $d_{BG} = d_B - t_l v_B$. Partial points: deduct 0.3 pt for each mistake. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer computes $N_{b \\rightarrow a} = d_{BG} / L_B$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer gives a symbolic expression for the ratio $N_{\\text{out}} / N_{\\text{in}}$ = $\\frac{(1+v/c) c}{(1-v_B/c)v} \\frac{16}{37}$ or $\\frac{(c+v) v_B \\Delta t_1}{v \\Delta t_0 (c-v)}$. Partial points: deduct 0.2 pt for each mistake. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the final numerical result is correct (ratio=18). Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{18}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            5.0
+        ],
+        "modality": "text-only",
+        "field": "Modern Physics",
+        "source": "EuPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "EuPhO_2024_3_1",
+        "context": "As shown in the figure, a Fabry-Pérot interferometer consists of two identical parallel planar mirrors separated by a distance $L$. The space between and outside the mirrors is filled with air. The mirrors are partially reflective; when light is aimed towards one of these mirrors along the normal direction, the reflected beam has intensity $R < 1$ times the intensity of the incident beam. Assume that the mirrors are symmetric, meaning they interact the same way with light incident from either side, and lossless. Assume also that they are highly reflective, meaning $1 - R \\ll 1$. A monochromatic laser beam of power $P$ is aimed towards the interferometer perpendicular to the mirrors. The distance $L$ is chosen so that the back-reflected beam vanishes, i.e., all the optical power is transmitted through the interferometer. \n\n[figure1]",
+        "question": "Show that the laser beam must acquire a nonzero phase shift $\\phi$ when it passes through either of the mirrors.",
+        "marking": [
+            [
+                "Award 0.3 pt if the answer shows understanding that some light is initially reflected without entering the interferometer. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer shows understanding that light bounces back and forth between the mirrors. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer uses one or two travelling waves in each region. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer writes equations relating amplitudes via $r$ and $t$, such as $B = tA + rC$ and $0 = rA + tC$. Otherwise, award 0 pt.",
+                "Award 0.6 pt if the answer solves the system to obtain the condition $e^{-2i k L} = r^2 - t^2$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer uses the relation $|r|^2 + |t|^2 = 1$ or $R + T = 1$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer states that $|r|^2 + |t|^2 = 1$ is a consequence of conservation of energy. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer indicates that the solutions $r$ and $t$ should be complex numbers. Otherwise, award 0 pt."
+            ],
+            [
+                "Award 0.3 pt if the answer shows understanding that some light is initially reflected without entering the interferometer. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer shows understanding that light bounces back and forth between the mirrors. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer refers to the idea of superposition of complex amplitudes. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly includes the effects on amplitudes from reflection, transmission, and propagation. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer sums up the complex amplitudes as a geometric series. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer derives the equation: $e^{-2i k L} = r^2 - t^2$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer uses the condition $|r|^2 + |t|^2 = 1$ or $R + T = 1$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer states that $|r|^2 + |t|^2 = 1$ is a consequence of conservation of energy. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer shows understanding that the solutions $r$ and $t$ should be complex. Otherwise, award 0 pt."
+            ],
+            [
+                "Award 0.3 pt if the answer shows understanding that some light is initially reflected without entering the interferometer. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer shows understanding that light bounces back and forth between the mirrors. Otherwise, award 0 pt.",
+                "Award 0.7 pt if the answer uses the relation $1 + r = t$. Otherwise, award 0 pt.",
+                "Award 0.8 pt if the answer justifies the relation $1 + r = t$ using continuity of the electric field or thin-mirror arguments. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer uses the condition $|r|^2 + |t|^2 = 1$ or $R + T = 1$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer states that $|r|^2 + |t|^2 = 1$ is a consequence of conservation of energy. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer shows understanding that the solutions $r$ and $t$ should be complex. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            ""
+        ],
+        "answer_type": [
+            "Open-Ended"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Optics",
+        "source": "EuPhO_2024",
+        "image_question": [
+            "images_question/EuPhO_2024_3_1_1.png"
+        ]
+    },
+    {
+        "id": "EuPhO_2024_3_2",
+        "context": "As shown in the figure, a Fabry-Pérot interferometer consists of two identical parallel planar mirrors separated by a distance $L$. The space between and outside the mirrors is filled with air. The mirrors are partially reflective; when light is aimed towards one of these mirrors along the normal direction, the reflected beam has intensity $R < 1$ times the intensity of the incident beam. Assume that the mirrors are symmetric, meaning they interact the same way with light incident from either side, and lossless. Assume also that they are highly reflective, meaning $1 - R \\ll 1$. A monochromatic laser beam of power $P$ is aimed towards the interferometer perpendicular to the mirrors. The distance $L$ is chosen so that the back-reflected beam vanishes, i.e., all the optical power is transmitted through the interferometer. \n\n[figure1]",
+        "question": "The laser beam must acquire a nonzero phase shift $\\phi$ when it passes through either of the mirrors. Find the magnitude of $\\phi$ (expressed in $^{\\circ}$).",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer explicitly concludes that the phase difference is $90^{\\circ}$. Otherwise, award 0 pt.",
+                "Award 0.7 pt if the answer takes the modulus of $e^{-2ikL} = r^2 - t^2$ to get the conditions involving only $r$ and $t$. Otherwise, award 0 pt.",
+                "Award 0.8 pt if the answer correctly manipulates $r, t, r^*, t^*$ to show that $r$ is an imaginary number times $t$. Otherwise, award 0 pt."
+            ],
+            [
+                "Award 0.5 pt if the answer explicitly concludes that the phase difference is $90^{\\circ}$. Otherwise, award 0 pt.",
+                "Award 0.7 pt if the answer takes the modulus of $e^{-2ikL} = r^2 - t^2$ to get the conditions involving only $r$ and $t$. Otherwise, award 0 pt.",
+                "Award 0.8 pt if the answer uses a geometric argument to show that $r$ and $t$ must make a right angle. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$\\pm 90$}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "$^{\\circ}$"
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Optics",
+        "source": "EuPhO_2024",
+        "image_question": [
+            "images_question/EuPhO_2024_3_1_1.png"
+        ]
+    },
+    {
+        "id": "EuPhO_2024_3_3",
+        "context": "As shown in the figure, a Fabry-Pérot interferometer consists of two identical parallel planar mirrors separated by a distance $L$. The space between and outside the mirrors is filled with air. The mirrors are partially reflective; when light is aimed towards one of these mirrors along the normal direction, the reflected beam has intensity $R < 1$ times the intensity of the incident beam. Assume that the mirrors are symmetric, meaning they interact the same way with light incident from either side, and lossless. Assume also that they are highly reflective, meaning $1 - R \\ll 1$. A monochromatic laser beam of power $P$ is aimed towards the interferometer perpendicular to the mirrors. The distance $L$ is chosen so that the back-reflected beam vanishes, i.e., all the optical power is transmitted through the interferometer. \n\n[figure1]",
+        "question": "At a certain moment, the incident laser beam is switched off rapidly. Find the total energy $E$ of the light that travels back from the interferometer towards the laser after the laser is switched off.",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer states that since $|t| \\ll |r|$, the amplitudes $|B|$ and $|C|$ are very large and their difference is small, thus, $|B|$ and $|C|$ are approximately equal. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer applies symmetry to show that $2E = U$ ($U$ is the initial energy stored inside the interferometer) or that half the energy is emitted towards the laser. Otherwise, award 0 pt.",
+                "Award 1.5 pt if the answer finds the relation between $P^{\\prime}$ and $P$ (suppose the power contained in each wave (forwards- and backwards-propagating) in region II is $P^{\\prime}$), i.e., $P^{\\prime} \\approx P / (1 - R)$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer gives the correct value for the initial energy stored inside the interferometer: $U \\approx \\frac{2}{1 - R} \\cdot \\frac{L P}{c}$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer gives the correct value for the total energy: $E \\approx \\frac{1}{1 - R} \\cdot \\frac{L P}{c}$. Otherwise, award 0 pt."
+            ],
+            [
+                "Award 1.5 pt if the answer shows, even by reasoning, that the power propagating out through the first mirror decreases by a factor $R^2$ every $\\Delta t$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer multiplies by $\\Delta t$ to convert power or intensity to energy. Otherwise, award 0 pt.",
+                "Award 1.5 pt if the answer sums a geometric series to find the total energy $E$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer gives the correct value for $E$, i.e., $E \\approx \\frac{1}{1 - R} \\cdot \\frac{L P}{c}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$E \\approx \\frac{1}{1 - R} \\frac{LP}{c}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            4.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Optics",
+        "source": "EuPhO_2024",
+        "image_question": [
+            "images_question/EuPhO_2024_3_1_1.png"
+        ]
+    },
+    {
+        "id": "EuPhO_2024_3_4",
+        "context": "As shown in the figure, a Fabry-Pérot interferometer consists of two identical parallel planar mirrors separated by a distance $L$. The space between and outside the mirrors is filled with air. The mirrors are partially reflective; when light is aimed towards one of these mirrors along the normal direction, the reflected beam has intensity $R < 1$ times the intensity of the incident beam. Assume that the mirrors are symmetric, meaning they interact the same way with light incident from either side, and lossless. Assume also that they are highly reflective, meaning $1 - R \\ll 1$. A monochromatic laser beam of power $P$ is aimed towards the interferometer perpendicular to the mirrors. The distance $L$ is chosen so that the back-reflected beam vanishes, i.e., all the optical power is transmitted through the interferometer. \n\n[figure1]",
+        "question": "Estimate the duration $T$ of the light pulse that travels back towards the laser.",
+        "marking": [
+            [
+                "Award 0.2 pt if the answer states that energy reduces by a factor $R^2$ each time a wave is removed. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer uses the fact that the reduction of energy occurs at intervals $\\Delta t$. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer provides a valid mathematical argument that leads to the correct result: $T \\approx \\frac{1}{1 - R} \\cdot \\frac{L}{c}$ when $1 - R \\ll 1$. Otherwise, award 0 pt."
+            ],
+            [
+                "Award 0.2 pt if the answer states that the decay of stored energy is roughly exponential like $e^{-2 \\log(1/R) t / (\\Delta t)}$. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer states the outwards energy flux. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer uses the energy decay equation to determine the decay constant and finds $T \\approx \\frac{1}{1 - R} \\cdot \\frac{L}{c}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$T \\approx \\frac{1}{1 - R} \\cdot \\frac{L}{c}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Optics",
+        "source": "EuPhO_2024",
+        "image_question": [
+            "images_question/EuPhO_2024_3_1_1.png"
+        ]
+    }
+]

data/EuPhO_2025.json

@@ -0,0 +1,208 @@
+[
+    {
+        "information": "None."
+    },
+    {
+        "id": "EuPhO_2025_1_1",
+        "context": "You are asked to study the features of the brightly lit circle and dark rings in the figures below. Make your calculations for an idealized situation: the chair leg is strictly cylindrical of radius $a$, strictly vertical, with a perfectly smooth, cylindrical, and perfectly reflecting surface. You may make any additional model assumptions and approximations you deem reasonable that will simplify your calculations.",
+        "question": "Determine how the illuminance surplus $I(r, \\theta)$ inside the brightly lit circle on the floor depends on the polar coordinates $r \\gg a$ and $\\theta$. The illuminance quantifies the amount of incoming light per area. By \"surplus\" we mean the additional illuminance introduced due to the presence of the cylinder. Express the answer in terms of $I_0$ defined as the illuminance difference between points $A$ and $B$ in the figure.",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer uses correct angles, $2\\alpha + \\theta > \\pi$. Partial points: award 0.2 pt if the answer only states $2\\alpha + \\theta = \\pi$ without justification. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer states the reflection law in any form. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer includes the equation $x = a \\sin \\alpha$. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer derives $I_0 \\delta x = 2I l \\delta \\alpha$. Partial points: subtract 0.3 pt if the factor of 2 is missing; subtract 0.3 pt if $r$ is used instead of $l$. If the answer does not derive such formula, award 0 pt.",
+                "Award 0.5 pt if the answer derives $I = -\\frac{I_0 a}{2l} \\cos(\\beta)$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer justifies the approximation $l \\approx r$. Partial points: award 0.2 pt if the answer states the approximation without justification. Otherwise, award 0 pt.",
+                "Award 1.5 pt if the answer correctly obtains the results $I \\approx \\frac{I_{0}a}{2r} \\sin (\\theta /2)$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$\\frac{I_{0}a}{2r} \\sin (\\theta /2)$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            5.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Optics",
+        "source": "EuPhO_2025",
+        "image_question": [
+            "image_question/EuPhO_2025_1_1_1.png"
+        ]
+    },
+    {
+        "id": "EuPhO_2025_1_2",
+        "context": "You are asked to study the features of the brightly lit circle and dark rings in the figures below. Make your calculations for an idealized situation: the chair leg is strictly cylindrical of radius $a$, strictly vertical, with a perfectly smooth, cylindrical, and perfectly reflecting surface. You may make any additional model assumptions and approximations you deem reasonable that will simplify your calculations.",
+        "question": "In the following figure, some fingers are blocking some of the light from reaching the chair leg. Let $R(\\theta)$ denote the radial distance of the middle dark ring as a function of the angle $\\theta$ and let $R_{\\min}$ be the minimal value of $R(\\theta)$. Determine $R(\\theta) - R_{\\min}$.",
+        "marking": [
+            [
+                "Award 1.0 pt if the answer includes a correct Cosine Law expression. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer correctly obtains $l = l_0 + a \\cos(\\alpha) = l_0 + a |\\cos(\\beta)|$ where $l_0$ is the horizontal distance when $\\alpha = \\pi/2$. Partial points: award 0.5 pt if the answer only provides a qualitative explanation of why $R(\\theta)$ varies with $\\theta$. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer justifies $l_0 \\approx R_{\\min}$ to leading order. Partial points: award 0.5 pt if the answer only states $l_0 \\approx R_{\\min}$ without justification. Otherwise, award 0 pt.",
+                "Award 2.0 pt if the answer correcly derives $R - R_{\\min} \\approx 2a \\sin(\\theta/2)$. Partial points: award 1.0 pt if the answer gives $R - R_{\\min} = a \\sin(\\theta/2)$; award 0.5 pt if the answer only states $R_{\\max} - R_{\\min} = 2a$; award 0.0 pt if the answer only states $R_{\\max} - R_{\\min} = a$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$R(\\theta) - R_{\\min} \\approx 2a \\sin(\\theta/2)$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            5.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Optics",
+        "source": "EuPhO_2025",
+        "image_question": [
+            "image_question/EuPhO_2025_1_2_1.png"
+        ]
+    },
+    {
+        "id": "EuPhO_2025_2_1",
+        "context": "A table is made by fastening a metal frame to a massive uniform plate (so they form a rigid body) and attaching it with chains to another frame that is fixed on the horizontal ground. The motion of the table is limited to the plane of the side view (right picture).\n\nThe masses of the chains and the frame can be neglected. The chains are frictionless, inextensible, and remain tensioned in oscillations. The grid step is $a = 0.100 m$, the acceleration of gravity $g = 9.81 m/s^2$.",
+        "question": "Show that in the configuration on the side view (right picture), the table is in a stable equilibrium.",
+        "marking": [
+            [
+                "Award 2.0 pt if the answer shows that the table is in equilibrium. Partial points: award 1.0 pt if the answer only gives a sketch of forces or an equation for forces. Otherwise, award 0 pt.",
+                "Award 2.0 pt if the answer shows that the equilibrium is stable. Partial points: award 1.0 pt if the answer gives a sketch with returning forces, but there is no proof that $\\omega_{0} = 0$; award 1.0 pt if the answer gives a statement that the stability comes from $y = k x^2$ if $k > 0$, but $k$ is not found correctly ($k = \\frac{1}{3a}$), which could potentially be negative. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            ""
+        ],
+        "answer_type": [
+            "Open-Ended"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            4.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "EuPhO_2025",
+        "image_question": [
+            "image_question/EuPhO_2025_2_1_1.png"
+        ]
+    },
+    {
+        "id": "EuPhO_2025_2_2",
+        "context": "A table is made by fastening a metal frame to a massive uniform plate (so they form a rigid body) and attaching it with chains to another frame that is fixed on the horizontal ground. The motion of the table is limited to the plane of the side view (right picture).\n\nThe masses of the chains and the frame can be neglected. The chains are frictionless, inextensible, and remain tensioned in oscillations. The grid step is $a = 0.100 m$, the acceleration of gravity $g = 9.81 m/s^2$.",
+        "question": "Find the period $T$ of the small oscillations: (1) write the formula for $T$, (2) calculate the number of $T$ (keep three significant figures, and express the unit in $s$).",
+        "marking": [
+            [
+                "Award 1.0 pt if the answer shows that the table can rotate, either explicitly in the sketch or by introducing $\\varphi$ in the equations. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer correctly argues that the table does not have immediate rotation, either by: geometric reasoning showing $\\varphi \\sim x^{2}$; or constraint equations leading to $\\varphi \\sim x^{2}$; or using immediate velocities to show no rotation. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer outlines a valid plan to find the oscillation period using any of the following ideas: second Newton's law: $\\ddot{x} \\sim -x$; or identifying kinetic and potential energies; or using curvature of the trajectory. Otherwise, award 0 pt.",
+                "Award 2.0 pt if the answer includes all necessary elements: small horizontal forces; correct approximations of kinetic and potential energies; accelerations/curvatures. Partial points: award 1.0 pt if not all elements are present or if the answer contains mistakes; award 1.0 pt if the answer misses the proof of $\\omega_0 = 0$ (or does not consider the rotation), but everything else is correct; award 0.0 pt if only partial elements are present with an unrelated approach (e.g., using energy but only writing force expressions). Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer provides both the correct formula ($T = 2 \\pi \\sqrt{\\frac{3a}{2g}}$) and numerical value ($T = 0.777 s$) for $T$. Partial points: award 0.5 pt the answer only gives the formula or only the number; award 0.5 pt if a simple mistake is made in the answer (like inverse formula under the root). Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$T = 2 \\pi \\sqrt{\\frac{3a}{2g}}$}",
+            "\\boxed{0.777}"
+        ],
+        "answer_type": [
+            "Expression",
+            "Numerical Value"
+        ],
+        "unit": [
+            null,
+            "s"
+        ],
+        "points": [
+            3.0,
+            3.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "EuPhO_2025",
+        "image_question": [
+            "image_question/EuPhO_2025_2_1_1.png"
+        ]
+    },
+    {
+        "id": "EuPhO_2025_3_2",
+        "context": "",
+        "question": "Now consider two infinite, straight, thin wires (wires $X$ and $Y$), each carrying a current $I$ as shown in the figure. The $x$-axis coincides with wire $X$, while wire $Y$ is parallel to the $y$-axis and passes through the point $(0, 0, -a)$. Let $P$ be the point $(3a, 0, r)$. Assuming $r \\ll a$, calculate $d$, the distance of closest approach of the magnetic field line that passes through $P$ to the wire $X$.",
+        "marking": [
+            [
+                "Award 0.4 pt if the answer correctly states that the magnetic field around an infinite, straight, thin wire carrying a current $I$ has magnitude $\\frac{\\mu_0 I}{2 \\pi \\rho}$, where $\\rho$ is the perpendicular distance to the wire. Partial points: award 0.2 pt if the direction is unclear. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer states that the magnetic field line is locally nearly circular. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer describes that the field line resembles helix tightly wound around wire. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer mentions that the radius of the helix is changing. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer introduces the idea of considering the funnel surface $S$. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer relizes and justifies that the $\\vec{B}_Y$ flux is conserved along the funnel. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly argues or shows that the radius of the flux tube is smallest at $x = 0$. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer approximates $\\vec{B}_Y$ as uniform across the flux tube cross-sections. Otherwise, award 0 pt.",
+                "Award 0.5 pt for correctly determining flux at $x = 3a$. Partial points: award 0.3 pt for correct projection; award 0.2 pt for correct area. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer determines the flux at $x = 0$ using $\\rho$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly calculates the final result for $d$: $d = r/\\sqrt{10}$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer checks the validity of approximations used in the considered region. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$d = r/\\sqrt{10}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            5.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Electromagnetism",
+        "source": "EuPhO_2025",
+        "image_question": [
+            "image_question/EuPhO_2025_3_2_1.png"
+        ]
+    },
+    {
+        "id": "EuPhO_2025_3_3",
+        "context": "Now consider two infinite, straight, thin wires (wires $X$ and $Y$), each carrying a current $I$ as shown in the figure. The $x$-axis coincides with wire $X$, while wire $Y$ is parallel to the $y$-axis and passes through the point $(0, 0, -a)$. Let $P$ be the point $(3a, 0, r)$. Assuming $r \\ll a$, calculate $d$, the distance of closest approach of the magnetic field line that passes through $P$ to the wire $X$.",
+        "question": "Let $L$ be the length of this field line between $P$ and its point of closest approach to wire $X$. Using values $a = 10 cm$ and $r = 1.0 mm$, calculate $L$ to within 20% relative error (express the unit in $m$).",
+        "marking": [
+            [
+                "Award 0.2 pt if the answer states or implicitly assumes that $L$ is much larger than $a$ and $r$. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer correctly equates $\\mathrm{d}x$ with $\\mathrm{d}L$ using $B$-field components or an angle. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer identifies that $B_{\\prep}$ is dominated by wire $X$. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer derives a correct integral expression for $L$: $L = \\int_Q^P \\mathrm{d}L = \\int_0^{3a} \\frac{a^2+x^2}{a \\rho} \\mathrm{d}x$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer provides an expression for $\\rho$ as a function of $a$, $r$, and $x$: $\\rho^2 = \\frac{r^2 (a^2 + x^2)}{10 a^2}$. Otherwise, award 0 pt.",
+                "Award 1.2 pt if the answer carries out reasonable numerical approximation ($L = \\int_0^{3a} \\frac{\\sqrt{10}}{r} \\sqrt{a^2 + x^2} \\mathrm{d}x = \\frac{\\sqrt{10} a^2}{r} \\int_0^3 \\sqrt{1+u^2} \\mathrm{d}u$) or rigorous calculation of integral ($\\int_0^3 \\sqrt{1+u^2} \\mathrm{d}u \\approx 6.24$). Otherwise, award 0 pt.",
+                "Award 1.0 pt if the final result of $L$ within the range of $140 m \\leq L \\leq 215 m$. Partial points: award 0.8 pt if the answer is within the correct range but has only 1 significant figure or more than 3. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{[140, 215]}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "m"
+        ],
+        "points": [
+            4.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Electromagnetism",
+        "source": "EuPhO_2025",
+        "image_question": [
+            "image_question/EuPhO_2025_3_2_1.png"
+        ]
+    }
+]

data/F=MA_2024.json

@@ -0,0 +1,565 @@
+[
+    {
+        "information": "Use $g = 10 \\mathrm{N}/\\mathrm{kg}$ throughout, unless otherwise specified."
+    },
+    {
+        "id": "F=MA_2024_01",
+        "context": "",
+        "question": "An archer fires an arrow from the ground so that it passes through two hoops, which are both a height $h$ above the ground. The arrow passes through the first hoop one second after the arrow is launched, and through the second hoop another second later. What is the value of $h$?\n\n(A) $5 m$. \n(B) $10 m$. \n(C) $12 m$. \n(D) $15 m$. \n(E) There is not enough information to decide.",
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2024_02",
+        "context": "",
+        "question": "An amusement park ride consists of a circular, horizontal room. A rider leans against its frictionless outer walls,which are angled back at $30^{\\circ}$ with respect to the vertical, so that the rider's center of mass is $5.0 m$ from the center of the room. When the room begins to spin about its center, at what angular velocity will the rider's feet first lift off the floor?\n\n(A) $1.9 \\mathrm{rad}/\\mathrm{s}$. \n(B) $2.3 \\mathrm{rad}/\\mathrm{s}$. \n(C) $3.5 \\mathrm{rad}/\\mathrm{s}$. \n(D) $4.0 \\mathrm{rad}/\\mathrm{s}$. \n(E) $5.6 \\mathrm{rad}/\\mathrm{s}$.",
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2024_03",
+        "context": "",
+        "question": "As shown in the figure, a simple bridge is made of five thin rods rigidly connected at four vertices. The ground is frictionless, so that it can only exert vertical normal forces at $B$ and $D$. The weight of the bridge is negligible, but a person stands at its middle, exerting a downward force $F$ at vertex $C$. In static equilibrium, each rod can be experiencing either tension or compression. Which of the following is true?\n\n(A) Only the vertical rod is in tension. \n(B) Only the horizontal rods are in tension. \n(C) Both the vertical rod and the diagonal rods are in tension. \n(D) Both the vertical rod and the horizontal rods are in tension. \n(E) All of the rods are in tension.",
+        "answer": [
+            "\\boxed{D}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_03_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2024_04",
+        "context": "",
+        "question": "A bouncy ball is thrown vertically upward from the ground. Air resistance is negligible, and the ball's collisions with the ground are perfectly elastic. Which of the following plots in the figure shows the kinetic energy of the ball as a function of time? Assume the collisions are too quick for their duration to be seen in the plot.",
+        "answer": [
+            "\\boxed{D}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+data figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_04_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2024_05",
+        "context": "",
+        "question": "As shown in the figure, a massless inclined plane with angle $30^{\\circ}$ to the horizontal is fixed to a scale. A block of mass $m$ is released from the top of the plane, which is frictionless. As the block slides down the plane, what is the reading on the scale? \n\n(A) $\\sqrt{3}{mg}/4$. \n(B) ${mg}/2$. \n(C) ${3mg}/4$. \n(D) $\\sqrt{3}{mg}/2$. \n(E) ${mg}$.",
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_05_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2024_06",
+        "context": "",
+        "question": "As shown in the figure, a pendulum is made with a string and a bucket full of water. When the string is vertical, the bottom of the bucket is near the ground. Then, the pendulum is set swinging with a small amplitude, and a very small hole is opened at the bottom of the bucket, which leaks water at a constant rate. After a few full swings, which of the following plots in the figure best shows the amount of water that has landed on the ground as a function of position?",
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+data figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_06_1.png",
+            "image_question/F=MA_2024_06_2.png"
+        ]
+    },
+    {
+        "id": "F=MA_2024_07",
+        "context": "",
+        "question": "A particle travels in a straight line. Its velocity as a function of time is shown in the figure. Which of the following plots shows the velocity as a function of distance $x$ from its initial position?",
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+data figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_07_1.png",
+            "image_question/F=MA_2024_07_2.png"
+        ]
+    },
+    {
+        "id": "F=MA_2024_08",
+        "context": "",
+        "question": "As shown in the figure, a rod of length $L$ is sliding down a frictionless wall. When the rod makes an angle of $45^{\\circ}$ to the horizontal, the bottom of the rod has speed $v$. At this moment, what is the speed of the middle of the rod?\n\n(A) $v/2$. \n(B) $v/\\sqrt{2}$. \n(C) $v$. \n(D) $\\sqrt{2}v$. \n(E) ${2v}$.",
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_08_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2024_09",
+        "context": "",
+        "question": "When a car's brakes are fully engaged, it takes $100 m$ to stop on a dry road, which has coefficient of kinetic friction $\\mu_{k} = 0.8$ with the tires. Now suppose only the first $50 m$ of the road is dry, and the rest is covered with ice, with $\\mu_{k} = 0.2$. What total distance does the car need to stop?\n\n(A) $150 m$. \n(B) $200 m$. \n(C) $250 m$. \n(D) $400 m$. \n(E) $850 m$.",
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2024_10",
+        "context": "",
+        "question": "As shown in the figure, a block of mass $m$ is connected to the walls of a frictionless box by two massless springs with relaxed lengths $\\ell$ and $2\\ell$, and spring constants $k$ and $2k$ respectively. The length of the box is $3\\ell$. The system rotates with a constant angular velocity $\\omega$ about one of its walls. Suppose the block stays at a constant distance $r$ from the axis of rotation, without touching either of the walls. What is the value of $r$?\n\n(A) $\\frac{2k\\ell}{2k - m\\omega^2}$. \n(B) $\\frac{2k\\ell}{2k + m\\omega^2}$. \n(C) $\\frac{2k\\ell}{3k + m\\omega^2}$. \n(D) $\\frac{3k\\ell}{3k - m\\omega^2}$. \n(E) $\\frac{3k\\ell}{3k + m\\omega^2}$.",
+        "answer": [
+            "\\boxed{D}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_10_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2024_11",
+        "context": "",
+        "question": "Two hemispherical shells can be pressed together to form a airtight sphere of radius $40 cm$. Suppose the shells are pressed together at a high altitude, where the air pressure is half its value at sea level. The sphere is then returned to sea level, where the air pressure is $10^{5}\\mathrm{Pa}$. What force $F$, applied directly outward to each hemisphere, is required to pull them apart?\n\n(A) $25,000 N$. \n(B) $50,000 N$. \n(C) $100,000 N$. \n(D) $200,000 N$. \n(E) $400,000 N$.",
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2024_12",
+        "context": "",
+        "question": "A space probe with mass $m$ at point P traverses through a cluster of three asteroids,at points A, B, and C. The masses and locations of the asteroids are shown in the figure. What is the torque on the probe about point C?\n\n(A) $\\frac{1}{2\\sqrt{2}} \\frac{GMm}{d}$. \n(B) $\\frac{1}{2} \\frac{GMm}{d}$. \n(C) $\\frac{1}{\\sqrt{2}}\\frac{GMm}{d}$. \n(D) $\\frac{GMm}{d}$. \n(E) $\\frac{\\sqrt{2}GMm}{d}$.",
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_12_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2024_13",
+        "context": "",
+        "question": "Two frictionless blocks of mass $m$ are connected by a massless string which passes through a fixed massless pulley, which is a height $h$ above the ground. Suppose the blocks are initially held with horizontal separation $x$, and the length of the string is chosen so that the right block hangs in the air as shown in the figure. If the blocks are relased, the tension in the string immediately afterward will be $T$. Which of the following plots in the figure shows a plot of $T$ versus $x$?",
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+data figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_13_1.png",
+            "image_question/F=MA_2024_13_2.png"
+        ]
+    },
+    {
+        "id": "F=MA_2024_14",
+        "context": "",
+        "question": "A bead of mass $m$ can slide frictionlessly on a vertical circular wire hoop of radius $20 cm$. The hoop is attached to a stand of mass $m$, which can slide frictionlessly on the ground. Initially, the bead is at the bottom of the hoop, the stand is at rest, and the bead has velocity $2 m/s$ to the right. At some point, the bead will stop moving with respect to the hoop. At that moment, through what angle along the hoop has the bead traveled?\n\n(A) $30^{\\circ}$. \n(B) $45^{\\circ}$. \n(C) $60^{\\circ}$. \n(D) $90^{\\circ}$. \n(E) $120^{\\circ}$.",
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_14_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2024_15",
+        "context": "",
+        "question": "The viscous force between two plates of area $A$, with relative speed $v$ and separation $d$, is $F = {\\eta Av}/d$, where $\\eta$ is the viscosity. In fluid mechanics,the Ohnesorge number is a dimensionless number proportional to $\\eta$ which characterizes the importance of viscous forces,in a drop of fluid of density $\\rho$, surface tension $\\gamma$, and length scale $\\ell$. Which of the following could be the definition of the Ohnesorge number?\n\n(A) $\\frac{\\eta \\ell}{\\sqrt{\\rho \\gamma}}$. \n(B) $\\eta \\ell \\sqrt{\\frac{\\rho}{\\gamma}}$. \n(C) $\\eta \\sqrt{\\frac{\\rho}{\\gamma \\ell}}$. \n(D) $\\eta \\sqrt{\\frac{\\rho \\ell}{\\gamma}}$. \n(E) $\\frac{\\eta}{\\sqrt{\\rho \\gamma \\ell}}$.",
+        "answer": [
+            "\\boxed{E}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2024_16",
+        "context": "",
+        "question": "A child of mass $m$ holds onto the end of a massless rope of length $\\ell$ ,which is attached to a pivot a height $H$ above the ground. The child is released from rest when the rope is straight and horizontal. As shown in the figure, at some point, the child lets go of the rope, flies through the air, and lands on the ground a horizontal distance $d$ from the pivot. On Earth,the maximum possible value of $d$ is $d_E$. If the setup is moved to the Moon, which has $1/6$ the gravitational acceleration, what is the new maximum possible value of $d$?\n\n(A) $d_E/6$. \n(B) $d_E/\\sqrt{6}$. \n(C) $d_E$. \n(D) $\\sqrt{6}d_E$. \n(E) $6d_E$.",
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_16_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2024_17",
+        "context": "",
+        "question": "As shown in the figure, consider the system of massless and frictionless pulleys, ropes, and springs. Initially, a block of mass $m$ is attached to the end of a rope, and the system is in equilibrium. Next the block is doubled in mass, and the system is allowed to come to equilibrium again. During the transition between these equilibria, how far does the end of the rope (where the block is suspended) move?\n\n(A) $\\frac{7}{12}\\frac{mg}{k}$. \n(B) $\\frac{11}{12}\\frac{mg}{k}$. \n(C) $\\frac{13}{12}\\frac{mg}{k}$. \n(D) $\\frac{7}{6}\\frac{mg}{k}$. \n(E) $\\frac{11}{6}\\frac{mg}{k}$.",
+        "answer": [
+            "\\boxed{E}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_17_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2024_18",
+        "context": "",
+        "question": "As shown in the figure, a satellite is initially in a circular orbit of radius $R$ around a planet of mass $M$. It fires its rockets to instantaneously increase its speed by $\\Delta v$, keeping the direction of its velocity the same, so that it enters an elliptical orbit whose maximum distance from the planet is $2R$. What is the value of $\\Delta v$? (Hint: when the satellite is in an elliptical orbit with semimajor axis $a$, its total energy per unit mass is $- {GM}/{2a}$.) \n\n(A) $0.08 \\sqrt{\\frac{GM}{R}}$. \n(B) $0.15 \\sqrt{\\frac{GM}{R}}$. \n(C) $0.22 \\sqrt{\\frac{GM}{R}}$. \n(D) $0.29 \\sqrt{\\frac{GM}{R}}$. \n(E) $0.41 \\sqrt{\\frac{GM}{R}}$.",
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_18_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2024_19",
+        "context": "",
+        "question": "A wheel of radius $R$ has a thin rim and four spokes,each of which have uniform density.As shown in the figure, the entire rim has mass $m$, three of the spokes each have mass $m$, and the fourth spoke has mass $3m$. The wheel is suspended on a horizontal frictionless axle passing through its center. If the wheel is slightly rotated from its equilibrium position, what is the angular frequency of small oscillations?\n\n(A) $\\sqrt{\\frac{g}{3R}}$. \n(B) $\\sqrt{\\frac{g}{2R}}$. \n(C) $\\sqrt{\\frac{2g}{3R}}$. \n(D) $\\sqrt{\\frac{g}{R}}$. \n(E) $\\sqrt{\\frac{7g}{6R}}$.",
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_19_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2024_20",
+        "context": "",
+        "question": "As shown in the figure, four massless rigid rods are connected into a quadrilateral by four hinges. The hinges have mass $m$, and allow the rods to freely rotate. A spring of spring constant $k$ is connected across each of the diagonals, so that the springs are at their relaxed length when the rods form a square. Assume the springs do not interfere with each other. If the square is slightly compressed along one of its diagonals, its shape will oscillate over time. What is the period of these oscillations?\n\n(A) $2\\pi \\sqrt{\\frac{m}{4k}}$. \n(B) $2\\pi \\sqrt{\\frac{m}{2k}}$. \n(C) $2\\pi \\sqrt{\\frac{m}{k}}$. \n(D) $2\\pi \\sqrt{\\frac{2m}{k}}$. \n(E) $2\\pi \\sqrt{\\frac{4m}{k}}$.",
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_20_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2024_21",
+        "context": "",
+        "question": "A syringe is filled with water of density $\\rho$ and negligible viscosity. Its body is a cylinder of cross-sectional area $A_1$, which gradually tapers into a needle with cross-sectional area $A_2 \\ll  A_1$. The syringe is held in place and its end is slowly pushed inward by a force $F$, so that it moves with constant speed $v$. Water shoots straight out of the needle's tip. What is the approximate value of $F$?\n\n(A) $\\rho {v}^2{A}_{1}$. \n(B) $\\frac{\\rho {v}^2{A}_{1}^2}{2{A}_{2}}$. \n(C) $\\frac{\\rho {v}^2{A}_{1}^2}{{A}_{2}}$. \n(D) $\\frac{\\rho {v}^2{A}_{1}^{3}}{2{A}_{2}^2}$. \n(E) $\\frac{\\rho {v}^2{A}_{1}^{3}}{{A}_{2}^2}$",
+        "answer": [
+            "\\boxed{D}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2024_22",
+        "context": "",
+        "question": "A spherical shell is made from a thin sheet of material with a mass per area of $\\sigma$. Consider two points, $P_1$ and $P_2$, which are close to each other, but just inside and outside the sphere, respectively. If the accelerations due to gravity at these points are $g_1$ and $g_2$, respectively, what is the value of $|g_1 - g_2|$?\n\n(A) $\\pi G\\sigma$. \n(B) $4\\pi G\\sigma /3$. \n(C) $2\\pi G\\sigma$. \n(D) $4\\pi G\\sigma$. \n(E) $8\\pi G\\sigma$.",
+        "answer": [
+            "\\boxed{D}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2024_23",
+        "context": "",
+        "question": "Collisions between ping pong balls and paddles are not perfectly elastic. Suppose that if a player holds a paddle still and drops a ball on top of it from any height $h$, it will bounce back up to height $h/2$. To keep the ball bouncing steadily, the player moves the paddle up and down, so that it is moving upward with speed $1.0 m/s$ whenever the ball hits it. What is the height to which the ball is bouncing?\n\n(A) $0.21 m$. \n(B) $0.45 m$. \n(C) $1.0 m$. \n(D) $1.7 m$. \n(E) There is not enough information to determine the height.",
+        "answer": [
+            "\\boxed{D}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2024_24",
+        "context": "",
+        "question": "When a projectile falls through a fluid, it experiences a drag force proportional to the product of its cross-sectional area, the fluid density $\\rho_{f}$, and the square of its speed. Suppose a sphere of density $\\rho_{s} \\gg \\rho_{f}$ of radius $R$ is dropped in the fluid from rest. When the projectile has reached half of its terminal velocity, which of the following is its displacement proportional to?\n\n(A) $R\\sqrt{\\rho_{s}/\\rho_{f}}$. \n(B) $R \\rho_{s}/\\rho_{f}$. \n(C) $R (\\rho_{s}/\\rho_{f})^{3/2}$. \n(D) $R (\\rho_{s}/\\rho_{f})^2$. \n(E) $R (\\rho_{s}/\\rho_{f})^{3}$",
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2024_25",
+        "context": "",
+        "question": "A yo-yo consists of two massive uniform disks of radius $R$ connected by a thin axle. As shown in the figure, a thick string is wrapped many times around the axle, so that the end of the string is initially a distance $R$ from the axle. Then, the end of the string is held in place and the yo-yo is dropped from rest. Assume that energy losses are negligible, and that the string has negligible mass and always remains vertical. Below, we show a cross-section of the yo-yo partway through its descent. Between the moment the yo-yo is released and the moment the string completely unwinds, which of the following is true regarding the yo-yo's acceleration?\n\n(A) It is always zero.\n(B) It points downward, but decreases in magnitude over time. \n(C) It points downward and has constant magnitude. \n(D) It points downward, but increases in magnitude over time. \n(E) None of the above.",
+        "answer": [
+            "\\boxed{E}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2024",
+        "image_question": [
+            "image_question/F=MA_2024_25_1.png"
+        ]
+    }
+]

data/F=MA_2025.json

@@ -0,0 +1,558 @@
+[
+    {
+        "information": "Use $g = 10 \\mathrm{N}/\\mathrm{kg}$ throughout, unless otherwise specified."
+    },
+    {
+        "id": "F=MA_2025_01",
+        "context": "",
+        "question": "A particle is moving on a plane at a constant speed of $1 m/s$, but not necessarily in a straight line. Which of the following plot pairs (as shown in the figure) could describe the particle's position over time, in rectilinear coordinates?\n\n(A) I only. \n(B) II only. \n(C) III only. \n(D) II and III only. \n(E) All three plots could describe the particle's position over time.",
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+data figure",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": [
+            "image_question/F=MA_2025_01_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2025_02",
+        "context": "As shown in the figure, three identical disks are placed on a frictionless table. Initially, two of the disks are at rest and in contact with each other. The third disk is launched with speed $v$ directly toward the midpoint of the two stationary disks along a path perpendicular to the line connecting their centers, as shown in the diagram. Analyze the motion of the disks after the collision, assuming all interactions are perfectly elastic and that when the disks collide, there is no friction or inelastic energy loss.",
+        "question": "Assume that all three disks collide simultaneously. What is the final velocity of the third disk?\n\n(A) $v/3$ in the opposite direction to the initial velocity. \n(B) $v/3$ in the same direction as the initial velocity. \n(C) $\\overrightarrow{0}$. \n(D) $v/5$ in the opposite direction to the initial velocity. \n(E) $v/5$ in the same direction as the initial velocity.",
+        "answer": [
+            "\\boxed{D}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": [
+            "image_question/F=MA_2025_02_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2025_03",
+        "context": "As shown in the figure, three identical disks are placed on a frictionless table. Initially, two of the disks are at rest and in contact with each other. The third disk is launched with speed $v$ directly toward the midpoint of the two stationary disks along a path perpendicular to the line connecting their centers, as shown in the diagram. Analyze the motion of the disks after the collision, assuming all interactions are perfectly elastic and that when the disks collide, there is no friction or inelastic energy loss.",
+        "question": "Assume that there is a little imperfection in disks' initial alignment so when the disks collide two collisions happen one at a time, rather than all three disks colliding simultaneously. What is the final speed of the third disk?\n\n (A) $v/2$. \n(B) $v/3$. \n(C) $v/4$. \n(D) $v/5$. \n(E) $0$.",
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": [
+            "image_question/F=MA_2025_03_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2025_04",
+        "context": "",
+        "question": "As shown in the figure, a mouse $M$ is running from $A$ to $A^{\\prime}$ with constant speed $u_1$ . A cat $C$ is chasing the mouse with constant speed $u_2$ and direction always toward the mouse. At a certain time $MC \\perp AA^\\prime$ and the length of $MC = L$ . What is the magnitude of the acceleration of the cat $C$?\n\n(A) $0$. \n(B) $(u_1 - u_2)^2/(2\\pi L)$. \n(C) $u_1 u_2/L$. \n(D) $u_1 u_2/( 2\\pi L)$. \n(E) none of the above.",
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": [
+            "image_question/F=MA_2025_04_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2025_05",
+        "context": "",
+        "question": "Three identical cylinders are used in this setup. Two of them are placed side by side on a horizontal surface, with a negligible distance between their surfaces so they do not touch. The third identical cylinder is placed on top of the first two, such that their centers form an equilateral triangle, as shown in the figure below.\n\nThe coefficients of friction are:\n\n$\\mu_1$: the coefficient of friction between the cylinders, and\n\n $\\mu_2$: the coefficient of friction between the cylinders and the ground.\n\nFor which of the following pairs $(\\mu_1, \\mu_2)$ will the system remain in equilibrium? \n\nPair 1: $(\\frac{1}{2}, \\frac{1}{12})$. Pair 2: $(\\frac{1}{3}, \\frac{1}{10})$. Pair 3: $(\\frac{1}{4}, \\frac{1}{8}})$.\n\n(A) Pair 1 only. \n(B) Pair 2 only. \n(C) Pair 3 only. \n(D) Pairs 1 and 2 only. \n(E) Pairs 1, Pair 2, and Pair 3.",
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": [
+            "image_question/F=MA_2025_05_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2025_06",
+        "context": "",
+        "question": "A ball rolls without slipping down a ramp, which turns horizontal at the bottom; at the bottom of the ramp,the ball falls through the air, as in the diagram. If the ball starts from the position marked $O$, it lands $10 cm$ away from the bottom of the ramp. Which starting position will get the ball to land closest to $25 cm$ away?\n\n(A) Position A. \n(B) Position B. \n(C) Position C. \n(D) Position D. \n(E) Position E.",
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": [
+            "image_question/F=MA_2025_06_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2025_07",
+        "context": "",
+        "question": "A mechanism consists of three point masses, each of mass $m$,connected by two massless rods of length $l$ and a torsion spring acting as a hinge. The potential energy of the torsion spring is given by $U_s$ . This system is designed to \"walk\" down a set of stairs, as shown in the figure. The angles $\\theta_1$ and $\\theta_2$ (see figure) represent the orientation of the rods,and their rates of change, $\\omega_1$ and $\\omega_2$, are the corresponding angular velocities. Assume that the mass on the surface is instantaneously at rest.\n\nWhich equation correctly describes the total energy of the system?\n\n(A) $E = m{l}^2\\omega_{1}^2 + \\frac{1}{2}m{l}^2\\omega_{2}^2 + m{l}^2\\omega_{1}\\omega_{2}\\cos \\left(\\theta_{1} + \\theta_{2}\\right) + {2mgl}\\sin \\theta_{1} + {mgl}\\sin \\theta_{2} + {U}_{s}$. \n(B) $E = \\frac{1}{2}m{l}^2\\omega_{1}^2 + \\frac{1}{2}m{l}^2\\omega_{2}^2 + {2mgl}\\sin \\theta_{1} + {mgl}\\sin \\theta_{2} - {U}_{s}$. \n(C) $E = m{l}^2\\omega_{1}^2 + \\frac{1}{2}m{l}^2\\omega_{2}^2 + m{l}^2\\omega_{1}\\omega_{2}\\cos \\left(\\theta_{1} - \\theta_{2}\\right) + {2mgl}\\sin \\theta_{1} + {mgl}\\sin \\theta_{2} + {U}_{s}$. \n(D) $E = m{l}^2\\omega_{1}^2 + \\frac{1}{2}m{l}^2\\omega_{2}^2 - m{l}^2\\omega_{1}\\omega_{2}\\cos \\left(\\theta_{1} - \\theta_{2}\\right) + {2mgl}\\sin \\theta_{1} + {mgl}\\sin \\theta_{2} + {U}_{s}$. \n(E) $E = \\frac{1}{2}m{l}^2\\omega_{1}^2 + \\frac{1}{2}m{l}^2\\omega_{2}^2 + {2mgl}\\sin \\theta_{1} + {mgl}\\sin \\theta_{2} + {U}_{s}$.",
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": [
+            "image_question/F=MA_2025_07_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2025_08",
+        "context": "",
+        "question": "A symmetric spinning top,rotating clockwise at an angular frequency $\\omega$, is placed upright in the center of a frictionless circular plate. The plate then begins to rotate counterclockwise at a constant angular velocity $\\omega$. Assume the top's axis remains perfectly vertical and stable without any precession. From the perspective of an observer rotating with the plate, how does the top appear to rotate?\n\n(A) The top appears stationary without any rotation. \n(B) The top appears to rotate in the clockwise direction at an angular frequency $\\omega$. \n(C) The top appears to rotate in the clockwise direction at an angular frequency $2\\omega$. \n(D) The top appears to rotate in the counterclockwise direction at an angular frequency $\\omega$. \n(E) The top appears to rotate in the counterclockwise direction at an angular frequency ${2\\omega}$.",
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2025_09",
+        "context": "",
+        "question": "$N$ circles in a plane, $C_{i}$, each rotate with frequency $\\omega$ relative to an inertial frame. The center of $C_{1}$ is fixed in the inertial frame, and the center of $C_{i}$ is fixed on $C_{i - 1}$ (for $i = 2, \\ldots, N$), as shown in the figure. Each circle has radius ${r}_{i} = \\lambda {r}_{i - 1}$, where $0 < \\lambda  < 1$. A mass is fixed on $C_{N}$. The position of the mass relative to the center of $C_{1}$ is $R\\left( t\\right)$. For the $N = 4$ case shown, which of the following statements is true?\n\n During the time interval from 0 to ${2\\pi}/\\omega$, the magnitude of acceleration of mass on $C_{4}$:\n\n(A) reached its maximum and minimum more than once. \n(B) reached its maximum and minimum exactly once. \n(C) reached its maximum only once but the minimum more than once. \n(D) reached its minimum only once but the maximum more than once. \n(E) was constant.",
+        "answer": [
+            "\\boxed{E}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": [
+            "image_question/F=MA_2025_09_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2025_10",
+        "context": "When two objects of very different masses collide, it is difficult to transfer a substantial fraction of the energy of one to the other. Consider two objects, of mass $m$ and $M \\gg m$.",
+        "question": "If the lighter object is initially at rest, and the heavier object collides elastically with it, what is the approximate maximum fraction of the heavier object's kinetic energy that could be transferred to the lighter object?\n\n(A) $m/M$. \n(B) ${2m}/M$. \n(C) ${4m}/M$. \n(D) ${m}^2/{M}^2$. \n(E) $2{m}^2/{M}^2$.",
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2025_11",
+        "context": "When two objects of very different masses collide, it is difficult to transfer a substantial fraction of the energy of one to the other. Consider two objects, of mass $m$ and $M \\gg m$.",
+        "question": "Now suppose that instead, the heavier object is initially at rest, and the lighter object collides elastically with it. What is the approximate maximum fraction of the lighter object's kinetic energy that could be transferred to the heavier object?\n\n(A) $m/M$. \n(B) ${2m}/M$. \n(C) ${4m}/M$. \n(D) ${m}^2/{M}^2$. \n(E) $2{m}^2/{M}^2$.",
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2025_12",
+        "context": "",
+        "question": "A $50 g$ piece of clay is thrown horizontally with a velocity of $20 m/s$ striking the bob of a stationary pendulum with length $l = 1 m$ and a bob mass of $200 g$. Upon impact, the clay sticks to the pendulum weight and the pendulum starts to swing. What is the maximum change in angle of the pendulum?\n\n(A) $\\arccos(1/5)$. \n(B) $\\arcsin(7/10)$. \n(C) $\\arccos(2/3)$. \n(D) $\\arcsin(3/10)$. \n(E) $\\arctan(4/5)$.",
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2025_13",
+        "context": "Angela the puppy loves chasing tennis balls, so her owners built a tennis ball launcher. It fires balls along the floor at some initial speed, applying no rotation to them. The balls initially slip along the floor, then start rolling without slipping. Ignore the potential deformation of the ball and floor during this process, as well as air resistance.",
+        "question": "Which of the following plot pairs (as shown in the figure) could show the linear speed $v$ and rotational speed $\\omega$ of one of the balls over time? Assume the floor has a constant roughness.",
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+data figure",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": [
+            "image_question/F=MA_2025_13_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2025_14",
+        "context": "Angela the puppy loves chasing tennis balls, so her owners built a tennis ball launcher. It fires balls along the floor at some initial speed, applying no rotation to them. The balls initially slip along the floor, then start rolling without slipping. Ignore the potential deformation of the ball and floor during this process, as well as air resistance.",
+        "question": "There are three kinds of balls that can be launched in this set-up, all having the same radius $R$:\nI. a regular tennis ball (a thin spherical shell of rubber) of mass $m_1$. \nII. a solid wooden ball of mass $m_2$. \nIII. a solid rubber ball of mass $m_3$.\n where $m_1 < m_2 < m_3$. All three types of ball emerge from the launcher with the same velocity. For which ball will the final velocity be highest?\n\n(A) Ball I. \n(B) Ball II. \n(C) Ball III. \n(D) Balls II and III. \n(E) The final velocity will be the same for all three balls.",
+        "answer": [
+            "\\boxed{D}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2025_15",
+        "context": "",
+        "question": "As shown in the figure, a uniform rigid rod of mass $M$ and length $2L$ is attached to a massless rod of length $L$, which is fixed at one end to the ceiling and free to rotate in a vertical plane. The massive rod is connected to the free end of the massless rod. Suppose an impulse $J$ is applied horizontally to the bottom of the massive rod.\n\nDetermine the relationship between the magnitudes of the angular velocity $\\omega$ of the massless rod and $\\Omega$ of the massive rod immediately after the impulse is applied. The moment of inertia of a uniform rod of length $d$ and mass $m$ about its center of mass is given by $I = \\frac{1}{12}m{d}^2$.\n\n(A) $\\Omega = \\frac{3}{4}\\omega$. \n(B) $\\Omega = \\frac{2}{3}\\omega$. \n(C) $\\Omega = \\frac{4}{3}\\omega$. \n(D) $\\Omega = \\frac{1}{12}\\omega$. \n(E) $\\Omega = \\frac{3}{2}\\omega$.",
+        "answer": [
+            "\\boxed{E}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": [
+            "image_question/F=MA_2025_15_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2025_16",
+        "context": "",
+        "question": "Two soap bubbles of radii $R_1 = 1 cm$ and $R_2 = 2 cm$ conjoin together in the air,such that a narrow bridge forms between them. Assuming the system starts in equilibrium, the bubbles are extremely thin, and that air can flow freely between the bubbles through the bridge, describe the evolution and final state of the bubbles.\n\n(A) The smaller bubble will shrink and the larger bubble will grow. \n(B) The larger bubble will shrink and the smaller bubble will grow. \n(C) The bubbles will maintain their sizes. \n(D) Air will oscillate between the two bubbles. \n(E) Both bubbles will simultaneously shrink.",
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2025_17",
+        "context": "",
+        "question": "A particle of mass $m$ moves in the ${xy}$ plane with potential energy $U(x,y) = -k\\frac{x^2 + y^2}{2}$. The closest point to the origin $(x = 0, y = 0)$ during its motion was at a distance $d$, and the particle's speed at that point was $v \\neq 0$. Which of the following statements is true regarding the path of the particle after a long time $t$ ($t \\gg d/v$)?\n\n(A) The particle's trajectory will be circular. \n(B) The particle's trajectory will be asymptotic to a straight line pointing away from the origin. \n(C) The particle will spiral outwards away from the origin. \n(D) The particle will travel on a parabolic trajectory. \n(E) The particle will spiral inwards towards the origin.",
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2025_18",
+        "context": "",
+        "question": "A particle of mass $m$ moves in the ${xy}$ plane with potential energy $U(x,y) = {kxy}/2$. If the particle begins at the origin, then it is possible to displace it slightly in some direction, so that the particle subsequently oscillates periodically. What is the period of this motion?\n\n(A) $2\\pi\\sqrt{m/{4k}}$. \n(B) $2\\pi\\sqrt{m/{2k}}$. \n(C) $2\\pi\\sqrt{m/k}$. \n(D) $2\\pi\\sqrt{{2m}/k}$. \n(E) $2\\pi\\sqrt{{4m}/k}$.",
+        "answer": [
+            "\\boxed{D}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2025_19",
+        "context": "",
+        "question": "Near the ground, wind speed can be modeled as proportional to height above the ground. (This is a reasonable assumption for small heights.) A wind turbine converts a constant fraction of the available kinetic energy into electricity. The conditions are such that when operating at $10 m$ above the ground,the turbine delivers $15 kW$ of power. How much power would the same windmill deliver if it were operating at $20 m$ above the ground?\n\n(A) $15 kW$. \n(B) $21 kW$. \n(C) $30 kW$. \n(D) $60 kW$. \n(E) $120 kW$.",
+        "answer": [
+            "\\boxed{E}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2025_20",
+        "context": "",
+        "question": "The International Space Station orbits the Earth in a circular orbit $400 km$ above the surface,and a full revolution takes 93 minutes. An astronaut on a space walk neglects safety precautions and tosses away a spanner at a speed of $1 m/s$ directly towards the Earth. You may assume that the Earth is a sphere of uniform density. At which of the following five times will the spanner be closest to the astronaut?\n\n(A) After 139.5 minutes. \n(B) After 131.5 minutes. \n(C) After 93 minutes. \n(D) After 46.5 minutes. \n(E) After 1 minute.",
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2025_21",
+        "context": "As shown in the figure, water flows through a pipe with a radius of $5 cm$ at a velocity of $10 cm/s$ before entering a narrower section of pipe with a radius of $2.5 cm$.",
+        "question": "What is the difference in the speed of water between the two pipes?\n\n(A) $20 cm/s$. \n(B) $30 cm/s$. \n(C) $40 cm/s$. \n(D) $50 cm/s$. \n(E) $60 cm/s$.",
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": [
+            "image_question/F=MA_2025_21_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2025_22",
+        "context": "As shown in the figure, water flows through a pipe with a radius of $5 cm$ at a velocity of $10 cm/s$ before entering a narrower section of pipe with a radius of $2.5 cm$.",
+        "question": "To measure this difference, two graduated cylinders are connected to the top of the pipe (one in the broad section and the other in the narrowed section). Water then flows up each pipe and the height the water reaches is measured. Estimate the difference in height between the two cylinders.\n\n(A) $6.3 mm$. \n(B) $7.5 mm$. \n(C) $8.2 mm$. \n(D) $12.2 mm$. \n(E) $12.7 mm$.",
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": [
+            "image_question/F=MA_2025_22_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2025_23",
+        "context": "",
+        "question": "A student conducted an experiment to determine the spring constant of a spring using a ruler and two different weighing scales. The measured elongation of the spring was $1.5 cm$ ,and the smallest division on the ruler was $1 mm$ . The mass of the attached weight was measured using two different scales in the school laboratory,yielding values of $198 g$ and $210 g$ . The student also found that the local acceleration due to gravity in her city is given as $(9.806 \\pm 0.001) m/s^2$. Calculate the percent error in measuring the spring constant.\n\n(A) $2\\%$. \n(B) $4\\%$. \n(C) $8\\%$. \n(D) ${11}\\%$. \n(E) ${14}\\%$.",
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": []
+    },
+    {
+        "id": "F=MA_2025_24",
+        "context": "",
+        "question": "As shown in the figure, a massive bead is attached to the end of a massless rigid rod of length $L$. The other end of the rod is attached to an ideal pivot, which allows it to rotate frictionlessly in any direction. The rod is initially at angle $\\theta$ to the horizontal, and there is no gravitational force. Next, the bead receives an impulse directly into the page, giving it a speed $v$. How long does it take for the bead to return to its original position?\n\n(A) $2\\pi L / v$. \n(B) $(2 \\pi L / v) \\sin \\theta$. \n(C) $(2 \\pi L / v) \\cos \\theta$. \n(D) $(2 \\pi L / v) \\cos^2 \\theta$. \n(E) $(2 \\pi L / v) \\cos^2 (2\\theta)$.",
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": [
+            "image_question/F=MA_2025_24_1.png"
+        ]
+    },
+    {
+        "id": "F=MA_2025_25",
+        "context": "",
+        "question": "As shown in the figure, a puck of mass $m$ can slide on a frictionless inclined plane (prism). The prism has a much greater mass compared to the puck and is itself sliding without friction on a horizontal surface. The velocity of the prism is $v = \\sqrt{2gh}$, where $g$ is the acceleration due to gravity and $h$ is the height from which the puck starts sliding on the prism. The transition from the prism to the horizontal surface is smooth. The puck starts from rest relative to the prism. Find the final velocity of the puck once it begins sliding on the horizontal surface.\n\n(A) $\\frac{v}{2}$. \n(B) $\\frac{v}{\\sqrt{2}}$. \n(C) $v$. \n(D) $\\sqrt{2}v$. \n(E) $2v$.",
+        "answer": [
+            "\\boxed{E}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "F=MA_2025",
+        "image_question": [
+            "image_question/F=MA_2025_25_1.png"
+        ]
+    }
+]

data/NBPhO_2024.json

@@ -0,0 +1,745 @@
+[
+    {
+        "information": "None."
+    },
+    {
+        "id": "NBPhO_2024_1_1",
+        "context": "[Four Charges] \n\nFour identical particles are initially in the corners of a square, as shown in the figure. All particles have the same charge $q$, mass $m$, and the same magnitude of initial velocity $v_0$. The directions of the initial velocities are indicated in the figure. You can assume $v \\ll c$ and ignore gravity.\n\n[figure1]",
+        "question": "After a long time has passed, what is the magnitude of the final velocity $v_f$ of the particles with respect to the center of mass of the system?",
+        "marking": [
+            [
+                "Award 0.2 pt if the answer shows the idea of using energy conservation to relate initial and final states. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer recognizes symmetry and equality of particle quantities (e.g., all four particles have same mass and charge). Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer expresses total energy as a sum of kinetic and electrostatic energy. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer writes the formula for electrostatic energy with two different distances: $\\frac{k q^2}{L}$ and $\\frac{k q^2}{\\sqrt{2}L}$. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer includes the factor $\\frac{1}{2}$ in the electrostatic energy to avoid double counting from pairings. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer correctly states that the final energy is purely kinetic: $E_{\\text{final}} = 4 \\cdot \\frac{1}{2} m v_f^2$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer gives the correct final expression for $v_f$: $v_f = \\sqrt{v_0^2 + \\frac{k q^2}{L m} \\cdot \\frac{4 + \\sqrt{2}}{2}}$. Partial points: award 0.2 pt if only dimensionless factors are missing but the final answer is reasonable. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$v_{f} = \\sqrt{v_0^{2} + \\frac{k{q}^{2}}{Lm} \\frac{4 + \\sqrt{2}}{2}}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            2.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Electromagnetism",
+        "source": "NBPhO_2024",
+        "image_question": [
+            "image_question/NBPhO_2024_1_1_1.png"
+        ]
+    },
+    {
+        "id": "NBPhO_2024_1_2",
+        "context": "[Four Charges] \n\nFour identical particles are initially in the corners of a square, as shown in the figure. All particles have the same charge $q$, mass $m$, and the same magnitude of initial velocity $v_0$. The directions of the initial velocities are indicated in the figure. You can assume $v \\ll c$ and ignore gravity.\n\n[figure1]\n\n(i) After a long time has passed, what is the magnitude of the final velocity $v_f$ of the particles with respect to the center of mass of the system?\n\nPart (i) is a preliminary question and should not be included in the final answer.",
+        "question": "What is the angle between the initial velocity $\\vec{v}_0$ and the final velocity $\\vec{v}_f$ of a particle?",
+        "marking": [
+            [
+                "Award 1.0 pt if the answer correctly states that the effective repulsive force $F = \\frac{A}{r^2}$ is pointing from the center of the masses (explicit expression or statement is required). Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer recognizes the motion as hyperbolic. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer identifies that the central charge $Q_e$ is located at the correct focus of the hyperbola. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer expresses the angle $\\phi$ or $\\alpha$ in terms of geometrical parameters, e.g., $\\cos \\alpha = a/c$, or $\\tan \\alpha = b/a$. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer applies the vis-viva equation or energy conservation to the hyperbolic orbit: $E = \\frac{k q Q_e}{2a} = \\frac{1}{2} m v_0^2 + \\frac{\\sqrt{2} k q Q_e}{L}$. Partial points for angular momentum conservation: award 0.3 pt if the answer mentions that angular momentum is conserved; award 0.3 pt if the answer writes the angular momentum around the center of the masses using $L$ and $v_0$; award 0.4 pt if the answer writes final angular momentum around the center of the masses using $v_f$. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer derives the final answer $\\sin \\phi = \\frac{1}{1 + \\frac{4 L m v_0^2}{k q^2 (4+\\sqrt{2})}}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$\\sin \\phi = \\frac{1}{1 + \\frac{4 L m v_0^2}{k q^2 (4+\\sqrt{2})}}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            5.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Electromagnetism",
+        "source": "NBPhO_2024",
+        "image_question": [
+            "image_question/NBPhO_2024_1_1_1.png"
+        ]
+    },
+    {
+        "id": "NBPhO_2024_2_1",
+        "context": "[Oklo Fission Reactor] \n\nBased on the ratio of the uranium isotopes $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$, as well as the abundances of the isotopes produced by nuclear reactors, researchers have established that self-sustaining natural nuclear reactors operated in Oklo ca $T_0 = 1.8 \\times {10}^{9}$ years ago in Gabon, central Africa. For such reactors to exist, two conditions must be met: (a) presence of deposits with high enough concentration of uranium; (b) sufficiently high abundance of $^{235}\\mathrm{U}$ in natural uranium. Rich uranium ores were created by floods: scattered uranium was dissolved in oxygen-rich water and transported by it to underground pools. Significant concentration of oxygen appeared in the atmosphere only around 2.5 billion years ago, so the first condition was not met earlier than that. You will learn below that the abundance of $^{235}\\mathrm{U}$ decreases relatively fast in time, so the second condition ceased to be satisfied soon after the operation of Oklo's reactor. \n\nWhat made the operation of Oklo's reactor possible was a stable influx of ground water that kept the uranium deposits sufficiently wet. Water is the so-called moderator for the fission reactor: it slows down neutrons emerging from fission reactions, dramatically enhancing the chances of a neutron triggering the fission of a next $^{235}\\mathrm{U}$ nucleus. \n\nIn what follows,in addition to $T_0$, you can use the following numerical values. Energy released by the fission of a single $^{235}\\mathrm{U}$ nucleus: $E_0 = 200 \\mathrm{MeV}$. \nHalf-life of $^{235} \\mathrm{U}$: $\\tau_5 \\approx 7 \\times 10^8$ years. \nHalf-life of $^{238}\\mathrm{U}$: $\\tau_8 \\approx 4.5 \\times 10^9$ years. \nLatent heat of evaporation of water: $L = 2260 \\mathrm{kJ} \\mathrm{kg}^{-1}$. \nSpecific heat of water $c = 4200 \\mathrm{J} \\mathrm{kg}^{-1} \\mathrm{K}^{-1}$. \nAbundance of $^{235}\\mathrm{U}$ in natural uranium today: $R = 0.72\\%$. We define abundance as the number of atoms of the isotope, normalized to the number of atoms of the given element.\n\nAverage abundance of $^{235}\\mathrm{U}$ in the uranium from Oklo's uranium ore today: $R_{O} = 0.62\\%$. \nThe total amount of uranium in Oklo's mine today: $M = 5 \\times {10}^{8}\\mathrm{kg}$. \n\nThe duration of the time period over which \nOklo's reactor operated: $T \\approx 1 \\times {10}^{5}$ year. \nElementary charge: $e = 1.6 \\times {10}^{-19}\\mathrm{C}$. \nAtomic mass unit: $u = 1.66 \\times {10}^{-27} \\mathrm{kg}$. \nAvogadro's number: $N_{A} = 6.02 \\times {10}^{23} \\mathrm{mol}^{-1}$. \n\nNote that: (a) the abundance of other isotopes of uranium besides $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$ is negligibly small; (b) $^{235}\\mathrm{U}$ is not among the decay products of $^{238}\\mathrm{U}$; and (c) fission channels other than the fission of $^{235}\\mathrm{U}$ (e.g., synthesis and fission of plutonium) can be neglected.",
+        "question": "What was the abundance of $^{235}\\mathrm{U}$ in natural uranium when the Oklo's reactor operated? Express your answer as a percentage.",
+        "marking": [
+            [
+                "Award 0.3 pt if the answer correctly gives the amount of $^{238}\\mathrm{U}$ during operation as $\\nu_8^{\\prime} = \\nu_8 2^{T_0 / \\tau_8}$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly gives the amount of $^{235}\\mathrm{U}$ during operation as $\\nu_5^{\\prime} = \\nu_5 2^{T_0 / \\tau_5}$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the abundance of $^{235}\\mathrm{U}$ is expressed in terms of $\\nu_5$ and $\\nu_8$, where $\\nu$ is the amount of $^{235} \\mathrm{U}$ at some point in time. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the ratio $\\nu_5 / \\nu_8$ is expressed in terms of $R$ as $\\nu_5 / \\nu_8 = R / (1 - R)$, where $\\nu$ is the amount of $^{235} \\mathrm{U}$ at some point in time. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the abundance of $^{235}\\mathrm{U}$ is correctly expressed in terms of $R$, and the final numerical answer $\\approx 3.16\\%$ is given. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$3.16 \\%$}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.5
+        ],
+        "modality": "text-only",
+        "field": "Modern Physics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_2_2",
+        "context": "[Oklo Fission Reactor] \n\nBased on the ratio of the uranium isotopes $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$, as well as the abundances of the isotopes produced by nuclear reactors, researchers have established that self-sustaining natural nuclear reactors operated in Oklo ca $T_0 = 1.8 \\times {10}^{9}$ years ago in Gabon, central Africa. For such reactors to exist, two conditions must be met: (a) presence of deposits with high enough concentration of uranium; (b) sufficiently high abundance of $^{235}\\mathrm{U}$ in natural uranium. Rich uranium ores were created by floods: scattered uranium was dissolved in oxygen-rich water and transported by it to underground pools. Significant concentration of oxygen appeared in the atmosphere only around 2.5 billion years ago, so the first condition was not met earlier than that. You will learn below that the abundance of $^{235}\\mathrm{U}$ decreases relatively fast in time, so the second condition ceased to be satisfied soon after the operation of Oklo's reactor. \n\nWhat made the operation of Oklo's reactor possible was a stable influx of ground water that kept the uranium deposits sufficiently wet. Water is the so-called moderator for the fission reactor: it slows down neutrons emerging from fission reactions, dramatically enhancing the chances of a neutron triggering the fission of a next $^{235}\\mathrm{U}$ nucleus. \n\nIn what follows,in addition to $T_0$, you can use the following numerical values. Energy released by the fission of a single $^{235}\\mathrm{U}$ nucleus: $E_0 = 200 \\mathrm{MeV}$. \nHalf-life of $^{235} \\mathrm{U}$: $\\tau_5 \\approx 7 \\times 10^8$ years. \nHalf-life of $^{238}\\mathrm{U}$: $\\tau_8 \\approx 4.5 \\times 10^9$ years. \nLatent heat of evaporation of water: $L = 2260 \\mathrm{kJ} \\mathrm{kg}^{-1}$. \nSpecific heat of water $c = 4200 \\mathrm{J} \\mathrm{kg}^{-1} \\mathrm{K}^{-1}$. \nAbundance of $^{235}\\mathrm{U}$ in natural uranium today: $R = 0.72\\%$. We define abundance as the number of atoms of the isotope, normalized to the number of atoms of the given element.\n\nAverage abundance of $^{235}\\mathrm{U}$ in the uranium from Oklo's uranium ore today: $R_{O} = 0.62\\%$. \nThe total amount of uranium in Oklo's mine today: $M = 5 \\times {10}^{8}\\mathrm{kg}$. \n\nThe duration of the time period over which \nOklo's reactor operated: $T \\approx 1 \\times {10}^{5}$ year. \nElementary charge: $e = 1.6 \\times {10}^{-19}\\mathrm{C}$. \nAtomic mass unit: $u = 1.66 \\times {10}^{-27} \\mathrm{kg}$. \nAvogadro's number: $N_{A} = 6.02 \\times {10}^{23} \\mathrm{mol}^{-1}$. \n\nNote that: (a) the abundance of other isotopes of uranium besides $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$ is negligibly small; (b) $^{235}\\mathrm{U}$ is not among the decay products of $^{238}\\mathrm{U}$; and (c) fission channels other than the fission of $^{235}\\mathrm{U}$ (e.g., synthesis and fission of plutonium) can be neglected. \n\n(i) What was the abundance of $^{235}\\mathrm{U}$ in natural uranium when the Oklo's reactor operated? \n\nPart (i) is a preliminary question and should not be included in the final answer.",
+        "question": "What was the average power of the Oklo's reactor? Express your answer in $W$.",
+        "marking": [
+            [
+                "Award 0.4 pt if the answer gives the final mass of $^{235}\\mathrm{U}$ by the end of operation as $M_5^{\\prime} = 2^{T_0 / \\tau_5} M R_0$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer gives the final mass of $^{238}\\mathrm{U}$ by the end of operation as $M_8^{\\prime} = 2^{T_0 / \\tau_8} M (1 - R)$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer gives the initial mass of $^{235}\\mathrm{U}$ at the beginning of operation as $2^{T_0 / \\tau_8} M (1 - R) \\cdot \\frac{R^{\\prime}}{1 - R^{\\prime}}$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the number of atoms that have undergone fission is computed as $N = N_A \\cdot \\Delta M / 0.235$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the total energy is given as $E = N E_0$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if all units (e.g., mass in $kg$, power in $W$) are correctly converted and used. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the power is calculated correctly using $P = \\nu E_0 / T$ (where $\\nu$ is the number of $^{235}U$ nuclei that reacted), and the numerical value is approximately $7.73 \\times 10^7$ W. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$7.73 \\times 10^7$}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "W"
+        ],
+        "point": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Modern Physics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_2_3",
+        "context": "[Oklo Fission Reactor] \n\nBased on the ratio of the uranium isotopes $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$, as well as the abundances of the isotopes produced by nuclear reactors, researchers have established that self-sustaining natural nuclear reactors operated in Oklo ca $T_0 = 1.8 \\times {10}^{9}$ years ago in Gabon, central Africa. For such reactors to exist, two conditions must be met: (a) presence of deposits with high enough concentration of uranium; (b) sufficiently high abundance of $^{235}\\mathrm{U}$ in natural uranium. Rich uranium ores were created by floods: scattered uranium was dissolved in oxygen-rich water and transported by it to underground pools. Significant concentration of oxygen appeared in the atmosphere only around 2.5 billion years ago, so the first condition was not met earlier than that. You will learn below that the abundance of $^{235}\\mathrm{U}$ decreases relatively fast in time, so the second condition ceased to be satisfied soon after the operation of Oklo's reactor. \n\nWhat made the operation of Oklo's reactor possible was a stable influx of ground water that kept the uranium deposits sufficiently wet. Water is the so-called moderator for the fission reactor: it slows down neutrons emerging from fission reactions, dramatically enhancing the chances of a neutron triggering the fission of a next $^{235}\\mathrm{U}$ nucleus. \n\nIn what follows,in addition to $T_0$, you can use the following numerical values. Energy released by the fission of a single $^{235}\\mathrm{U}$ nucleus: $E_0 = 200 \\mathrm{MeV}$. \nHalf-life of $^{235} \\mathrm{U}$: $\\tau_5 \\approx 7 \\times 10^8$ years. \nHalf-life of $^{238}\\mathrm{U}$: $\\tau_8 \\approx 4.5 \\times 10^9$ years. \nLatent heat of evaporation of water: $L = 2260 \\mathrm{kJ} \\mathrm{kg}^{-1}$. \nSpecific heat of water $c = 4200 \\mathrm{J} \\mathrm{kg}^{-1} \\mathrm{K}^{-1}$. \nAbundance of $^{235}\\mathrm{U}$ in natural uranium today: $R = 0.72\\%$. We define abundance as the number of atoms of the isotope, normalized to the number of atoms of the given element.\n\nAverage abundance of $^{235}\\mathrm{U}$ in the uranium from Oklo's uranium ore today: $R_{O} = 0.62\\%$. \nThe total amount of uranium in Oklo's mine today: $M = 5 \\times {10}^{8}\\mathrm{kg}$. \n\nThe duration of the time period over which \nOklo's reactor operated: $T \\approx 1 \\times {10}^{5}$ year. \nElementary charge: $e = 1.6 \\times {10}^{-19}\\mathrm{C}$. \nAtomic mass unit: $u = 1.66 \\times {10}^{-27} \\mathrm{kg}$. \nAvogadro's number: $N_{A} = 6.02 \\times {10}^{23} \\mathrm{mol}^{-1}$. \n\nNote that: (a) the abundance of other isotopes of uranium besides $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$ is negligibly small; (b) $^{235}\\mathrm{U}$ is not among the decay products of $^{238}\\mathrm{U}$; and (c) fission channels other than the fission of $^{235}\\mathrm{U}$ (e.g., synthesis and fission of plutonium) can be neglected. \n\n(i) What was the abundance of $^{235}\\mathrm{U}$ in natural uranium when the Oklo's reactor operated? \n\n(ii) What was the average power of the Oklo's reactor? \n\nParts (i)–(ii) are preliminary questions and should not be included in the final answer.",
+        "question": "Qualitatively explain why was Oklo's reactor operating in a stable regime and did not blow up. Water inflow rate varied over time; what happened to the reactor when the water inflow rate increased two times?",
+        "marking": [
+            [
+                "Award 0.3 pt if the answer states that neutrons from fission are unlikely to cause further fission unless slowed down (moderation). Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer identifies water as a moderator for slowing down neutrons. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer explains that the reactor was self-regulating because increased power would vaporize water, reducing moderation and slowing the reaction. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer states that when water inflow doubles, the reactor power also doubles. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            ""
+        ],
+        "answer_type": [
+            "Open-Ended"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.5
+        ],
+        "modality": "text-only",
+        "field": "Modern Physics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_2_4",
+        "context": "[Oklo Fission Reactor] \n\nBased on the ratio of the uranium isotopes $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$, as well as the abundances of the isotopes produced by nuclear reactors, researchers have established that self-sustaining natural nuclear reactors operated in Oklo ca $T_0 = 1.8 \\times {10}^{9}$ years ago in Gabon, central Africa. For such reactors to exist, two conditions must be met: (a) presence of deposits with high enough concentration of uranium; (b) sufficiently high abundance of $^{235}\\mathrm{U}$ in natural uranium. Rich uranium ores were created by floods: scattered uranium was dissolved in oxygen-rich water and transported by it to underground pools. Significant concentration of oxygen appeared in the atmosphere only around 2.5 billion years ago, so the first condition was not met earlier than that. You will learn below that the abundance of $^{235}\\mathrm{U}$ decreases relatively fast in time, so the second condition ceased to be satisfied soon after the operation of Oklo's reactor. \n\nWhat made the operation of Oklo's reactor possible was a stable influx of ground water that kept the uranium deposits sufficiently wet. Water is the so-called moderator for the fission reactor: it slows down neutrons emerging from fission reactions, dramatically enhancing the chances of a neutron triggering the fission of a next $^{235}\\mathrm{U}$ nucleus. \n\nIn what follows,in addition to $T_0$, you can use the following numerical values. Energy released by the fission of a single $^{235}\\mathrm{U}$ nucleus: $E_0 = 200 \\mathrm{MeV}$. \nHalf-life of $^{235} \\mathrm{U}$: $\\tau_5 \\approx 7 \\times 10^8$ years. \nHalf-life of $^{238}\\mathrm{U}$: $\\tau_8 \\approx 4.5 \\times 10^9$ years. \nLatent heat of evaporation of water: $L = 2260 \\mathrm{kJ} \\mathrm{kg}^{-1}$. \nSpecific heat of water $c = 4200 \\mathrm{J} \\mathrm{kg}^{-1} \\mathrm{K}^{-1}$. \nAbundance of $^{235}\\mathrm{U}$ in natural uranium today: $R = 0.72\\%$. We define abundance as the number of atoms of the isotope, normalized to the number of atoms of the given element.\n\nAverage abundance of $^{235}\\mathrm{U}$ in the uranium from Oklo's uranium ore today: $R_{O} = 0.62\\%$. \nThe total amount of uranium in Oklo's mine today: $M = 5 \\times {10}^{8}\\mathrm{kg}$. \n\nThe duration of the time period over which \nOklo's reactor operated: $T \\approx 1 \\times {10}^{5}$ year. \nElementary charge: $e = 1.6 \\times {10}^{-19}\\mathrm{C}$. \nAtomic mass unit: $u = 1.66 \\times {10}^{-27} \\mathrm{kg}$. \nAvogadro's number: $N_{A} = 6.02 \\times {10}^{23} \\mathrm{mol}^{-1}$. \n\nNote that: (a) the abundance of other isotopes of uranium besides $^{235}\\mathrm{U}$ and $^{238}\\mathrm{U}$ is negligibly small; (b) $^{235}\\mathrm{U}$ is not among the decay products of $^{238}\\mathrm{U}$; and (c) fission channels other than the fission of $^{235}\\mathrm{U}$ (e.g., synthesis and fission of plutonium) can be neglected. \n\n(i) What was the abundance of $^{235}\\mathrm{U}$ in natural uranium when the Oklo's reactor operated? \n\n(ii) What was the average power of the Oklo's reactor? \n\n(iii) Qualitatively explain why was Oklo's reactor operating in a stable regime and did not blow up. Water inflow rate varied over time; what happened to the reactor when the water inflow rate increased two times? \n\nParts (i)–(iii) are preliminary questions and should not be included in the final answer.",
+        "question": "Estimate the total mass of water that flowed into the Oklo's reactor during its operation period. Express your answer in $kg$.",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer assumes that all energy from the reactor went into heating and vaporizing water. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer approximates $\\Delta T \\approx 100\\ ^\\circ\\mathrm{C}$, e.g., by considering water flows in at $0^\\circ\\mathrm{C}$ and leaves at $100^\\circ\\mathrm{C}$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer considers both heating and vaporization. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the amount of energy absorbed by 1kg of water is $E_w = 2.68 \\times 10^6 \\mathrm{J}$ or similar results. Otherwise, award 0 pt.",
+                "Award 0.6 pt if the total mass of water is correctly computed as $\\nu E_0 / E_w = 9.09 \\times 10^{13} \\mathrm{kg}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$9.09 \\times 10^{13}$}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "kg"
+        ],
+        "point": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Modern Physics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_3_1",
+        "context": "",
+        "question": "[Sticky Ball] \n\nA glass ball of radius $R$ rests on a flat glass plate. A tiny droplet of water (of surface tension $\\sigma$) is injected with a syringe so that water forms a small thin neck between the ball and the plate. Both the ball and the plate are perfectly hydrophilic, i.e. the contact angle of water is $0^{\\circ}$. Find the increase of the normal force ($\\Delta F$) between the plate and the ball caused by the presence of the neck of water.",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer states that the meniscus is (roughly) inverse spherical or uses the constant $r$ to characterise the meniscus as inverse spherical. Otherwise, award 0 pt.",
+                "Award 0.1 pt if the answer states or uses $r \\ll \\rho \\ll R$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer states that surface tension force at the contact is negligible or that $\\Delta F$ comes from pressure difference (either explicitly or implicitly). Otherwise, award 0 pt.",
+                "Award 0.7 pt if the answer uses the relation $2r \\cdot 2R \\approx \\rho^2$. If this relation is not found, partial points can be earned as below. (1) Award 0.1 pt if the relation $CT \\approx 2r$ is used or mentioned. (2) Award 0.1 pt if the relation $R + r \\approx R$ is used. (3) Award 0.2 pt if $AC \\approx \\rho$ is stated or used. (4) Award 0.3 pt if the answer uses $AC^2 \\approx CT \\cdot CD$ or states the interesecting secants theorems. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the pressure difference is given as $\\Delta p = \\sigma / r$. If this equation is not found, partial points can be earned as below. (1) Award 0.2 pt for attempting to use a Laplace-Young formula such as $\\Delta p = \\sigma (1/r_1 + 1/r_2)$. (2) Award 0.3 pt if $\\Delta p = \\sigma / (1/\\rho - 1/r)$ is used. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the vertical cross-sectional area is given as $S = \\pi \\rho^2$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the force difference is expressed as $\\Delta F = \\Delta p \\cdot S$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the final expression $\\Delta F \\approx 4 \\pi \\sigma R$ is obtained. Partial points: (1) Award 0.1 pt if the answer states that $\\Delta F > 0$ or notes that the contact force increases. (2) Award 0.1 pt if the correct dimensionless factor $4\\pi$ and correct dimensions are used (only if the approach is correct). Otherwise, award 0 pt."
+            ],
+            [
+                "Award 0.5 pt if the answer states that the meniscus is (roughly) inverse spherical or uses constant $r$ to characterize the meniscus as inverse spherical. Otherwise, award 0 pt.",
+                "Award 0.1 pt if the answer uses the small-angle approximation $|\\theta| \\ll 1$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the neck radius is approximated as $\\rho \\approx R \\theta$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the distance $AC \\approx \\rho$ is used. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the curvature radius is correctly derived as $r \\approx R \\theta^2 / 4$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the surface tension force $\\Delta F_\\sigma$ is stated to be negligible or $\\Delta F = \\Delta F_p$ is used. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the pressure difference is given as $\\Delta p = \\sigma / r$. If this equation is not found, partial points can be earned as below: (1) Award 0.2 pt if a Laplace-like equation is attempted, such as $\\Delta p = \\sigma (1/r_1 + 1/r_2)$. (2) Award 0.3 pt if the pressure difference is given as $\\Delta p = \\sigma (1/\\rho - 1/r)$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the correct effective area $S = \\pi \\rho^2$ is used for computing the force. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the force from pressure is computed using $\\Delta F_p = - S \\Delta p$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the final result $\\Delta F \\approx 4 \\pi \\sigma R$ is obtained. Partial points: (1) Award 0.1 pt if the answer states that $\\Delta F > 0$ or notes that the contact force increases. (2) Award 0.1 pt if the correct dimensionless factor $4\\pi$ and correct dimensions are used (only if the approach is correct). Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$\\Delta F \\approx 4 \\pi \\sigma R$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            4.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_4_1",
+        "context": "[Totality] \n\nTotal solar eclipses are a rare phenomenon which occur when the Moon completely covers the disk of the Sun for some parts of the Earth. This doesn't happen during every solar eclipse because the Moon's apparent size in the sky is sometimes too small to fully cover the Sun, but also because the Moon's shadow usually misses the Earth due its orbital inclination. As a result, total solar eclipses occur on average every 18 months. \n\nLet us consider a total solar eclipse where during the peak, the centre-points of Earth, the Moon and the Sun lie on a line on the same plane as the equator. We measure that right before the total solar eclipse ends at latitude $\\lambda = 28.5^{\\circ}$, the totality lasts for $t_0 = 2 \\mathrm{min}$. Earth's radius is $r_{e} = 6370 \\mathrm{km}$, Moon's radius is $r_{m} = 1740 \\mathrm{km}$, orbital period of the Moon $T_{m} = 27.3 \\mathrm{d}$,orbital radius of the Moon ${R}_{m} = 384000 \\mathrm{km}$. One day on Earth is $T_0 = 24 \\mathrm{hrs}$.",
+        "question": "For how long is there a place on Earth where the total solar eclipse is observable? Express your answer in hours.",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer explains that the Moon's shadow speed on Earth can be approximated by the Moon's position. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the Moon's speed is correctly calculated using $v_m = \\frac{2 \\pi R_m}{T_m}$ and the value $v_m \\approx 1.02\\ \\mathrm{km/s}$ is obtained. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the final eclipse duration expression $T_{\\mathrm{ecl}} = \\frac{2 r_e}{v_m} = \\frac{r_e}{\\pi R_m} T_m$ is correctly obtained, where the numerical result of $T_{\\mathrm{ecl}}$ is $3.46 h$. Partial points: Deduct 0.2 pt for a minor mistake in the final expression (e.g., missing constant or slight dimensional inconsistency). Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{3.46}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "h"
+        ],
+        "point": [
+            1.5
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_4_2",
+        "context": "[Totality] \n\nTotal solar eclipses are a rare phenomenon which occur when the Moon completely covers the disk of the Sun for some parts of the Earth. This doesn't happen during every solar eclipse because the Moon's apparent size in the sky is sometimes too small to fully cover the Sun, but also because the Moon's shadow usually misses the Earth due its orbital inclination. As a result, total solar eclipses occur on average every 18 months. \n\nLet us consider a total solar eclipse where during the peak, the centre-points of Earth, the Moon and the Sun lie on a line on the same plane as the equator. We measure that right before the total solar eclipse ends at latitude $\\lambda = 28.5^{\\circ}$, the totality lasts for $t_0 = 2 \\mathrm{min}$. Earth's radius is $r_{e} = 6370 \\mathrm{km}$, Moon's radius is $r_{m} = 1740 \\mathrm{km}$, orbital period of the Moon $T_{m} = 27.3 \\mathrm{d}$,orbital radius of the Moon ${R}_{m} = 384000 \\mathrm{km}$. One day on Earth is $T_0 = 24 \\mathrm{hrs}$. \n\n(i) For how long is there a place on Earth where the total solar eclipse is observable? \n\nPart (i) is a preliminary question and should not be included in the final answer.",
+        "question": "How many degrees in longitude on Earth does the total solar eclipse cover?",
+        "marking": [
+            [
+                "Award 0.3 pt if the answer identifies that the eclipse would cover $\\pi$ radians without the Earth's rotation. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly explains that the angle becomes smaller than $\\pi$ due to Earth's rotation. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the final formula $\\left|180 \\left(1 - \\frac{2 T_{\\mathrm{ecl}}}{T_0} \\right)\\right|$ and answer $\\approx 128$ degrees is given correctly. Partial points: Deduct 0.3 pt if the answer incorrectly assumes Earth’s rotation goes against the Moon’s shadow and obtains an angle greater than $\\pi$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{128}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "degree"
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_4_3",
+        "context": "[Totality] \n\nTotal solar eclipses are a rare phenomenon which occur when the Moon completely covers the disk of the Sun for some parts of the Earth. This doesn't happen during every solar eclipse because the Moon's apparent size in the sky is sometimes too small to fully cover the Sun, but also because the Moon's shadow usually misses the Earth due its orbital inclination. As a result, total solar eclipses occur on average every 18 months. \n\nLet us consider a total solar eclipse where during the peak, the centre-points of Earth, the Moon and the Sun lie on a line on the same plane as the equator. We measure that right before the total solar eclipse ends at latitude $\\lambda = 28.5^{\\circ}$, the totality lasts for $t_0 = 2 \\mathrm{min}$. Earth's radius is $r_{e} = 6370 \\mathrm{km}$, Moon's radius is $r_{m} = 1740 \\mathrm{km}$, orbital period of the Moon $T_{m} = 27.3 \\mathrm{d}$,orbital radius of the Moon ${R}_{m} = 384000 \\mathrm{km}$. One day on Earth is $T_0 = 24 \\mathrm{hrs}$. \n\n(i) For how long is there a place on Earth where the total solar eclipse is observable? \n\n(ii) How many degrees in longitude on Earth does the total solar eclipse cover? \n\nParts (i)–(ii) are preliminary questions and should not be included in the final answer.",
+        "question": "What is the width of the path of totality near the equator? Express your answer in $km$.",
+        "marking": [
+            [
+                "Award 0.4 pt if the answer realizes that the velocities are perpendicular at the end of the eclipse. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer computes the width of the eclipse at some point on Earth as $w_\\lambda = v_m t_0 = \\frac{2\\pi R_m t_0}{T_m} \\approx 123 \\mathrm{km}$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly explains how to translate the width to the width at the equator using angular diameter $\\alpha = 2r_m / R_m$ and computes $\\alpha r_e \\approx 57.7 \\mathrm{km}$. Otherwise, award 0 pt.",
+                "Award 0.1 pt if the final expression $w_\\text{eq} = v_m t_0 + \\alpha r_e = 180 \\mathrm{km}$ is correct. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{180}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "km"
+        ],
+        "point": [
+            1.5
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_4_4",
+        "context": "[Totality] \n\nTotal solar eclipses are a rare phenomenon which occur when the Moon completely covers the disk of the Sun for some parts of the Earth. This doesn't happen during every solar eclipse because the Moon's apparent size in the sky is sometimes too small to fully cover the Sun, but also because the Moon's shadow usually misses the Earth due its orbital inclination. As a result, total solar eclipses occur on average every 18 months. \n\nLet us consider a total solar eclipse where during the peak, the centre-points of Earth, the Moon and the Sun lie on a line on the same plane as the equator. We measure that right before the total solar eclipse ends at latitude $\\lambda = 28.5^{\\circ}$, the totality lasts for $t_0 = 2 \\mathrm{min}$. Earth's radius is $r_{e} = 6370 \\mathrm{km}$, Moon's radius is $r_{m} = 1740 \\mathrm{km}$, orbital period of the Moon $T_{m} = 27.3 \\mathrm{d}$,orbital radius of the Moon ${R}_{m} = 384000 \\mathrm{km}$. One day on Earth is $T_0 = 24 \\mathrm{hrs}$. \n\n(i) For how long is there a place on Earth where the total solar eclipse is observable? \n\n(ii) How many degrees in longitude on Earth does the total solar eclipse cover? \n\n(iii) What is the width of the path of totality near the equator? \n\nParts (i)–(iii) are preliminary questions and should not be included in the final answer.",
+        "question": "What is the longest amount of time the total eclipse is visible for a single location on Earth? Express your answer in minutes.",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer shows that the eclipse is observable for the longest time at the equator. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly finds the velocity of the surface of the Earth as $v_e = \\frac{2\\pi r_e}{T_0} = 0.46 \\mathrm{km/s}$ and the relative speed of the Moon's shadow as $v_{\\text{rel}} = \\sqrt{v_m^2 + v_e^2 - 2 v_m v_e \\cos \\lambda} = 0.654 \\mathrm{km/s}$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer derives the formula for the maximum duration of the eclipse: $t_\\text{eq} = \\frac{w_\\text{eq}}{v_\\text{rel}} = 276 \\mathrm{s} = 4.6 \\mathrm{min}$. Partial points: Deduct 0.3 pt if the angle $\\lambda$ between the two velocities is not taken into account in the relative velocity calculation. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{4.6}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "min"
+        ],
+        "point": [
+            1.5
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_4_5",
+        "context": "[Totality] \n\nTotal solar eclipses are a rare phenomenon which occur when the Moon completely covers the disk of the Sun for some parts of the Earth. This doesn't happen during every solar eclipse because the Moon's apparent size in the sky is sometimes too small to fully cover the Sun, but also because the Moon's shadow usually misses the Earth due its orbital inclination. As a result, total solar eclipses occur on average every 18 months. \n\nLet us consider a total solar eclipse where during the peak, the centre-points of Earth, the Moon and the Sun lie on a line on the same plane as the equator. We measure that right before the total solar eclipse ends at latitude $\\lambda = 28.5^{\\circ}$, the totality lasts for $t_0 = 2 \\mathrm{min}$. Earth's radius is $r_{e} = 6370 \\mathrm{km}$, Moon's radius is $r_{m} = 1740 \\mathrm{km}$, orbital period of the Moon $T_{m} = 27.3 \\mathrm{d}$,orbital radius of the Moon ${R}_{m} = 384000 \\mathrm{km}$. One day on Earth is $T_0 = 24 \\mathrm{hrs}$. \n\n(i) For how long is there a place on Earth where the total solar eclipse is observable? \n\n(ii) How many degrees in longitude on Earth does the total solar eclipse cover? \n\n(iii) What is the width of the path of totality near the equator? \n\n(iv) What is the longest amount of time the total eclipse is visible for a single location on Earth? \n\nParts (i)–(iv) are preliminary questions and should not be included in the final answer.",
+        "question": "For how long is the total eclipse near the location described in (iii), at the distance of $a = 50 km$ from the centreline of the eclipse path? Express your answer in minutes.",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer explains that the relative velocity $v_\\text{rel}$ is approximately as $v_{\\text{rel}} = \\sqrt{v_m^2 + v_e^2 - 2 v_m v_e \\cos \\lambda} = 0.654 \\mathrm{km/s}$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly applies the geometry and gives the correct time as $\\frac{1}{v_\\text{rel}}\\sqrt{w_\\text{eq}^2 - 4a^2} = 230 \\mathrm{s} = 3.8 \\mathrm{min}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{3.8}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "min"
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_4_6",
+        "context": "[Totality] \n\nTotal solar eclipses are a rare phenomenon which occur when the Moon completely covers the disk of the Sun for some parts of the Earth. This doesn't happen during every solar eclipse because the Moon's apparent size in the sky is sometimes too small to fully cover the Sun, but also because the Moon's shadow usually misses the Earth due its orbital inclination. As a result, total solar eclipses occur on average every 18 months. \n\nLet us consider a total solar eclipse where during the peak, the centre-points of Earth, the Moon and the Sun lie on a line on the same plane as the equator. We measure that right before the total solar eclipse ends at latitude $\\lambda = 28.5^{\\circ}$, the totality lasts for $t_0 = 2 \\mathrm{min}$. Earth's radius is $r_{e} = 6370 \\mathrm{km}$, Moon's radius is $r_{m} = 1740 \\mathrm{km}$, orbital period of the Moon $T_{m} = 27.3 \\mathrm{d}$,orbital radius of the Moon ${R}_{m} = 384000 \\mathrm{km}$. One day on Earth is $T_0 = 24 \\mathrm{hrs}$. \n\n(i) For how long is there a place on Earth where the total solar eclipse is observable? \n\n(ii) How many degrees in longitude on Earth does the total solar eclipse cover? \n\n(iii) What is the width of the path of totality near the equator? \n\n(iv) What is the longest amount of time the total eclipse is visible for a single location on Earth? \n\n(v) For how long is the total eclipse near the location described in (iii), at the distance of $a = 50 km$ from the centreline of the eclipse path? \n\nParts (i)–(v) are preliminary questions and should not be included in the final answer.",
+        "question": "Find the average time interval (in years) between two total solar eclipses for a given location on Earth by making the following simplifying assumptions:\n\n(a) the average width of the full eclipse path is equal to the arithmetic average of its smallest and largest width; \n(b) typical width of a full eclipse path is half of the average width of the eclipse studied above; \n(c) typical length of a full eclipse path is equal to the length of the eclipse path studied above if the Earth were not rotating; \n(d) total solar eclipses occur with equal likelihood anywhere on Earth.",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer explains that the probability per eclipse is equal to the area covered by the eclipse divided by the total area of the earth. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly calculates the area covered by one eclipse as $\\pi r_e (w_\\lambda + w_\\text{eq})/4$ using the given assumptions. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly multiplies the inverse probability with the duration between eclipses and finds the correct answer, i.e. $\\frac{16r_e}{w_\\lambda + w_\\text{eq}} \\cdot 18 \\text{months} \\approx 500 \\text{years}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{500}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "years"
+        ],
+        "point": [
+            1.5
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_6_1",
+        "context": "",
+        "question": "[Cones] \n\nThe photo below shows a self-anamorphic drawing - a red heart in green background. The reflection of the red heart in the conical mirror is a reduced green heart. What is the apex angle of the conical mirror? Express your answer in degrees. You can take measurements from the photo. The distance where the photo was taken was much larger than the diameter of the red heart.\n\n[figure1]",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer states that the image is formed by vertical rays because the camera is far away. Otherwise, award 0 pt.",
+                "Award 1 pt if the answer gives a correct explicit expression for $\\theta$ in terms of $r$ or another measurable quantity. If this is not found, partial points can be earned as below: (1) Award 0.2 pt if the answer provides a correct geometrical figure of the setup. (2) Award 0.5 pt if the answer derives $r = \\tan \\theta / \\tan 2\\theta$ or equivalent or a correct implicit equation for $\\theta$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the final numerical result is correct and within $2\\theta \\in [65^\\circ, 76^\\circ]$. Partial points: Deduct 0.2 pt if only $\\theta$ is given and $2\\theta$ is not reported. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{71}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "degrees"
+        ],
+        "point": [
+            2.0
+        ],
+        "modality": "text+data figure",
+        "field": "Optics",
+        "source": "NBPhO_2024",
+        "image_question": [
+            "image_question/NBPhO_2024_6_1_1.png"
+        ]
+    },
+    {
+        "id": "NBPhO_2024_6_2",
+        "context": "",
+        "question": "[Cones] \n\nA point-like puck of mass $m$ can slide frictionlessly along the internal surface of a cone of half apex angle $\\theta$. The gravitational acceleration is $g$ and points along the symmetry axis of the cone at the apex. The puck starts sliding from a point $P$ on the surface of the cone with such a horizontal velocity that it will stay moving at the same fixed height while performing uniform circular motion of radius $R$. What is its speed $v$?",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer states the correct balance of forces. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer derives the correct expression for $v$ as $v = \\sqrt{Rg \\cot \\theta}$. Partial points: Deduct 0.1 pt if there are mistakes in trigonometry. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$v = \\sqrt{Rg \\cot \\theta}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_6_3",
+        "context": "[Cones] \n\n(ii) A point-like puck of mass $m$ can slide frictionlessly along the internal surface of a cone of half apex angle $\\theta$. The gravitational acceleration is $g$ and points along the symmetry axis of the cone at the apex. The puck starts sliding from a point $P$ on the surface of the cone with such a horizontal velocity that it will stay moving at the same fixed height while performing uniform circular motion of radius $R$. What is its speed $v$? \n\nPart (ii) is a preliminary question and should not be included in the final answer.",
+        "question": "Now the puck starts sliding horizontally from the same point $P$ as before, but its initial speed is reduced to $v/2$. What is the smallest distance between the puck and the cone's axis during the subsequent motion?",
+        "marking": [
+            [
+                "Award 0.1 pt if the answer mentions using energy conservation. Otherwise, award 0 pt.",
+                "Award 0.1 pt if the answer mentions using angular momentum conservation. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the energy conservation equation is correctly written as $E = \\frac{v^2}{8} + g h_f = \\frac{v_f^2}{2}$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the angular momentum conservation equation is correctly written as $\\frac{v}{2} R = v_f R_f$. Otherwise, award 0 pt.",
+                "Award 0.1 pt if the relation between $h_f$ and $R_f$ is correctly given as $h_f = (R - R_f) \\cot \\theta$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the correct third degree polynomial in $R_f$ is derived: $8R_f^3 - 9R R_f^2 + R^3 = 0$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the physically meaningful root $R_f = \\frac{1 + \\sqrt{33}}{16}R$ is correctly selected. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the final answer $R_f \\approx 0.42 R$ is correctly given. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$0.42 R$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            2.5
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_6_4",
+        "context": "[Cones] \n\n(ii) A point-like puck of mass $m$ can slide frictionlessly along the internal surface of a cone of half apex angle $\\theta$. The gravitational acceleration is $g$ and points along the symmetry axis of the cone at the apex. The puck starts sliding from a point $P$ on the surface of the cone with such a horizontal velocity that it will stay moving at the same fixed height while performing uniform circular motion of radius $R$. What is its speed $v$? \n\n(iii) Now the puck starts sliding horizontally from the same point $P$ as before, but its initial speed is reduced to $v/2$. What is the smallest distance between the puck and the cone's axis during the subsequent motion? \n\nParts (ii)–(iii) are preliminary questions and should not be included in the final answer.",
+        "question": "Now, the cone and the puck are moved to weightlessness. The puck starts again from the point $P$ with the same velocity as in part (ii). By how many degrees will the radius vector drawn from the cone's axis to the puck rotate during the subsequent motion? Assume that the cone is infinitely long. Express your answer in radians.",
+        "marking": [
+            [
+                "Award 0.2 pt if the answer mentions the idea of unfolding the cone. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer states that the puck’s trajectory is a straight line in the folded plane. Otherwise, award 0 pt.",
+                "Award 0.6 pt if the answer correctly describes a $90^{\\circ}$ ($\\pi/2$) rotation in the folded diagram, or provides a correct corresponding figure. Otherwise, award 0 pt.",
+                "Award 0.7 pt if the answer correctly relates the rotation in the folded plane to the rotation of the radius vector, noting that a $360^{\\circ}$ rotation of the radius vector corresponds to a $2\\phi$ rotation in the drawing, and correctly derives the number of radians of rotation as $\\frac{\\pi}{2 \\sin \\theta}$. Partial points: if the answer incorrectly uses $\\theta$ instead of $2 \\theta$ as the apex angle, award 0.4 pt. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$\\frac{\\pi}{2 \\sin \\theta}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            "radians"
+        ],
+        "point": [
+            2.5
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_7_1",
+        "context": "The dispersion relation (i.e. the dependence of the circular frequency $\\omega$ on the wave vector $k = \\frac{2\\pi}{\\lambda}$) of capillary-gravity waves is $\\omega^2 = g k^{\\alpha} + \\frac{\\sigma}{\\rho} k^{\\beta}$, where $\\sigma$ denotes the surface tension, $g = 9.81 m/s^2$, and $\\rho = 1000 \\mathrm{kg} m^{-3}$.",
+        "question": "(1) Determine the value of the exponent $\\alpha$. \n(2) Determine the value of the exponent $\\beta$.",
+        "marking": [
+            [
+                "Award 0.2 pt if the answer applies dimensional analysis (or an equivalent technique) to determine the exponents. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly states the units of surface tension $\\sigma$ as $\\mathrm{kg/s^2}$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly finds $\\alpha = 1$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly finds $\\beta = 3$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$\\alpha = 1$}",
+            "\\boxed{$\\beta = 3$}"
+        ],
+        "answer_type": [
+            "Numerical Value",
+            "Numerical Value"
+        ],
+        "unit": [
+            null,
+            null
+        ],
+        "point": [
+            0.5,
+            0.5
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_7_2",
+        "context": "The dispersion relation (i.e. the dependence of the circular frequency $\\omega$ on the wave vector $k = \\frac{2\\pi}{\\lambda}$) of capillary-gravity waves is $\\omega^2 = g k^{\\alpha} + \\frac{\\sigma}{\\rho} k^{\\beta}$, where $\\sigma$ denotes the surface tension, $g = 9.81 m/s^2$, and $\\rho = 1000 \\mathrm{kg} m^{-3}$.\n\n[figure1] \n\n(i) Determine the values of the exponents $\\alpha$ and $\\beta$. \n\nPart (i) is a preliminary question and should not be included in the final answer.",
+        "question": "In the figure, we can see how an object moving with a constant speed $U = 60 \\mathrm{cm} \\mathrm{s}^{-1}$ generates a wake - a set of waves of different wavelengths. Pay attention to the short-wavelength waves whose wave crest extends from the object almost up to the edges of the photo: the presence of a very long wavefront testifies that for these particular waves, the phase and group velocities are equal.\n\n(1) Determine the expression of the surface tension $\\sigma$ of water. \n(2) Estimate the numerical value of $\\sigma$ in $g/s^2$. \nYou can take measurements from the photo. Note that while phase speed is the speed of a constant phase of the wave, the group speed $v_g = \\frac{\\mathrm{d} \\omega}{\\mathrm{d} k}$ is the speed of a wave packet (a train of waves).",
+        "marking": [
+            [
+                "Award 0.6 pt if the answer correctly identifies $\\frac{d\\omega}{dk} = \\omega / k$. Otherwise, award 0 pt.",
+                "Award 0.6 pt if the answer gives the correct angle $\\sin(\\mu) = \\omega / (U k)$. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer measures $\\mu$ from the picture. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer provides the measured $\\mu$ within the range $[20^{\\circ}, 30^{\\circ}]$. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer correctly derivatives $d\\omega/dk$ from the dispersion relation and obtains $2 \\omega \\frac{d\\omega}{dk} = g + 3 \\frac{\\sigma}{\\rho} k^2$. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer correctly derives the expression for $\\sigma$ as $\\sigma = \\frac{\\rho U^4 \\sin^4 \\mu}{4 g}$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly obtains the final value $\\sigma \\approx 60 g s^{-2}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$\\sigma = \\frac{\\rho U^4 {\\sin}^{4} \\mu}{4g}$}",
+            "\\boxed{60}"
+        ],
+        "answer_type": [
+            "Expression",
+            "Numerical Value"
+        ],
+        "unit": [
+            null,
+            "g/s^2"
+        ],
+        "point": [
+            1.5,
+            1.5
+        ],
+        "modality": "text+data figure",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": [
+            "image_question/NBPhO_2024_7_2_1.png"
+        ]
+    },
+    {
+        "id": "NBPhO_2024_8_1",
+        "context": "Two airplanes pass each other while flying at a constant altitudes. While they have identical airspeeds, their ground speeds are $v_1$ and $v_2$, respectively, and the angle between the velocity vectors is $\\alpha$.",
+        "question": "Based on the above knowledge, what is the minimal possible value of the airspeed of the planes?",
+        "marking": [
+            [
+                "Award 0.2 pt if the answer represents the problem using vectors and correctly adds the velocities as $\\vec{v}_1 = \\vec{w} + \\vec{u}_1$ and $\\vec{v}_2 = \\vec{w} + \\vec{u}_2$, where $\\vec{w}$ is the speed of the wind at the airplanes' altitude, $\\vec{u}_1$ and $\\vec{u}_2$ are the planes' respective speeds in absence of wind. Partial points: award 0.1 pt if the answer has the wrong order or sign in vector addition. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer identifies that the wind velocity vector $\\vec{w}$ lies on the perpendicular bisector of segment $(AB)$, where the segment $(OA)$ corresponds to the vector $\\vec{v}_1$ ($\\vec{v}_1 = \\vec{w} + \\vec{u}_1$), the segment $(OB)$ corresponds to the vector $\\vec{v}_2$ ($\\vec{v}_2 = \\vec{w} + \\vec{u}_2$), $\\vec{w}$ is the speed of the wind at the airplanes' altitude, $\\vec{u}_1$ and $\\vec{u}_2$ are the planes' respective speeds in absence of wind. Otherwise, award 0 pt.",
+                "Award 0.1 pt if the answer states that the minimum airspeed requires $\\vec{w}$ to lie on the intersection of $AB$ and the perpendicular bisector, where the segment $(OA)$ corresponds to the vector $\\vec{v}_1$ ($\\vec{v}_1 = \\vec{w} + \\vec{u}_1$), the segment $(OB)$ corresponds to the vector $\\vec{v}_2$ ($\\vec{v}_2 = \\vec{w} + \\vec{u}_2$), $\\vec{w}$ is the speed of the wind at the airplanes' altitude, $\\vec{u}_1$ and $\\vec{u}_2$ are the planes' respective speeds in absence of wind. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer calculates the correct formula for the minimum airspeed as $|\\vec{u}_1|_{\\min} = \\frac{|\\vec{v}_1 - \\vec{v}_2|}{2} = \\frac{1}{2} \\sqrt{v_1^2 + v_2^2 - 2 v_1 v_2 \\cos \\alpha}$. Partial points: award 0.2 pt if the answer has a small error but is otherwise reasonable with correct units, or if the answer is not expanded when it can be simply expanded. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$\\frac{1}{2} \\sqrt{v_1^{2} + v_2^{2} - 2 v_1 v_2 \\cos \\alpha}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_8_2",
+        "context": "Two airplanes pass each other while flying at a constant altitudes. While they have identical airspeeds, their ground speeds are $v_1$ and $v_2$, respectively, and the angle between the velocity vectors is $\\alpha$.",
+        "question": "Based on the above knowledge, what is the minimal possible value of the wind speed $|\\vec{w}|$ at the altitudes of the planes?",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer represents the problem using vectors and correctly adds the velocities as $\\vec{v}_1 = \\vec{w} + \\vec{u}_1$ and $\\vec{v}_2 = \\vec{w} + \\vec{u}_2$, where $\\vec{w}$ is the speed of the wind at the airplanes' altitude, $\\vec{u}_1$ and $\\vec{u}_2$ are the planes' respective speeds in absence of wind. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer states that the wind velocity vector $\\vec{w}$ lies on the perpendicular bisector of segment $(AB)$, where the segment $(OA)$ corresponds to the vector $\\vec{v}_1$ ($\\vec{v}_1 = \\vec{w} + \\vec{u}_1$), the segment $(OB)$ corresponds to the vector $\\vec{v}_2$ ($\\vec{v}_2 = \\vec{w} + \\vec{u}_2$), $\\vec{w}$ is the speed of the wind at the airplanes' altitude, $\\vec{u}_1$ and $\\vec{u}_2$ are the planes' respective speeds in absence of wind. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer states that the wind velocity $\\vec{w}$ is smallest when it is perpendicular to $l$, where $l$ is the perpendicular bisector of the segment $(AB)$, the segment $(OA)$ corresponds to the vector $\\vec{v}_1$ ($\\vec{v}_1 = \\vec{w} + \\vec{u}_1$), the segment $(OB)$ corresponds to the vector $\\vec{v}_2$ ($\\vec{v}_2 = \\vec{w} + \\vec{u}_2$), $\\vec{w}$ is the speed of the wind at the airplanes' altitude, $\\vec{u}_1$ and $\\vec{u}_2$ are the planes' respective speeds in absence of wind. Otherwise, award 0 pt.",
+                "Award 1.5 pt if the answer correctly calculates the minimal wind speed as $|\\vec{w}|_{\\min} = \\frac{|v_1^2 - v_2^2|}{2\\sqrt{v_1^2 + v_2^2 - 2 v_1 v_2 \\cos \\alpha}}$. Partial points: award 1.0 pt if there is a small error but the answer is otherwise reasonable with correct units, or if the answer is not expanded when it can be simply expanded. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$\\frac{\\left| v_1^{2} - v_2^{2} \\right|}{2 \\sqrt{v_1^{2} + v_2^{2} - 2 v_1 v_2 \\cos \\alpha}}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            3.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_8_3",
+        "context": "Two airplanes pass each other while flying at a constant altitudes. While they have identical airspeeds, their ground speeds are $v_1$ and $v_2$, respectively, and the angle between the velocity vectors is $\\alpha$.",
+        "question": "If now $v_1 = v_2 = v$, but it is known that the airspeed of one of the planes is two times bigger than that of the other. What is the minimal possible value of the wind speed at the altitudes of the planes?",
+        "marking": [
+            [
+                "Award 0.2 pt if the answer represents the problem using vectors and correctly adds the velocities as $\\vec{v}_1 = \\vec{w} + \\vec{u}_1$ and $\\vec{v}_2 = \\vec{w} + \\vec{u}_2$, where $\\vec{w}$ is the speed of the wind at the airplanes' altitude, $\\vec{u}_1$ and $\\vec{u}_2$ are the planes' respective speeds in absence of wind. Otherwise, award 0 pt.",
+                "Award 1.3 pt if the answer states that the wind velocity vector $\\vec{w}$ lies on the Apollonius circle (name not required). Partial points: award 0.5 pt if the answer realizes it is a circle but provides an incorrect one. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly calculates the radius $R_A$ of the Apollonius circle as $R_A = \\frac{2|\\vec{v}_1 - \\vec{v}_2|}{3}$, where $R_A$ is the distance from the circle's center to its intersection point with the line $AB$. Otherwise, award 0 pt.",
+                "Award 1 pt if the answer correctly calculates the minimal wind speed as $|\\vec{w}|_{\\min} = \\left| \\frac{4}{3}\\vec{v}_2 - \\frac{1}{3}\\vec{v}_1 \\right| - \\frac{2}{3}|\\vec{v}_2 - \\vec{v}_1| = \\frac{\\sqrt{17 - 8 \\cos \\alpha} - 4 \\sin(\\frac{\\alpha}{2})}{3} v$. Partial points: award 0.5 pt if there is a small error but the answer is otherwise reasonable with correct units, or if the answer is not expanded when it can be simply expanded. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$\\frac{\\sqrt{17 - 8 \\cos \\alpha} - 4 \\sin(\\frac{\\alpha}{2})}{3} v$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            3.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2024_9_1",
+        "context": "Three identical small iron balls were initially arranged in an equilateral triangle formation, connected by massless nonstretchable threads. Upon being thrown into the air, the system experienced the following conditions: (a) all threads were taut initially; (b) all balls possessed different velocities; (c) all velocities were confined to the plane of the triangle as the system underwent free fall within Earth's gravitational field. At a certain moment $t = 0$, two threads ruptured, leaving two balls tethered together while the third ball separated from the rest of the system. The accompanying diagram depicts the positions of all three balls and the remaining thread within the plane of their initial arrangement at a later moment of time $t = T$ when all the balls were still continuing their free fall. To answer the questions below, you can take measurements from the figure.\n\n[figure1]",
+        "question": "By how many degrees did the remaining thread rotate during the time period from $t = 0$ to $t = T$? Express your answer in radians.",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer shows understanding of the situation, i.e., recognizes that the motion is rotating in a plane perpendicular to the ground. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly determines the center of mass $G$ and explains that $|GA| = 2|GM|$, where $A$ is the position of the detached ball and $M$ is the midpoint of the remaining string. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer draws or clearly describes the trajectory after separation and does so correctly, i.e., the lines of motion of $A$ and $M$ are parallel but not colinear, representing the motion of the center of mass of each part. Otherwise, award 0 pt.",
+                "Award 1.0 pt if the answer correctly shows that $\\omega_1 = \\omega_2$ based on the triangle configuration and the two connected balls, where $\\omega_1$ is the angular velocity of the detached ball and $\\omega_2$ is the angular velocity of the two balls that remain connected. Otherwise, award 0 pt.",
+                "Award 0.6 pt for the answer correctly writing each of the following formulas (0.2 pt each): $v_1 = \\omega_1 r$, $s_1 = v_1 T$, and $\\alpha = \\omega_2 T$, where $v_1$ is the linear speed of the detached ball, $\\omega_1$ is the angular velocity of the detached ball, $r$ is the radius of its circular trajectory, $s_1$ is the arc length traveled by the detached ball after separation, $T$ is the travel time after separation, $\\alpha$ is the rotation angle of the two connected balls, and $\\omega_2$ is the angular velocity of the two connected balls. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer obtains the correct final expression for $\\alpha$ as $\\alpha = \\frac{s_1}{r}$, where $\\alpha$ is the rotation angle of the two connected balls, $s_1$ is the arc length traveled by the detached ball after separation, and $r$ is the radius of its circular trajectory. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer obtains the correct numerical final value $\\alpha \\approx 5.6 \\text{rad}$ (allowing for some tolerance for small measurement errors). Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{5.6}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "radian"
+        ],
+        "point": [
+            4.0
+        ],
+        "modality": "text+data figure",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": [
+            "image_question/NBPhO_2024_9_1_1.png"
+        ]
+    },
+    {
+        "id": "NBPhO_2024_9_2",
+        "context": "Three identical small iron balls were initially arranged in an equilateral triangle formation, connected by massless nonstretchable threads. Upon being thrown into the air, the system experienced the following conditions: (a) all threads were taut initially; (b) all balls possessed different velocities; (c) all velocities were confined to the plane of the triangle as the system underwent free fall within Earth's gravitational field. At a certain moment $t = 0$, two threads ruptured, leaving two balls tethered together while the third ball separated from the rest of the system. The accompanying diagram depicts the positions of all three balls and the remaining thread within the plane of their initial arrangement at a later moment of time $t = T$ when all the balls were still continuing their free fall. To answer the questions below, you can take measurements from the figure.\n\n[figure1] \n\n(i) By how many degrees did the remaining thread rotate during the time period from $t = 0$ to $t = T$? \n\nPart (i) is a preliminary question and should not be included in the final answer.",
+        "question": "Was the rotation clockwise or counterclockwise?\n\n(A) Clockwise \n(B) Counterclockwise",
+        "marking": [
+            [
+                "Award 1.0 pt if the answer correctly determines that the rotation is counterclockwise or selects option B as the final answer. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "point": [
+            1.0
+        ],
+        "modality": "text+data figure",
+        "field": "Mechanics",
+        "source": "NBPhO_2024",
+        "image_question": [
+            "image_question/NBPhO_2024_9_1_1.png"
+        ]
+    }
+]

data/NBPhO_2025.json

@@ -0,0 +1,686 @@
+[
+    {
+        "information": "None."
+    },
+    {
+        "id": "NBPhO_2025_1_1",
+        "context": "[Flying Dumbbell] \n\nIn this problem, we shall study the dynamics of a dumbbell consisting of two steel balls, each with radius $r = 1 \\mathrm{cm}$, connected by a steel rod with diameter $d = 1 \\mathrm{mm}$ and length $l = 10 \\mathrm{cm}$, in the absence of gravity. Unless instructed otherwise, assume steel is perfectly elastic. You may simplify your calculations by assuming $l \\gg r$.",
+        "question": "Given that the Young's modulus of steel is $Y = 2 \\times 10^{11} \\mathrm{Pa}$ and the density of steel is $\\rho = 7800 \\mathrm{kg}\\mathrm{m}^{-3}$, determine the period $T$ of free longitudinal oscillations of the dumbbell. (Do not consider oscillations with standing waves in the rod where the balls remain almost at rest.) Young's modulus is the ratio of a material's stress (force per unit area) to its strain (relative deformation). Express your answer in $\\mathrm{ms}$.",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer explains that the oscillation is symmetric around the centre of the rod (or invokes Newton's third law). Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly expresses the stiffness of the half-rod as $k = Y \\frac{\\pi}{2} d^2 / l$. Partial points: award 0.3 pt if there is a minor mistake in the stiffness expression. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly gives the mass of the ball as $m = \\frac{4}{3} \\pi r^3 \\rho$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer realises that the system can be treated as a spring. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly obtains the oscillation period formula as $T = 2 \\pi \\sqrt{m/k}$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly obtains the final answer for the oscillation period as $T \\approx 0.64 \\mathrm{ms}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{0.64}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "ms"
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2025",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2025_1_2",
+        "context": "[Flying Dumbbell] \n\nIn this problem, we shall study the dynamics of a dumbbell consisting of two steel balls, each with radius $r = 1 \\mathrm{cm}$, connected by a steel rod with diameter $d = 1 \\mathrm{mm}$ and length $l = 10 \\mathrm{cm}$, in the absence of gravity. Unless instructed otherwise, assume steel is perfectly elastic. You may simplify your calculations by assuming $l \\gg r$. \n\n(i) Given that the Young's modulus of steel is $Y = 2 \\times 10^{11} \\mathrm{Pa}$ and the density of steel is $\\rho = 7800 \\mathrm{kg}\\mathrm{m}^{-3}$, determine the period $T$ of free longitudinal oscillations of the dumbbell. (Do not consider oscillations with standing waves in the rod where the balls remain almost at rest.) Young's modulus is the ratio of a material's stress (force per unit area) to its strain (relative deformation). \n\nPart (i) is a preliminary question and should not be included in the final answer.",
+        "question": "Estimate the impact time $\\tau$ when a steel ball bounces off a steel wall. Express your answer in $\\mathrm{\\mu s}$.",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer realises that the compressed ball is essentially a compression wave. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly gives the formula for the speed of sound as $c_s = \\sqrt{Y / \\rho}$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly gives the relation between time, radius and speed as $\\tau \\approx 2 r / c_s = 2r \\sqrt{\\rho / Y}$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly obtains the final answer $\\tau \\approx 4 \\mathrm{\\mu s}$. Otherwise, award 0 pt."
+            ],
+            [
+                "Award 0.5 pt if the answer realises that the ball can be thought of as a spring. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly approximates the ball as a spring of stiffness $\\kappa \\sim Y r$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly gives the relation between the spring constant and the time of frequency as $\\tau \\approx 2 \\pi \\sqrt{m / \\kappa}$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly obtains the final answer $\\tau \\approx 4 \\mathrm{\\mu s}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{4}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "$\\mu s$"
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2025",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2025_1_4",
+        "context": "[Flying Dumbbell] \n\nIn this problem, we shall study the dynamics of a dumbbell consisting of two steel balls, each with radius $r = 1 \\mathrm{cm}$, connected by a steel rod with diameter $d = 1 \\mathrm{mm}$ and length $l = 10 \\mathrm{cm}$, in the absence of gravity. Unless instructed otherwise, assume steel is perfectly elastic. You may simplify your calculations by assuming $l \\gg r$. \n\n(i) Given that the Young's modulus of steel is $Y = 2 \\times 10^{11} \\mathrm{Pa}$ and the density of steel is $\\rho = 7800 \\mathrm{kg}\\mathrm{m}^{-3}$, determine the period $T$ of free longitudinal oscillations of the dumbbell. (Do not consider oscillations with standing waves in the rod where the balls remain almost at rest.) Young's modulus is the ratio of a material's stress (force per unit area) to its strain (relative deformation).\n\n(ii) Estimate the impact time $\\tau$ when a steel ball bounces off a steel wall. \n\n(iii) The dumbbell moves toward a steel wall with velocity $\\vec{v} = -v \\hat{x}$, with its axis perpendicular to the wall, and bounces back. Here, $\\hat{x}$ denotes a unit vector along the axis perpendicular to the wall. Sketch how the $x$-component $v_x$ of the front ball's velocity (the ball that collides with the wall) depends on time. \n\nParts (i)–(iii) are preliminary questions and should not be included in the final answer.",
+        "question": "Now, the dumbbell moves toward a steel wall with velocity $\\vec{v} = -v \\hat{x}$ as before, but its axis forms an angle $\\alpha$ with the $x$-axis. The interaction of the front ball with the wall depends qualitatively on the angle $\\alpha$, with a transition from one type of interaction to another occurring at $\\alpha = \\alpha_{0}$. Determine the value of $\\alpha_{0}$. Express your answer in degrees. Hint: $\\min \\left(\\frac{\\sin x}{x}\\right) \\approx -0.217$.",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer realises it behaves as in the previous question (balls at velocity $-v$ and $v$, center of mass at rest), but it now also rotates and oscillates around the centre of mass. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly gives the expression for the angular speed of rotation as $\\Omega = 2 v \\sin \\alpha / l$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly gives the expression for the oscillation amplitude as $a = v \\cos \\alpha \\sqrt{m/k} = v \\cos \\alpha / \\omega$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer realises that the difference in interaction is whether the first ball bounces once or twice. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly gives the formula for the distance of the front ball to the wall over time as $\\frac{l}{2} \\cos \\alpha - \\left[ \\frac{l}{2} - a \\sin(\\omega t) \\right] \\cos(\\alpha + \\Omega t)$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer realises that if the distance of the front ball to the wall over time is over 0 for all $t > 0$, the first ball does not hit the wall twice. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly finds the critical angle $\\alpha_0 \\approx 25^{\\circ}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{25^{\\circ}}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "degrees"
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2025",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2025_1_5",
+        "context": "[Flying Dumbbell] \n\nIn this problem, we shall study the dynamics of a dumbbell consisting of two steel balls, each with radius $r = 1 \\mathrm{cm}$, connected by a steel rod with diameter $d = 1 \\mathrm{mm}$ and length $l = 10 \\mathrm{cm}$, in the absence of gravity. Unless instructed otherwise, assume steel is perfectly elastic. You may simplify your calculations by assuming $l \\gg r$. \n\n(i) Given that the Young's modulus of steel is $Y = 2 \\times 10^{11} \\mathrm{Pa}$ and the density of steel is $\\rho = 7800 \\mathrm{kg}\\mathrm{m}^{-3}$, determine the period $T$ of free longitudinal oscillations of the dumbbell. (Do not consider oscillations with standing waves in the rod where the balls remain almost at rest.) Young's modulus is the ratio of a material's stress (force per unit area) to its strain (relative deformation).\n\n(ii) Estimate the impact time $\\tau$ when a steel ball bounces off a steel wall. \n\n(iii) The dumbbell moves toward a steel wall with velocity $\\vec{v} = -v \\hat{x}$, with its axis perpendicular to the wall, and bounces back. Here, $\\hat{x}$ denotes a unit vector along the axis perpendicular to the wall. Sketch how the $x$-component $v_x$ of the front ball's velocity (the ball that collides with the wall) depends on time. \n\n(iv) Now, the dumbbell moves toward a steel wall with velocity $\\vec{v} = -v \\hat{x}$ as before, but its axis forms an angle $\\alpha$ with the $x$-axis. The interaction of the front ball with the wall depends qualitatively on the angle $\\alpha$, with a transition from one type of interaction to another occurring at $\\alpha = \\alpha_{0}$. Determine the value of $\\alpha_{0}$. Hint: $\\min \\left(\\frac{\\sin x}{x}\\right) \\approx -0.217$. \n\nParts (i)–(iv) are preliminary questions and should not be included in the final answer.",
+        "question": "Under the assumptions of the previous task, let $\\alpha > \\alpha_{0}$. Additionally, assume that while steel is highly elastic, it is not infinitely so: any oscillations excited in the rod will decay by the time the rear ball collides with the wall. Determine the speed with which the centre of mass of the dumbbell departs from the wall.",
+        "marking": [
+            [
+                "Award 0.2 pt if the answer realises that the dumbbell rotates around its centre of mass after the first collision. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer realises that the longitudinal oscillations have decayed by the time of the second collision. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly gives the expression for the velocity of the ball at the moment of the second collision as $v \\sin \\alpha$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly gives the expression for the component of the ball's velocity in the direction of the surface normal at the moment of the second collision as $v \\sin^2 \\alpha$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer realises that the component of velocity of the second ball in the direction of the surface normal is also $v \\sin^2 \\alpha$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer realises that the speed of the centre of mass is $v \\sin^2 \\alpha$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$v \\sin^{2} \\alpha$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2025",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2025_2_1",
+        "context": "[Evaporation] \n\nFor the subsequent tasks, the graph shows how the density of saturated water vapour in $\\mathrm{g} \\mathrm{m}^{-3}$ depends on the temperature in $^\\circ C$.\n\n[figure1]\n\nYou may also use the following characteristics of water. Specific heat capacity $c = 4200 \\mathrm{J} \\mathrm{kg}^{-1} \\mathrm{K}^{-1}$; latent heat of vaporization $L = 2260 \\mathrm{kJ} \\mathrm{kg}^{-1}$; density $\\rho = 1000 \\mathrm{kg} \\mathrm{m}^{-3}$; molar mass of water $\\mu = 18 \\mathrm{g} \\mathrm{mol}^{-1}$. You may also assume water vapour to behave as an ideal gas. The universal gas constant is $R = 8.31 \\mathrm{J} \\mathrm{mol}^{-1} \\mathrm{K}^{-1}$.",
+        "question": "A cylinder contains a certain amount of water at temperature $T_{0} = 90^{\\circ} \\mathrm{C}$, as shown in the figure. The cross-sectional area of the piston is $S = 1 \\mathrm{dm}^{2}$. What is the minimum pulling force required to move the piston? Express your answer in $N$. The pressure of the surrounding air is $p_{0} = 100 \\mathrm{kPa}$.\n\n[figure2]",
+        "marking": [
+            [
+                "Award 0.8 pt if the answer realises that the pressure inside the cylinder equals the saturated vapour pressure of water at temperature $T_0$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer reads the density $\\rho$ from the graph in the range $[400, 440] \\mathrm{g m^{-3}}$. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer uses the ideal gas law to obtain an expression for the pressure $p_1$ at temperature $T_0$ as $p_1 = \\rho R T / \\mu$. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer correctly gives the expression for the force needed to pull the piston as $S (p_0 - p_1)$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly obtains the numerical value of the minimum pulling force required to move the piston as $F \\approx 300 \\mathrm{N}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{300}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "N"
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+data figure",
+        "field": "Thermodynamics",
+        "source": "NBPhO_2025",
+        "image_question": [
+            "image_question/NBPhO_2025_2_1_1.png",
+            "image_question/NBPhO_2025_2_1_2.png"
+        ]
+    },
+    {
+        "id": "NBPhO_2025_2_2",
+        "context": "[Evaporation] \n\nFor the subsequent tasks, the graph shows how the density of saturated water vapour in $\\mathrm{g} \\mathrm{m}^{-3}$ depends on the temperature in $^\\circ C$.\n\n[figure1]\n\nYou may also use the following characteristics of water. Specific heat capacity $c = 4200 \\mathrm{J} \\mathrm{kg}^{-1} \\mathrm{K}^{-1}$; latent heat of vaporization $L = 2260 \\mathrm{kJ} \\mathrm{kg}^{-1}$; density $\\rho = 1000 \\mathrm{kg} \\mathrm{m}^{-3}$; molar mass of water $\\mu = 18 \\mathrm{g} \\mathrm{mol}^{-1}$. You may also assume water vapour to behave as an ideal gas. The universal gas constant is $R = 8.31 \\mathrm{J} \\mathrm{mol}^{-1} \\mathrm{K}^{-1}$. \n\n(i) A cylinder contains a certain amount of water at temperature $T_{0} = 90^{\\circ} \\mathrm{C}$, as shown in the figure. The cross-sectional area of the piston is $S = 1 \\mathrm{dm}^{2}$. What is the minimum pulling force required to move the piston? The pressure of the surrounding air is $p_{0} = 100 \\mathrm{kPa}$.\n\n[figure2]\n\nPart (i) is a preliminary question and should not be included in the final answer.",
+        "question": "If the piston is pulled so that it shifts by $a = 3 \\mathrm{dm}$, the water cools to a temperature of $T_{1} = 89^{\\circ} \\mathrm{C}$; what is the mass of the water under the piston? Express your answer in $g$.",
+        "marking": [
+            [
+                "Award 0.2 pt if the answer reads the vapour density $\\rho_1$ from the graph in the range $[390, 420] \\mathrm{g m^{-3}}$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly gives the expression for the mass of water vapour as $m_v = S a \\rho_1$, where $S$ is the cross-sectional area of the piston, and $\\rho_1$ is the vapour density. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly gives the expression for the latent heat as $m_v L$, where $m_v$ is the mass of vapour. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly gives the expression for the heat lost by water as $(m - m_v) \\cdot c \\cdot (T_0 - T_1)$, where $m$ is the mass of the water, $m_v$ is the mass of vapour, and the water cools from $T_0$ to $T_1$. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer correctly applies the energy conservation equation $m_v L = (m - m_v) \\cdot c \\cdot (T_0 - T_1)$, where $m$ is the mass of the water, $m_v$ is the mass of vapour, and the water cools from $T_0$ to $T_1$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly obtains the expression for the mass of water $m = \\frac{\\rho_1 S a L}{c (T_0 - T_1)}$, where $\\rho_1$ is the vapour density, and the water cools from $T_0$ to $T_1$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly obtains the numerical value of the mass of water $m \\in [630, 680] \\mathrm{g}$ with the correct dimension. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{[630, 680]}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "g"
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+data figure",
+        "field": "Thermodynamics",
+        "source": "NBPhO_2025",
+        "image_question": [
+            "image_question/NBPhO_2025_2_1_1.png",
+            "image_question/NBPhO_2025_2_1_2.png"
+        ]
+    },
+    {
+        "id": "NBPhO_2025_2_3",
+        "context": "[Evaporation] \n\nFor the subsequent tasks, the graph shows how the density of saturated water vapour in $\\mathrm{g} \\mathrm{m}^{-3}$ depends on the temperature in $^\\circ C$.\n\n[figure1]\n\nYou may also use the following characteristics of water. Specific heat capacity $c = 4200 \\mathrm{J} \\mathrm{kg}^{-1} \\mathrm{K}^{-1}$; latent heat of vaporization $L = 2260 \\mathrm{kJ} \\mathrm{kg}^{-1}$; density $\\rho = 1000 \\mathrm{kg} \\mathrm{m}^{-3}$; molar mass of water $\\mu = 18 \\mathrm{g} \\mathrm{mol}^{-1}$. You may also assume water vapour to behave as an ideal gas. The universal gas constant is $R = 8.31 \\mathrm{J} \\mathrm{mol}^{-1} \\mathrm{K}^{-1}$. \n\n(i) A cylinder contains a certain amount of water at temperature $T_{0} = 90^{\\circ} \\mathrm{C}$, as shown in the figure. The cross-sectional area of the piston is $S = 1 \\mathrm{dm}^{2}$. What is the minimum pulling force required to move the piston? The pressure of the surrounding air is $p_{0} = 100 \\mathrm{kPa}$.\n\n[figure2]\n\n(ii) If the piston is pulled so that it shifts by $a = 3 \\mathrm{dm}$, the water cools to a temperature of $T_{1} = 89^{\\circ} \\mathrm{C}$; what is the mass of the water under the piston? \n\nWater evaporation has a cooling effect the intensity of which depends on the relative humidity and air convection intensity. It appears, however, that once a dynamical thermal equilibrium is reached, the equilibrium temperature of a wet surface depends only on the relative humidity and the temperature of air and does not depend on the convection speed (as long as the convection is not too weak). This is so because the two competing processes determining the equilibrium state both depend on the thickness of the laminar (non-turbulent) surface layer exactly in the same way. In what follows we shall use the following assumptions. (a) Atop a wet surface (such as a sweating bare skin), there is a layer with a laminar flow of a certain thickness $d$. (b) Atop the laminar layer, the surrounding turbulent flow keeps a constant temperature $T$ and relative humidity $r$, both equal to the respective values in the bulk of the surrounding air. (c) Heat flux from beneath the wet surface (e.g. through the skin) can be neglected. (d) The heat conductivity of air $\\kappa = 30 \\mathrm{mW} \\mathrm{m}^{-1} \\mathrm{K}^{-1}$ at $T = 70^{\\circ} \\mathrm{C}$ (neglect the temperature dependence), and the diffusivity of water molecules in air $D = 26 \\mathrm{mm}^{2} \\mathrm{s}^{-1}$. Neglect the dependence of $D$ on the temperature. Note that the particle flux (net number of molecules passing a cross-section in y-z-plane per second and per cross-sectional area) can be found as $J = D \\frac{\\mathrm{d} n}{\\mathrm{d} x}$, where $n$ denotes the number density (number of molecules per volume). \n\nParts (i)–(ii) are preliminary questions and should not be included in the final answer.",
+        "question": "Determine the temperature of sweating human skin in a sauna if the air temperature $T = 110^{\\circ}\\mathrm{C}$ and $r = 3\\%$. Express your answer in $^{\\circ}\\mathrm{C}$.",
+        "marking": [
+            [
+                "Award 0.4 pt if the answer states that at equilibrium the heat going away from the skin (up) is equal to the heat going to the skin (down) due to evaporation. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly gives the expression for the heat flux down as $\\kappa \\frac{dT}{dx}$, where $\\kappa$ is the heat conductivity of air and $T$ is the temperature. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly gives the expression for the heat flux up as $\\frac{D L \\mu}{R} \\frac{d}{dx} \\frac{P}{T}$, where $D$ is the diffusivity of water molecules in air, $L$ is the latent heat of vaporization, $\\mu$ is the molar mass of water, $R$ is the universal gas constant, $P$ is the vapour pressure, and $T$ is the temperature. Partial points: award 0.3 pt if the answer only correctly gives the magnitude of the heat flux up as $L J m$, where $J$ is the particle flux and $m$ is the mass of one molecule; award 0.1 pt if the answer correctly gives the expression for the mass of one molecule as $m = \\mu / N_A$, where $N_A$ is Avogadro's number; award 0.1 pt if the answer correctly gives the expression $n = P / (T k_B)$. Otherwise, award 0 pt.",
+                "Award 0.1 pt if the answer deduces that the direction of the heat flow is opposite to $\\frac{dn}{dx}$, explicitly mentioning or implying the existence of the minus sign in the equations. Otherwise, award 0 pt.",
+                "Award 0.1 pt if the answer correctly states the relation between the pressure of the water $P$ and the relative humidity $r$ as $P = r p$, where $p$ is the saturation pressure of vapour. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer correctly integrates to obtain $\\kappa (T - T_s) = \\frac{D L \\mu}{R} \\left[ \\frac{p(T_s)}{T_s} - \\frac{r p(T)}{T} \\right]$, where the index $s$ denotes quantities evaluated at the skin surface and $r$ is the relative humidity. Alternatively, if a change from $d$ to $\\Delta$ in the derivatives is made, it must be properly justified: for heat conductivity, no explicit explanation is needed, but for Fick's law, the answer must state that $J$ is constant because the number of particles is conserved. Otherwise, award 0 pt.",
+                "Award 0.1 pt if the answer correctly substitutes $\\rho = p \\mu / (R T)$ for the density. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer reads the density $\\rho$ correctly from the graph as $\\rho_1 \\in [800, 815] \\mathrm{g m^{-3}}$. Otherwise, award 0 pt.",
+                "Award 0.8 pt if the answer correctly applies the graphical method to find that $\\rho(T_s) = r p(T) + \\frac{\\kappa}{D L} (T - T_s)$ defines a straight line in $(T, \\rho)$ graph, where the index $s$ denotes quantities evaluated at the skin surface. Alternatively, any other valid numerical method that is explained is accepted. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly gives the numerical final result of the temperature of sweating human skin in a sauna as $T_s \\in [36, 47] ^\\circ \\mathrm{C}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{[36, 47]}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "$^{\\circ}\\mathrm{C}$"
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+data figure",
+        "field": "Thermodynamics",
+        "source": "NBPhO_2025",
+        "image_question": [
+            "image_question/NBPhO_2025_2_1_1.png",
+            "image_question/NBPhO_2025_2_1_2.png"
+        ]
+    },
+    {
+        "id": "NBPhO_2025_3_1",
+        "context": "[Nuclear Reactors] \n\nIn order to maintain a chain reaction in a modern thermal-neutron nuclear reactor one needs three things: 1. nuclear fuel (e.g. $\\mathrm{U}^{235}$), 2. moderator (e.g. water) and 3. coolant. In most cases the moderator acts as the coolant as well. Neutrons released from a thermal fission of $\\mathrm{U}^{235}$ have a mean kinetic energy of approximately $E_{0} = 2 \\mathrm{MeV}$. However, neutrons which are that fast are inefficient in triggering fission of $\\mathrm{U}^{235}$: neutrons need to be slowed down to an average kinetic energy of $E_{f} = 0.025 \\mathrm{eV}$. In what follows, justify why non-relativistic approximations can be used unless explicitly instructed otherwise.",
+        "question": "The rest energy of neutrons $m_{\\mathrm{n}} c^{2} = 940 \\mathrm{MeV}$, the Boltzmann constant $k_{\\mathrm{B}} = 1.38 \\times  10^{-23} \\mathrm{J} \\mathrm{K}^{-1}$, and the elementary charge $e = 1.602 \\times 10^{-19} \\mathrm{C}$. \n\n(1) What is the required speed of neutrons, i.e. the speed $v_f$ of neutrons with kinetic energy $E_{f}$? Express your answer in $m/s$. \n(2) What is the temperature $T_{f}$ of a neutron gas where the average kinetic energy of neutrons is $E_{f}$? Express your answer in $K$. \n(3) What is the initial speed of neutrons, i.e. the speed $v_0$ of neutrons with energy $E_{0}$? Express your answer in $m/s$.",
+        "marking": [
+            [
+                "Award 0.3 pt if the answer correctly expresses the required velocity of the neutrons with the kinetic energy $E_f$ as $v_f = \\sqrt{2 E_f / m}$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly calculates both $v_f = 2.2 \\times 10^3 \\mathrm{m/s}$ ($v_f$ is the velocity with $E_f$) and $v_0 = 2.0 \\times 10^7 \\mathrm{m/s}$ ($v_0$ is the velocity with $E_0$) for the given cases. Partial points: award 0.3 pt if only one numerical value is correct. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly uses $E_f = \\frac{3}{2} k_B T$, where $k_B$ is the Boltzmann constant and $T$ is the temperature. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer uses $E_f = \\frac{3}{2} k_B T$ to correctly calculate the temperature of a neutron gas with $E_f$ as $T_f = 193 \\mathrm{K}$. Otherwise, award 0 pt.",
+                "Award 0.6 pt if the answer justifies the validity of the classical (non-relativistic) approach for both cases, for example by showing that the speed is much less than the speed of light or that the kinetic energy is significantly less than the rest energy $E_f \\ll m c^2$. Partial points: award 0.3 pt if the answer uses $E_f = k_B T$ without justification and finds $T = 290 \\mathrm{K}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$2.2 \\times 10^3$}",
+            "\\boxed{193}",
+            "\\boxed{$2.0 \\times 10^7$}"
+        ],
+        "answer_type": [
+            "Numerical Value",
+            "Numerical Value",
+            "Numerical Value"
+        ],
+        "unit": [
+            "m/s",
+            "K",
+            "m/s"
+        ],
+        "points": [
+            0.5,
+            0.5,
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Thermodynamics",
+        "source": "NBPhO_2025",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2025_3_3",
+        "context": "[Nuclear Reactors] \n\nIn order to maintain a chain reaction in a modern thermal-neutron nuclear reactor one needs three things: 1. nuclear fuel (e.g. $\\mathrm{U}^{235}$), 2. moderator (e.g. water) and 3. coolant. In most cases the moderator acts as the coolant as well. Neutrons released from a thermal fission of $\\mathrm{U}^{235}$ have a mean kinetic energy of approximately $E_{0} = 2 \\mathrm{MeV}$. However, neutrons which are that fast are inefficient in triggering fission of $\\mathrm{U}^{235}$: neutrons need to be slowed down to an average kinetic energy of $E_{f} = 0.025 \\mathrm{eV}$. In what follows, justify why non-relativistic approximations can be used unless explicitly instructed otherwise.\n\n(i) The rest energy of neutrons $m_{\\mathrm{n}} c^{2} = 940 \\mathrm{MeV}$, the Boltzmann constant $k_{\\mathrm{B}} = 1.38 \\times  10^{-23} \\mathrm{J} \\mathrm{K}^{-1}$, and the elementary charge $e = 1.602 \\times 10^{-19} \\mathrm{C}$. What is the required speed of neutrons, i.e. the speed $v_f$ of neutrons with kinetic energy $E_{f}$? What is the temperature $T_{f}$ of a neutron gas where the average kinetic energy of neutrons is $E_{f}$? \n\n(ii) What is the initial speed of neutrons, i.e. the speed $v_0$ of neutrons with energy $E_{0}$? \n\nParts (i)–(ii) are preliminary questions and should not be included in the final answer.",
+        "question": "(1) From a completely nonrelativistic point of view, what should be the mass $M$ of the moderator's atoms to slow down the fast neutrons as efficiently as possible? Express $M$ in terms of $m_{\\mathrm{n}}$. \n(2) If the mass of the moderator's atoms were to be $M = 135 m_{\\mathrm{n}}$, how many collisions with such atoms at a temperature much lower than $T_{f}$ would a fast neutron need to experience to slow down from $E_{0}$ to $E_{f}$? Assume that all collisions are elastic and central.",
+        "marking": [
+            [
+                "Award 0.3 pt if the answer states that $T \\ll T_f$, so the moderator atoms are essentially at rest. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer justifies that the maximum momentum transfer occurs when $m_n = M$, where $m_n$ is the neutron mass and $M$ is the mass of the moderator atom. Otherwise, award 0 pt.",
+                "Award 0.6 pt if the answer applies both momentum conservation $m_1 (v_{1,f} - v_{1,i}) = m_2 (v_{2,i} - v_{2,f})$ and kinetic energy conservation $m_1 (v_{1,f}^2 - v_{1,i}^2) = m_2 (v_{2,i}^2 - v_{2,f}^2)$, where $m_1$ and $m_2$ are the particle masses, $v_{1,i}$ and $v_{1,f}$ are the initial and final velocities of particle 1, and $v_{2,i}$ and $v_{2,f}$ are the initial and final velocities of particle 2. Partial points: award 0.3 pt if the answer applies only momentum conservation or only energy conservation. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer expresses $u = v \\frac{m_1 - m_2}{m_1 + m_2}$, where $u$ is the speed of the neutron after collision and $v$ is the speed of the neutron before collision. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer expresses $v_f = v_0 \\left( \\frac{m_1 - m_2}{m_1 + m_2} \\right)^N$, where $v_f$ is the final velocity after $N$ collisions, $v_0$ is the initial velocity of the neutron, $m_1$ and $m_2$ are the masses, and $N$ is the number of collisions. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly calculates the number of needed collisions from $E_0$ to $E_f$ as $N = 614$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$M = m_n$}",
+            "\\boxed{614}"
+        ],
+        "answer_type": [
+            "Expression",
+            "Numerical Value"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            0.7,
+            1.8
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2025",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2025_3_4",
+        "context": "[Nuclear Reactors] \n\nIn order to maintain a chain reaction in a modern thermal-neutron nuclear reactor one needs three things: 1. nuclear fuel (e.g. $\\mathrm{U}^{235}$), 2. moderator (e.g. water) and 3. coolant. In most cases the moderator acts as the coolant as well. Neutrons released from a thermal fission of $\\mathrm{U}^{235}$ have a mean kinetic energy of approximately $E_{0} = 2 \\mathrm{MeV}$. However, neutrons which are that fast are inefficient in triggering fission of $\\mathrm{U}^{235}$: neutrons need to be slowed down to an average kinetic energy of $E_{f} = 0.025 \\mathrm{eV}$. In what follows, justify why non-relativistic approximations can be used unless explicitly instructed otherwise.\n\n(i) The rest energy of neutrons $m_{\\mathrm{n}} c^{2} = 940 \\mathrm{MeV}$, the Boltzmann constant $k_{\\mathrm{B}} = 1.38 \\times  10^{-23} \\mathrm{J} \\mathrm{K}^{-1}$, and the elementary charge $e = 1.602 \\times 10^{-19} \\mathrm{C}$. What is the required speed of neutrons, i.e. the speed $v_f$ of neutrons with kinetic energy $E_{f}$? What is the temperature $T_{f}$ of a neutron gas where the average kinetic energy of neutrons is $E_{f}$? \n\n(ii) What is the initial speed of neutrons, i.e. the speed $v_0$ of neutrons with energy $E_{0}$? \n\n(iii) From a completely nonrelativistic point of view, what should be the mass of the moderator's atoms to slow down the fast neutrons as efficiently as possible? If the mass of the moderator's atoms were to be $M = 135 m_{\\mathrm{n}}$, how many collisions with such atoms at a temperature much lower than $T_{f}$ would a fast neutron need to experience to slow down from $E_{0}$ to $E_{f}$? Assume that all collisions are elastic and central. \n\nParts (i)–(iii) are preliminary questions and should not be included in the final answer.",
+        "question": "Nuclear fuel, i.e. $\\mathrm{U}^{235}$, is placed inside metal rods and pressurized with helium gas to $p_{0} = 2.5 \\mathrm{MPa}$. During operation, as $\\mathrm{U}^{235}$ keeps on fissioning inside the fuel rods, there is a build up of gas inside the rods. With a non-invasive ultrasound measurement we can measure that the gas pressure inside the rod after it is finally picked out from the core is $p = 6.5 \\mathrm{MPa}$. \n\n(1) Assuming that the gas released inside the rods is completely made of xenon isotope ${}_{54}^{135}\\mathrm{Xe}$ and that the initial gas volume drops from $V_{0} = 18 \\mathrm{cm}^{3}$ to $V = 9 \\mathrm{cm}^{3}$ due to the swelling of the fuel pellets, how many moles of xenon are released from fission? \n(2) What is the ratio of helium to xenon inside the rod? The measurements are done at $T_{0} = 20^{\\circ}\\mathrm{C}$; the universal gas constant $R = 8.31 \\mathrm{J} \\mathrm{mol}^{-1} \\mathrm{K}^{-1}$.",
+        "marking": [
+            [
+                "Award 0.3 pt if the answer correctly applies Boyle's law. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer correctly applies Dalton's law. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer expresses $n_{\\mathrm{Xe}} = p_{\\mathrm{Xe}} V / (R T_0)$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly calculates $n_{\\mathrm{Xe}} = 5.5 \\times 10^{-3} \\mathrm{mol}$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly expresses $n_{\\mathrm{He}} = p_{\\mathrm{He}} V_0 / (R T_0)$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly calculates $\\frac{n_{\\mathrm{He}}}{n_{\\mathrm{Xe}}} = 3.3$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$5.5 \\times 10^{-3}$}",
+            "\\boxed{3.3}"
+        ],
+        "answer_type": [
+            "Numerical Value",
+            "Numerical Value"
+        ],
+        "unit": [
+            "mol",
+            null
+        ],
+        "points": [
+            1.1,
+            0.4
+        ],
+        "modality": "text-only",
+        "field": "Thermodynamics",
+        "source": "NBPhO_2025",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2025_5_1",
+        "context": "",
+        "question": "[Throwing] \n\nA drone starts from the origin at rest and accelerates horizontally with an acceleration $g$ to the $+x$ direction. Simultaneously, a ball is thrown from the point with coordinates $(x, y)  = (-h, -h)$. What is the minimum initial speed $v_0$ the ball needs to hit the drone? The free fall acceleration $g$ is antiparallel to the $y$-axis.",
+        "marking": [
+            [
+                "Award 1.0 pt if the answer identifies the idea of switching to the coaccelerating frame, where the drone is at rest and an effective gravitational field is present. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer states that the ball gains a horizontal acceleration $g$, where $g$ is the free-fall acceleration magnitude. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly gives the effective gravitational field magnitude as $g \\sqrt{2}$, pointing at a $45^{\\circ}$ angle from the drone to the throwing point. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer applies the energy conservation equation $\\frac{1}{2} m v_0^2 = m (g \\sqrt{2}) (h \\sqrt{2})$, where $m$ is the ball mass, $v_0$ is the initial speed, and $h$ is the given distance parameter. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly obtains the minimum initial speed $v_0 = 2 \\sqrt{g h}$. Otherwise, award 0 pt."
+            ],
+            [
+                "Award 0.6 pt if the answer correctly writes the three kinematic equations: (1) In time $t$, the drone travels a distance $s = \\frac{1}{2} gt^2$; (2) For a collision to occur at time $t$, the ball must travel a vertical distance $h$, giving $h = v t \\sin \\alpha - \\frac{1}{2} g t^2$; (3) The horizontal distance $h + s$ gives $h + s = v t \\cos \\alpha$, where $\\alpha$ is the launch angle, $v$ is the initial speed, and $t$ is the flight time. Partial points: award 0.2 pt for each of the three equations. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer derives the condition $\\sin \\alpha = \\cos \\alpha$ from the kinematic equations. Otherwise, award 0 pt.",
+                "Award 0.1 pt if the answer explicitly states $\\alpha = 45^{\\circ}$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer uses $v_x = v_y$ to argue that the highest point of the trajectory must be as low as possible to minimize the initial speed. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer applies energy conservation or the respective kinematical equation. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly obtains the minimum initial speed $v_0 = 2 \\sqrt{g h}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$v_{0} = 2 \\sqrt{gh}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2025",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2025_5_2",
+        "context": "",
+        "question": "[Throwing] \n\nA stone is thrown from point $S$ (shown in the figure below) with an initial speed $v$. A boy at point $B$ wishes to hit the stone in midair by throwing a ball simultaneously with the stone's release. He wants to use the minimum possible speed $u$ that will still allow the ball to hit the stone in midair. After calculating the stone's trajectory, he determines the optimal trajectory for the ball and throws it according to his calculations. The collision point $C$ is shown in the figure. Using the scale provided and necessary measurements from the figure: \n\n(1) Find the initial speed $v$ of the stone. Express your answer in $m/s$. \n(2) Find the initial speed $u$ of the ball. Express your answer in $m/s$. \nThe free fall acceleration is $g = 9.8 m s^{-2}$.\n\n[figure1]",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer identifies the idea of switching to the free-falling frame. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer states explicitly or implicitly that the stone and the ball travel in straight lines in the free-falling frame. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly writes $SC' = v t$ and $BC' = u t$, where $v$ is the initial speed of the stone, $u$ is the initial speed of the ball, $t$ is the collision time, and $C'$ is the point obtained in the free-falling frame by shifting the collision point $C$ vertically upward by $h = \\frac{1}{2} g t^2$. Partial points: award 0.1 pt for each correct equation. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer notices that we need to minimize $\\frac{|SC'|}{|BC'|}$ (or maximizing its reciprocal), where $C'$ is defined as the point obtained by shifting $C$ vertically upward by $h = \\frac{1}{2} g t^2$ in the free-falling frame. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer correctly applies the sine theorem to minimize $\\frac{|SC'|}{|BC'|}$, where $C'$ is defined as the point obtained by shifting $C$ vertically upward by $h = \\frac{1}{2} g t^2$ in the free-falling frame. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer states that the maximum of $\\frac{|SC'|}{|BC'|}$ occurs when $\\angle C'BS = 90^{\\circ}$, where $C'$ is defined as the point obtained by shifting $C$ vertically upward by $h = \\frac{1}{2} g t^2$ in the free-falling frame. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer states that in the free-falling frame the collision point $C'$ is shifted upwards with respect to $S$, $B$, and $C$ by a distance $h = \\frac{1}{2} g t^2$, where $g$ is the gravitational acceleration. Otherwise, award 0 pt.",
+                "Award 0.1 pt if the answer correctly expresses $|CC'| = \\frac{1}{2} g t^2$, where $C'$ is defined as the point obtained by shifting $C$ vertically upward by $h = \\frac{1}{2} g t^2$ in the free-falling frame. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer provides a well-drawn and correct geometrical construction showing $S$, $B$, $C$, and $C'$ (where $C'$ is defined as the point obtained by shifting $C$ vertically upward by $h = \\frac{1}{2} g t^2$ in the free-falling frame). Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly gives $v = \\sqrt{g} |SC'| / \\sqrt{2 |CC'|}$ and $u = \\sqrt{g} |BC'| / \\sqrt{2 |CC'|}$, where $|SC'|$ and $|BC'|$ are distances from $S$ and $B$ to $C'$ in the free-falling frame, and $C'$ is defined as the point obtained by shifting $C$ vertically upward by $h = \\frac{1}{2} g t^2$. Partial points: award 0.1 pt for each correct formula. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly calculates $v \\in [11.8, 12.7] \\mathrm{m/s}$ and $u \\in [10.5, 11.4] \\mathrm{m/s}$. Partial points: award 0.1 pt for each correct value (only if the method is correct). Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{[11.8, 12.7]}",
+            "\\boxed{[10.5, 11.4]}"
+        ],
+        "answer_type": [
+            "Numerical Value",
+            "Numerical Value"
+        ],
+        "unit": [
+            "m/s",
+            "m/s"
+        ],
+        "points": [
+            2.0,
+            2.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "NBPhO_2025",
+        "image_question": [
+            "image_question/NBPhO_2025_5_2_1.png"
+        ]
+    },
+    {
+        "id": "NBPhO_2025_6_1",
+        "context": "",
+        "question": "[Birds] \n\nA long and thin homogenous beam with uniform thickness and square cross-section floats horizontally in water with its top surface parallel to the water surface. A bird lands on one end of the beam, and as a result, the beam sinks so that the edge of the upper face on the bird's side is exactly at the same height as the water surface, while at the other end of the beam the lower face does not rise above the water. What is the maximum number of such birds that this beam can hold above water?",
+        "marking": [
+            [
+                "Award 0.6 pt if the answer uses any correct torque balance to solve the problem. Partial points: award 0.3 pt if the answer uses any correct force balance with the bird present. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly identifies the moment arm for the force of the bird as $\\frac{1}{2} L$, where $L$ is the beam length. Otherwise, award 0 pt.",
+                "Award 0.6 pt if the answer explicitly states or derives that the centre of mass of a triangle is located at the intersection of its medians, a distance $\\frac{2}{3} L$ away from the bird. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly gives the moment arm for the buoyancy force as $\\frac{1}{6} L$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer justifies the numerical result for the maximum number of birds is 4. Otherwise, award 0 pt.",
+                "Award 2.0 pts if the answer explicitly states that the final result for the maximum number of birds does not depend on fixing any unknown parameters. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{4}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            4.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2025",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2025_7_1",
+        "context": "[Charged Rod] \n\nA rod of mass $m$ carries a charge $q$; both the charge and the mass are homogeneously distributed over its entire length $l$. The system is in homogeneous magnetic field of strength $B$, parallel to the $z$-axis whereas the rod is in the x-y-plane. Neglect any forces except for the Lorentz force. One end of the rod is painted red, and the other - blue.",
+        "question": "Consider the case when the rod rotates around its centre of mass. What should be the angular speed $\\omega$ for the mechanical tension force at the centre of the rod to be zero?",
+        "marking": [
+            [
+                "Award 0.4 pt if the answer considers forces on an infinitesimal part of the rod. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer equates the Lorentz force and the centrifugal force with justification, i.e., writes $\\mathrm{d}q v B = \\mathrm{d}m \\omega^2 r$, where $\\mathrm{d}q$ is the infinitesimal charge, $v$ is the tangential velocity, $B$ is the magnetic field strength, $\\mathrm{d}m$ is the infinitesimal mass, $\\omega$ is the angular speed, and $r$ is the distance from the axis of rotation. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer uses the relation $\\omega = \\frac{v}{r}$, where $v$ is the tangential velocity and $r$ is the distance from the axis of rotation. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer uses the ratio $\\frac{\\mathrm{d}q}{\\mathrm{d}m} = \\frac{q}{m}$, where $q$ is the total charge of the rod and $m$ is the total mass of the rod. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer expresses the angular speed as $\\omega = \\frac{qB}{m}$, where $q$ is the total charge, $B$ is the magnetic field strength, and $m$ is the total mass of the rod. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$\\omega = \\frac{Bq}{m}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Electromagnetism",
+        "source": "NBPhO_2025",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2025_7_2",
+        "context": "[Charged Rod] \n\nA rod of mass $m$ carries a charge $q$; both the charge and the mass are homogeneously distributed over its entire length $l$. The system is in homogeneous magnetic field of strength $B$, parallel to the $z$-axis whereas the rod is in the x-y-plane. Neglect any forces except for the Lorentz force. One end of the rod is painted red, and the other - blue. \n\n(i) Consider the case when the rod rotates around its centre of mass. What should be the angular speed $\\omega$ for the mechanical tension force at the centre of the rod to be zero? \n\nPart (i) is a preliminary question and should not be included in the final answer.",
+        "question": "Consider now a case when initially the blue end of the rod is at the origin $(x = y = 0)$, and the red end at $x = l$. The blue end's initial speed is zero while the red end's speed is $v$, parallel to the y-axis. It turns out that after a certain time $t$, the red end passes through the origin. \n\n(1) Find the smallest possible value for $t$. \n(2) Express the corresponding value of $v$ in terms of $m$, $q$ and $l$.",
+        "marking": [
+            [
+                "Award 0.5 pt if the answer deduces, with justification, that the net force on the rod is $\\vec{F} = q \\vec{v}_C \\times \\vec{B}$, where $q$ is the total charge, $\\vec{v}_C$ is the velocity of the center of mass, and $\\vec{B}$ is the magnetic field. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer obtains that the velocity of the center of mass is $v_C = v/2$, where $v$ is the initial velocity of the red end. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer justifies that the center of mass moves on a circular path. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer expresses the radius of the circular path of the center of mass as $R = \\frac{m v}{2 q B}$, where $m$ is the total mass, $q$ is the total charge, and $B$ is the magnetic field strength. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer concludes that the angular velocity of the center of mass is $\\omega = \\frac{q B}{m}$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer expresses the angular velocity of the rotation of the rod around the center of mass as $\\Omega = v/l$, where $l$ is the length of the rod. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer justifies that $\\Omega$ is conserved, where $\\Omega$ is the angular velocity of the rotation of the rod around the center of mass. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer argues that $t < 2\\pi/\\omega$ is possible only if $R = l/2$. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer justifies that in this case, the red end will never end up at the origin. Otherwise, award 0 pt.",
+                "Award 0.4 pt if the answer justifies that the condition for the red end to reach the origin after time $T$ is $\\Omega T = \\pi + 2\\pi k$ with $k \\in \\mathbb{Z}_{\\geq 0}$. Otherwise, award 0 pt.",
+                "Award 0.6 pt if the answer expresses the final result as $v = \\frac{q B l}{m} \\left( \\frac{1}{2} + k \\right)$, where $k$ is a non-negative integer. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$\\frac{2 \\pi}{\\omega}$}",
+            "\\boxed{$v = \\frac{lBq}{2m}$}"
+        ],
+        "answer_type": [
+            "Expression",
+            "Expression"
+        ],
+        "unit": [
+            null,
+            null
+        ],
+        "points": [
+            2.0,
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Electromagnetism",
+        "source": "NBPhO_2025",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2025_8_1",
+        "context": "[Phase Spiral] \n\nHere we shall study the motion of Milky Way stars in the Solar neighbourhood in the direction of the $z$-axis, i.e. perpendicular to the galactic plane. For our purposes, we can model the galactic gravity field as being created by a continuous mass density $\\rho$ (that accounts for the masses of stars, dark matter, gas, interstellar dust, etc), and assume that this mass forms an infinite mirror-symmetric plate, i.e. $\\rho \\equiv \\rho(z)$ and $\\rho(z) = \\rho(-z)$ is independent of $x$ and $y$. Throughout the problem, you may assume that each star's total energy is conserved over the entire considered time period. Gravitational constant $G = 6.67 \\times 10^{-11} \\mathrm{m}^{3} \\mathrm{kg}^{-1} \\mathrm{s}^{-2} = 4.30 \\times 10^{-3} \\mathrm{pc} \\mathrm{M}_{\\odot}^{-1} (\\mathrm{km}/\\mathrm{s})^{2}$.",
+        "question": "Assuming that the mass density is constant over the plate's thickness. i.e. $\\rho(z) = \\rho_{0}$, what is the acceleration $a_{z}$ of a star at a distance $z$ from the mid-plane?",
+        "marking": [
+            [
+                "Award 0.3 pt if the answer identifies that the gravitational acceleration obeys Gauss' law, i.e., the number of field lines passing through a closed surface is proportional to the enclosed mass. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer correctly writes the formula relating the mass inside with gravitational flus, e.g., $\\iint g \\cdot \\mathrm{d}A = -4\\pi G M$, where $g$ is the gravitational field (acceleration), $G$ is the gravitational constant, and $M$ is a point mass. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer applies Gauss' law to a cuboid of area $A$ and half-thickness $z$, obtaining $-2 a_z A = -4\\pi G (2A z \\rho_0)$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer gives the final result for the acceleration at distance $z$ from the mid-plane as $a_z = -4\\pi G \\rho_0 z$. Otherwise, award 0 pt."
+            ],
+            [
+                "Award 0.5 pt if the answer finds the acceleration of a thin disk by integrating the surface contribution over the plate. Partial points: award 0.3 pt if the answer writes the correct integral; award 0.2 pt if the answer gives the correct evaluation, including finding that the acceleration is independent of the displacement from the surface. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer infers that only the layers within $-a < z < a$ contribute to the final acceleration. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer gives the final result for the acceleration at distance $z$ from the mid-plane as $a_z = -4\\pi G \\rho_0 z$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$a_{z} = -4 \\pi G \\rho_{0} z$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            1.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2025",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2025_8_2",
+        "context": "[Phase Spiral] \n\nHere we shall study the motion of Milky Way stars in the Solar neighbourhood in the direction of the $z$-axis, i.e. perpendicular to the galactic plane. For our purposes, we can model the galactic gravity field as being created by a continuous mass density $\\rho$ (that accounts for the masses of stars, dark matter, gas, interstellar dust, etc), and assume that this mass forms an infinite mirror-symmetric plate, i.e. $\\rho \\equiv \\rho(z)$ and $\\rho(z) = \\rho(-z)$ is independent of $x$ and $y$. Throughout the problem, you may assume that each star's total energy is conserved over the entire considered time period. Gravitational constant $G = 6.67 \\times 10^{-11} \\mathrm{m}^{3} \\mathrm{kg}^{-1} \\mathrm{s}^{-2} = 4.30 \\times 10^{-3} \\mathrm{pc} \\mathrm{M}_{\\odot}^{-1} (\\mathrm{km}/\\mathrm{s})^{2}$. \n\n(i) Assuming that the mass density is constant over the plate's thickness. i.e. $\\rho(z) = \\rho_{0}$, what is the acceleration $a_{z}$ of a star at a distance $z$ from the mid-plane? \n\nPart (i) is a preliminary question and should not be included in the final answer.",
+        "question": "Consider a star that starts with zero velocity at a distance of $z = a$ from the mid-plane. With what period does it start oscillating around the mid-plane?",
+        "marking": [
+            [
+                "Award 0.3 pt if the answer notices that the movement is that of a harmonic oscillator. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer correctly gives the oscillation period as $T = \\sqrt{\\frac{\\pi}{G \\rho_0}}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$\\sqrt{\\frac{\\pi}{G \\rho_0}}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            0.5
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "NBPhO_2025",
+        "image_question": []
+    },
+    {
+        "id": "NBPhO_2025_8_4",
+        "context": "[Phase Spiral] \n\nHere we shall study the motion of Milky Way stars in the Solar neighbourhood in the direction of the $z$-axis, i.e. perpendicular to the galactic plane. For our purposes, we can model the galactic gravity field as being created by a continuous mass density $\\rho$ (that accounts for the masses of stars, dark matter, gas, interstellar dust, etc), and assume that this mass forms an infinite mirror-symmetric plate, i.e. $\\rho \\equiv \\rho(z)$ and $\\rho(z) = \\rho(-z)$ is independent of $x$ and $y$. Throughout the problem, you may assume that each star's total energy is conserved over the entire considered time period. Gravitational constant $G = 6.67 \\times 10^{-11} \\mathrm{m}^{3} \\mathrm{kg}^{-1} \\mathrm{s}^{-2} = 4.30 \\times 10^{-3} \\mathrm{pc} \\mathrm{M}_{\\odot}^{-1} (\\mathrm{km}/\\mathrm{s})^{2}$. \n\n(i) Assuming that the mass density is constant over the plate's thickness. i.e. $\\rho(z) = \\rho_{0}$, what is the acceleration $a_{z}$ of a star at a distance $z$ from the mid-plane? \n\n(ii) Consider a star that starts with zero velocity at a distance of $z = a$ from the mid-plane. With what period does it start oscillating around the mid-plane? \n\nIn reality, density decreases with growing $|z|$. Measuring density has been a great challenge because of contributions from dark and other difficult-to-see matter. Here, we consider a breakthrough method of doing it. Consider the distributions of the stars in our neighbourhood on the $z$-$v_{z}$ phase plane, where each star is a dot with coordinates $(v_{z}, z)$; $v_{z}$ denotes the $z$-component the star's velocity, and $z$ - the vertical coordinate. Initially, these dots were distributed nearly homogeneously, but some time ago, the Milky Way was perturbed externally, probably by a passing-by dwarf galaxy; this shuffled the positions and velocities of stars, creating a bar-shaped overdensity region. When moving within that bar-shaped region from the centre to the periphery, the total energy per mass of stars increased monotonously. Over time, this overdensity region started \"winding up\", due to the oscillation periods of stars in the vertical plane depending on their oscillation amplitude $z_{\\mathrm{m}}$, and evolved into a spiral pictured below (Antoja et al. 2018, Nature 561, 360). An observation that you need to exploit below is that the ordering of stars by energies along the spiral today remains the same as it was at the time of perturbation. \n\nThe oscillation period of stars depends on the amplitude $z_{\\mathrm{m}}$ because the gravitational potential (the potential energy per mass) $\\Phi(z)$ is not parabolic. In such a case, the period can be approximately found by substituting the real $\\Phi(z)$ with a $k z^{2}$ matching $\\Phi(z)$ at $z = z_{\\mathrm{m}}$, i. e. with $k = \\Phi(z_{\\mathrm{m}}) / z_{\\mathrm{m}}^{2}$. \n\n[figure1] \n\n(iii) At the intersection points of the spiral with $v_z = 0$, calculate $\\Phi(z)$ by interpolating data linearly where appropriate; plot your results (this follows the analysis of Guo et al. 2024, ApJ, 960, 133) \n\nParts (i)–(iii) are preliminary questions and should not be included in the final answer.",
+        "question": "Assuming that the mass density is almost constant for $|z| \\leq 0.3 \\mathrm{kpc}$, what is the mass density near $z = 0$? Express your answer in $\\mathrm{kg} / \\mathrm{m}^3$.",
+        "marking": [
+            [
+                "Award 0.8 pt if the answer connects the first value of $\\Phi(z_1)$ with $\\rho_0$ by assuming constant mass density. Partial points: award 0.6 pt if the answer obtains correct values of $\\Phi(z)$ but does not use the first data point. Otherwise, award 0 pt.",
+                "Award 0.1 pt if the answer correctly obtains the final expression for $\\rho_0$ as $\\rho_0 = \\frac{\\Phi(z_1)}{2\\pi G z_1^2}$. Otherwise, award 0 pt.",
+                "Award 0.1 pt if the answer correctly obtains the numerical value of $\\rho_0$ within the range $[5.5 \\times 10^{-21}, 6.7 \\times 10^{-21}]$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{[5.5 \\times 10^{-21}, 6.7 \\times 10^{-21}]}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "$\\mathrm{kg} / \\mathrm{m}^3$."
+        ],
+        "points": [
+            1.0
+        ],
+        "modality": "text+data figure",
+        "field": "Modern Physics",
+        "source": "NBPhO_2025",
+        "image_question": [
+            "image_question/NBPhO_2025_8_3_1.png"
+        ]
+    },
+    {
+        "id": "NBPhO_2025_8_5",
+        "context": "[Phase Spiral] \n\nHere we shall study the motion of Milky Way stars in the Solar neighbourhood in the direction of the $z$-axis, i.e. perpendicular to the galactic plane. For our purposes, we can model the galactic gravity field as being created by a continuous mass density $\\rho$ (that accounts for the masses of stars, dark matter, gas, interstellar dust, etc), and assume that this mass forms an infinite mirror-symmetric plate, i.e. $\\rho \\equiv \\rho(z)$ and $\\rho(z) = \\rho(-z)$ is independent of $x$ and $y$. Throughout the problem, you may assume that each star's total energy is conserved over the entire considered time period. Gravitational constant $G = 6.67 \\times 10^{-11} \\mathrm{m}^{3} \\mathrm{kg}^{-1} \\mathrm{s}^{-2} = 4.30 \\times 10^{-3} \\mathrm{pc} \\mathrm{M}_{\\odot}^{-1} (\\mathrm{km}/\\mathrm{s})^{2}$. \n\n(i) Assuming that the mass density is constant over the plate's thickness. i.e. $\\rho(z) = \\rho_{0}$, what is the acceleration $a_{z}$ of a star at a distance $z$ from the mid-plane? \n\n(ii) Consider a star that starts with zero velocity at a distance of $z = a$ from the mid-plane. With what period does it start oscillating around the mid-plane? \n\nIn reality, density decreases with growing $|z|$. Measuring density has been a great challenge because of contributions from dark and other difficult-to-see matter. Here, we consider a breakthrough method of doing it. Consider the distributions of the stars in our neighbourhood on the $z$-$v_{z}$ phase plane, where each star is a dot with coordinates $(v_{z}, z)$; $v_{z}$ denotes the $z$-component the star's velocity, and $z$ - the vertical coordinate. Initially, these dots were distributed nearly homogeneously, but some time ago, the Milky Way was perturbed externally, probably by a passing-by dwarf galaxy; this shuffled the positions and velocities of stars, creating a bar-shaped overdensity region. When moving within that bar-shaped region from the centre to the periphery, the total energy per mass of stars increased monotonously. Over time, this overdensity region started \"winding up\", due to the oscillation periods of stars in the vertical plane depending on their oscillation amplitude $z_{\\mathrm{m}}$, and evolved into a spiral pictured below (Antoja et al. 2018, Nature 561, 360). An observation that you need to exploit below is that the ordering of stars by energies along the spiral today remains the same as it was at the time of perturbation. \n\nThe oscillation period of stars depends on the amplitude $z_{\\mathrm{m}}$ because the gravitational potential (the potential energy per mass) $\\Phi(z)$ is not parabolic. In such a case, the period can be approximately found by substituting the real $\\Phi(z)$ with a $k z^{2}$ matching $\\Phi(z)$ at $z = z_{\\mathrm{m}}$, i. e. with $k = \\Phi(z_{\\mathrm{m}}) / z_{\\mathrm{m}}^{2}$. \n\n[figure1] \n\n(iii) At the intersection points of the spiral with $v_z = 0$, calculate $\\Phi(z)$ by interpolating data linearly where appropriate; plot your results (this follows the analysis of Guo et al. 2024, ApJ, 960, 133) \n\n(iv) Assuming that the mass density is almost constant for $|z| \\leq 0.3 \\mathrm{kpc}$, what is the mass density near $z = 0$? \n\nParts (i)–(iv) are preliminary questions and should not be included in the final answer.",
+        "question": "Dark matter is an \"invisible\" form of matter that only interacts by gravity. In general, it is found that dark matter forms halos that extend significantly farther than visible matter structures. By assuming that the dark matter density doesn't vary significantly within the volume of interest and that it starts dominating far away from the galactic plane, from around $z = 0.7 \\mathrm{kpc}$, estimate the local dark matter density $\\rho_{\\mathrm{DM}}$. Express your answer in $\\mathrm{kg} / \\mathrm{m}^3$.",
+        "marking": [
+            [
+                "Award 0.7 pt if the answer obtains an expression for the total mass per unit area (surface density) enclosed within height $z$ using the constant density approximation, e.g., $\\Sigma(z) = \\rho_0 z = \\frac{\\Phi(z)}{2\\pi G z}$, where $\\Sigma(z)$ is the surface density, $\\rho_0$ is the assumed constant mass density over the plate's thickness, $\\Phi(z)$ is the gravitational potential per unit mass, and $G$ is the gravitational constant. Otherwise, award 0 pt.",
+                "Award 0.9 pt if the answer takes the difference between the total surface densities at two heights $z_6$ and $z_5$ where dark matter dominates at $z_6$ (and not at $z_5$), and sets $\\Sigma(z_6) - \\Sigma(z_5) = \\rho_{\\mathrm{DM}} (z_6 - z_5)$ to isolate the dark matter contribution, where $\\rho_{\\mathrm{DM}}$ is the local dark matter density. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer expresses $\\rho_{\\mathrm{DM}}$ in terms of $\\Phi(z)$ at the two heights as $\\rho_{\\mathrm{DM}} \\approx \\frac{1}{2\\pi G (z_6 - z_5)} \\left(\\frac{\\Phi(z_6)}{z_6} - \\frac{\\Phi(z_5)}{z_5}\\right)$. Otherwise, award 0 pt.",
+                "Award 0.1 pt if the answer gives a numerical value of the local dark matter density $\\rho_{\\mathrm{DM}}$ within the range $[6.9 \\times 10^{-22} \\mathrm{kg/m^3}, 8.5 \\times 10^{-22} \\mathrm{kg/m^3}]$. Otherwise, award 0 pt."
+            ],
+            [
+                "Award 1.6 pt if the answer uses the previous relation between density and potential to express $(z_6 - z_5) \\rho_{\\mathrm{DM}} = z_6 \\rho(z_6) - z_5 \\rho(z_5)$, where the difference between the total surface densities at two heights $z_6$ and $z_5$. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer presents a final explicit expression for $\\rho_{\\mathrm{DM}}$ based on $z_6$ and $z_5$, e.g., $\\rho_{\\mathrm{DM}} = \\frac{1}{2\\pi G (z_6 - z_5)} \\left(\\frac{\\Phi(z_6)}{z_6} - \\frac{\\Phi(z_5)}{z_5}\\right).$ Otherwise, award 0 pt.",
+                "Award 0.1 pt if the answer gives a numerical value of the local dark matter density $\\rho_{\\mathrm{DM}}$ within the range $[6.9 \\times 10^{-22} \\mathrm{kg/m^3}, 8.5 \\times 10^{-22} \\mathrm{kg/m^3}]$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{[6.9 \\times 10^{-22}, 8.5 \\times 10^{-22}]}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "$\\mathrm{kg} / \\mathrm{m}^3$"
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+data figure",
+        "field": "Modern Physics",
+        "source": "NBPhO_2025",
+        "image_question": [
+            "image_question/NBPhO_2025_8_3_1.png"
+        ]
+    },
+    {
+        "id": "NBPhO_2025_8_6",
+        "context": "[Phase Spiral] \n\nHere we shall study the motion of Milky Way stars in the Solar neighbourhood in the direction of the $z$-axis, i.e. perpendicular to the galactic plane. For our purposes, we can model the galactic gravity field as being created by a continuous mass density $\\rho$ (that accounts for the masses of stars, dark matter, gas, interstellar dust, etc), and assume that this mass forms an infinite mirror-symmetric plate, i.e. $\\rho \\equiv \\rho(z)$ and $\\rho(z) = \\rho(-z)$ is independent of $x$ and $y$. Throughout the problem, you may assume that each star's total energy is conserved over the entire considered time period. Gravitational constant $G = 6.67 \\times 10^{-11} \\mathrm{m}^{3} \\mathrm{kg}^{-1} \\mathrm{s}^{-2} = 4.30 \\times 10^{-3} \\mathrm{pc} \\mathrm{M}_{\\odot}^{-1} (\\mathrm{km}/\\mathrm{s})^{2}$. \n\n(i) Assuming that the mass density is constant over the plate's thickness. i.e. $\\rho(z) = \\rho_{0}$, what is the acceleration $a_{z}$ of a star at a distance $z$ from the mid-plane? \n\n(ii) Consider a star that starts with zero velocity at a distance of $z = a$ from the mid-plane. With what period does it start oscillating around the mid-plane? \n\nIn reality, density decreases with growing $|z|$. Measuring density has been a great challenge because of contributions from dark and other difficult-to-see matter. Here, we consider a breakthrough method of doing it. Consider the distributions of the stars in our neighbourhood on the $z$-$v_{z}$ phase plane, where each star is a dot with coordinates $(v_{z}, z)$; $v_{z}$ denotes the $z$-component the star's velocity, and $z$ - the vertical coordinate. Initially, these dots were distributed nearly homogeneously, but some time ago, the Milky Way was perturbed externally, probably by a passing-by dwarf galaxy; this shuffled the positions and velocities of stars, creating a bar-shaped overdensity region. When moving within that bar-shaped region from the centre to the periphery, the total energy per mass of stars increased monotonously. Over time, this overdensity region started \"winding up\", due to the oscillation periods of stars in the vertical plane depending on their oscillation amplitude $z_{\\mathrm{m}}$, and evolved into a spiral pictured below (Antoja et al. 2018, Nature 561, 360). An observation that you need to exploit below is that the ordering of stars by energies along the spiral today remains the same as it was at the time of perturbation. \n\nThe oscillation period of stars depends on the amplitude $z_{\\mathrm{m}}$ because the gravitational potential (the potential energy per mass) $\\Phi(z)$ is not parabolic. In such a case, the period can be approximately found by substituting the real $\\Phi(z)$ with a $k z^{2}$ matching $\\Phi(z)$ at $z = z_{\\mathrm{m}}$, i. e. with $k = \\Phi(z_{\\mathrm{m}}) / z_{\\mathrm{m}}^{2}$. \n\n[figure1] \n\n(iii) At the intersection points of the spiral with $v_z = 0$, calculate $\\Phi(z)$ by interpolating data linearly where appropriate; plot your results (this follows the analysis of Guo et al. 2024, ApJ, 960, 133) \n\n(iv) Assuming that the mass density is almost constant for $|z| \\leq 0.3 \\mathrm{kpc}$, what is the mass density near $z = 0$? \n\n(v) Dark matter is an \"invisible\" form of matter that only interacts by gravity. In general, it is found that dark matter forms halos that extend significantly farther than visible matter structures. By assuming that the dark matter density doesn't vary significantly within the volume of interest and that it starts dominating far away from the galactic plane, from around $z = 0.7 \\mathrm{kpc}$, estimate the local dark matter density $\\rho_{\\mathrm{DM}}$. \n\nParts (i)–(v) are preliminary questions and should not be included in the final answer.",
+        "question": "How long ago did the perturbation occur? Express your answer in $s$.",
+        "marking": [
+            [
+                "Award 1.0 pt if the answer includes the idea of using differences in the winding rate between two points on the spiral. Otherwise, award 0 pt.",
+                "Award 0.5 pt if the answer expresses the angular frequency $\\omega$ in terms of $\\Phi(z)$ by assuming a harmonic oscillator, i.e., $\\omega(z) = \\sqrt{\\frac{2\\Phi(z)}{z^2}}$, where $G$ is the gravitational constant, $\\rho_0$ is the constant mass density, $\\Phi(z)$ is the gravitational potential per unit mass at height $z$, and $z$ is the distance from the mid-plane. Otherwise, award 0 pt.",
+                "Award 0.3 pt if the answer picks two labeled points on the spiral (e.g., $z_1$ and $z_6$) and connects the age of the spiral, $\\omega$ and the winding amount via $T_0 = 2.5 \\frac{2\\pi}{\\omega(z_6) - \\omega(z_1)}$. Otherwise, award 0 pt.",
+                "Award 0.2 pt if the answer gives a numerical value for the time of the perturbation that falls within the range $[1.7 \\times 10^{16} \\mathrm{s}, 2.1 \\times 10^{16} \\mathrm{s}$. Otherwise, award 0 pt."
+            ]
+        ],
+        "answer": [
+            "\\boxed{$[1.7 \\times 10^{16} \\mathrm{s}, 2.1 \\times 10^{16} \\mathrm{s}$}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "s"
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+data figure",
+        "field": "Modern Physics",
+        "source": "NBPhO_2025",
+        "image_question": [
+            "image_question/NBPhO_2025_8_3_1.png"
+        ]
+    }
+]

data/PanMechanics_2024.json

@@ -0,0 +1,704 @@
+[
+    {
+        "information": "若有需要，取重力加速度 $g = 10 m/s^2$ 及 重力常数 $G = 6.67 \\times 10^{-11} N m^2/\\mathrm{kg}^2$（若没有特别注明，取所有摩擦力为零）。"
+    },
+    {
+        "id": "PanMechanics_2024_1",
+        "context": "",
+        "question": "由三根质量为 $M$ 、长度为 $L$ 的相同均匀杆组成一个三角形。它通过顶部的枢轴铰接在垂直平面上，如图所示。这个物理摆的小振荡周期是多少？杆子通过其质心的转动惯量为 $I_{\\mathrm{CM}} = \\frac{1}{12} ML^2$。\n\n[figure1]\n\n(A) $\\sqrt{\\frac{3L}{2g}}$ \n(B) $2\\pi \\sqrt{\\frac{3L}{g}}$ \n(C) $\\pi \\sqrt{\\frac{3L}{g}}$ \n(D) $\\pi \\sqrt{\\frac{3ML}{g}}$ \n(E) $\\pi \\sqrt{\\frac{2\\sqrt{3}L}{g}}$",
+        "marking": [],
+        "answer": [
+            "\\boxed{E}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_1_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_2",
+        "context": "",
+        "question": "一辆以 $36 m/s$ 速度行驶的卡车经过一辆以 $45 m/s$ 速度朝相反方向行驶的警车。如果警笛相对于警车的频率为 $500 Hz$,那么当警车接近卡车时,卡车内的观察者听到的频率是多少？(空气中的声速为 $343 m/s$) \n\n(A) $396 \\mathrm{Hz}$ \n(B) $636 \\mathrm{Hz}$ \n(C) $361 \\mathrm{Hz}$ \n(D) $393 \\mathrm{Hz}$ \n(E) $617 \\mathrm{Hz}$",
+        "marking": [],
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2024_3",
+        "context": "",
+        "question": "一颗卫星绕 X 行星做圆形轨道运行，且轨道距离行星表面非常近。要估计行星 X 的密度，我们只需测量：\n\n(A) 卫星的周期 \n(B) 轨道半径 \n(C) 卫星的速度 \n(D) 行星 X 的质量 \n(E) 卫星的质量",
+        "marking": [],
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2024_4",
+        "context": "质量为 $M$ 的三角楔子置于水平无摩擦的地面上。将质量为 $m$ 的木块放在楔子上，如图所示。木块和楔子之间没有摩擦力。系统从静止状态释放。给定 $M = 3m$ 和 $\\alpha = 45^{\\circ}$。",
+        "question": "求三角楔子加速度的大小。\n\n(A) $g/6$ \n(B) $g/7$ \n(C) $g/4$ \n(D) $g$ \n(E) 0",
+        "marking": [],
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_4_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_5",
+        "context": "质量为 $M$ 的三角楔子置于水平无摩擦的地面上。将质量为 $m$ 的木块放在楔子上，如图所示。木块和楔子之间没有摩擦力。系统从静止状态释放。给定 $M = 3m$ 和 $\\alpha = 45^{\\circ}$。",
+        "question": "若当木块滑到地面时，楔子相对地面的速度为 $1 m/s$。求木块在楔子上离地面的初始高度（假设木块为无体积的重点）。 \n\n(A) $0.60 m$ \n(B) $0.82 m$ \n(C) $1.00 m$ \n(D) $1.05 m$ \n(E) $1.40 m$",
+        "marking": [],
+        "answer": [
+            "\\boxed{E}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_4_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_6",
+        "context": "",
+        "question": "在一维运动中，力 $F = -(m/b) v^2$ 作用在质量为 $m$ 的粒��上，其中 $v$ 是粒子的速度，$b$ 是常数。在 $t = 0 s$ 时，该粒子位于 $x = 0 m$。哪一个是粒子随时间变化的可能位置？\n\n(A) $x(t) = b \\ln (\\frac{t}{1 \\mathrm{s}})$ \n(B) $x(t) = b \\ln (\\frac{t}{1 \\mathrm{s}} + 1)$ \n(C) $x(t) = b \\frac{t / 1\\mathrm{s}}{2 + ( t/1\\mathrm{s})^2}$ \n(D) $x(t) = \\frac{b}{t/1\\mathrm{s}}$ \n(E) $x(t) = b \\sin (t/1\\mathrm{s})$",
+        "marking": [],
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2024_7",
+        "context": "一个体重 $60 \\mathrm{kg}$ 的人以 $5 m/s$ 的初始速度沿着半径为 $3 m$、质量为 $100 \\mathrm{kg}$ 的固定均匀圆形平台的切线跑步，如图所示。平台本来静止，当人跳上及静止在平台上后，平台绕中心的垂直轴旋转。圆形平台通过其质心的转动惯量为 $I_{\\mathrm{CM}} = \\frac{1}{2} M R^2$。",
+        "question": "求该人跳上平台后系统的角速度。\n\n(A) 0.500 rad/s \n(B) 0.250 rad/s \n(C) 1.33 rad/s \n(D) 0.909 rad/s \n(E) 1.705 rad/s",
+        "marking": [],
+        "answer": [
+            "\\boxed{D}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_7_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_8",
+        "context": "一个体重 $60 \\mathrm{kg}$ 的人以 $5 m/s$ 的初始速度沿着半径为 $3 m$、质量为 $100 \\mathrm{kg}$ 的固定均匀圆形平台的切线跑步，如图所示。平台本来静止，当人跳上及静止在平台上后，平台绕中心的垂直轴旋转。圆形平台通过其质心的转动惯量为 $I_{\\mathrm{CM}} = \\frac{1}{2} M R^2$。",
+        "question": "找出总机械能的损失。\n\n(A) $150 J$ \n(B) $341 J$ \n(C) $257 J$ \n(D) $457 J$ \n(E) $0 J$",
+        "marking": [],
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_7_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_9",
+        "context": "两个 $1.0 \\mathrm{kg}$ 的粒子以 $(40.0 m/s) \\hat{l}$ 和 $(-20.0 m/s) \\hat{l}$ 的速度沿直线相互移动并發生碰撞。碰撞后，其中一个粒子以 $30.0 m/s$ 的速度离开。在碰撞过程中，两颗粒子共损失了 $100 \\mathrm{J}$ 的能量。",
+        "question": "求碰撞后另一个粒子的速度。\n\n(A) $33.2 m/s$ \n(B) $36.1 m/s$ \n(C) $17.3 m/s$ \n(D) $26.8 m/s$ \n(E) $30.0 m/s$",
+        "marking": [],
+        "answer": [
+            "\\boxed{E}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2024_10",
+        "context": "两个 $1.0 \\mathrm{kg}$ 的粒子以 $(40.0 m/s) \\hat{l}$ 和 $(-20.0 m/s) \\hat{l}$ 的速度沿直线相互移动并發生碰撞。碰撞后，其中一个粒子以 $30.0 m/s$ 的速度离开。在碰撞过程中，两颗粒子共损失了 $100 \\mathrm{J}$ 的能量。",
+        "question": "求碰撞后粒子速度之间的夹角。\n\n(A) $141^{\\circ}$ \n(B) $105^{\\circ}$ \n(C) $70.5^{\\circ}$ \n(D) $96.4^{\\circ}$ \n(E) $48.2^{\\circ}$",
+        "marking": [],
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2024_11",
+        "context": "",
+        "question": "如图所示，粒子在 $x = a$ 点从静止状态释放，并根据图中所示的势能函数 $U(x)$ 沿 $x$ 轴移动。图中 $U(a) = U(e)$。粒子其后的运动为：\n\n(A) 移动到 $x = e$ 左侧的点，停止并保持静止。\n(B) 在 $x = a$ 及 $x = e$ 之间来回移动。\n(C) 以不同的速度移动到无穷大 $(x \\rightarrow \\infty)$。\n(D) 移动到 $x = b$，并保持静止状态。\n(E) 移动到 $x = e$，然后移动到 $x = d$，并保持静止状态。",
+        "marking": [],
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+data figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_11_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_12",
+        "context": "",
+        "question": "逃离木星引力的最低速度为 60 公里/秒。假设木星的半径为 70,000 公里，那么 80 公斤重的宇航员在木星上的重量是多少？\n\n(A) $1029 N$ \n(B) $1371 N$ \n(C) $2057 N$ \n(D) $2742 N$ \n(E) $4114 N$",
+        "marking": [],
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2024_13",
+        "context": "",
+        "question": "下列哪一个人必须是非惯性观察者？将地面视为惯性系。应考虑空气摩擦力。\nI. 一个人的位置被另一个观察者描述为 $y(t) = -\\frac{g}{2} t^2$。\nII. 坐在固定在地面上旋转的旋转木马边缘的人。\nIII. 一个人垂直向上跳跃。而此刻，当人处于最高位置时。\nIV. 一个人垂直向上跳跃。而此刻，人还在上升的时候。\nV. 一个戴着打开的降落伞进行跳伞的人。\n\n(A) 只有 I, IV 和 V \n(B) 只有 I 和 II \n(C) 只有 I,II,IV 和 V \n(D) 只有 II, III 和 IV \n(E) I, II, III, IV 和 V",
+        "marking": [],
+        "answer": [
+            "\\boxed{D}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2024_14",
+        "context": "",
+        "question": "如图所示一个 3.0 kg 的三角体，求推动三角体的力 $F$，使在三角块上的 1.0 kg 方形块不会沿斜面移动。假设所有表面都是无摩擦的。\n\n(A) $15 N$ \n(B) $20 N$ \n(C) $25 N$ \n(D) $40 N$ \n(E) $45 N$",
+        "marking": [],
+        "answer": [
+            "\\boxed{D}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_14_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_15",
+        "context": "",
+        "question": "一个边长为 $L$ 的正方体平稳地漂浮在容器内静止的水中。此时有一半的立方体位于水面以下。再将密度为水四分之一的液体添加到容器中，使立方体完全浸没在液体的表面下，而液体和水不混合，液体留在水上面。添加液体后，立方体从原來的水面上升了多少？\n\n(A) $L/6$ \n(B) $L/3$ \n(C) $L/2$ \n(D) $L/4$ \n(E) $L/5$",
+        "marking": [],
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2024_16",
+        "context": "",
+        "question": "一个盒子由两根具有相同线性质量密度的绳子悬挂在天花板上，如图所示。求弦 1 的基频 $f_1$ 与弦 2 的基频 $f_2$ 之比，$f_1 / f_2$。\n\n(A) $\\sqrt{3 \\sqrt{3}}$ \n(B) 3 \n(C) $3 \\sqrt{3}$ \n(D) $\\sqrt{6}$ \n(E) $\\sqrt{3}/4$",
+        "marking": [],
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_16_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_17_1",
+        "context": "质量为 $m$ 的质点附着在力常数为 $k$ 的弹簧上，在粗糙表面上沿 X 轴移动。以原点为弹簧自然长度时的位置。",
+        "question": "当 $t = 0$ 时，粒子在 $x_0 \\neq 0$ 及静止。假设弹簧力足够大，使得粒子在恒定的摩擦力 $f$ 下移动。在时间 $0 \\leq t \\leq \\tau$ 内，求 $x(t)$，其中 $\\tau$ 是 $t = 0$ 后粒子第一次停止的时间。用 $k$、$m$、$f$ 和 $x_0$ 表示 $x(t)$。设静摩擦系数为 0.03，动摩擦系数为 0.01，$m = 1 \\mathrm{kg}$，$k = 10 \\mathrm{N}/\\mathrm{m}$，重力加速度 $g = 10 m/s^2$。",
+        "marking": [],
+        "answer": [
+            "\\boxed{$x(t) = (x_0 - \\frac{f}{k}) \\cos(\\sqrt{\\frac{k}{m}}t) + \\frac{f}{k}$}",
+            "\\boxed{$x(t) = (x_0 + \\frac{f}{k}) \\cos(\\sqrt{\\frac{k}{m}}t) - \\frac{f}{k}$}"
+        ],
+        "answer_type": [
+            "Expression",
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            8.0,
+            8.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2024_17_2",
+        "context": "质量为 $m$ 的质点附着在力常数为 $k$ 的弹簧上，在粗糙表面上沿 X 轴移动。以原点为弹簧自然长度时的位置。\n(a) 当 $t = 0$ 时，粒子在 $x_0 \\neq 0$ 及静止。假设弹簧力足够大，使得粒子在恒定的摩擦力 $f$ 下移动。在时间 $0 \\leq t \\leq \\tau$ 内，求 $x(t)$，其中 $\\tau$ 是 $t = 0$ 后粒子第一次停止的时间。用 $k$、$m$、$f$ 和 $x_0$ 表示 $x(t)$。设静摩擦系数为 0.03，动摩擦系数为 0.01，$m = 1 \\mathrm{kg}$，$k = 10 \\mathrm{N}/\\mathrm{m}$，重力加速度 $g = 10 m/s^2$。\n注意：（a）是前置问题，请不要写入最终答案中。",
+        "question": "设 $x_0 = 1 m$。使用 (a) 或其他方式，找到粒子永久停止时的最终位置。（单位用 $m$ 表示）",
+        "marking": [],
+        "answer": [
+            "\\boxed{-0.02}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "m"
+        ],
+        "points": [
+            8.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2024_17_3",
+        "context": "质量为 $m$ 的质点附着在力常数为 $k$ 的弹簧上，在粗糙表面上沿 X 轴移动。以原点为弹簧自然长度时的位置。\n(a) 当 $t = 0$ 时，粒子在 $x_0 \\neq 0$ 及静止。假设弹簧力足够大，使得粒子在恒定的摩擦力 $f$ 下移动。在时间 $0 \\leq t \\leq \\tau$ 内，求 $x(t)$，其中 $\\tau$ 是 $t = 0$ 后粒子第一次停止的时间。用 $k$、$m$、$f$ 和 $x_0$ 表示 $x(t)$。设静摩擦系数为 0.03，动摩擦系数为 0.01，$m = 1 \\mathrm{kg}$，$k = 10 \\mathrm{N}/\\mathrm{m}$，重力加速度 $g = 10 m/s^2$。\n(b) 设 $x_0 = 1 m$。使用 (a) 或其他方式，找到粒子永久停止时的最终位置。（单位用 $m$ 表示）。\n注意：(a) 和 (b) 都是前置问题，请不要写入最终答案中。",
+        "question": "求出粒子的总移动距离。（单位用 $m$ 表示）",
+        "marking": [],
+        "answer": [
+            "\\boxed{49.98}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "m"
+        ],
+        "points": [
+            6.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2024_17_4",
+        "context": "质量为 $m$ 的质点附着在力常数为 $k$ 的弹簧上，在粗糙表面上沿 X 轴移动。以原点为弹簧自然长度时的位置。\n(a) 当 $t = 0$ 时，粒子在 $x_0 \\neq 0$ 及静止。假设弹簧力足够大，使得粒子在恒定的摩擦力 $f$ 下移动。在时间 $0 \\leq t \\leq \\tau$ 内，求 $x(t)$，其中 $\\tau$ 是 $t = 0$ 后粒子第一次停止的时间。用 $k$、$m$、$f$ 和 $x_0$ 表示 $x(t)$。设静摩擦系数为 0.03，动摩擦系数为 0.01，$m = 1 \\mathrm{kg}$，$k = 10 \\mathrm{N}/\\mathrm{m}$，重力加速度 $g = 10 m/s^2$。\n(b) 设 $x_0 = 1 m$。使用 (a) 或其他方式，找到粒子永久停止时的最终位置。（单位用 $m$ 表示）。\n(c) 求出粒子的总移动距离。（单位用 $m$ 表示）\n注意：(a)、(b) 和 (c) 都是前置问题，请不要写入最终答案中。",
+        "question": "求粒子永久停止之前所经过的总时间。（单位用 $s$ 表示）",
+        "marking": [],
+        "answer": [
+            "\\boxed{48.68}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "s"
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2024_18_1",
+        "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。",
+        "question": "求 $k_N$ 。答案以 $K$ 表示。",
+        "marking": [],
+        "answer": [
+            "\\boxed{$k_N = (N-1) K$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_18_1.png",
+            "image_question/PanMechanics_2024_18_2.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_18_2",
+        "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。",
+        "question": "求释放前处于平衡状态的弹簧的总长度 $L_0$。答案以 $M$，$g$ 和 $K$ 表示。",
+        "marking": [],
+        "answer": [
+            "\\boxed{$L_0 = \\frac{Mg}{2K}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_18_1.png",
+            "image_question/PanMechanics_2024_18_2.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_18_3",
+        "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。",
+        "question": "应用牛顿第二定律，写出顶部 $x_N$、底部 $x_1$ 和第 $n$ 个质量 $x_n$ 的运动方程，而 $1 < n < N$，答案以 $m_N$、$k_N$、$g$ 以及其他质量的坐标 $x_2, x_3, \\ldots$（如果需要）表示。",
+        "marking": [],
+        "answer": [
+            "\\boxed{$m_N \\ddot{x}_1 = -k_N (x_1 - x_2) + m_N g$}",
+            "\\boxed{$m_N \\ddot{x}_n = k_N (x_{n+1} - 2x_n + x_{n-1}) + m_N g$}",
+            "\\boxed{$m_N \\ddot{x}_N = k_N (x_{N-1} - x_N) + m_N g$}"
+        ],
+        "answer_type": [
+            "Equation",
+            "Equation",
+            "Equation"
+        ],
+        "unit": [
+            null,
+            null,
+            null
+        ],
+        "points": [
+            2.0,
+            2.0,
+            2.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_18_1.png",
+            "image_question/PanMechanics_2024_18_2.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_18_4",
+        "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 \\mathrm{s}$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 \\mathrm{s}$ 时，$x_N = 0 \\mathrm{m}$。现在考虑 $N = 2$、$m_N = 1 \\mathrm{kg}$ 且 $k_N = 1 \\mathrm{N}/\\mathrm{m}$ 的情况（$g = 10 m/s^2$）。",
+        "question": "求系统质心的加速度。（向下为正，单位用 $m/s^2$ 表示）",
+        "marking": [],
+        "answer": [
+            "\\boxed{10}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "m/s^2"
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_18_1.png",
+            "image_question/PanMechanics_2024_18_2.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_18_5",
+        "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。现在考虑 $N = 2$、$m_N = 1 \\mathrm{kg}$ 且 $k_N = 1 \\mathrm{N}/\\mathrm{m}$ 的情况（$g = 10 m/s^2$）。",
+        "question": "求出两个质量随时间变化的距离函数：$d(t) = x_1(t) - x_2(t)$。(表达式中重力加速度用 $g$ 表示)",
+        "marking": [],
+        "answer": [
+            "\\boxed{$d(t) = g \\cos(\\sqrt{2} t)$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            5.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_18_1.png",
+            "image_question/PanMechanics_2024_18_2.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_18_6",
+        "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。现在考虑 $N = 2$、$m_N = 1 \\mathrm{kg}$ 且 $k_N = 1 \\mathrm{N}/\\mathrm{m}$ 的情况（$g = 10 m/s^2$）。",
+        "question": "当两个质量碰撞时（设碰撞时间为 $\\tau$），设底部质量从 $t = 0 \\mathrm{s}$ 的下降距离为 $D_2 = x_1(\\tau) - x_1(0) = \\gamma L_0$。求 $\\gamma$ 的数值。",
+        "marking": [],
+        "answer": [
+            "\\boxed{0.117}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            6.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_18_1.png",
+            "image_question/PanMechanics_2024_18_2.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_18_7",
+        "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。现在考虑 $N = 3$ 的情况。",
+        "question": "为了使弹簧的总质量和总弹簧常数 $K$ 与 $N = 2$、$m_N = 1 kg$ 且 $k_N = 1 N/m$ 的情况相同，（1）求出对应的 $m_N$（单位用 $kg$ 表示），（2）求出对应的 $k_N$（单位用 $N/m$ 表示）。",
+        "marking": [],
+        "answer": [
+            "\\boxed{2/3}",
+            "\\boxed{2}"
+        ],
+        "answer_type": [
+            "Numerical Value",
+            "Numerical Value"
+        ],
+        "unit": [
+            "kg",
+            "N/m"
+        ],
+        "points": [
+            1.0,
+            1.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_18_1.png",
+            "image_question/PanMechanics_2024_18_2.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_18_8",
+        "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。现在考虑 $N = 3$ 的情况。",
+        "question": "求解底部质量随时间变化的位置：$x_1(t)$。提示：尝试先找出质心的运动方程，$x_1$ 和 $x_3$ 之间的差的运动方程，及另一个由 $x_1, x_2$ 及 $x_3$ 的线性组合组成的量的运动方程。",
+        "marking": [],
+        "answer": [
+            "\\boxed{$x_1(t) = \\frac{5}{9}g + \\frac{1}{2}gt^2 + \\frac{g}{2} \\cos(\\sqrt{3}t) - \\frac{g}{18} \\cos(3t)$}"
+        ],
+        "answer_type": [
+            "Equation"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            8.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_18_1.png",
+            "image_question/PanMechanics_2024_18_2.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2024_18_9",
+        "context": "总质量为 $M$ 的弹性弹簧在未拉伸时具有均匀的质量分布。其弹簧常数为 $K$，为简单起见，假设其自然长度为零。现在它从顶端悬挂起来，并在恒定重力 $g$ 下垂直悬挂并达至静止状态。\n\n[figure1]\n\n如图 1 所示，在 $t = 0 s$ 时，顶端从静止状态释放，弹簧落下。为了理解它的下落运动，我们可以将弹簧建模为一系列 $N$ 个质量为 $m_N$ 的相同质量，与 $N - 1$ 个具有弹簧常数 $k_N$ 和零自然长度的相同弹簧连接。\n\n[figure2]\n\n如图 2 所示，坐标 $x_1, x_2, \\cdots x_N$ 分别是距离底部 $(x_1)$ 和顶部 $(x_N)$ 位置的质量，从天花板开始测量（向下为正）。在 $t = 0 s$ 时，$x_N = 0 m$。\n\n(f) 考虑 $N = 2$、$m_N = 1 \\mathrm{kg}$ 且 $k_N = 1 \\mathrm{N}/\\mathrm{m}$ 的情况（$g = 10 m/s^2$）。当两个质量碰撞时（设碰撞时间为 $\\tau$），设底部质量从 $t = 0 \\mathrm{s}$ 的下降距离为 $D_2 = x_1(\\tau) - x_1(0) = \\gamma L_0$。求 $\\gamma$ 的数值。\n注意：(f) 是前置问题，请不要写入最终答案中。\n\n现在考虑 $N = 3$ 的情况。",
+        "question": "底部质量经过 (f) 部分中的 $\\tau$ 时间后：\n\n（1）求下降的距离 $D_3 = x_1(\\tau) - x_1(0)$（用 $L_0$ 表示）。\n（2）比较 $D_3$ 及在 (f) 部分所得距离 $D_2$，看看哪一个比较小，在���案中写上 $D_3$ 或 $D_2$。",
+        "marking": [],
+        "answer": [
+            "\\boxed{$D_3 = 0.054 L_0$}",
+            "\\boxed{$D_3$}"
+        ],
+        "answer_type": [
+            "Expression",
+            "Open-Ended"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            1.0,
+            1.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2024",
+        "image_question": [
+            "image_question/PanMechanics_2024_18_1.png",
+            "image_question/PanMechanics_2024_18_2.png"
+        ]
+    }
+]

data/PanMechanics_2025.json

@@ -0,0 +1,548 @@
+[
+    {
+        "information": "若有需要，取重力加速度 $g = 9.80 m/s^2$ 及 重力常数 $G = 6.67 \\times 10^{-11} N m^2/\\mathrm{kg}^2$（若没有特别注明，取所有摩擦力为零）。"
+    },
+    {
+        "id": "PanMechanics_2025_1",
+        "context": "",
+        "question": "一辆重量为 800 公斤的赛车沿着半径为 $R = 100$ 米的圆形赛道行驶。赛道倾斜角度为 $\\beta = 30.0^{\\circ}$，如图所示。轮胎与赛道之间的静摩擦系数和动摩擦系数分别为 $\\mu_s = 0.300$ 和 $\\mu_k = 0.200$。保留 3 位有效数字，在什么速度范围内，轮胎不会相对于赛道打滑？您可以忽略空气阻力和滚动摩擦，这样轮胎和赛道之间在赛道切线方向上就没有摩擦力。\n\n[figure1]\n\n(A) $v \\leq 32.2 m/s$. \n(B) $15.2 m/s \\leq v$. \n(C) $v \\leq 38.4 m/s$. \n(D) $15.2 m/s \\leq v \\leq 32.2 m/s$. \n(E) $20.7 m/s \\leq v \\leq 38.4 m/s$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{D}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": [
+            "image_question/PanMechanics_2025_1_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2025_2",
+        "context": "",
+        "question": "一辆重量为 800 公斤的赛车沿着半径为 $R = 100$ 米的圆形赛道行驶。赛道倾斜角度为 $\\beta = 30.0^{\\circ}$，如图所示。轮胎与赛道之间的静摩擦系数和动摩擦系数分别为 $\\mu_s = 0.600$ 和 $\\mu_k = 0.400$。保留 3 位有效数字，在什么速度范围内，轮胎不会相对于赛道打滑？您可以忽略空气阻力和滚动摩擦，这样轮胎和赛道之间在赛道切线方向上就没有摩擦力。\n\n[figure1]\n\n(A) $10.4 m/s \\leq v$. \n(B) $v \\leq 35.1 m/s$. \n(C) $v \\leq 42.0 m/s$. \n(D) $10.4 m/s \\leq v \\leq 42.0 m/s$. \n(E) $18.5 m/s \\leq v \\leq 35.1 m/s$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": [
+            "image_question/PanMechanics_2025_1_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2025_3",
+        "context": "",
+        "question": "一个装有弹簧的玩具静止地放在水平、无摩擦的表面上。当弹簧松开时，玩具会分裂成三块，A、B 和 C，且各自沿着表面滑动。A、B 和 C 的质量分别为 $m_A$、$m_B$ 和 $m_C$。若已知 A 和 B 的速度之间的角度为 $120^{\\circ}$，问必须满足以下哪个条件才能确保 C 的速率为三者中最快？\n\n(A) $m_C < m_A + m_B$. \n(B) $m_C < \\frac{m_A + m_B}{2}$. \n(C) $m_C < \\frac{\\sqrt{3}}{2} (m_A + m_B)$. \n(D) $m_C < \\frac{\\min{m_A, m_B}}{2}$. \n(E) $m_C < \\frac{\\sqrt{3}}{2} \\min{m_A, m_B}$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{E}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2025_4",
+        "context": "",
+        "question": "质量为 $4.00 \\times 10^{30}$ 千克的恒星 A 正朝某个方向移动，质量为 $2.00 \\times 10^{30}$ 千克的恒星 B 位于其前方并正朝同一方向移动。恒星 A 和恒星 B 的初速度分别为 10 公里/秒和 40 公里/秒，它们的初始距离为 $2.00 \\times 10^8$ 公里。求它们的最大距离。\n\n(A) $2.58 \\times 10^8$ 公里. \n(B) $3.33 \\times 10^8$ 公里. \n(C) $4.75 \\times 10^8$ 公里. \n(D) $5.29 \\times 10^8$ 公里. \n(E) $6.14 \\times 10^8$ 公里.",
+        "marking": [],
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2025_5",
+        "context": "",
+        "question": "在一个半径为 $R$ 的均匀球体上挖出一个直径为 $R$ 的洞，使原球体的中心 $O$ 位于洞的表面，如图所示。洞的中心和 $O$ 都位于 $y$ 轴上。求该物体绕通过 $O$ 并与洞表面相切���轴（图中的 $x$ 轴）的转动惯量，以 $R$ 和物体的质量 $M$ 表达。\n\n[figure1]\n\n(A) $\\frac{1}{3} MR^2$. \n(B) $\\frac{57}{140} MR^2$. \n(C) $\\frac{463}{1120} MR^2$. \n(D) $\\frac{31}{70} MR^2$. \n(E) $\\frac{249}{560} MR^2$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": [
+            "image_question/PanMechanics_2025_5_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2025_6",
+        "context": "",
+        "question": "一个均匀的矩形板，其一边固定在地面上，且板面与垂直方向成角度 $\\theta_0$，其中 $0^{\\circ} \\leq \\theta_0 \\leq 90^{\\circ}$。如果 $\\theta_0$ 大于某个角度，则在放开板体使其在重力作用下绕固定边缘旋转的那一刻，板体的另一端的加速度的垂直向下分量大于 $g$。求该角度。\n\n(A) $\\sin^{-1} \\frac{1}{3}$. \n(B) $\\sin^{-1} \\sqrt{\\frac{1}{3}}$. \n(C) $\\sin^{-1} \\frac{2}{3}$. \n(D) $45^{\\circ}$. \n(E) $\\sin^{-1} \\sqrt{\\frac{2}{3}}$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{E}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2025_7",
+        "context": "",
+        "question": "一个均匀的矩形板，其一边固定在地面上，且板面与垂直方向成角度 $\\theta_0$，其中 $0^{\\circ} \\leq \\theta_0 \\leq 90^{\\circ}$。如果 $\\theta_0$ 大于某个角度，则在放开板体使其在重力作用下绕固定边缘旋转的那一刻，板体的另一端的加速度的垂直向下分量大于 $g$。当板体以上述角度被放开的瞬间，地面所施加的反作用力的大小和垂直分力分别是多少？\n\n(A) $\\sqrt{\\frac{3}{8}} Mg$, $\\frac{1}{2} Mg$ 向上. \n(B) $\\sqrt{\\frac{3}{8}} Mg$, $\\frac{1}{2} Mg$ 向下. \n(C) $\\frac{\\sqrt{3}}{2} Mg$, $\\frac{1}{2} Mg$ 向上. \n(D) $\\frac{\\sqrt{3}}{2} Mg$, $\\frac{1}{2} Mg$ 向下. \n(E) 此瞬间没有任何反作用力.",
+        "marking": [],
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2025_8",
+        "context": "",
+        "question": "一个均匀的矩形板，其一边固定在地面上，且板面与垂直方向成角度 $\\theta_0$，其中 $0^{\\circ} \\leq \\theta_0 \\leq 90^{\\circ}$。如果 $\\theta_0$ 大于某个角度，则在放开板体使其在重力作用下绕固定边缘旋转的那一刻，板体的另一端的加速度的垂直向下分量大于 $g$。如果 $\\theta_0$ 小于上述临界角，则当板旋转到某个角度时，板体的另一端的加速度的垂直向下分量大于 $g$。如果 $\\theta_0 = 0^{\\circ}$，求该角度。\n\n(A) $\\sin^{-1} \\frac{\\sqrt{2}-1}{3}$. \n(B) $\\cos^{-1} \\frac{1+\\sqrt{3}}{3}$. \n(C) $\\cos^{-1} \\frac{1+\\sqrt{2}}{3}$. \n(D) $\\sin^{-1} \\frac{1+\\sqrt{2}}{3}$. \n(E) $\\cos^{-1} \\frac{\\sqrt{2}-1}{3}$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2025_9",
+        "context": "",
+        "question": "一根均匀的细杆，质量为 $M$，长度为 $a$，最初静止在光滑的地板上。它的一端被一个点质量 $m$ 撞击，该点质量的速率为 $v$，垂直于杆。碰撞后，点质量嵌入杆中，如图所示。取 $M = 6.00$ 千克，$m = 2.00$ 千克，$a = 10.0$ 米，$v = 20.0$ 米/秒。求刚碰撞后瞬间杆另一端的速度。\n\n[figure1]\n\n(A) 0 米/秒. \n(B) 向右 6.67 米/秒. \n(C) 向右 10.2 米/秒. \n(D) 向左 3.45 米/秒. \n(E) 向左 5.71 米/秒.",
+        "marking": [],
+        "answer": [
+            "\\boxed{E}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": [
+            "image_question/PanMechanics_2025_9_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2025_10",
+        "context": "",
+        "question": "大炮 A 和大炮 B 可以以相同的速率 $u$ 发射炮弹。大炮 B 位于高为 $h$ 的悬崖顶。大炮 A 位于悬崖左方地面上，与悬崖的距离为 $d$。在某一时刻，大炮 B 向左水平发射一枚炮弹。大炮 A 同时以一定的仰角 $\\theta$ 发射一枚炮弹，如图所示。求出使得两枚炮弹可能相撞的 $\\theta$。\n\n[figure1]\n\n(A) $\\tan^{-1} \\frac{h}{d}$. \n(B) $\\tan^{-1} \\frac{d}{h}$. \n(C) $2 \\tan^{-1} \\frac{h}{d}$. \n(D) $2 \\tan^{-1} \\frac{d}{h}$. \n(E) $90^{\\circ}$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{C}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": [
+            "image_question/PanMechanics_2025_10_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2025_11",
+        "context": "",
+        "question": "大炮 A 和大炮 B 可以以相同的速率 $u$ 发射炮弹。大炮 B 位于高为 $h$ 的悬崖顶。大炮 A 位于悬崖左方地面上，与悬崖的距离为 $d$。在某一时刻，大炮 B 向左水平发射一枚炮弹。大炮 A 同时以一定的仰角 $\\theta$ 发射一枚炮弹，如图所示。当速度 $u$ 低于多少时，两颗炮弹不可能相撞？\n\n[figure1]\n\n(A) $\\frac{(h^2+d^2)}{hd} \\sqrt{gh}$. \n(B) $\\frac{1}{2} \\frac{(h^2+d^2)}{hd} \\sqrt{gh}$. \n(C) $\\frac{1}{\\sqrt{2}} \\frac{(h^2+d^2)}{hd} \\sqrt{gh}$. \n(D) $\\frac{1}{2\\sqrt{2}} \\frac{(h^2+d^2)}{hd} \\sqrt{gh}$. \n(E) $\\frac{1}{4} \\frac{(h^2+d^2)}{hd} \\sqrt{gh}$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{D}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": [
+            "image_question/PanMechanics_2025_10_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2025_12",
+        "context": "",
+        "question": "如图所示，一根刚性均匀杆，其右端受到向右 $2F$ 的力，左端受到向右 $F$ 的力。杆中点处的张力是多少？\n\n[figure1]\n\n(A) 0. \n(B) $F/2$. \n(C) $F$. \n(D) $3F/2$. \n(E) $2F$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": [
+            "image_question/PanMechanics_2025_12_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2025_13",
+        "context": "",
+        "question": "考虑一个牛顿摆，其中只有两个球 A 和 B，它们的质量分别为 $m_A$ 和 $m_B$，且这两个质量可能不同。每个球都用一根绳子悬挂在一水平杆上固定的点。B 最初处于静止状态，A 最初在一定高度开始向下摆动并以速率 $u$ 撞击 B，如图所示。此时，两根绳子都是垂直的。碰撞是理想的弹性碰撞。碰撞后，两个球向上摆动，然后再次向下摆动，并在相同的最低位置再次碰撞。求 $m_B / m_A$。\n\n[figure1]\n\n(A) 1/3. \n(B) 1/2. \n(C) 1. \n(D) 2. \n(E) 3.",
+        "marking": [],
+        "answer": [
+            "\\boxed{E}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": [
+            "image_question/PanMechanics_2025_13_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2025_14",
+        "context": "",
+        "question": "考虑一个牛顿摆，其中只有两个球 A 和 B，它们的质量分别为 $m_A$ 和 $m_B$，且这两个质量可能不同。每个球都用一根绳子悬挂在一水平杆上固定的点。B 最初处于静止状态，A 最初在一定高度开始向下摆动并以速率 $u$ 撞击 B，如图所示。此时，两根绳子都是垂直的。碰撞是理想的弹性碰撞。碰撞后，两个球向上摆动，然后再次向下摆动，并在相同的最低位置再次碰撞。第二次碰撞后 A 和 B 的速度分别是多少？取向右的速度为正。\n\n[figure1]\n\n(A) A: $-u$, B: 0. \n(B) A: $u$, B: 0. \n(C) A: $\\frac{m_A-m_B}{m_A+m_B} u$, B: $\\frac{2m_A}{m_A+m_B} u$. \n(D) A: 0, B: $\\sqrt{\\frac{m_A}{m_B}} u$. \n(E) A: 0, B: $\\frac{m_A}{m_B} u$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{A}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": [
+            "image_question/PanMechanics_2025_13_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2025_15",
+        "context": "",
+        "question": "考虑如图所示的装置。4 个金属球完全相同，质量均为 $m$。磁铁的质量为 $M = 5m$。整个装置放置在气垫轨道上，这样可以忽略所有摩擦力。最初，球 A 从左侧很远的地方以初速率 $u$ 向右移动。球 A 撞到磁铁后，球 D 被射向右侧，并在很远的地方达到最终速率 $v = 9u$。碰撞后，磁铁与球 A、B 和 C 粘在一起。由于没有摩擦，球在轨道上滑动而不旋转。分别用 $K_i$ 和 $K_f$ 表示整个系统的初动能和终动能。求 $K_f / K_i$。\n\n[figure1]\n\n(A) 81. \n(B) 89. \n(C) 105. \n(D) 121. \n(E) 137.",
+        "marking": [],
+        "answer": [
+            "\\boxed{B}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": [
+            "image_question/PanMechanics_2025_15_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2025_16",
+        "context": "",
+        "question": "一根 $2.00$ 米长的均匀细绳的一端连接到振荡器上，另一端固定。当振荡器设置为进行振幅为 $1.00$ 毫米、频率为 $10.0$ 赫兹的简谐运动时，会产生横向驻波。沿弦传播的波速为 $41.0$ 米/秒。由于振荡幅度很小，可以忽略由于振荡引起的弦长变化，故振荡器到固定端的距离可以取为 $2.00$ 米。求产生的驻波的最大振幅。\n\n(A) 1.00 毫米. \n(B) 6.53 毫米. \n(C) 9.35 毫米. \n(D) 13.1 毫米. \n(E) 18.7 毫米.",
+        "marking": [],
+        "answer": [
+            "\\boxed{D}"
+        ],
+        "answer_type": [
+            "Multiple Choice"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2025_17_1",
+        "context": "",
+        "question": "如图所示，两个均匀薄圆盘的质量分别为 $m$ 和 $4m$，半径分别为 $a$ 和 $2a$，由一根穿过它们中心的无质量的刚性杆牢固地固定住，该杆长度为 $l = \\sqrt{24} a$。该组件放在坚固平坦的表面上，并使其在表面上滚动而不滑动。绕杆轴的角速率为 $\\omega$。\n\n[figure1]\n\n求杆与水平面的夹角 $\\theta$。（单位用 $\\mathrm{rad}$ 表示）",
+        "marking": [],
+        "answer": [
+            "\\boxed{0.201}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "$\\mathrm{rad}$"
+        ],
+        "points": [
+            4.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": [
+            "image_question/PanMechanics_2025_17_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2025_17_2",
+        "context": "",
+        "question": "如图所示，两个均匀薄圆盘的质量分别为 $m$ 和 $4m$，半径分别为 $a$ 和 $2a$，由一根穿过它们中心的无质量的刚性杆牢固地固定住，该杆长度为 $l = \\sqrt{24} a$。该组件放在坚固平坦的表面上，并使其在表面上滚动而不滑动。绕杆轴的角速率为 $\\omega$。\n\n[figure1]\n\n求组件质心绕 $z$ 轴的角速率。",
+        "marking": [],
+        "answer": [
+            "\\boxed{$\\frac{\\omega}{5}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            7.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": [
+            "image_question/PanMechanics_2025_17_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2025_17_3",
+        "context": "",
+        "question": "如图所示，两个均匀薄圆盘的质量分别为 $m$ 和 $4m$，半径分别为 $a$ 和 $2a$，由一根穿过它们中心的无质量的刚性杆牢固地固定住，该杆长度为 $l = \\sqrt{24} a$。该组件放在坚固平坦的表面上，并使其在表面上滚动而不滑动。绕杆轴的角速率为 $\\omega$。\n\n[figure1]\n\n求绕点 $O$ 的轨道角动量的大小，即忽略在质心参考系中观察到的由于自转而产生的自旋角动量而只计算由于组件质心运动而产生的角动量。",
+        "marking": [],
+        "answer": [
+            "\\boxed{$\\frac{1944 \\sqrt{24}}{125} m \\omega a^2$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            11.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": [
+            "image_question/PanMechanics_2025_17_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2025_18_1",
+        "context": "",
+        "question": "考虑两个质量分别为 $m$ 和 $M$ 的物体在光滑水平面上作一维运动运动并发生弹性碰撞。$m$ 和 $M$ 的初速度分别为 $u$ 和 $U$，碰撞后的速度分别为 $v$ 和 $V$。\n\n(1) 求 $v$ 的表达式，用 $m$, $M$, $u$ 和 $U$ 表示。\n(2) 求 $V$ 的表达式，用 $m$, $M$, $u$ 和 $U$ 表示。",
+        "marking": [],
+        "answer": [
+            "\\boxed{$v = \\frac{m-M}{m+M} u + \\frac{2M}{m+M} U$}",
+            "\\boxed{$V = \\frac{2m}{m+M} u - \\frac{m-M}{m+M} U$}"
+        ],
+        "answer_type": [
+            "Expression",
+            "Expression"
+        ],
+        "unit": [
+            null,
+            null
+        ],
+        "points": [
+            2.0,
+            2.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": []
+    },
+    {
+        "id": "PanMechanics_2025_18_2",
+        "context": "考虑两个质量分别为 $m$ 和 $M$ 的物体在光滑水平面上作一维运动运动并发生弹性碰撞。$m$ 和 $M$ 的初速度分别为 $u$ 和 $U$，碰撞后的速度分别为 $v$ 和 $V$。\n\n已知 $m < M$ 且 $m$ 最初处于静止状态，而 $M$ 在右侧以速度 $U$ 向 $m$ 移动。将向右的方向视为正方向，因此 $U < 0$。初次碰撞后，$m$ 向左移动并在一定距离处撞到一墙壁，然后反弹回来再次向右移动。然后它可能再次撞到 $M$ 并再反弹回来撞到墙壁。将 $m$ 和 $M$ 之间的初次碰撞称为第 0 次碰撞。我们假设所有碰撞都是弹性碰撞。设 $m$ 和 $M$ 在第 $n$ 次碰撞之前的速度分别为 $v_n$ 和 $V_n$，因此 $V_0 = U$ 和 $v_0 = 0$。\n\n[figure1]",
+        "question": "用相空间中的一个点 $(\\sqrt{M}V, \\sqrt{m}v)$ 表示两个物体的运动状态。因此，在第一次碰撞之前，状态为 $(\\sqrt{M}V_0, \\sqrt{m}v_0) = (\\sqrt{M}U, 0)$。在第二次碰撞之前，状态为 $(\\sqrt{M}V_1, \\sqrt{m}v_1)$。用 $m$ 和 $M$ 表示如图所示的角度 $\\theta$。",
+        "marking": [],
+        "answer": [
+            "\\boxed{$\\theta = \\tan^{-1} \\frac{2 \\sqrt{Mm}}{M-m}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            6.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": [
+            "image_question/PanMechanics_2025_18_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2025_18_3",
+        "context": "考虑两个质量分别为 $m$ 和 $M$ 的物体在光滑水平面上作一维运动运动并发生弹性碰撞。$m$ 和 $M$ 的初速度分别为 $u$ 和 $U$，碰撞后的速度分别为 $v$ 和 $V$。\n\n已知 $m < M$ 且 $m$ 最初处于静止状态，而 $M$ 在右侧以速度 $U$ 向 $m$ 移动。将向右的方向视为正方向，因此 $U < 0$。初次碰撞后，$m$ 向左移动并在一定距离处撞到一墙壁，然后反弹回来再次向右移动。然后它可能再次撞到 $M$ 并再反弹回来撞到墙壁。将 $m$ 和 $M$ 之间的初次碰撞称为第 0 次碰撞。我们假设所有碰撞都是弹性碰撞。设 $m$ 和 $M$ 在第 $n$ 次碰撞之前的速度分别为 $v_n$ 和 $V_n$，因此 $V_0 = U$ 和 $v_0 = 0$。\n\n[figure1]\n\n用相空间中的一个点 $(\\sqrt{M}V, \\sqrt{m}v)$ 表示两个物体的运动状态。因此，在第一次碰撞之前，状态为 $(\\sqrt{M}V_0, \\sqrt{m}v_0) = (\\sqrt{M}U, 0)$。在第二次碰撞之前，状态为 $(\\sqrt{M}V_1, \\sqrt{m}v_1)$。",
+        "question": "如上所述，或以其他方法：(1) 求 $v_n$ 的表���式，用 $m$, $M$, 和 $U$ 表示。\n(2) 求 $V_n$ 的表达式，用 $m$, $M$, 和 $U$ 表示。",
+        "marking": [],
+        "answer": [
+            "\\boxed{$v_n = -\\sqrt{\\frac{M}{m}} U \\sin\\left( n \\tan^{-1} \\frac{2\\sqrt{Mm}}{M-m} \\right)$}",
+            "\\boxed{$V_n = U \\cos\\left( n \\tan^{-1} \\frac{2\\sqrt{Mm}}{M-m} \\right)$}"
+        ],
+        "answer_type": [
+            "Expression",
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            6.5,
+            6.5
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": [
+            "image_question/PanMechanics_2025_18_1.png"
+        ]
+    },
+    {
+        "id": "PanMechanics_2025_18_4",
+        "context": "考虑两个质量分别为 $m$ 和 $M$ 的物体在光滑水平面上作一维运动运动并发生弹性碰撞。$m$ 和 $M$ 的初速度分别为 $u$ 和 $U$，碰撞后的速度分别为 $v$ 和 $V$。\n\n已知 $m < M$ 且 $m$ 最初处于静止状态，而 $M$ 在右侧以速度 $U$ 向 $m$ 移动。将向右的方向视为正方向，因此 $U < 0$。初次碰撞后，$m$ 向左移动并在一定距离处撞到一墙壁，然后反弹回来再次向右移动。然后它可能再次撞到 $M$ 并再反弹回来撞到墙壁。将 $m$ 和 $M$ 之间的初次碰撞称为第 0 次碰撞。我们假设所有碰撞都是弹性碰撞。设 $m$ 和 $M$ 在第 $n$ 次碰撞之前的速度分别为 $v_n$ 和 $V_n$，因此 $V_0 = U$ 和 $v_0 = 0$。上述过程重复一定次数，直到系统达到不再发生碰撞的状态。",
+        "question": "假设两物体之间以及 $m$ 与墙壁之间的碰撞总次数为 $N$。如果画出 $N$ 与 $\\sqrt{M/m}$ 的关系图，那么当$M \\gg m$ 时，这些点渐近位于一条直线上。求这条直线的斜率是多少？",
+        "marking": [],
+        "answer": [
+            "\\boxed{$\\pi$}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            7.0
+        ],
+        "modality": "text-only",
+        "field": "Mechanics",
+        "source": "PanMechanics_2025",
+        "image_question": []
+    }
+]

data/PanPhO_2024.json

@@ -0,0 +1,850 @@
+[
+    {
+        "information": "None."
+    },
+    {
+        "id": "PanPhO_2024_1_1",
+        "context": "We consider an inhomogeneous cylinder whose have are mode of two materials of different densities. The cylinder has radius $r$ and length $L$ and its cross section is shown in the Figure 1. The bottom half made of a material whose density is $1 kg / m^3$ while the upper half is mode of a material whose density is $c kg / m^3$, where $c$ is a parameter $0 < c < 1$. In the problem, we can use the fact that for a half-cylinder of radius $r$, the center of mass (CM) is located at a distance of $\\frac{4 r}{3 \\pi}$ from the axis of the half-cylinder, as shown in the Figure 2.\n\n[figure1]\n\n[figure2]",
+        "question": "(1) Compute the total mass $M$ of the entire inhomogeneous cylinder. Express the answer in terms of $r, L, c$. \n(2) Compute the distance $d$ between the geometrical center and the center of mass of the entire cylinder. Express the answer in terms of $r, L, c$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$M = \\frac{\\pi r^2 L}{2}(1+c)$}",
+            "\\boxed{$d = \\frac{4 r}{3 \\pi} \\left(\\frac{1-c}{1+c}\\right)$}"
+        ],
+        "answer_type": [
+            "Expression",
+            "Expression"
+        ],
+        "unit": [
+            null,
+            null
+        ],
+        "points": [
+            1.0,
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_1_1_1.png",
+            "image_question/PanPhO_2024_1_1_2.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_1_2",
+        "context": "We consider an inhomogeneous cylinder whose have are mode of two materials of different densities. The cylinder has radius $r$ and length $L$ and its cross section is shown in the Figure 1. The bottom half made of a material whose density is $1 kg / m^3$ while the upper half is mode of a material whose density is $c kg / m^3$, where $c$ is a parameter $0 < c < 1$. In the problem, we can use the fact that for a half-cylinder of radius $r$, the center of mass (CM) is located at a distance of $\\frac{4 r}{3 \\pi}$ from the axis of the half-cylinder, as shown in the Figure 2.\n\n[figure1]\n\n[figure2]",
+        "question": "Compute the moment of inertia $I$ of the entire inhomogeneous cylinder with respect to its geometrical axis. Express the answers in terms of $r, L, c$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$I = \\frac{\\pi r^4 L}{4} (1+c)$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_1_1_1.png",
+            "image_question/PanPhO_2024_1_1_2.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_1_3",
+        "context": "We consider an inhomogeneous cylinder whose have are mode of two materials of different densities. The cylinder has radius $r$ and length $L$ and its cross section is shown in the Figure 1. The bottom half made of a material whose density is $1 kg / m^3$ while the upper half is mode of a material whose density is $c kg / m^3$, where $c$ is a parameter $0 < c < 1$. In the problem, we can use the fact that for a half-cylinder of radius $r$, the center of mass (CM) is located at a distance of $\\frac{4 r}{3 \\pi}$ from the axis of the half-cylinder, as shown in the Figure 2.\n\n[figure1]\n\n[figure2]\n\nDenote $M$ as the total mass of the entire inhomogeneous cylinder, $d$ as the distance between the geometrical center and the center of mass of the entire cylinder, $I$ as the moment of inertia of the entire inhomogeneous cylinder with respect to its geometrical axis.",
+        "question": "The geometrical axis of the cylinder is fixed in a horizontal position, but the cylinder is free to rotate without any friction around the axis. If the cylinder oscillates around its stable equilibrium position with small amplitude, calculate the period $T$ of oscillation of the cylinder. Express the answer in terms of $M, I, r, d$ and the gravitational acceleration $g$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$T = 2 \\pi \\sqrt{\\frac{I}{M g d}}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_1_1_1.png",
+            "image_question/PanPhO_2024_1_1_2.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_1_4",
+        "context": "We consider an inhomogeneous cylinder whose have are mode of two materials of different densities. The cylinder has radius $r$ and length $L$ and its cross section is shown in the Figure 1. The bottom half made of a material whose density is $1 kg / m^3$ while the upper half is mode of a material whose density is $c kg / m^3$, where $c$ is a parameter $0 < c < 1$. In the problem, we can use the fact that for a half-cylinder of radius $r$, the center of mass (CM) is located at a distance of $\\frac{4 r}{3 \\pi}$ from the axis of the half-cylinder, as shown in the Figure 2.\n\n[figure1]\n\n[figure2]\n\nDenote $M$ as the total mass of the entire inhomogeneous cylinder, $d$ as the distance between the geometrical center and the center of mass of the entire cylinder, $I$ as the moment of inertia of the entire inhomogeneous cylinder with respect to its geometrical axis.\n\nNow we assume that the cylinder is completely free to move on a horizontal table under the gravity. We assume that the coefficient of static friction between the cylinder and the table is infinite, such that the cylinder cannot slide. Suppose that at time $t=0$, the cylinder is in its equilibrium position with an initial angular velocity $\\omega_0$.",
+        "question": "If $\\omega_0$ is sufficiently small, the cylinder will undergo a period motion around its stable equilibrium. What is the period $T$ of oscillation if the amplitude of the oscillation is small? Express the answer in terms of $M, l, r, d$ and the gravitational acceleration $g$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$T = 2 \\pi \\sqrt{\\frac{I + M(r^2 - 2 r d)}{M g d}}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_1_1_1.png",
+            "image_question/PanPhO_2024_1_1_2.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_1_5",
+        "context": "We consider an inhomogeneous cylinder whose have are mode of two materials of different densities. The cylinder has radius $r$ and length $L$ and its cross section is shown in the Figure 1. The bottom half made of a material whose density is $1 kg / m^3$ while the upper half is mode of a material whose density is $c kg / m^3$, where $c$ is a parameter $0 < c < 1$. In the problem, we can use the fact that for a half-cylinder of radius $r$, the center of mass (CM) is located at a distance of $\\frac{4 r}{3 \\pi}$ from the axis of the half-cylinder, as shown in the Figure 2.\n\n[figure1]\n\n[figure2]\n\nDenote $M$ as the total mass of the entire inhomogeneous cylinder, $d$ as the distance between the geometrical center and the center of mass of the entire cylinder, $I$ as the moment of inertia of the entire inhomogeneous cylinder with respect to its geometrical axis.\n\nNow we assume that the cylinder is completely free to move on a horizontal table under the gravity. We assume that the coefficient of static friction between the cylinder and the table is infinite, such that the cylinder cannot slide. Suppose that at time $t=0$, the cylinder is in its equilibrium position with an initial angular velocity $\\omega_0$. If $\\omega_0$ is sufficiently small, the cylinder will undergo a period motion around its stable equilibrium.",
+        "question": "What is the minimum value of $\\omega_0$ that allows the cylinder to roll forever in the same direction. Express the answer in terms of $M, l, r, d$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$\\omega_0 = \\sqrt{\\frac{4 M g d}{M (r-d)^2 + (I - M d^2)}}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_1_1_1.png",
+            "image_question/PanPhO_2024_1_1_2.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_2_1",
+        "context": "A closed container is divided into three compartments, A, B, and C, by two partitions, $D_1$ and $D_2$, as shown in the figure. Each compartment is filled with the same monoatomic ideal gas with pressure $P$, volume $V$, and absolute temperature $T$ as shown in the figure. The mass of partition $D_1$ is $m$, which can slide freely without friction, while partition $D_2$ is fixed and has a small valve on it. Now, the valve on partition $D_2$ is opened, allowing the gases in compartments B and C to mix and the entire system to reach equilibrium while maintaining a constant temperature $T_0$.\n\n[figure1]",
+        "question": "After the entire system reaches equilibrium: \n(1) What is the pressure $P_A$ of the gas in compartment A? Express your answer in terms of $P_0$. \n(2) What is the pressure $P_B$ of the gas in compartment B? Express your answer in terms of $P_0$. \n(3) What is the pressure $P_C$ of the gas in compartment C? Express your answer in terms of $P_0$. \n(4) What is the volume $V_A$ of the gas in compartment A? Express your answer in terms of $V_0$. \n(5) What is the volume $V_B$ of the gas in compartment B? Express your answer in terms of $V_0$. \n(6) What is the volume $V_C$ of the gas in compartment C? Express your answer in terms of $V_0$. \nNote: The definitions of $P_0$ and $V_0$ are given in the figure.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$P_A = \\frac{4}{3} P_0$}",
+            "\\boxed{$P_B = \\frac{4}{3} P_0$}",
+            "\\boxed{$P_C = \\frac{4}{3} P_0$}",
+            "\\boxed{$V_A = \\frac{3}{4} V_0$}",
+            "\\boxed{$V_B = \\frac{5}{4} V_0$}",
+            "\\boxed{$V_C = V_0$}"
+        ],
+        "answer_type": [
+            "Expression",
+            "Expression",
+            "Expression",
+            "Expression",
+            "Expression",
+            "Expression"
+        ],
+        "unit": [
+            null,
+            null,
+            null,
+            null,
+            null,
+            null
+        ],
+        "points": [
+            0.5,
+            0.5,
+            0.5,
+            0.5,
+            0.5,
+            0.5
+        ],
+        "modality": "text+variable figure",
+        "field": "Thermodynamics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_2_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_2_2",
+        "context": "A closed container is divided into three compartments, A, B, and C, by two partitions, $D_1$ and $D_2$, as shown in the figure. Each compartment is filled with the same monoatomic ideal gas with pressure $P$, volume $V$, and absolute temperature $T$ as shown in the figure. The mass of partition $D_1$ is $m$, which can slide freely without friction, while partition $D_2$ is fixed and has a small valve on it. Now, the valve on partition $D_2$ is opened, allowing the gases in compartments B and C to mix and the entire system to reach equilibrium while maintaining a constant temperature $T_0$.\n\n[figure1]",
+        "question": "How much total heat is absorbed by the gases in compartments B and C during the process of the entire system reaching equilibrium? Express your answer in terms of $P_0$ and $V_0$. \nNote: The definitions of $P_0$ and $V_0$ are given in the figure.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$0.288 P_0 V_0$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Thermodynamics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_2_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_2_3",
+        "context": "A closed container is divided into three compartments, A, B, and C, by two partitions, $D_1$ and $D_2$, as shown in the figure. Each compartment is filled with the same monoatomic ideal gas with pressure $P$, volume $V$, and absolute temperature $T$ as shown in the figure. The mass of partition $D_1$ is $m$, which can slide freely without friction, while partition $D_2$ is fixed and has a small valve on it. Now, the valve on partition $D_2$ is opened, allowing the gases in compartments B and C to mix and the entire system to reach equilibrium while maintaining a constant temperature $T_0$.\n\n[figure1]",
+        "question": "Calculate the change in entropy $\\Delta S$ of the entire system during the process of reaching equilibrium. Express your answer in terms of $P_0$, $V_0$, and $T_0$. \nNote: The definitions of $P_0$, $V_0$, and $T_0$ are given in the figure.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$\\Delta S \\approx 0.236 \\frac{P_0 V_0}{T_0}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            4.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Thermodynamics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_2_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_3_2",
+        "context": "[Trapped Ball] \n\nAs shown in the figure, a ball (modelled as a point charge of magnitude $q > 0$) of mass $m$ is trapped in a spherical cavity of radius $R$ carved from an infinite grounded conductor. The charge is at a distance $z$ from the center. In the problem, the gravity can be ignored.\n\n[figure1]",
+        "question": "Find the magnitude of the electric force $F(z)$ acting on the ball in terms of $q$, $z$, $R$ and the vacuum permittivity $\\varepsilon_0$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$\\frac{1}{4 \\pi \\varepsilon_0} \\frac{q^2 R z}{(R^2 - z^2)^2}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            4.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Electromagnetism",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_3_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_3_3",
+        "context": "[Trapped Ball] \n\nAs shown in the figure, a ball (modelled as a point charge of magnitude $q > 0$) of mass $m$ is trapped in a spherical cavity of radius $R$ carved from an infinite grounded conductor. The charge is at a distance $z$ from the center. In the problem, the gravity can be ignored.\n\n[figure1]",
+        "question": "If the ball is released at the center with a very small speed, find the speed of the ball $v$ when it is at a distance $R / 2$ from the center of the conductor. Express the answer in terms of $q, m, R$ and the vacuum permittivity $\\varepsilon_0$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$v = \\sqrt{\\frac{1}{12 \\pi \\varepsilon_0} \\frac{q^2}{m R}}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            4.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Electromagnetism",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_3_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_4_1",
+        "context": "[In this question, all answers cannot be expressed in terms of any trigonometrical functions.] \n\nAn ice hemisphere with radius $R$ and refractive index $n$ lies on a warm flat table and melts slowly. The rate of heat transfer from the table to the ice is proportional to the area of contact between them. It is known that the ice hemisphere completely melts in time $T_{0}$. Throughout the process, a laser beam incident on the ice from above. The beam is vertically incident at a distance of $R / 2$ from the axis of symmetry (see figure).\n\nAssume that the temperature of the ice and the surrounding atmosphere are $0^{\\circ} \\mathrm{C}$ and remains constant during the melting process. The laser beam does not transfer energy to the ice. The melting water immediately flows off the table, and the ice does not move along the table.\n\n[figure1]",
+        "question": "What is the position of the point on the table, $x_{0} = x(t=0)$, where the beam hit at time $t = 0$? Express the answer in terms of $n$ and $R$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$x_0 = \\frac{2 R}{\\sqrt{12 n^2 - 3} + 1}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            4.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Optics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_4_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_4_2",
+        "context": "[In this question, all answers cannot be expressed in terms of any trigonometrical functions.] \n\nAn ice hemisphere with radius $R$ and refractive index $n$ lies on a warm flat table and melts slowly. The rate of heat transfer from the table to the ice is proportional to the area of contact between them. It is known that the ice hemisphere completely melts in time $T_{0}$. Throughout the process, a laser beam incident on the ice from above. The beam is vertically incident at a distance of $R / 2$ from the axis of symmetry (see figure).\n\nAssume that the temperature of the ice and the surrounding atmosphere are $0^{\\circ} \\mathrm{C}$ and remains constant during the melting process. The laser beam does not transfer energy to the ice. The melting water immediately flows off the table, and the ice does not move along the table.\n\n[figure1]",
+        "question": "What is the height of the ice $z(t)$ as a function of time $t$? Express the answer in terms of $R$ and $T_{0}$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$z(t) = R (1 - \\frac{t}{T_0})$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Thermodynamics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_4_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_4_3",
+        "context": "[In this question, all answers cannot be expressed in terms of any trigonometrical functions.] \n\nAn ice hemisphere with radius $R$ and refractive index $n$ lies on a warm flat table and melts slowly. The rate of heat transfer from the table to the ice is proportional to the area of contact between them. It is known that the ice hemisphere completely melts in time $T_{0}$. Throughout the process, a laser beam incident on the ice from above. The beam is vertically incident at a distance of $R / 2$ from the axis of symmetry (see figure).\n\nAssume that the temperature of the ice and the surrounding atmosphere are $0^{\\circ} \\mathrm{C}$ and remains constant during the melting process. The laser beam does not transfer energy to the ice. The melting water immediately flows off the table, and the ice does not move along the table.\n\n[figure1]",
+        "question": "What is the position of the point on the table, $x(t)$, where the beam hit for $t \\geq 0$? Express the answer in terms of $n, R, T_{0}$ and $t$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$x(t) = \\frac{R}{2}$ for $t \\geq \\frac{\\sqrt{3}}{2} T_0$}",
+            "\\boxed{$x(t) = \\frac{R}{2} - \\frac{R}{2} \\left(1 - \\frac{4}{\\sqrt{12 n^2 - 3} + 1}\\right) \\left(1 - \\frac{2}{\\sqrt{3}} \\frac{t}{T_0}\\right)$ for $t < \\frac{\\sqrt{3}}{2} T_0$}"
+        ],
+        "answer_type": [
+            "Expression",
+            "Expression"
+        ],
+        "unit": [
+            null,
+            null
+        ],
+        "points": [
+            1.5,
+            1.5
+        ],
+        "modality": "text+variable figure",
+        "field": "Optics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_4_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_5_1",
+        "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part A: Gravitational fluctuations on the orbit of the satellite]",
+        "question": "Here we only consider gravity from the earth and consider the earth as a homogeneous ideal ball. Give the periodicity $T$ of a satellite rotating around the earth. Please use second as the unit and give three significant figures.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$3.15 \\times 10^{5}$}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "s"
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_5_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_5_2",
+        "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part A: Gravitational fluctuations on the orbit of the satellite]\n\n[figure2]",
+        "question": "Since the shape and density of the earth is inhomogeneous, the satellite will feel an additional acceleration $\\delta a$ in addition to the uniform circular motion. To simplify the calculation, let us model the inhomogeneity of the earth as follows: Consider an ideal ball with mass $M - 2m$. Two additional point masses (each has mass $m$) are put to diametrically opposite points on the equator of the earth. Assume that the satellite orbit and the earth are in the same plane, with angle $\\theta$ between them. Give the precise formula to calculate $\\delta a$: \n(1) the $x$-component of the acceleration $\\delta a_{x}$; \n(2) the $y$-component of the acceleration $\\delta a_{y}$. \nPlease express your answer in terms of $G$, $M$, $m$, $R$, $r$, and $\\theta$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$\\delta a_{x} = -\\frac{G(M - 2m)}{R^2} - \\frac{G m (R - r \\cos \\theta)}{(R^2 + r^2 - 2 R r \\cos \\theta)^{3/2}} - \\frac{G m (R + r \\cos \\theta)}{(R^2 + r^2 + 2 R r \\cos \\theta)^{3/2}}$}",
+            "\\boxed{$\\delta a_{y} = \\frac{G m r \\sin \\theta}{(R^2 + r^2 - 2 R r \\cos \\theta)^{3/2}} - \\frac{G m r \\sin \\theta}{(R^2 + r^2 + 2 R r \\cos \\theta)^{3/2}}$}"
+        ],
+        "answer_type": [
+            "Expression",
+            "Expression"
+        ],
+        "unit": [
+            null,
+            null
+        ],
+        "points": [
+            1.0,
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_5_1_1.png",
+            "image_question/PanPhO_2024_5_2_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_5_3",
+        "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part A: Gravitational fluctuations on the orbit of the satellite]\n\n[figure2]\n\nSince the shape and density of the earth is inhomogeneous, the satellite will feel an additional acceleration $\\delta a$ in addition to the uniform circular motion. To simplify the calculation, let us model the inhomogeneity of the earth as follows: Consider an ideal ball with mass $M - 2m$. Two additional point masses (each has mass $m$) are put to diametrically opposite points on the equator of the earth. Assume that the satellite orbit and the earth are in the same plane, with angle $\\theta$ between them.",
+        "question": "Calculate all the possible periodicities for $\\delta a$. Please use second as the unit and give three significant figures.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$3.39 \\times 10^{4}$}",
+            "\\boxed{$5.95 \\times 10^{4}$}"
+        ],
+        "answer_type": [
+            "Numerical Value",
+            "Numerical Value"
+        ],
+        "unit": [
+            "s",
+            "s"
+        ],
+        "points": [
+            1.0,
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_5_1_1.png",
+            "image_question/PanPhO_2024_5_2_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_5_4",
+        "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part A: Gravitational fluctuations on the orbit of the satellite]\n\n[figure2]\n\nSince the shape and density of the earth is inhomogeneous, the satellite will feel an additional acceleration $\\delta a$ in addition to the uniform circular motion. To simplify the calculation, let us model the inhomogeneity of the earth as follows: Consider an ideal ball with mass $M - 2m$. Two additional point masses (each has mass $m$) are put to diametrically opposite points on the equator of the earth. Assume that the satellite orbit and the earth are in the same plane, with angle $\\theta$ between them.",
+        "question": "In the $R \\gg r$ limit, give the leading order expression (the lowest nonzero order in the Taylor expansion of $r / R$) for $\\delta a$: \n(1) the $x$-component of the acceleration $\\delta a_{x}$; \n(2) the $y$-component of the acceleration $\\delta a_{y}$. \nPlease express your answer in terms of $G$, $m$, $r$, $R$ and $\\theta$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$\\delta a_{x} \\simeq \\frac{3 G m r^{2} (1 - 3 \\cos^2 \\theta)}{R^4}$}",
+            "\\boxed{$\\delta a_{y} \\simeq \\frac{3 G m r^{2} \\sin 2 \\theta}{R^4}$}"
+        ],
+        "answer_type": [
+            "Expression",
+            "Expression"
+        ],
+        "unit": [
+            null,
+            null
+        ],
+        "points": [
+            1.0,
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_5_1_1.png",
+            "image_question/PanPhO_2024_5_2_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_5_5",
+        "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part A: Gravitational fluctuations on the orbit of the satellite]\n\n[figure2]\n\nSince the shape and density of the earth is inhomogeneous, the satellite will feel an additional acceleration $\\delta a$ in addition to the uniform circular motion. To simplify the calculation, let us model the inhomogeneity of the earth as follows: Consider an ideal ball with mass $M - 2m$. Two additional point masses (each has mass $m$) are put to diametrically opposite points on the equator of the earth. Assume that the satellite orbit and the earth are in the same plane, with angle $\\theta$ between them.",
+        "question": "Estimate the typical value of $\\delta a$ with the unit of $m/s^2$ (an error within two orders of magnitude will be considered as correct).",
+        "marking": [],
+        "answer": [
+            "\\boxed{$[10^{-11}, 10^{-7}]$}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "m/s^2"
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_5_1_1.png",
+            "image_question/PanPhO_2024_5_2_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_5_6",
+        "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part A: Gravitational fluctuations on the orbit of the satellite]\n\n[figure2]\n\nSince the shape and density of the earth is inhomogeneous, the satellite will feel an additional acceleration $\\delta a$ in addition to the uniform circular motion. To simplify the calculation, let us model the inhomogeneity of the earth as follows: Consider an ideal ball with mass $M - 2m$. Two additional point masses (each has mass $m$) are put to diametrically opposite points on the equator of the earth. Assume that the satellite orbit and the earth are in the same plane, with angle $\\theta$ between them.",
+        "question": "In satellite experiments, we are interested in the gravitational waves with a particular periodicity (such as periods between 1-1000 seconds). Thus, if the periodicity of the gravitational fluctuation is too long, it will not interfere the gravitational wave measurement. Assume the satellite is co-rotating in the same direction with the spinning direction of the earth. In the Taylor expansion of $\\delta a$, calculate the component with period closest to 1000s. Denote this component as $\\delta a_{1000}$. Estimate the value of $\\delta a_{1000} / \\delta a$ for $\\theta = \\pi / 3$. (an error within two orders of magnitude will be considered as correct)",
+        "marking": [],
+        "answer": [
+            "\\boxed{$[10^{-143}, 10^{-139}]$}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Mechanics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_5_1_1.png",
+            "image_question/PanPhO_2024_5_2_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_5_7",
+        "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part B: Free electrons from the solar wind]\n\nConsider the laser signal between the satellites. Although the space between satellites is close to the vacuum, but it is not the absolute vacuum. In particular, solar wind will introduce free electrons. Let the number density of the free electrons be $N_{e}$, the electric charge of an electron be $e$, the electron mass be $m_{e}$. And we ignore other media apart from these electrons.",
+        "question": "Assume that the electrons move freely in the electric field produced by the laser. Calculate the acceleration of the electron $d \\mathbf{v}_{e} / d t$ as a function of the electric field $\\mathbf{E}$ produced by the laser.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$\\frac{d \\mathbf{v}_{e}}{d t} = -e \\frac{\\mathbf{E}}{m_{e}}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Electromagnetism",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_5_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_5_8",
+        "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part B: Free electrons from the solar wind]\n\nConsider the laser signal between the satellites. Although the space between satellites is close to the vacuum, but it is not the absolute vacuum. In particular, solar wind will introduce free electrons. Let the number density of the free electrons be $N_{e}$, the electric charge of an electron be $e$, the electron mass be $m_{e}$. And we ignore other media apart from these electrons.\n\nAssume that the electrons move freely in the electric field produced by the laser. Denote the velocity of the electron as $\\mathbf{v}_e$, and the electric field produced by the laser as $\\mathbf{E}$.",
+        "question": "Calculate the time dependence of the electric current, $\\frac{d \\mathbf{J}}{d t}$, from the free electrons.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$\\frac{d \\mathbf{J}}{d t} = \\frac{N_{e} e^{2} \\mathbf{E}}{m_{e}}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Electromagnetism",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_5_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_5_9",
+        "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part B: Free electrons from the solar wind]\n\nConsider the laser signal between the satellites. Although the space between satellites is close to the vacuum, but it is not the absolute vacuum. In particular, solar wind will introduce free electrons. Let the number density of the free electrons be $N_{e}$, the electric charge of an electron be $e$, the electron mass be $m_{e}$. And we ignore other media apart from these electrons.\n\nAssume that the electrons move freely in the electric field produced by the laser. Denote the velocity of the electron as $\\mathbf{v}_e$, and the electric field produced by the laser as $\\mathbf{E}$.",
+        "question": "Calculate the phase speed of the laser $v_{p}$ in the environment of the free electrons (since $v_{p}$ is very close to the speed of light, the higher order difference between $v_{p}$ and the speed of light can be ignored). \n\nHint: from the Maxwell equations, one can derive that $\\frac{\\partial^{2} \\mathbf{E}}{\\partial t^{2}} - c^{2} \\nabla^{2} \\mathbf{E} + c^{2} \\mu_{0} \\frac{d \\mathbf{J}}{d t} = 0$, where $c$ is the speed of light in vacuum.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$v_p \\simeq c \\left(1 + \\frac{\\mu_{0} N_{e} e^{2} \\lambda^{2}}{8 \\pi^{2} m_{e}}\\right)$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Electromagnetism",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_5_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_5_10",
+        "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part B: Free electrons from the solar wind]\n\nConsider the laser signal between the satellites. Although the space between satellites is close to the vacuum, but it is not the absolute vacuum. In particular, solar wind will introduce free electrons. Let the number density of the free electrons be $N_{e}$, the electric charge of an electron be $e$, the electron mass be $m_{e}$. And we ignore other media apart from these electrons.",
+        "question": "Let $N_{e} = 10 \\mathrm{cm}^{-3}$. Calculate the phase error of the laser between two satellites. In other words, if there were no free electrons, the laser waveform arrived at a satellite is $\\cos \\theta$. Now with free electrons, the same wave form at the same moment changes into $\\cos (\\theta + \\delta \\theta)$. Calculate the value of $\\delta \\theta$. Express your answer in radians, and give the numerical value with three significant figures.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$5.19 \\times 10^{-6}$}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Optics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_5_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_5_11",
+        "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part C: Shot noise]\n\nAny precision measurements are limited by the uncertainty principle of quantum mechanics. Assume that every photon's arrival time at the detector can be considered as independent stochastic processes. Also, in actual experiments, phase error of the laser is more important. But here for simplicity, here we only estimate photon number errors.",
+        "question": "During a certain period of time, the average photon number in the laser is $N$. In this case, the error in the photon number measurement in the laser is $\\Delta N = N^{\\alpha}$. Find $\\alpha$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$1/2$}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            1.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Modern Physics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_5_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_5_12",
+        "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part C: Shot noise]\n\nAny precision measurements are limited by the uncertainty principle of quantum mechanics. Assume that every photon's arrival time at the detector can be considered as independent stochastic processes. Also, in actual experiments, phase error of the laser is more important. But here for simplicity, here we only estimate photon number errors.",
+        "question": "If we request that in one second, the relative error of photon number measurement is $\\frac{\\Delta N}{N} < 3 \\times 10^{-6}$. Calculate the minimal power of laser $P_{\\text{rec}}$ that the satellite should receive. Express your answer in the unit of $W$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$2 \\times 10^{-8}$}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "W"
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Modern Physics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_5_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_5_13",
+        "context": "[Spatiotemporal varying electric permittivity] \n\nThe discovery of gravitational waves initiated an era of gravitational wave astronomy. In addition to the ground-based gravitational wave observatories, gravitational wave observatories based on laser interference between satellites are also planned, for example, the Taiji and Tianqin programs in China and LISA in Europe. Here, we study a simplified version similar to the Tianqin program.\n\n[figure1]\n\nAs illustrated in this figure, we consider three satellites surrounding the earth following circular orbits. They form an equilateral triangle. They form an interferometry in the nearly vacuum environment near the earth. From the change of interference patterns, the change of space distance is measured to detect gravitational waves. Here we will study the error sources for Tianqin to reach its desired measurement precision.\n\nIn this problem, we will use the physical constants and satellite parameters including:\n\nNewton's gravitational constant $G = 6.67 \\times 10^{-11} m^3 / (\\mathrm{kg} s^2)$ \nPlanck's constant $h = 6.626 \\times 10^{-34} m^2 \\mathrm{kg} / s$ \nVacuum Permeability $\\mu_0 = 1.257 \\times 10^{-6} \\mathrm{kg} m s^{-2} A^{-2}$ \nThe mass of the earth $M = 5.97 \\times 10^{24} \\mathrm{kg}$ \nThe radius of the earth $r = 6.37 \\times 10^6 m$ \nThe distance from a satellite to the center of the earth $R = 10^8 m$ \nThe laser wavelength used by the satellite $\\lambda = 1064 \\mathrm{nm}$ \nThe size of the optical system of the satellite $D = 0.1 m$ \n\n[Part C: Shot noise]\n\nAny precision measurements are limited by the uncertainty principle of quantum mechanics. Assume that every photon's arrival time at the detector can be considered as independent stochastic processes. Also, in actual experiments, phase error of the laser is more important. But here for simplicity, here we only estimate photon number errors.",
+        "question": "Assume the laser arrived at a satellite is emitted from the other satellite from the three-satellite system. Estimate: in an ideal case, what is the minimal emission power of laser $P_{\\text{emit}}$ from the other satellite (can be considered to be correct if the order-of-magnitude is correct). Express your answer in the unit of $W$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$[1, 9]$}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "W"
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+illustration figure",
+        "field": "Modern Physics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_5_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_6_1",
+        "context": "[Metric-modified geodesic and heat conduction] \n\nSolving physics, such as wave propagation, geodesics, and thermal conduction, on a curved surface in 3D requires a thorough understanding of metrics and differential geometry. However, there can be significant simplifications for systems with spatial symmetry or by adopting coordinate transformation. In this question, we will go through two problems for physics on a curved surface. The first one is light propagating on a curved surface. Figure 1 (a) shows a circular cone with height 5 mm and a base diameter of $2 \\rho_{0} = 10 \\mathrm{mm}$, joining to a flat surface. The flat surface has a circular hole of the same diameter so that as a whole, there is only one single surface with the cone part indicating the 'curved space.' The entire surface, including the flat surface and the cone, have a very thin surface so that light can be effectively confined on such a surface. We have assumed the cone is joint smoothly to the flat surface. \n\n[figure1]\n\nFigure 1(a) depicts a curved surface created by connecting a circular cone to a flat surface with a hole of the same size as the cone's base. In Figure 1(b), we present a top view of this surface. Light confined to such a surface originates on the flat surface at the bottom, undergoes bending due to the cone, and exits in a different direction. \n\n[Part A: Geodesic on A Rotational Symmetric Curved Surface]\n\nIn mechanics, we are aware that when a system exhibits rotational symmetry, we can simplify the derivation of dynamics by applying the conservation of angular momentum. For instance, we can employ the conservation of angular momentum to derive Kepler's laws. In the current scenario, we consider a normalized angular momentum, denoted as L, which is defined as: \n$L = \\hat{z} \\cdot \\mathbf{\\rho} \\times \\frac{d \\mathbf{\\rho}}{d s} = \\rho^{2} \\frac{d \\phi}{d s}$ \nHere, $\\mathbf{\\rho}$ represents the projected position vector on the two-dimensional $x-y$ plane, given by $\\mathbf{\\rho} = x \\hat{x} + y \\hat{y} = \\hat{x} \\rho \\cos \\phi + \\hat{y} \\rho \\sin \\phi$, with the projected cone center as the origin. $\\rho$ is the magnitude of vector $\\mathbf{\\rho}$ and $s$ is the arc length along the path of light on the surface.",
+        "question": "Given that the infinitesimal arc length on the cone satisfies $d s^{2} = d x^{2} + d y^{2} + d z^{2}$ and $z = z(\\rho)$ is the height at that point, find the equation that the geodesic on the cone satisfies, in terms of $\\rho^{\\prime}(\\phi), \\rho, L$, and $\\theta$. Instead of using arc length $s$ to parametrize the geodesic, we have used $\\phi$ for parametrization.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$\\rho^{\\prime}(\\phi)^{2} = \\frac{\\sin^{2} \\theta}{L^{2}} \\rho^{2} (\\rho^{2} - L^{2})$}"
+        ],
+        "answer_type": [
+            "Equation"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            3.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_6_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_6_2",
+        "context": "[Metric-modified geodesic and heat conduction] \n\nSolving physics, such as wave propagation, geodesics, and thermal conduction, on a curved surface in 3D requires a thorough understanding of metrics and differential geometry. However, there can be significant simplifications for systems with spatial symmetry or by adopting coordinate transformation. In this question, we will go through two problems for physics on a curved surface. The first one is light propagating on a curved surface. Figure 1 (a) shows a circular cone with height 5 mm and a base diameter of $2 \\rho_{0} = 10 \\mathrm{mm}$, joining to a flat surface. The flat surface has a circular hole of the same diameter so that as a whole, there is only one single surface with the cone part indicating the 'curved space.' The entire surface, including the flat surface and the cone, have a very thin surface so that light can be effectively confined on such a surface. We have assumed the cone is joint smoothly to the flat surface. \n\n[figure1]\n\nFigure 1(a) depicts a curved surface created by connecting a circular cone to a flat surface with a hole of the same size as the cone's base. In Figure 1(b), we present a top view of this surface. Light confined to such a surface originates on the flat surface at the bottom, undergoes bending due to the cone, and exits in a different direction. \n\n[Part A: Geodesic on A Rotational Symmetric Curved Surface]\n\nIn mechanics, we are aware that when a system exhibits rotational symmetry, we can simplify the derivation of dynamics by applying the conservation of angular momentum. For instance, we can employ the conservation of angular momentum to derive Kepler's laws. In the current scenario, we consider a normalized angular momentum, denoted as L, which is defined as: \n$L = \\hat{z} \\cdot \\mathbf{\\rho} \\times \\frac{d \\mathbf{\\rho}}{d s} = \\rho^{2} \\frac{d \\phi}{d s}$ \nHere, $\\mathbf{\\rho}$ represents the projected position vector on the two-dimensional $x-y$ plane, given by $\\mathbf{\\rho} = x \\hat{x} + y \\hat{y} = \\hat{x} \\rho \\cos \\phi + \\hat{y} \\rho \\sin \\phi$, with the projected cone center as the origin. $\\rho$ is the magnitude of vector $\\mathbf{\\rho}$ and $s$ is the arc length along the path of light on the surface.",
+        "question": "For a light starting on the flat surface with a perpendicular distance of $\\rho_{0} / 2$ to the origin, what will be minimal $\\rho$ the light can go.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$\\frac{\\rho_{0}}{2}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Mechanics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_6_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_6_3",
+        "context": "[Metric-modified geodesic and heat conduction] \n\nSolving physics, such as wave propagation, geodesics, and thermal conduction, on a curved surface in 3D requires a thorough understanding of metrics and differential geometry. However, there can be significant simplifications for systems with spatial symmetry or by adopting coordinate transformation. In this question, we will go through two problems for physics on a curved surface. The first one is light propagating on a curved surface. Figure 1 (a) shows a circular cone with height 5 mm and a base diameter of $2 \\rho_{0} = 10 \\mathrm{mm}$, joining to a flat surface. The flat surface has a circular hole of the same diameter so that as a whole, there is only one single surface with the cone part indicating the 'curved space.' The entire surface, including the flat surface and the cone, have a very thin surface so that light can be effectively confined on such a surface. We have assumed the cone is joint smoothly to the flat surface. \n\n[figure1]\n\nFigure 1(a) depicts a curved surface created by connecting a circular cone to a flat surface with a hole of the same size as the cone's base. In Figure 1(b), we present a top view of this surface. Light confined to such a surface originates on the flat surface at the bottom, undergoes bending due to the cone, and exits in a different direction. \n\n[Part A: Geodesic on A Rotational Symmetric Curved Surface]\n\nIn mechanics, we are aware that when a system exhibits rotational symmetry, we can simplify the derivation of dynamics by applying the conservation of angular momentum. For instance, we can employ the conservation of angular momentum to derive Kepler's laws. In the current scenario, we consider a normalized angular momentum, denoted as L, which is defined as: \n$L = \\hat{z} \\cdot \\mathbf{\\rho} \\times \\frac{d \\mathbf{\\rho}}{d s} = \\rho^{2} \\frac{d \\phi}{d s}$ \nHere, $\\mathbf{\\rho}$ represents the projected position vector on the two-dimensional $x-y$ plane, given by $\\mathbf{\\rho} = x \\hat{x} + y \\hat{y} = \\hat{x} \\rho \\cos \\phi + \\hat{y} \\rho \\sin \\phi$, with the projected cone center as the origin. $\\rho$ is the magnitude of vector $\\mathbf{\\rho}$ and $s$ is the arc length along the path of light on the surface.",
+        "question": "What is the deflection angle $\\gamma$ by comparing the entering and exit rays on the flat surface? Express the numerical answer in degrees. \nHint: you may need to use $\\int \\frac{d x}{x \\sqrt{x^{2}-1}} = \\tan^{-1} \\sqrt{x^{2}-1} + c$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{49.7}"
+        ],
+        "answer_type": [
+            "Numerical Value"
+        ],
+        "unit": [
+            "degree"
+        ],
+        "points": [
+            7.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Optics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_6_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_6_4",
+        "context": "[Metric-modified geodesic and heat conduction] \n\nSolving physics, such as wave propagation, geodesics, and thermal conduction, on a curved surface in 3D requires a thorough understanding of metrics and differential geometry. However, there can be significant simplifications for systems with spatial symmetry or by adopting coordinate transformation. In this question, we will go through two problems for physics on a curved surface. The first one is light propagating on a curved surface. Figure 1 (a) shows a circular cone with height 5 mm and a base diameter of $2 \\rho_{0} = 10 \\mathrm{mm}$, joining to a flat surface. The flat surface has a circular hole of the same diameter so that as a whole, there is only one single surface with the cone part indicating the 'curved space.' The entire surface, including the flat surface and the cone, have a very thin surface so that light can be effectively confined on such a surface. We have assumed the cone is joint smoothly to the flat surface. \n\n[figure1]\n\nFigure 1(a) depicts a curved surface created by connecting a circular cone to a flat surface with a hole of the same size as the cone's base. In Figure 1(b), we present a top view of this surface. Light confined to such a surface originates on the flat surface at the bottom, undergoes bending due to the cone, and exits in a different direction.\n\n[Part B: Heat Conduction on A Spherical Surface]\n\nA usual trick is to search for a coordinate transform from the curved surface (represented by the Cartesian coordinates $(x, y, z)$) to a 2-dimensional coordinates system $X-Y$ plane so that the physics on the $(X, Y)$ just looks like a flat plane. For a unit spherical surface, such a map is the stereographic projection \n$(X, Y) = \\left(\\frac{x}{z+1}, \\frac{y}{z+1}\\right) = (\\rho \\cos \\phi, \\rho \\sin \\phi)$ \nSuppose now we consider heat conduction problem on such a spherical surface, i.e. a very thin shell of spherical surface. The steady-state heat conduction has the temperature profile satisfying the Laplace equation \n$\\nabla^{2} T(\\theta, \\phi) = 0$ \nwhile temperature profile is independent of radial distance $r$ in spherical coordinate $(r, \\theta, \\phi)$. The spherical surface is at radius $r=1$.",
+        "question": "Write down the Laplace equation that $T$ satisfies on the $(X, Y)$ coordinate and prove it. \nHint: For convenience, we are given the Laplacian in spherical and cylindrical coordinates as \nIn spherical coordinate $(r, \\theta, \\phi)$: \n$\\nabla^{2} f = \\frac{1}{r^{2}} \\partial_{r}\\left(r^{2} \\partial_{r} f\\right) + \\frac{1}{r^{2} \\sin \\theta} \\partial_{\\theta} \\left(\\sin \\theta \\partial_{\\theta} f\\right) + \\frac{1}{r^{2} \\sin^{2} \\theta} \\partial_{\\phi}^{2} f$  \ncylindrical coordinate $(\\rho, \\phi, z)$: \n$\\nabla^{2} f = \\frac{1}{\\rho} \\partial_{\\rho} \\left(\\rho \\partial_{\\rho} f\\right) + \\frac{1}{\\rho^{2}} \\partial_{\\phi}^{2} f + \\partial_{z}^{2} f$.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$\\frac{1}{\\rho} \\partial_{\\rho} \\left(\\rho \\partial_{\\rho} T\\right) + \\frac{1}{\\rho^{2}} \\partial_{\\phi}^{2} T = 0$}"
+        ],
+        "answer_type": [
+            "Equation"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            4.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Thermodynamics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_6_1_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_6_5",
+        "context": "[Metric-modified geodesic and heat conduction] \n\nSolving physics, such as wave propagation, geodesics, and thermal conduction, on a curved surface in 3D requires a thorough understanding of metrics and differential geometry. However, there can be significant simplifications for systems with spatial symmetry or by adopting coordinate transformation. In this question, we will go through two problems for physics on a curved surface. The first one is light propagating on a curved surface. Figure 1 (a) shows a circular cone with height 5 mm and a base diameter of $2 \\rho_{0} = 10 \\mathrm{mm}$, joining to a flat surface. The flat surface has a circular hole of the same diameter so that as a whole, there is only one single surface with the cone part indicating the 'curved space.' The entire surface, including the flat surface and the cone, have a very thin surface so that light can be effectively confined on such a surface. We have assumed the cone is joint smoothly to the flat surface. \n\n[figure1]\n\nFigure 1(a) depicts a curved surface created by connecting a circular cone to a flat surface with a hole of the same size as the cone's base. In Figure 1(b), we present a top view of this surface. Light confined to such a surface originates on the flat surface at the bottom, undergoes bending due to the cone, and exits in a different direction.\n\n[Part B: Heat Conduction on A Spherical Surface]\n\nA usual trick is to search for a coordinate transform from the curved surface (represented by the Cartesian coordinates $(x, y, z)$) to a 2-dimensional coordinates system $X-Y$ plane so that the physics on the $(X, Y)$ just looks like a flat plane. For a unit spherical surface, such a map is the stereographic projection \n$(X, Y) = \\left(\\frac{x}{z+1}, \\frac{y}{z+1}\\right) = (\\rho \\cos \\phi, \\rho \\sin \\phi)$ \nSuppose now we consider heat conduction problem on such a spherical surface, i.e. a very thin shell of spherical surface. The steady-state heat conduction has the temperature profile satisfying the Laplace equation \n$\\nabla^{2} T(\\theta, \\phi) = 0$ \nwhile temperature profile is independent of radial distance $r$ in spherical coordinate $(r, \\theta, \\phi)$. The spherical surface is at radius $r=1$.\n\n[figure2]\n\nNow, the figure 2 gives the thin shell in the shape of lamp shade, which is in a hemi-spherical surface with a circular opening at the top. The whole shape still has a rotational symmetry about the vertical z-axis. The bottom of the lamp shade is kept at temperature $T_{b}$ (at $\\theta = \\frac{\\pi}{2}$ for spherical polar coordinate) and the top is kept at temperature $T_{t}$ (at $\\theta = \\theta_{t}$).",
+        "question": "Solve the temperature profile $T(\\theta)$, as a function of $\\theta$, with such rotational symmetry.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$T(\\theta) = (T_{t}-T_{b}) \\frac{\\ln \\left(\\tan \\frac{\\theta}{2}\\right)}{\\ln \\left(\\tan \\frac{\\theta_t}{2}\\right)} + T_{b}$}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            4.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Thermodynamics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_6_1_1.png",
+            "image_question/PanPhO_2024_6_5_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_6_6",
+        "context": "[Metric-modified geodesic and heat conduction] \n\nSolving physics, such as wave propagation, geodesics, and thermal conduction, on a curved surface in 3D requires a thorough understanding of metrics and differential geometry. However, there can be significant simplifications for systems with spatial symmetry or by adopting coordinate transformation. In this question, we will go through two problems for physics on a curved surface. The first one is light propagating on a curved surface. Figure 1 (a) shows a circular cone with height 5 mm and a base diameter of $2 \\rho_{0} = 10 \\mathrm{mm}$, joining to a flat surface. The flat surface has a circular hole of the same diameter so that as a whole, there is only one single surface with the cone part indicating the 'curved space.' The entire surface, including the flat surface and the cone, have a very thin surface so that light can be effectively confined on such a surface. We have assumed the cone is joint smoothly to the flat surface. \n\n[figure1]\n\nFigure 1(a) depicts a curved surface created by connecting a circular cone to a flat surface with a hole of the same size as the cone's base. In Figure 1(b), we present a top view of this surface. Light confined to such a surface originates on the flat surface at the bottom, undergoes bending due to the cone, and exits in a different direction.\n\n[Part C: Heat Conduction on A Spherical Surface]\n\n[figure2]\n\nNow, we consider the top opening is tilted about the $y$-axis in breaking rotational symmetry. Suppose the top opening is still a circle on the spherical surface passing through $(x, y, z) = (0, 0, 1)$ and $(\\sin \\alpha, 0, \\cos \\alpha)$ as diameter and its normal is on the $x-z$ plane.",
+        "question": "Determine where the top opening is mapped on $(X, Y)$ plane through the stereographic projection map. Write down the equation of the circle that $X$ and $Y$ satisfy. \nHint: the answer is still a circle in $X$ and $Y$ coordinates.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$\\left(X - \\frac{1}{2} \\tan \\frac{\\alpha}{2}\\right)^{2} + Y^{2} = \\frac{1}{4} \\tan^{2} \\frac{\\alpha}{2}$}"
+        ],
+        "answer_type": [
+            "Equation"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            2.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Thermodynamics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_6_1_1.png",
+            "image_question/PanPhO_2024_6_6_1.png"
+        ]
+    },
+    {
+        "id": "PanPhO_2024_6_7",
+        "context": "[Metric-modified geodesic and heat conduction] \n\nSolving physics, such as wave propagation, geodesics, and thermal conduction, on a curved surface in 3D requires a thorough understanding of metrics and differential geometry. However, there can be significant simplifications for systems with spatial symmetry or by adopting coordinate transformation. In this question, we will go through two problems for physics on a curved surface. The first one is light propagating on a curved surface. Figure 1 (a) shows a circular cone with height 5 mm and a base diameter of $2 \\rho_{0} = 10 \\mathrm{mm}$, joining to a flat surface. The flat surface has a circular hole of the same diameter so that as a whole, there is only one single surface with the cone part indicating the 'curved space.' The entire surface, including the flat surface and the cone, have a very thin surface so that light can be effectively confined on such a surface. We have assumed the cone is joint smoothly to the flat surface. \n\n[figure1]\n\nFigure 1(a) depicts a curved surface created by connecting a circular cone to a flat surface with a hole of the same size as the cone's base. In Figure 1(b), we present a top view of this surface. Light confined to such a surface originates on the flat surface at the bottom, undergoes bending due to the cone, and exits in a different direction.\n\n[Part C: Heat Conduction on A Spherical Surface]\n\n[figure2]\n\nNow, we consider the top opening is tilted about the $y$-axis in breaking rotational symmetry. Suppose the top opening is still a circle on the spherical surface passing through $(x, y, z) = (0, 0, 1)$ and $(\\sin \\alpha, 0, \\cos \\alpha)$ as diameter and its normal is on the $x-z$ plane.\n\nThe top opening of the surface is mapped onto the (X, Y) plane through the stereographic projection, resulting in a circle. The coordinates $X$ and $Y$ satisfy the equation of this circle.",
+        "question": "Solve the temperature profile $T(X, Y)$ when the bottom opening is kept at temperature $T_{b}$ and the top is kept at temperature $T_{t}$. You can leave your answer in terms of $X$ and $Y$ coordinates. \nHint: In the stereographic projected domain X-Y plane, Laplace equation is satisfied and you can further use general method of image, like the one used in solving electrostatic problem by putting two point charges on the $X$-axis with undetermined charges.",
+        "marking": [],
+        "answer": [
+            "\\boxed{$T = \\frac{T_{t}-T_{b}}{2 \\ln \\mu} \\ln \\left(\\frac{(X-\\mu)^{2} + Y^{2}}{\\left(X-\\mu^{-1}\\right)^{2} + Y^{2}}\\right) + 2 T_{b} - T_{t}$ or $T = \\frac{T_t - T_b}{2 \\ln \\mu} \\ln \\left( \\frac{\\tan^2(\\theta/2) - 2\\mu \\tan(\\theta/2) \\cos \\phi + \\mu^2}{\\tan^2(\\theta/2) - 2\\mu^{-1} \\tan(\\theta/2) \\cos \\phi + \\mu^{-2}} \\right) + 2T_b - T_t$, where $\\mu = \\frac{1+\\sqrt{1-4a^2}}{2a}$ and $a = \\frac{1}{2} \\tan \\frac{\\alpha}{2}$.}"
+        ],
+        "answer_type": [
+            "Expression"
+        ],
+        "unit": [
+            null
+        ],
+        "points": [
+            8.0
+        ],
+        "modality": "text+variable figure",
+        "field": "Thermodynamics",
+        "source": "PanPhO_2024",
+        "image_question": [
+            "image_question/PanPhO_2024_6_1_1.png",
+            "image_question/PanPhO_2024_6_6_1.png"
+        ]
+    }
+]