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[
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|
|
{
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"information": "None."
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|
},
|
|
|
{
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"id": "APhO_2025_1_A_1",
|
|
|
"context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents.",
|
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"question": "Find the values of exponents: (1) $\\beta$, (2) $\\gamma$, and (3) $\\delta$.",
|
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"marking": [
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|
[
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"Award 0.2 pt if the answer correctly expresses the dimension of $G$ as $[G] = L^3 M^{-1} T^{-2}$, where $L$ is the base dimensions length, $M$ is mass, and $T$ is time. Otherwise, award 0 pt.",
|
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|
"Award 0.1 pt if the answer correctly sets up the exponent equation $0 = 2 - \\beta$. Otherwise, award 0 pt.",
|
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"Award 0.1 pt if the answer correctly sets up the exponent equation $0 = \\gamma + 1$. Otherwise, award 0 pt.",
|
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|
"Award 0.1 pt if the answer correctly sets up the exponent equation $1 = \\delta - 3$. Otherwise, award 0 pt.",
|
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"Award 0.1 pt if the answer obtains the correct value $\\beta = 2$. Otherwise, award 0 pt.",
|
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"Award 0.1 pt if the answer obtains the correct value $\\gamma = -1$. Otherwise, award 0 pt.",
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|
"Award 0.1 pt if the answer obtains the correct value $\\delta = 4$. Otherwise, award 0 pt."
|
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]
|
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|
],
|
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"answer": [
|
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"\\boxed{$\\beta = 2$}",
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"\\boxed{$\\gamma = -1$}",
|
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"\\boxed{$\\delta = 4$}"
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],
|
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"answer_type": [
|
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|
"Numerical Value",
|
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|
"Numerical Value",
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"Numerical Value"
|
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|
],
|
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"unit": [
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null,
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null,
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null
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],
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"points": [
|
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0.3,
|
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0.3,
|
|
|
0.2
|
|
|
],
|
|
|
"modality": "text+illustration figure",
|
|
|
"field": "Mechanics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_1_a_1.png"
|
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|
]
|
|
|
},
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{
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|
"id": "APhO_2025_1_A_2",
|
|
|
"context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$.",
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"question": "Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1.",
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"marking": [
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[
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"Award 0.1 pt if the answer correctly calculates $\\omega = \\frac{2\\pi}{24 h} = 7.27 \\times 10^{-5} s^{-1}$, where $\\omega$ is the angular speed of the Earth's rotation. Otherwise, award 0 pt.",
|
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"Award 0.1 pt if the answer correctly gives $h_{\\max} = 21.9 \\mathrm{km}$ from the relation $h_{\\max} \\propto \\frac{\\omega^2 R^4}{G M_E}$, where $R$ is the Earth's radius and $M_E$ is the Earth's mass. Otherwise, award 0 pt."
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]
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|
],
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"answer": [
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"\\boxed{21.9}"
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],
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"answer_type": [
|
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"Numerical Value"
|
|
|
],
|
|
|
"unit": [
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|
"km"
|
|
|
],
|
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|
"points": [
|
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0.2
|
|
|
],
|
|
|
"modality": "text+illustration figure",
|
|
|
"field": "Mechanics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_1_a_1.png"
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|
|
]
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|
},
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{
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|
"id": "APhO_2025_1_B_1",
|
|
|
"context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis.",
|
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"question": "Find (1) the direction and (2) magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$.",
|
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"marking": [
|
|
|
[
|
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|
"Award 0.2 pt if the answer correctly expresses the magnitude of $U(z) = -G \\frac{M_S}{\\sqrt{z^2 + d_{SE}^2}}$, where $M_S$ is the Sun's mass, $z$ is the axis coordinate, $d_{SE}$ is the Earth-Sun distance, $G$ is the gravitational constant. Otherwise, award 0 pt.",
|
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|
"Award 0.1 pt if the answer includes the correct negative sign in $U(z) = -G \\frac{M_S}{\\sqrt{z^2 + d_{SE}^2}}$. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer expresses $g_z(z)$ as a derivative $g_z(z) = - \\frac{dU}{dz}$. Partial points: award 0.1 pt if the answer does not include the negative sign. Otherwise, award 0 pt.",
|
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|
"Award 0.2 pt if the derivative is calculated correctly as $g_z(z) = -G M_S \\frac{z}{(z^2 + d_{SE}^2)^{3/2}}$. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer correctly obtains the approximate form for small $z$, $g_z(z) \\approx - \\frac{G M_S}{d_{SE}^3} z$. Otherwise, award 0 pt.",
|
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|
"Award 0.2 pt if the answer indicates that the negative sign means $g_z$ points towards the center of the Sun ring (correct direction in figure). Otherwise, award 0 pt."
|
|
|
],
|
|
|
[
|
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|
"Award 0.2 pt if the answer correctly writes the gravitational field of an element of the ring as $\\mathrm{d}g = G \\frac{\\mathrm{d}M}{z^2 + d_{SE}^2}$, where $\\mathrm{d}M$ is the ring element mass, $z$ is the axis coordinate, $d_{SE}$ is the Earth–Sun distance, and $G$ is the gravitational constant. Otherwise, award 0 pt.",
|
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|
"Award 0.1 pt if the answer includes a figure with correct geometry showing $\\mathrm{d}g$, $\\mathrm{d}g_z$, angle $\\theta$, $z$, and $d_{SE}$. Otherwise, award 0 pt.",
|
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|
"Award 0.1 pt if the answer takes only the $z$ component for symmetry reasons, writing $\\mathrm{d}g_z = - \\mathrm{d}g \\cos \\theta$. Otherwise, award 0 pt.",
|
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|
"Award 0.1 pt if the answer sums or integrates over the whole ring to obtain the total field. Otherwise, award 0 pt.",
|
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|
"Award 0.2 pt if the answer correctly calculates $g_z$ at arbitrary $z$ as $g_z = -G M_S \\frac{z}{(z^2 + d_{SE}^2)^{3/2}}$. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the approximate form for small $z$, $g_z \\approx - G M_S \\frac{z}{d_{SE}^3}$, is correctly given. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer indicates that the negative sign means $g_z$ points towards the center of the Sun ring (correct direction in figure). Otherwise, award 0 pt."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{The direction of the gravitational field is toward the center of the Sun ring.}",
|
|
|
"\\boxed{$g_z(z) \\approx -\\frac{G M_S}{d_{SE}^3} z$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Open-Ended",
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null,
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.2,
|
|
|
0.8
|
|
|
],
|
|
|
"modality": "text+illustration figure",
|
|
|
"field": "Mechanics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_1_a_1.png",
|
|
|
"image_question/APhO_2025_1_b_1.png",
|
|
|
"image_question/APhO_2025_1_b_2.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_1_B_2",
|
|
|
"context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis. \n\n(B.1) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$.",
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"question": "Find (1) the direction and (2) magnitude of the gravitational field generated by the Sun ring at a point in the ecliptic plane whose distance from the origin is $r$. Assume $r \\ll d_{SE}$.",
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"marking": [
|
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|
[
|
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|
"Award 0.5 pt if the answer presents the idea of using Gauss's law to calculate the gravitational field in the plane of the Sun ring. Otherwise, award 0 pt.",
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"Award 0.4 pt if the answer correctly identifies a cylindrical Gaussian surface with axis along $z$ near the center of the Sun ring. Otherwise, award 0 pt.",
|
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|
"Award 0.6 pt if the answer writes Gauss's law correctly as $g_r 2z \\times 2\\pi r + g_z \\cdot 2r^2 \\pi = 0$, where $g_r$ is the radial component and $g_z$ the axial component. Partial points: award 0.3 pt if there is a mistake in areas; award 0 pt if the error is dimensional. Otherwise, award 0 pt.",
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|
"Award 0.3 pt if the answer correctly obtains the final result that $g_r$ is proportional to $r$, i.e. $g_r(r) \\propto r$. Otherwise, award 0 pt.",
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|
"Award 0.2 pt if the answer gives the correct proportionality constant $g_r(r) = \\frac{G M_S}{2 d_{SE}^3} r$. Partial points: award 0.1 pt if the prefactor is wrong but dimensionally consistent; award 0 pt if the error is dimensional. Otherwise, award 0 pt.",
|
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|
"Award 0.2 pt if the answer indicates that the field points radially outwards (correct direction in the figure). Otherwise, award 0 pt."
|
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|
],
|
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[
|
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|
"Award 0.2 pt if the answer correctly expresses the distance $s = \\sqrt{d_{SE}^2 + r^2 - 2 d_{SE} r \\cos \\varphi}$ from trigonometry, where $r$ is the distance of point $P$ from the center, $d_{SE}$ is the Earth-Sun distance, and $\\varphi$ is the angle. Otherwise, award 0 pt.",
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|
"Award 0.1 pt if the answer correctly writes the potential generated by a small element of the ring as $\\mathrm{d}U = - G M_S \\frac{\\mathrm{d}\\varphi}{2 \\pi s}$. Otherwise, award 0 pt.",
|
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"Award 0.1 pt if the answer writes the total potential $U(r)$ as an integral $U(r) = - \\frac{G M_S}{2 \\pi} \\int_0^{2\\pi} (d_{SE}^2 + r^2 - 2 d_{SE} r \\cos \\varphi)^{-1/2} \\mathrm{d}\\varphi$. Otherwise, award 0 pt.",
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"Award 0.6 pt if the answer expands the integrand to second order and obtains $(1 + \\varepsilon)^{-1/2} \\approx 1 - \\frac{r^{2}}{2 d_{SE}^{2}} + \\frac{r \\cos \\varphi}{d_{SE}} + \\frac{3 r^{2} \\cos^{2}\\varphi}{2 d_{SE}^{2}}.$, where $\\varepsilon = \\frac{r^2}{d_{SE}^2} - \\frac{2r \\cos \\varphi}{d_{SE}}$ and $(1+\\varepsilon)^{-1/2} \\approx 1 - \\varepsilon/2 + 3\\varepsilon^2/8$. Partial points: award 0.1 pt if only first order is calculated; award 0.4 pt if the $\\cos^2 \\varphi$ term is missing. Otherwise, award 0 pt.",
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"Award 0.2 pt if the answer integrates over $\\varphi$ correctly: $U(r) = -\\frac{G M_{S}}{2\\pi d_{SE}} \\int_{0}^{2\\pi} \\left( 1 - \\frac{r^{2}}{2 d_{SE}^{2}} + \\frac{3 r^{2} \\cos^{2} \\varphi}{2 d_{SE}^{2}} \\right) \\mathrm{d}\\varphi$. Partial points: award 0.1 pt if the $\\cos^2 \\varphi$ term is missing. Otherwise, award 0 pt.",
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"Award 0.2 pt if the answer correctly expresses $g_z$ as the derivative of $U(z)$: $g_r(r) = - \\frac{dU}{dr}$. Partial points: aAward 0.1 pt if the negative sign is omitted. Otherwise, award 0 pt.",
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"Award 0.1 pt if the derivative $\\frac{dU}{dr}$ is calculated correctly, yielding $g_r(r) = \\frac{G M_S}{2 d_{SE}^3} r$. Otherwise, award 0 pt.",
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"Award 0.3 pt if the final result shows $g_r \\propto r$. Otherwise, award 0 pt.",
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"Award 0.2 pt if the proportionality constant is correct, $g_r(r) = \\frac{G M_S}{2 d_{SE}^3} r$. Partial points: award 0.1 pt if only the prefactor is wrong but dimensionally consistent; award 0 pt if dimensional error. Otherwise, award 0 pt.",
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"Award 0.2 pt if the answer indicates that the field points radially outwards (correct direction in the figure). Otherwise, award 0 pt."
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]
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],
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"answer": [
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"\\boxed{The direction of the gravitational field is outward along the radial direction.}",
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"\\boxed{$g_r(r) = \\frac{G M_S}{2 d_{SE}^3} r$}"
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],
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"answer_type": [
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"Open-Ended",
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"Expression"
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],
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"unit": [
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null,
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null
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],
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"points": [
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0.2,
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2.0
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],
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"modality": "text+illustration figure",
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"field": "Mechanics",
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"source": "APhO_2025",
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|
"image_question": [
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"image_question/APhO_2025_1_a_1.png",
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"image_question/APhO_2025_1_b_1.png",
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"image_question/APhO_2025_1_b_2.png"
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]
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},
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{
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"id": "APhO_2025_1_C_1",
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|
"context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis. \n\n(B.1) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$. \n\n(B.2) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point in the ecliptic plane whose distance from the origin is $r$. Assume $r \\ll d_{SE}$. \n\n[Part C: The torque acting on the Earth] \n\nIn this section, you are asked to determine the torque exerted on the Earth due to the gravitational field obtained in Part B. For simplicity, consider the Earth as a rigid body with homogeneous mass distribution. Let us take into account that the rotational ellipsoid can be imagined as if we removed excess parts from a sphere with the equatorial radius of Earth $R_{e}$ (see Figure C.1). \n\n[figure4] \nFigure C.1. The ellipsoidal shape of the Earth can be imagined as if the excess parts were removed from a complete sphere of radius $R_{e}$.",
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"question": "Find the mass $m$ of one of the two excess regions indicated in Figure C.1. Express your answer in terms of $h_{\\text{max}}$, the mass of the Earth $M_{E}$, and its polar radius $R_{p}$.",
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"marking": [
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[
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"Award 0.2 pt if the answer includes the idea of transforming the ellipsoid of revolution into a perfect sphere of radius $R_e$ by stretching uniformly along the polar diameter with factor $R_e/R_p$. Otherwise, award 0 pt.",
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"Award 0.3 pt if the answer correctly computes the volume of one of the excess regions as $V = \\frac{1}{2} \\left( \\frac{4\\pi}{3} R_e^3 - \\frac{4\\pi}{3} R_e^2 R_p \\right) = \\frac{2\\pi}{3} R_e^2 h_{max}$. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer writes the correct expression for the Earth's density as $\\rho = \\frac{3 M_E}{4 \\pi R_e^2 R_p}$, where $M_E$ is the Earth's mass. Otherwise, award 0 pt.",
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"Award 0.2 pt if the answer obtains the final expression for the mass of one excess region as $m = \\frac{h_{max}}{2 R_p} M_E$. Otherwise, award 0 pt."
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]
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],
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"answer": [
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"\\boxed{$m = \\frac{h_{\\max}}{2 R_p} M_E$}"
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],
|
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"answer_type": [
|
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"Expression"
|
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|
],
|
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|
"unit": [
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null
|
|
|
],
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"points": [
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0.8
|
|
|
],
|
|
|
"modality": "text+variable figure",
|
|
|
"field": "Mechanics",
|
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|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_1_a_1.png",
|
|
|
"image_question/APhO_2025_1_b_1.png",
|
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"image_question/APhO_2025_1_b_2.png",
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"image_question/APhO_2025_1_c_1.png"
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]
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},
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{
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|
"id": "APhO_2025_1_C_2",
|
|
|
"context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis. \n\n(B.1) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$. \n\n(B.2) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point in the ecliptic plane whose distance from the origin is $r$. Assume $r \\ll d_{SE}$. \n\n[Part C: The torque acting on the Earth] \n\nIn this section, you are asked to determine the torque exerted on the Earth due to the gravitational field obtained in Part B. For simplicity, consider the Earth as a rigid body with homogeneous mass distribution. Let us take into account that the rotational ellipsoid can be imagined as if we removed excess parts from a sphere with the equatorial radius of Earth $R_{e}$ (see Figure C.1). \n\n[figure4] \nFigure C.1. The ellipsoidal shape of the Earth can be imagined as if the excess parts were removed from a complete sphere of radius $R_{e}$. \n\n(C.1) Find the mass $m$ of one of the two excess regions indicated in Figure C.1. Express your answer in terms of $h_{\\text{max}}$, the mass of the Earth $M_{E}$, and its polar radius $R_{p}$. \n\nIt can be shown that the torque acting on the excess regions is equivalent to the torque acting on two point masses, each with a mass equal to $2m / 5$, positioned at the endpoints $A$ and $B$ of the polar diameter (see Figure C.1).",
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"question": "Given this idea, find the torque $\\tau$ exerted by the Sun ring on the Earth. Express your answer in terms of $M_{E}, M_{S}, d_{SE}, R$ (the average radius), $h_{\\max}$ and the angle $\\alpha$. You can use that $h_{\\max} \\ll R$.",
|
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"marking": [
|
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|
[
|
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|
"Award 0.1 pt if the answer mentions that the net torque acting on a perfect sphere of radius $R_e$ is zero due to symmetry. Otherwise, award 0 pt.",
|
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|
"Award 0.2 pt if the answer states the idea that the torque on the ellipsoid-shaped Earth is given by $\\vec{\\tau} = - \\vec{\\tau}'$. Otherwise, award 0 pt.",
|
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|
"Award 0.4 pt if the answer correctly includes the torque contribution from $F_z$, where $F_z = \\frac{2}{5} m |g_z| = \\frac{2}{5} m G M_S \\dfrac{R \\cos \\alpha}{d_{SE}^3}$. Otherwise, award 0 pt.",
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|
"Award 0.4 pt if the answer correctly includes the torque contribution from $F_r$, where $F_r = \\frac{2}{5} m |g_r| = \\frac{2}{5} m G M_S \\dfrac{R \\sin \\alpha}{2 d_{SE}^3}$. Otherwise, award 0 pt.",
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"Award 0.2 pt if the answer correctly adds the two torque contributions with the right sign in $|\\tau'| = 2 F_z R \\sin \\alpha + 2 F_r R \\cos \\alpha$. Otherwise, award 0 pt.",
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|
"Award 0.3 pt if the calculation is carried through to obtain the correct net torque $|\\tau'| = \\frac{6}{5} \\dfrac{G m M_S}{d_{SE}^3} R^2 \\sin \\alpha \\cos \\alpha = \\frac{3}{5} \\dfrac{G M_E M_S}{d_{SE}^3} R h_{max} \\sin \\alpha \\cos \\alpha$. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer gives the correct direction of $\\vec{\\tau}$: pointing into the plane (opposite to $\\vec{\\tau}'$). Otherwise, award 0 pt."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
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|
"\\boxed{$|\\tau| = \\frac{3}{5} \\cdot \\frac{G M_E M_S}{d_{SE}^3} \\cdot R h_{\\max} \\sin \\alpha \\cos \\alpha$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
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|
"points": [
|
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|
1.8
|
|
|
],
|
|
|
"modality": "text+variable figure",
|
|
|
"field": "Mechanics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_1_a_1.png",
|
|
|
"image_question/APhO_2025_1_b_1.png",
|
|
|
"image_question/APhO_2025_1_b_2.png",
|
|
|
"image_question/APhO_2025_1_c_1.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_1_D_1",
|
|
|
"context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis. \n\n(B.1) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$. \n\n(B.2) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point in the ecliptic plane whose distance from the origin is $r$. Assume $r \\ll d_{SE}$. \n\n[Part C: The torque acting on the Earth] \n\nIn this section, you are asked to determine the torque exerted on the Earth due to the gravitational field obtained in Part B. For simplicity, consider the Earth as a rigid body with homogeneous mass distribution. Let us take into account that the rotational ellipsoid can be imagined as if we removed excess parts from a sphere with the equatorial radius of Earth $R_{e}$ (see Figure C.1). \n\n[figure4] \nFigure C.1. The ellipsoidal shape of the Earth can be imagined as if the excess parts were removed from a complete sphere of radius $R_{e}$. \n\n(C.1) Find the mass $m$ of one of the two excess regions indicated in Figure C.1. Express your answer in terms of $h_{\\text{max}}$, the mass of the Earth $M_{E}$, and its polar radius $R_{p}$. \n\nIt can be shown that the torque acting on the excess regions is equivalent to the torque acting on two point masses, each with a mass equal to $2m / 5$, positioned at the endpoints $A$ and $B$ of the polar diameter (see Figure C.1). \n\n(C.2) Given this idea, find the torque $\\tau$ exerted by the Sun ring on the Earth. Express your answer in terms of $M_{E}, M_{S}, d_{SE}, R$ (the average radius), $h_{\\max}$ and the angle $\\alpha$. You can use that $h_{\\max} \\ll R$. \n\n[Part D: Angular speed of the precession of the Earth's axis] \n\nThe Earth's axis of rotation moves very slowly around the $z$ axis in a conical motion. That is, it precesses.",
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"question": "Give an expression for the period $T_{1}$ of precession of the Earth's axis. Express your answer in terms of $M_{S}, d_{SE}$, the angular speed $\\omega$ of the Earth's rotation, $h_{\\text{max}}, R$ and $\\alpha$.",
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"marking": [
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[
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"Award 0.2 pt if the answer applies Newton's second law for rotational motion, $\\vec{\\tau} = \\dfrac{d \\vec{L}}{dt}$, where $\\tau$ is torque and $\\vec{L}$ is angular momentum. Otherwise, award 0 pt.",
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"Award 0.2 pt if the answer expresses angular momentum as $|\\vec{L}| = I \\omega$, where $I$ is the moment of inertia and $\\omega$ is the angular velocity of Earth's rotation. Otherwise, award 0 pt.",
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"Award 0.2 pt if the answer writes the correct moment of inertia for a uniform sphere as $I = \\frac{2}{5} M_E R^2$. Partial points: award 0.1 pt if the prefactor is wrong but dimensions are correct; award 0 pt if there is a dimensional error. Otherwise, award 0 pt.",
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"Award 0.8 pt if the answer correctly relates the time derivative of angular momentum to precession angular speed: $| \\frac{d \\vec{L}}{dt} | = \\Omega_1 |\\vec{L}| \\sin \\alpha$, where $\\Omega_1$ is the angular speed of precession and $\\alpha$ is the half-apex angle. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer uses the relation $\\Omega_1 = \\dfrac{2\\pi}{T_1}$ between precession angular velocity and precession period. Otherwise, award 0 pt.",
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"Award 0.3 pt if the answer finds the correct precession period $T_1 = \\dfrac{4 \\pi}{3} \\dfrac{d_{SE}^3 R \\omega}{G M_S h_{\\max} \\cos \\alpha}$. Otherwise, award 0 pt."
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]
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],
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"answer": [
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"\\boxed{$T_1 = \\frac{4 \\pi}{3} \\cdot \\frac{d_{SE}^3 R \\omega}{G M_S h_{\\max} \\cos \\alpha}$}"
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],
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"answer_type": [
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"Expression"
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],
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"unit": [
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null
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],
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"points": [
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1.8
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],
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"modality": "text+variable figure",
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"field": "Mechanics",
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"source": "APhO_2025",
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"image_question": [
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"image_question/APhO_2025_1_a_1.png",
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"image_question/APhO_2025_1_b_1.png",
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"image_question/APhO_2025_1_b_2.png",
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"image_question/APhO_2025_1_c_1.png"
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]
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},
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{
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"id": "APhO_2025_1_D_2",
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"context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis. \n\n(B.1) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$. \n\n(B.2) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point in the ecliptic plane whose distance from the origin is $r$. Assume $r \\ll d_{SE}$. \n\n[Part C: The torque acting on the Earth] \n\nIn this section, you are asked to determine the torque exerted on the Earth due to the gravitational field obtained in Part B. For simplicity, consider the Earth as a rigid body with homogeneous mass distribution. Let us take into account that the rotational ellipsoid can be imagined as if we removed excess parts from a sphere with the equatorial radius of Earth $R_{e}$ (see Figure C.1). \n\n[figure4] \nFigure C.1. The ellipsoidal shape of the Earth can be imagined as if the excess parts were removed from a complete sphere of radius $R_{e}$. \n\n(C.1) Find the mass $m$ of one of the two excess regions indicated in Figure C.1. Express your answer in terms of $h_{\\text{max}}$, the mass of the Earth $M_{E}$, and its polar radius $R_{p}$. \n\nIt can be shown that the torque acting on the excess regions is equivalent to the torque acting on two point masses, each with a mass equal to $2m / 5$, positioned at the endpoints $A$ and $B$ of the polar diameter (see Figure C.1). \n\n(C.2) Given this idea, find the torque $\\tau$ exerted by the Sun ring on the Earth. Express your answer in terms of $M_{E}, M_{S}, d_{SE}, R$ (the average radius), $h_{\\max}$ and the angle $\\alpha$. You can use that $h_{\\max} \\ll R$. \n\n[Part D: Angular speed of the precession of the Earth's axis] \n\nThe Earth's axis of rotation moves very slowly around the $z$ axis in a conical motion. That is, it precesses. \n\n(D.1) Give an expression for the period $T_{1}$ of precession of the Earth's axis. Express your answer in terms of $M_{S}, d_{SE}$, the angular speed $\\omega$ of the Earth's rotation, $h_{\\text{max}}, R$ and $\\alpha$.",
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"question": "Calculate the precession period $T_{1}$ in years.",
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"marking": [
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[
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"Award 0.2 pt if the answer gives the correct numerical result for the precession period as $T_1 \\approx 80600$ years, obtained by correctly substituting the given data into a dimensionally correct formula. Partial points: award 0 pt if the substitution is incorrect or if the formula used has a dimensional error."
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]
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],
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"answer": [
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"\\boxed{80600}"
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],
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"answer_type": [
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"Numerical Value"
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],
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"unit": [
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"years"
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],
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"points": [
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0.2
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|
],
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"modality": "text+variable figure",
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"field": "Mechanics",
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"source": "APhO_2025",
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|
"image_question": [
|
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|
"image_question/APhO_2025_1_a_1.png",
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"image_question/APhO_2025_1_b_1.png",
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"image_question/APhO_2025_1_b_2.png",
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"image_question/APhO_2025_1_c_1.png"
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]
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},
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{
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"id": "APhO_2025_1_E_1",
|
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|
"context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis. \n\n(B.1) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$. \n\n(B.2) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point in the ecliptic plane whose distance from the origin is $r$. Assume $r \\ll d_{SE}$. \n\n[Part C: The torque acting on the Earth] \n\nIn this section, you are asked to determine the torque exerted on the Earth due to the gravitational field obtained in Part B. For simplicity, consider the Earth as a rigid body with homogeneous mass distribution. Let us take into account that the rotational ellipsoid can be imagined as if we removed excess parts from a sphere with the equatorial radius of Earth $R_{e}$ (see Figure C.1). \n\n[figure4] \nFigure C.1. The ellipsoidal shape of the Earth can be imagined as if the excess parts were removed from a complete sphere of radius $R_{e}$. \n\n(C.1) Find the mass $m$ of one of the two excess regions indicated in Figure C.1. Express your answer in terms of $h_{\\text{max}}$, the mass of the Earth $M_{E}$, and its polar radius $R_{p}$. \n\nIt can be shown that the torque acting on the excess regions is equivalent to the torque acting on two point masses, each with a mass equal to $2m / 5$, positioned at the endpoints $A$ and $B$ of the polar diameter (see Figure C.1). \n\n(C.2) Given this idea, find the torque $\\tau$ exerted by the Sun ring on the Earth. Express your answer in terms of $M_{E}, M_{S}, d_{SE}, R$ (the average radius), $h_{\\max}$ and the angle $\\alpha$. You can use that $h_{\\max} \\ll R$. \n\n[Part D: Angular speed of the precession of the Earth's axis] \n\nThe Earth's axis of rotation moves very slowly around the $z$ axis in a conical motion. That is, it precesses. \n\n(D.1) Give an expression for the period $T_{1}$ of precession of the Earth's axis. Express your answer in terms of $M_{S}, d_{SE}$, the angular speed $\\omega$ of the Earth's rotation, $h_{\\text{max}}, R$ and $\\alpha$. \n\n(D.2) Calculate the precession period $T_{1}$ in years. \n\n[Part E: The effect of the Moon] \n\nThe value obtained in Part D is much larger than the observed value. The reason for this is that so far we have only considered the torque exerted by the Sun, and neglected the effect of the Moon. In the following calculations, assume that the Moon's orbit is in the ecliptic plane, and that the orbit of the Moon around the Earth is a circle of radius $d_{ME}$. Let us denote the mass of the Moon by $M_{M}$ and the period of precession in this modified model by $T_{2}$.",
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"question": "By what factor $T_{2} / T_{1}$ does the period of precession of the Earth's axis change if we also take into account the torque exerted by the Moon? Give your answer in terms of $d_{ME}, d_{SE}, M_{S}$ and $M_{M}$.",
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"marking": [
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[
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"Award 0.3 pt if the answer explicitly states that the torques exerted by the Sun and the Moon add up. Otherwise, award 0 pt.",
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"Award 0.4 pt if the answer correctly calculates the torque exerted by the Moon or states that it is proportional to $M_M/d_{ME}^3$, where $M_M$ is the Moon's mass and $d_{ME}$ is the Earth-Moon distance. Otherwise, award 0 pt.",
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"Award 0.3 pt if the answer expresses the ratio $T_2/T_1$ correctly as $\\dfrac{T_2}{T_1} = \\dfrac{M_S/d_{SE}^3}{M_M/d_{ME}^3 + M_S/d_{SE}^3}$. Partial points: award 0 pt if the expression implies $T_1 < T_2$. Otherwise, award 0 pt."
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]
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],
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"answer": [
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"\\boxed{$T_2 / T_1 = \\frac{M_S / d_{SE}^3}{M_M / d_{ME}^3 + M_S / d_{SE}^3}$}"
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],
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"answer_type": [
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"Expression"
|
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],
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"unit": [
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null
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|
],
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"points": [
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1.0
|
|
|
],
|
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|
"modality": "text+variable figure",
|
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|
"field": "Mechanics",
|
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|
"source": "APhO_2025",
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|
"image_question": [
|
|
|
"image_question/APhO_2025_1_a_1.png",
|
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|
"image_question/APhO_2025_1_b_1.png",
|
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|
"image_question/APhO_2025_1_b_2.png",
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"image_question/APhO_2025_1_c_1.png"
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|
]
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},
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{
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"id": "APhO_2025_1_E_2",
|
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|
"context": "[Precession of the Earth's axis] \n\n[Introduction] \n\nIt has been known since ancient times that the Earth's axis of rotation precesses. That is, the axis itself rotates around the line perpendicular to the ecliptic plane, i.e., the plane containing the Earth's orbit around the Sun. Ancient Greek astronomer Hipparchus concluded that the annual angular displacement of the axis was approximately 45'' (seconds of arc), which would imply that the period of axial precession is around 29000 years. Modern measurements indicate that the period is approximately 25800 years. In this problem, you are asked to investigate this phenomenon using Newtonian mechanics. \n\nYou may need the following constants: \ngravitational constant: $G = 6.67 \\times 10^{-11} \\mathrm{Nm}^{2} / \\mathrm{kg}^{2}$ \naverage radius of Earth: $R = 6.371 \\times 10^{6} \\mathrm{m}$ \nmass of the Earth: $M_{E} = 5.972 \\times 10^{24} \\mathrm{kg}$ \naverage distance of the Sun from the Earth: $d_{SE} = 1.496 \\times 10^{11} \\mathrm{m}$ \nmass of the Sun: $M_{S} = 1.989 \\times 10^{30} \\mathrm{kg}$ \naverage distance of the Moon from the Earth: $d_{ME} = 3.844 \\times 10^{8} \\mathrm{m}$ \nmass of the Moon: $M_{M} = 7.348 \\times 10^{22} \\mathrm{kg}$ \nEarth's axial tilt: $\\alpha = 23.5^{\\circ}$ \n\n[Part A: The shape of the Earth] \n\nThe Sun and the Moon exert nonzero torques on the Earth because of its non-spherical shape, giving rise to its axial precession. The main reason behind the Earth's non-spherical shape is the centrifugal force caused by the Earth's rotation about its axis. The tectonic plates located on the Earth's surface have deformed over millions of years to minimize stress within them. Therefore, as an approximation, let us model the Earth as a large liquid droplet of uniform density whose shape is determined by centrifugal and gravitational forces. In this model, the Earth's surface is an oblate spheroid (ellipsoid of revolution) characterized by the polar radius $R_{p}$ and the equatorial radius $R_{e}$ (see Figure A.1). \n\n[figure1] \nFigure A.1. The ellipsoidal shape of the Earth. The polar and equatorial radii are indicated. $\\alpha = 23.5^{\\circ}$ is the angle between the Earth's axis of rotation and the normal of the ecliptic plane. \n\nThe difference between the equatorial and polar radii of the Earth, $h_{\\max} = R_{e} - R_{p}$ is much smaller than the average radius $R = (R_{e} + R_{p}) / 2$. Up to a dimensionless factor, the value of $h_{\\max}$ can be expressed in terms of the angular speed of the Earth's rotation $\\omega$, its mass $M_{E}$ and average radius $R$ as \n$h_{\\max} \\propto G^{-1} \\omega^{\\beta} M_E^{\\gamma} R^{\\delta}$ \nwhere $G$ is the gravitational constant, and $\\beta, \\gamma$ and $\\delta$ are constant exponents. \n\n(A.1) Find the values of exponents: $\\beta$, $\\gamma$, and $\\delta$. \n\n(A.2) Calculate the numerical value of $h_{\\text{max}}$ in $km$ assuming that the dimensionless factor in the relation given above equals 1. \n\nRegardless of whether you were able to find $h_{\\text{max}}$ in part A.2., use the empirical value $h_{\\text{max}} = 21 \\mathrm{km}$ in the following questions. \n\n[Part B: The time-averaged gravitational field of the Sun] \n\nTo see why the Sun exerts a nonzero torque (with respect to the center of the Earth) on our planet, consider Figure B.1 below. The difference in distance from the Sun causes the gravitational force $F_{1}$ to be greater than its counterpart $F_{2}$. \n\n[figure2] \nFigure B.1. Explanation of the nonzero torques exerted by the Sun (right side of the figure) on the Earth (left side). \n\nThe magnitude of this torque acting on the Earth varies continuously during the year. In the position shown in Figure B.1, the torque is maximal, a quarter of a year later, due to symmetry, the torque becomes zero. After half a year, it reaches the maximum again, three-quarters of a year later it is zero once again, and so on. Since the period of axial precession is much larger than one year, this time-dependent torque can be approximated well by its one-year average. \n\nTo calculate the average torque exerted by the Sun on the Earth, let us determine first the time average of the gravitational field generated by the Sun in the vicinity of the Earth. This average can be calculated as the field of a uniformly dense mass ring, a Sun ring, whose mass equals the mass of the Sun $M_{S}$ and whose radius equals the average distance between the Sun and the Earth $d_{SE}$ (see Figure B.2). \n\n[figure3] \nFigure B.2. Time averaging is equivalent to uniformly spreading the Sun along the circle of radius $d_{SE}$. \n\nLet our cylindrical coordinate system have the origin at the center of the Earth, and let the $z$ axis be perpendicular to the ecliptic plane (i.e. the plane of the ring). The axis of rotation of the Earth makes an angle of $\\alpha = 23.5^{\\circ}$ with the $z$ axis. \n\n(B.1) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point on the $z$ axis. Write your answer in terms of $M_{S}, d_{SE}$, and the coordinate $z$. Assume that $|z| \\ll d_{SE}$. \n\n(B.2) Find the direction and magnitude of the gravitational field generated by the Sun ring at a point in the ecliptic plane whose distance from the origin is $r$. Assume $r \\ll d_{SE}$. \n\n[Part C: The torque acting on the Earth] \n\nIn this section, you are asked to determine the torque exerted on the Earth due to the gravitational field obtained in Part B. For simplicity, consider the Earth as a rigid body with homogeneous mass distribution. Let us take into account that the rotational ellipsoid can be imagined as if we removed excess parts from a sphere with the equatorial radius of Earth $R_{e}$ (see Figure C.1). \n\n[figure4] \nFigure C.1. The ellipsoidal shape of the Earth can be imagined as if the excess parts were removed from a complete sphere of radius $R_{e}$. \n\n(C.1) Find the mass $m$ of one of the two excess regions indicated in Figure C.1. Express your answer in terms of $h_{\\text{max}}$, the mass of the Earth $M_{E}$, and its polar radius $R_{p}$. \n\nIt can be shown that the torque acting on the excess regions is equivalent to the torque acting on two point masses, each with a mass equal to $2m / 5$, positioned at the endpoints $A$ and $B$ of the polar diameter (see Figure C.1). \n\n(C.2) Given this idea, find the torque $\\tau$ exerted by the Sun ring on the Earth. Express your answer in terms of $M_{E}, M_{S}, d_{SE}, R$ (the average radius), $h_{\\max}$ and the angle $\\alpha$. You can use that $h_{\\max} \\ll R$. \n\n[Part D: Angular speed of the precession of the Earth's axis] \n\nThe Earth's axis of rotation moves very slowly around the $z$ axis in a conical motion. That is, it precesses. \n\n(D.1) Give an expression for the period $T_{1}$ of precession of the Earth's axis. Express your answer in terms of $M_{S}, d_{SE}$, the angular speed $\\omega$ of the Earth's rotation, $h_{\\text{max}}, R$ and $\\alpha$. \n\n(D.2) Calculate the precession period $T_{1}$ in years. \n\n[Part E: The effect of the Moon] \n\nThe value obtained in Part D is much larger than the observed value. The reason for this is that so far we have only considered the torque exerted by the Sun, and neglected the effect of the Moon. In the following calculations, assume that the Moon's orbit is in the ecliptic plane, and that the orbit of the Moon around the Earth is a circle of radius $d_{ME}$. Let us denote the mass of the Moon by $M_{M}$ and the period of precession in this modified model by $T_{2}$. \n\n(E.1) By what factor $T_{2} / T_{1}$ does the period of precession of the Earth's axis change if we also take into account the torque exerted by the Moon? Give your answer in terms of $d_{ME}, d_{SE}, M_{S}$ and $M_{M}$.",
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"question": "By substituting the data, calculate the period of precession $T_{2}$ in years.",
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"marking": [
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[
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"Award 0.2 pt if the answer gives the correct numerical result for the precession period $T_2 \\approx 25400$ years, obtained by substituting values into a dimensionally correct formula. Partial points: award 0 pt if the result does not come from explicit substitution (e.g. if it is taken from the introduction), if the substitution is incorrect, or if the formula has a dimensional error."
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]
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],
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"answer": [
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"\\boxed{25400}"
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],
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"answer_type": [
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"Numerical Value"
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],
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"unit": [
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"years"
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],
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"points": [
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0.2
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],
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"modality": "text+variable figure",
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"field": "Mechanics",
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"source": "APhO_2025",
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"image_question": [
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"image_question/APhO_2025_1_a_1.png",
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"image_question/APhO_2025_1_b_1.png",
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"image_question/APhO_2025_1_b_2.png",
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"image_question/APhO_2025_1_c_1.png"
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]
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},
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{
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"id": "APhO_2025_2_A_1",
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"context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass.",
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"question": "It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$.",
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"marking": [
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[
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"Award 0.1 pt if the answer gives the correct result for the angular momentum of the planar loop, $\\vec{L} = M R^2 \\vec{\\omega}$, where $M$ is the mass, $R$ the radius, and $\\vec{\\omega}$ the angular velocity (Award 0.1 pt if only the magnitude is given in the answer). Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer gives the correct result for the magnetic dipole moment, $\\vec{\\mu} = \\frac{Q}{2} R^2 \\vec{\\omega}$, where $Q$ is the charge (Award 0.1 pt if only the magnitude is given in the answer). Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer gives the correct result for the gyromagnetic ratio, $\\gamma = \\frac{Q}{2M}$. Otherwise, award 0 pt."
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]
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],
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"answer": [
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"\\boxed{$\\gamma = \\frac{Q}{2M}$}"
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],
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"answer_type": [
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"Expression"
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],
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"unit": [
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null
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],
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"points": [
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0.3
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],
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"modality": "text-only",
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"field": "Electromagnetism",
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"source": "APhO_2025",
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"image_question": []
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},
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{
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"id": "APhO_2025_2_A_2",
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"context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass. \n\n(A.1) It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$. \n\nThe ring is placed in a weak uniform magnetic field $\\vec{B} = B \\hat{z}$, making an angle $\\theta$ with $\\vec{\\omega}$, see Figure A.1. \n\n[figure1] \nFigure A.1.",
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"question": "Find the angular frequency $\\omega_{L}$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$.",
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"marking": [
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[
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"Award 0.1 pt if the answer writes the torque equation for a magnetic dipole correctly as $\\vec{\\tau} = \\vec{\\mu} \\times \\vec{B} = \\dfrac{d \\vec{L}}{dt}$, where $\\mu$ is the magnetic dipole moment, $\\vec{B}$ the magnetic field, and $\\vec{L}$ the angular momentum. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer notes that only the perpendicular component of angular momentum, $L \\sin \\theta$, changes during precession. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer obtains the correct magnitude of the Larmor frequency as $|\\omega_L| = \\frac{\\mu}{L} B$. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer gives the correct sign for the Larmor frequency, $\\omega_L = - \\frac{\\mu}{L} B = - \\gamma B$, where $\\gamma$ is the gyromagnetic ratio. Otherwise, award 0 pt."
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]
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],
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"answer": [
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"\\boxed{$\\omega_L = -\\gamma B$}"
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],
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"answer_type": [
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"Expression"
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],
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"unit": [
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null
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],
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"points": [
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0.4
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],
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"modality": "text+illustration figure",
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"field": "Electromagnetism",
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"source": "APhO_2025",
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"image_question": [
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"image_question/APhO_2025_2_a_1.png"
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]
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},
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{
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"id": "APhO_2025_2_A_3",
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"context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass. \n\n(A.1) It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$. \n\nThe ring is placed in a weak uniform magnetic field $\\vec{B} = B \\hat{z}$, making an angle $\\theta$ with $\\vec{\\omega}$, see Figure A.1. \n\n[figure1] \nFigure A.1. \n\n(A.2) Find the angular frequency $\\omega_{L}$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$. \n\nNow we turn off the external magnetic field and place an identical ring at a horizontal distance $d \\gg R$ from the original ring such that the magnetic moment of the new ring $\\vec{\\mu}_{2}$ makes an angle $\\theta$ with $\\vec{\\mu}_{1}$, see Figure A.2. \n\n[figure2]\nFigure A.2.",
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"question": "The magnetic interaction energy between the two rings can be written as $U = J_{0} \\vec{L}_{1} \\cdot \\vec{L}_{2}$, where $J_{0}$ is a constant and $\\vec{L}_{i}$ is the angular momentum of the $i$-th ring. Find $J_{0}$ in terms of $\\gamma$, $d$ and fundamental constants.",
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"marking": [
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[
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"Award 0.1 pt if the answer writes the interaction energy correctly as $U = - \\vec{\\mu}_2 \\cdot \\vec{B}_{1 on 2} = - \\frac{\\mu_0}{4 \\pi d^3} \\mu_1 \\mu_2 \\cos(\\pi - \\theta)$. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer gives the correct magnetic field magnitude, $|\\vec{B}_{1 on 2}| = \\frac{\\mu_0}{4 \\pi d^3} \\mu_1$. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer gives the correct magnetic field direction (with the correct minus sign in $\\vec{B}_{1 on 2}$). Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer writes the correct magnitude for $J_0$ as $J_0 = \\frac{\\mu_0 \\gamma^2}{4 \\pi d^3}$. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer gives the correct sign for $J_0$ in the expression $U = J_0 \\vec{L}_1 \\cdot \\vec{L}_2$. Otherwise, award 0 pt."
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]
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],
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"answer": [
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"\\boxed{$J_0 = \\frac{\\mu_0 \\gamma^2}{4 \\pi d^3}$}"
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],
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"answer_type": [
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"Expression"
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],
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"unit": [
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null
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],
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"points": [
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0.5
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],
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"modality": "text+illustration figure",
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"field": "Electromagnetism",
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"source": "APhO_2025",
|
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"image_question": [
|
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"image_question/APhO_2025_2_a_1.png",
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"image_question/APhO_2025_2_a_2.png"
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]
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},
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{
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"id": "APhO_2025_2_B_1",
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"context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass. \n\n(A.1) It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$. \n\nThe ring is placed in a weak uniform magnetic field $\\vec{B} = B \\hat{z}$, making an angle $\\theta$ with $\\vec{\\omega}$, see Figure A.1. \n\n[figure1] \nFigure A.1. \n\n(A.2) Find the angular frequency $\\omega_{L}$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$. \n\nNow we turn off the external magnetic field and place an identical ring at a horizontal distance $d \\gg R$ from the original ring such that the magnetic moment of the new ring $\\vec{\\mu}_{2}$ makes an angle $\\theta$ with $\\vec{\\mu}_{1}$, see Figure A.2. \n\n[figure2]\nFigure A.2. \n\n(A.3) The magnetic interaction energy between the two rings can be written as $U = J_{0} \\vec{L}_{1} \\cdot \\vec{L}_{2}$, where $J_{0}$ is a constant and $\\vec{L}_{i}$ is the angular momentum of the $i$-th ring. Find $J_{0}$ in terms of $\\gamma$, $d$ and fundamental constants. \n\n[Part B: Spin Waves] \n\nIn what follows we investigate the dynamics of spins. A spin is a particle with intrinsic angular momentum $\\vec{S}$, which has an associated magnetic moment $\\vec{\\mu}$ related to $\\vec{S}$ via the gyromagnetic ratio as in Part A.1, $\\vec{\\mu} = \\gamma \\vec{S}$. \n\nThe magnetic dipoles of two spins interact with each other. However, this interaction is negligible compared to another interaction arising from a quantum mechanical origin, which is not present in classical systems. Interestingly, the energy associated with this quantum interaction has the same form which we found in Part A.3, scaling with $\\vec{S}_{1} \\cdot \\vec{S}_{2}$, albeit with the opposite sign. \n\nNow we will look at a very long chain of spins. The positions of the spins are fixed along the $x$-axis, with a distance $a$ separating them, see Figure B.1. We will approximate the total energy of the system by considering the interactions between nearest neighbors only, so that the energy can be written as \n$E = -J \\sum_i \\vec{S}_i \\cdot \\vec{S}_{i+1}$ \nwhere $J > 0$ is the interaction strength, and $\\vec{S}_{i}$ is the spin angular momentum vector of the $i$-th dipole, with magnitude $S$. The spin vectors are free to rotate in three dimensions. Notice that the sign of the energy is different from the last part. This interaction is purely quantum mechanical. \n\n[figure3] \nFigure B.1.",
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"question": "The energy terms containing $\\vec{S}_{i}$ in the sum above can be viewed as the interaction energy between an effective magnetic field $\\vec{B}_{i, \\text{eff}}$ and the magnetic moment of $\\vec{S}_{i}$. Find $\\vec{B}_{i, \\text{eff}}$ and express your answer in terms of $J$, the gyromagnetic ratio $\\gamma$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$)",
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"marking": [
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[
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"Award 0.2 pt if the answer explicitly shows the understanding that spins $i-1$ and $i+1$ are the contributors to the interaction energy of spin $i$. Otherwise, award 0 pt.",
|
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"Award 0.1 pt if the answer gives the correct effective magnetic field as $\\vec{B}{i, \\text{eff}} = \\frac{J}{\\gamma}( \\vec{S}{i-1} + \\vec{S}_{i+1})$, where $J$ is the coupling constant and $\\gamma$ the gyromagnetic ratio. Otherwise, award 0 pt."
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]
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],
|
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"answer": [
|
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|
"\\boxed{$\\vec{B}_{i,\\text{eff}} = \\frac{J}{\\gamma} (\\vec{S}_{i-1} + \\vec{S}_{i+1})$}"
|
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],
|
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"answer_type": [
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"Expression"
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],
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"unit": [
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null
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],
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"points": [
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0.3
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|
|
],
|
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"modality": "text+variable figure",
|
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|
"field": "Modern Physics",
|
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|
"source": "APhO_2025",
|
|
|
"image_question": [
|
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"image_question/APhO_2025_2_a_1.png",
|
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"image_question/APhO_2025_2_a_2.png",
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"image_question/APhO_2025_2_b_1.png"
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|
]
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|
},
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{
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|
"id": "APhO_2025_2_B_2",
|
|
|
"context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass. \n\n(A.1) It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$. \n\nThe ring is placed in a weak uniform magnetic field $\\vec{B} = B \\hat{z}$, making an angle $\\theta$ with $\\vec{\\omega}$, see Figure A.1. \n\n[figure1] \nFigure A.1. \n\n(A.2) Find the angular frequency $\\omega_{L}$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$. \n\nNow we turn off the external magnetic field and place an identical ring at a horizontal distance $d \\gg R$ from the original ring such that the magnetic moment of the new ring $\\vec{\\mu}_{2}$ makes an angle $\\theta$ with $\\vec{\\mu}_{1}$, see Figure A.2. \n\n[figure2]\nFigure A.2. \n\n(A.3) The magnetic interaction energy between the two rings can be written as $U = J_{0} \\vec{L}_{1} \\cdot \\vec{L}_{2}$, where $J_{0}$ is a constant and $\\vec{L}_{i}$ is the angular momentum of the $i$-th ring. Find $J_{0}$ in terms of $\\gamma$, $d$ and fundamental constants. \n\n[Part B: Spin Waves] \n\nIn what follows we investigate the dynamics of spins. A spin is a particle with intrinsic angular momentum $\\vec{S}$, which has an associated magnetic moment $\\vec{\\mu}$ related to $\\vec{S}$ via the gyromagnetic ratio as in Part A.1, $\\vec{\\mu} = \\gamma \\vec{S}$. \n\nThe magnetic dipoles of two spins interact with each other. However, this interaction is negligible compared to another interaction arising from a quantum mechanical origin, which is not present in classical systems. Interestingly, the energy associated with this quantum interaction has the same form which we found in Part A.3, scaling with $\\vec{S}_{1} \\cdot \\vec{S}_{2}$, albeit with the opposite sign. \n\nNow we will look at a very long chain of spins. The positions of the spins are fixed along the $x$-axis, with a distance $a$ separating them, see Figure B.1. We will approximate the total energy of the system by considering the interactions between nearest neighbors only, so that the energy can be written as \n$E = -J \\sum_i \\vec{S}_i \\cdot \\vec{S}_{i+1}$ \nwhere $J > 0$ is the interaction strength, and $\\vec{S}_{i}$ is the spin angular momentum vector of the $i$-th dipole, with magnitude $S$. The spin vectors are free to rotate in three dimensions. Notice that the sign of the energy is different from the last part. This interaction is purely quantum mechanical. \n\n[figure3] \nFigure B.1. \n\n(B.1) The energy terms containing $\\vec{S}_{i}$ in the sum above can be viewed as the interaction energy between an effective magnetic field $\\vec{B}_{i, \\text{eff}}$ and the magnetic moment of $\\vec{S}_{i}$. Find $\\vec{B}_{i, \\text{eff}}$ and express your answer in terms of $J$, the gyromagnetic ratio $\\gamma$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$)",
|
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"question": "Using the concept of effective magnetic field, express the rate of change of the $i$-th spin vector, $d \\vec{S}_{i} / d t$, in terms of $J$, $\\vec{S}_{i}$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$).",
|
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"marking": [
|
|
|
[
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|
"Award 0.1 pt if the answer writes the rate of change of spin using the effective magnetic field, i.e. $\\frac{d \\vec{S}_i}{dt} = \\vec{\\mu}_i \\times \\vec{B}_{i,\\text{eff}}$. Otherwise, award 0 pt.",
|
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|
"Award 0.2 pt if the answer gives the correct explicit equation $\\frac{d \\vec{S}_i}{dt} = J \\vec{S}_i \\times ( \\vec{S}_{i-1} + \\vec{S}_{i+1})$, where $J$ is the coupling constant. Otherwise, award 0 pt."
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]
|
|
|
],
|
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|
"answer": [
|
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|
"\\boxed{$\\frac{d \\vec{S}_i}{dt} = J \\vec{S}_i \\times (\\vec{S}_{i-1} + \\vec{S}_{i+1})$}"
|
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|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
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0.3
|
|
|
],
|
|
|
"modality": "text+variable figure",
|
|
|
"field": "Modern Physics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_2_a_1.png",
|
|
|
"image_question/APhO_2025_2_a_2.png",
|
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"image_question/APhO_2025_2_b_1.png"
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|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_2_B_3",
|
|
|
"context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass. \n\n(A.1) It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$. \n\nThe ring is placed in a weak uniform magnetic field $\\vec{B} = B \\hat{z}$, making an angle $\\theta$ with $\\vec{\\omega}$, see Figure A.1. \n\n[figure1] \nFigure A.1. \n\n(A.2) Find the angular frequency $\\omega_{L}$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$. \n\nNow we turn off the external magnetic field and place an identical ring at a horizontal distance $d \\gg R$ from the original ring such that the magnetic moment of the new ring $\\vec{\\mu}_{2}$ makes an angle $\\theta$ with $\\vec{\\mu}_{1}$, see Figure A.2. \n\n[figure2]\nFigure A.2. \n\n(A.3) The magnetic interaction energy between the two rings can be written as $U = J_{0} \\vec{L}_{1} \\cdot \\vec{L}_{2}$, where $J_{0}$ is a constant and $\\vec{L}_{i}$ is the angular momentum of the $i$-th ring. Find $J_{0}$ in terms of $\\gamma$, $d$ and fundamental constants. \n\n[Part B: Spin Waves] \n\nIn what follows we investigate the dynamics of spins. A spin is a particle with intrinsic angular momentum $\\vec{S}$, which has an associated magnetic moment $\\vec{\\mu}$ related to $\\vec{S}$ via the gyromagnetic ratio as in Part A.1, $\\vec{\\mu} = \\gamma \\vec{S}$. \n\nThe magnetic dipoles of two spins interact with each other. However, this interaction is negligible compared to another interaction arising from a quantum mechanical origin, which is not present in classical systems. Interestingly, the energy associated with this quantum interaction has the same form which we found in Part A.3, scaling with $\\vec{S}_{1} \\cdot \\vec{S}_{2}$, albeit with the opposite sign. \n\nNow we will look at a very long chain of spins. The positions of the spins are fixed along the $x$-axis, with a distance $a$ separating them, see Figure B.1. We will approximate the total energy of the system by considering the interactions between nearest neighbors only, so that the energy can be written as \n$E = -J \\sum_i \\vec{S}_i \\cdot \\vec{S}_{i+1}$ \nwhere $J > 0$ is the interaction strength, and $\\vec{S}_{i}$ is the spin angular momentum vector of the $i$-th dipole, with magnitude $S$. The spin vectors are free to rotate in three dimensions. Notice that the sign of the energy is different from the last part. This interaction is purely quantum mechanical. \n\n[figure3] \nFigure B.1. \n\n(B.1) The energy terms containing $\\vec{S}_{i}$ in the sum above can be viewed as the interaction energy between an effective magnetic field $\\vec{B}_{i, \\text{eff}}$ and the magnetic moment of $\\vec{S}_{i}$. Find $\\vec{B}_{i, \\text{eff}}$ and express your answer in terms of $J$, the gyromagnetic ratio $\\gamma$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$) \n\n(B.2) Using the concept of effective magnetic field, express the rate of change of the $i$-th spin vector, $d \\vec{S}_{i} / d t$, in terms of $J$, $\\vec{S}_{i}$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$). \n\nFor the rest of Part B, assume that the system is highly magnetized along the $z$ direction, so we can use the approximations $S_{i, z} \\approx S$ and $d S_{i, z} / d t \\approx 0$ for each spin, see Figure B.2. In this regime, the set of equations describing the spins time evolution is satisfied by a traveling wave solution for $S_{i, x}$ and $S_{i, y}$ characterized by a wave vector $k$ and angular frequency $\\omega$. \n\n[figure4]\nFigure B.2.",
|
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|
"question": "Find the relationship between $\\omega$ and $k$ (known as the dispersion relation, $\\omega(k)$) for the spin waves in terms of $J$, $S$ and $a$. Hint: express the position of the $i$-th spin as $x = a \\cdot i$.",
|
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|
"marking": [
|
|
|
[
|
|
|
"Award 0.25 pt if the answer writes the traveling wave as a function of $kx \\pm \\omega t$ (either sign and either trigonometric functions $\\cos$ or $\\sin$ or complex exponentials are acceptable). Otherwise, award 0 pt.",
|
|
|
"Award 0.25 pt if the answer explicitly shows that the amplitudes of $S_x$ and $S_y$ are equal, i.e., $\\delta S_x = \\delta S_y$, where $\\delta$ is the wave amplitude. Otherwise, award 0 pt.",
|
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|
"Award 0.25 pt if the answer correctly identifies the phase relation between $S_x$ and $S_y$ as a difference of $\\pi/2$, e.g., $S_{i,x} = \\delta S \\cos(kx - \\omega t)$ and $S_{i,y} = \\delta S \\sin(kx - \\omega t)$. Otherwise, award 0 pt.",
|
|
|
"Award 0.5 pt if the answer writes the explicit equation of motion for either $S_x$ or $S_y$, such as $\\frac{d S_{i,x}}{dt} \\approx JS(2 S_{i,y} - S_{i-1,y} - S_{i+1,y})$ or $\\frac{d S_{i,y}}{dt} \\approx -JS(2 S_{i,x} - S_{i-1,x} - S_{i+1,x})$, where $J$ is the exchange coupling constant and $S$ is the spin magnitude. Otherwise, award 0 pt.",
|
|
|
"Award 0.25 pt if the answer explicitly uses the approximation $S_{i,z} \\approx S$, where $S$ is the spin magnitude. Otherwise, award 0 pt.",
|
|
|
"Award 0.5 pt if the final dispersion relation is correctly given as $\\omega(k) = 2JS [1 - \\cos(ka)]$ (with $\\pm$ accepted), where $a$ is the lattice spacing. Otherwise, award 0 pt."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{$\\omega(k) = 2JS[1 - \\cos(ka)]$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
2.0
|
|
|
],
|
|
|
"modality": "text+variable figure",
|
|
|
"field": "Modern Physics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_2_a_1.png",
|
|
|
"image_question/APhO_2025_2_a_2.png",
|
|
|
"image_question/APhO_2025_2_b_1.png",
|
|
|
"image_question/APhO_2025_2_b_2.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_2_B_4",
|
|
|
"context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass. \n\n(A.1) It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$. \n\nThe ring is placed in a weak uniform magnetic field $\\vec{B} = B \\hat{z}$, making an angle $\\theta$ with $\\vec{\\omega}$, see Figure A.1. \n\n[figure1] \nFigure A.1. \n\n(A.2) Find the angular frequency $\\omega_{L}$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$. \n\nNow we turn off the external magnetic field and place an identical ring at a horizontal distance $d \\gg R$ from the original ring such that the magnetic moment of the new ring $\\vec{\\mu}_{2}$ makes an angle $\\theta$ with $\\vec{\\mu}_{1}$, see Figure A.2. \n\n[figure2]\nFigure A.2. \n\n(A.3) The magnetic interaction energy between the two rings can be written as $U = J_{0} \\vec{L}_{1} \\cdot \\vec{L}_{2}$, where $J_{0}$ is a constant and $\\vec{L}_{i}$ is the angular momentum of the $i$-th ring. Find $J_{0}$ in terms of $\\gamma$, $d$ and fundamental constants. \n\n[Part B: Spin Waves] \n\nIn what follows we investigate the dynamics of spins. A spin is a particle with intrinsic angular momentum $\\vec{S}$, which has an associated magnetic moment $\\vec{\\mu}$ related to $\\vec{S}$ via the gyromagnetic ratio as in Part A.1, $\\vec{\\mu} = \\gamma \\vec{S}$. \n\nThe magnetic dipoles of two spins interact with each other. However, this interaction is negligible compared to another interaction arising from a quantum mechanical origin, which is not present in classical systems. Interestingly, the energy associated with this quantum interaction has the same form which we found in Part A.3, scaling with $\\vec{S}_{1} \\cdot \\vec{S}_{2}$, albeit with the opposite sign. \n\nNow we will look at a very long chain of spins. The positions of the spins are fixed along the $x$-axis, with a distance $a$ separating them, see Figure B.1. We will approximate the total energy of the system by considering the interactions between nearest neighbors only, so that the energy can be written as \n$E = -J \\sum_i \\vec{S}_i \\cdot \\vec{S}_{i+1}$ \nwhere $J > 0$ is the interaction strength, and $\\vec{S}_{i}$ is the spin angular momentum vector of the $i$-th dipole, with magnitude $S$. The spin vectors are free to rotate in three dimensions. Notice that the sign of the energy is different from the last part. This interaction is purely quantum mechanical. \n\n[figure3] \nFigure B.1. \n\n(B.1) The energy terms containing $\\vec{S}_{i}$ in the sum above can be viewed as the interaction energy between an effective magnetic field $\\vec{B}_{i, \\text{eff}}$ and the magnetic moment of $\\vec{S}_{i}$. Find $\\vec{B}_{i, \\text{eff}}$ and express your answer in terms of $J$, the gyromagnetic ratio $\\gamma$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$) \n\n(B.2) Using the concept of effective magnetic field, express the rate of change of the $i$-th spin vector, $d \\vec{S}_{i} / d t$, in terms of $J$, $\\vec{S}_{i}$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$). \n\nFor the rest of Part B, assume that the system is highly magnetized along the $z$ direction, so we can use the approximations $S_{i, z} \\approx S$ and $d S_{i, z} / d t \\approx 0$ for each spin, see Figure B.2. In this regime, the set of equations describing the spins time evolution is satisfied by a traveling wave solution for $S_{i, x}$ and $S_{i, y}$ characterized by a wave vector $k$ and angular frequency $\\omega$. \n\n[figure4]\nFigure B.2. \n\n(B.3) Find the relationship between $\\omega$ and $k$ (known as the dispersion relation, $\\omega(k)$) for the spin waves in terms of $J$, $S$ and $a$. Hint: express the position of the $i$-th spin as $x = a \\cdot i$. \n\nThe spin wave described above carries energy and momentum. At low energies, the relation between its energy and momentum resembles that of a massive classical particle with an effective mass $m_{\\text{eff}}$, a concept known as a quasi-particle.",
|
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"question": "For small $k$ ($k \\ll 1 / a$), find the effective mass $m_{\\text{eff}}$ of the spin wave. Express your answer in terms of $J, S, a$ and fundamental constants.",
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|
"marking": [
|
|
|
[
|
|
|
"Award 0.2 pt if the answer gives the correct Taylor expansion for small $k$, namely $\\omega(k) \\approx 2JS \\left[1 - 1 + \\frac{1}{2}(ka)^2 \\right] = JSa^2 k^2$, where $J$ is the exchange coupling constant, $S$ is the spin magnitude, and $a$ is the lattice spacing. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer uses the correct relation between momentum and wave vector, $p = \\hbar k$, where $p$ is the momentum and $\\hbar$ is the reduced Planck constant. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer uses the correct relation between energy and angular frequency, $E = \\hbar \\omega$, where $E$ is the energy and $\\omega$ is the angular frequency. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer correctly identifies the effective mass as $m_{\\text{eff}} = \\frac{\\hbar}{2JSa^2}$, where $m_{\\text{eff}}$ is the effective mass. Otherwise, award 0 pt."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{$m_{\\text{eff}} = \\frac{\\hbar}{2 J S a^2}$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.6
|
|
|
],
|
|
|
"modality": "text+variable figure",
|
|
|
"field": "Modern Physics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_2_a_1.png",
|
|
|
"image_question/APhO_2025_2_a_2.png",
|
|
|
"image_question/APhO_2025_2_b_1.png",
|
|
|
"image_question/APhO_2025_2_b_2.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_2_B_5",
|
|
|
"context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part A. Precession and interactions of magnetic dipoles] \n\nConsider a ring of radius $R$, total mass $M$, and charge $Q > 0$ distributed uniformly. The ring rotates with an angular speed $\\omega$ around a perpendicular axis that passes through its center of mass. \n\n(A.1) It is possible to write the ring's magnetic moment $\\vec{\\mu}$ in terms of its angular momentum $\\vec{L}$ as $\\vec{\\mu} = \\gamma \\vec{L}$. Find the constant $\\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$. \n\nThe ring is placed in a weak uniform magnetic field $\\vec{B} = B \\hat{z}$, making an angle $\\theta$ with $\\vec{\\omega}$, see Figure A.1. \n\n[figure1] \nFigure A.1. \n\n(A.2) Find the angular frequency $\\omega_{L}$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$. \n\nNow we turn off the external magnetic field and place an identical ring at a horizontal distance $d \\gg R$ from the original ring such that the magnetic moment of the new ring $\\vec{\\mu}_{2}$ makes an angle $\\theta$ with $\\vec{\\mu}_{1}$, see Figure A.2. \n\n[figure2]\nFigure A.2. \n\n(A.3) The magnetic interaction energy between the two rings can be written as $U = J_{0} \\vec{L}_{1} \\cdot \\vec{L}_{2}$, where $J_{0}$ is a constant and $\\vec{L}_{i}$ is the angular momentum of the $i$-th ring. Find $J_{0}$ in terms of $\\gamma$, $d$ and fundamental constants. \n\n[Part B: Spin Waves] \n\nIn what follows we investigate the dynamics of spins. A spin is a particle with intrinsic angular momentum $\\vec{S}$, which has an associated magnetic moment $\\vec{\\mu}$ related to $\\vec{S}$ via the gyromagnetic ratio as in Part A.1, $\\vec{\\mu} = \\gamma \\vec{S}$. \n\nThe magnetic dipoles of two spins interact with each other. However, this interaction is negligible compared to another interaction arising from a quantum mechanical origin, which is not present in classical systems. Interestingly, the energy associated with this quantum interaction has the same form which we found in Part A.3, scaling with $\\vec{S}_{1} \\cdot \\vec{S}_{2}$, albeit with the opposite sign. \n\nNow we will look at a very long chain of spins. The positions of the spins are fixed along the $x$-axis, with a distance $a$ separating them, see Figure B.1. We will approximate the total energy of the system by considering the interactions between nearest neighbors only, so that the energy can be written as \n$E = -J \\sum_i \\vec{S}_i \\cdot \\vec{S}_{i+1}$ \nwhere $J > 0$ is the interaction strength, and $\\vec{S}_{i}$ is the spin angular momentum vector of the $i$-th dipole, with magnitude $S$. The spin vectors are free to rotate in three dimensions. Notice that the sign of the energy is different from the last part. This interaction is purely quantum mechanical. \n\n[figure3] \nFigure B.1. \n\n(B.1) The energy terms containing $\\vec{S}_{i}$ in the sum above can be viewed as the interaction energy between an effective magnetic field $\\vec{B}_{i, \\text{eff}}$ and the magnetic moment of $\\vec{S}_{i}$. Find $\\vec{B}_{i, \\text{eff}}$ and express your answer in terms of $J$, the gyromagnetic ratio $\\gamma$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$) \n\n(B.2) Using the concept of effective magnetic field, express the rate of change of the $i$-th spin vector, $d \\vec{S}_{i} / d t$, in terms of $J$, $\\vec{S}_{i}$, and other spins $\\vec{S}_{j}$ (specify the indices $j$ in relation to $i$). \n\nFor the rest of Part B, assume that the system is highly magnetized along the $z$ direction, so we can use the approximations $S_{i, z} \\approx S$ and $d S_{i, z} / d t \\approx 0$ for each spin, see Figure B.2. In this regime, the set of equations describing the spins time evolution is satisfied by a traveling wave solution for $S_{i, x}$ and $S_{i, y}$ characterized by a wave vector $k$ and angular frequency $\\omega$. \n\n[figure4]\nFigure B.2. \n\n(B.3) Find the relationship between $\\omega$ and $k$ (known as the dispersion relation, $\\omega(k)$) for the spin waves in terms of $J$, $S$ and $a$. Hint: express the position of the $i$-th spin as $x = a \\cdot i$. \n\nThe spin wave described above carries energy and momentum. At low energies, the relation between its energy and momentum resembles that of a massive classical particle with an effective mass $m_{\\text{eff}}$, a concept known as a quasi-particle. \n\n(B.4) For small $k$ ($k \\ll 1 / a$), find the effective mass $m_{\\text{eff}}$ of the spin wave. Express your answer in terms of $J, S, a$ and fundamental constants. \n\nSpin waves can be experimentally probed using inelastic neutron scattering. Although neutrons have zero net charge, they have a finite spin, allowing them to interact with other spins. \n\n[figure5] \nFigure B.3.",
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"question": "Suppose that initially, all the spins in the chain are pointing along the $z$ direction. A neutron with low energy travels on the $x-y$ plane making an incident angle $\\theta_{\\text{in}}$ with the chain and scatters with an angle $\\theta_{\\text{out}}$ as shown in Figure B.3. Assuming the neutron excites a single low wave vector spin wave, find the effective mass $m_{\\text{eff}}$ of the spin wave, in terms of $\\theta_{\\text{in}}, \\theta_{\\text{out}}$ and the neutron mass $m_{n}$. Assume that the chain stays at rest.",
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"marking": [
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[
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"Award 0.4 pt if the answer applies conservation of momentum along the $y$-axis, i.e., $p_{\\text{in}} \\cos \\theta_{\\text{in}} = p_{\\text{out}} \\cos \\theta_{\\text{out}}$, where $p_{\\text{in}}$ and $p_{\\text{out}}$ are the incident and outgoing neutron momenta, and $\\theta_{\\text{in}}$, $\\theta_{\\text{out}}$ are their respective scattering angles. Otherwise, award 0 pt.",
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"Award 0.3 pt if the answer applies conservation of momentum along the $x$-axis, i.e., $p_s = p_{\\text{in}} \\sin \\theta_{\\text{in}} - p_{\\text{out}} \\sin \\theta_{\\text{out}}$, where $p_s$ is the spin-wave momentum. Otherwise, award 0 pt.",
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"Award 0.2 pt if the answer uses conservation of energy, i.e., $E_s = E_{\\text{in}} - E_{\\text{out}}$, where $E_s$ is the spin-wave energy, $E_{\\text{in}}$ and $E_{\\text{out}}$ are the neutron energies before and after scattering. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer gives the correct relation between $E_{\\text{out}}$ and $E_{\\text{in}}$, namely $E_{\\text{out}} = \\left( \\frac{\\cos \\theta_{\\text{in}}}{\\cos \\theta_{\\text{out}}} \\right)^2 E_{\\text{in}}$. Otherwise, award 0 pt.",
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"Award 0.3 pt if the answer derives the correct final expression for the effective mass, either as $m_{\\text{eff}} = \\frac{\\sin^2(\\theta_{\\text{in}} - \\theta_{\\text{out}})}{\\cos^2 \\theta_{\\text{out}} - \\cos^2 \\theta_{\\text{in}}} m_n$ or equivalently $m_{\\text{eff}} = \\frac{\\sin(\\theta_{\\text{in}} - \\theta_{\\text{out}})}{\\sin(\\theta_{\\text{in}} + \\theta_{\\text{out}})} m_n$, where $m_n$ is the neutron mass. Otherwise, award 0 pt (0.2 pt if partially correct)."
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]
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],
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"answer": [
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"\\boxed{$m_{\\text{eff}} = \\frac{\\sin(\\theta_{\\text{in}} - \\theta_{\\text{out}})}{\\sin(\\theta_{\\text{in}} + \\theta_{\\text{out}})} m_n$}"
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],
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"answer_type": [
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"Expression"
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],
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"unit": [
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null
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],
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"points": [
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1.3
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],
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"modality": "text+variable figure",
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"field": "Modern Physics",
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"source": "APhO_2025",
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"image_question": [
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"image_question/APhO_2025_2_a_1.png",
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"image_question/APhO_2025_2_a_2.png",
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"image_question/APhO_2025_2_b_1.png",
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"image_question/APhO_2025_2_b_2.png",
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"image_question/APhO_2025_2_b_3.png"
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]
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},
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{
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"id": "APhO_2025_2_C_1",
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"context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part C: Phase transitions in spin chains] \n\nNext we consider the same chain made of $N$ spins from Part B, except the spin vectors are now restricted to point either up or down along the $z$-axis, so that the spin component along $z$ can be written as $S_{i, z} = s_{i} S$, where $s_{i} = \\pm 1$, see Figure C.1. In addition to the nearest neighbor interactions, we could have an external magnetic field pointing along the $z$-axis so that the total energy of the system is given by \n$E = -\\tilde{J} \\sum_i s_i s_{i+1} - h \\sum_i s_i$ \nWe assume $\\tilde{J} \\geq 0$, and $h$ is a constant dependent on the magnetic field. The spin system is at equilibrium with a heat bath at temperature $T$. Ignore the edges of the chain. \n\n[figure1] \nFigure C.1.",
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"question": "Assume first that $\\tilde{J} = 0$, what is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_{\\uparrow}$ to being anti-aligned to the magnetic field $p_{\\downarrow}$? Express $p_{\\uparrow} / p_{\\downarrow}$ in terms of $h, T$ and fundamental constants.",
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"marking": [
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[
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"Award 0.2 pt if the answer uses the Boltzmann factor $p_i \\propto \\exp(-\\varepsilon_i / k_B T)$, where $\\varepsilon_i$ is the energy of state $i$, $k_B$ is the Boltzmann constant, and $T$ is the temperature. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer gives the correct Boltzmann factor for the spin-up state, $p_{\\uparrow} \\sim e^{h / k_B T}$, where $h$ is the Zeeman energy. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer gives the correct Boltzmann factor for the spin-down state, $p_{\\downarrow} \\sim e^{-h / k_B T}$. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer gives the correct ratio $\\frac{p_{\\uparrow}}{p_{\\downarrow}} = e^{2h / k_B T}$. Otherwise, award 0 pt."
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]
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],
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"answer": [
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"\\boxed{$p_{\\uparrow} / p_{\\downarrow} = e^{2h/k_B T}$}"
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],
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"answer_type": [
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"Expression"
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],
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"unit": [
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null
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],
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"points": [
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0.5
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],
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"modality": "text+illustration figure",
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"field": "Thermodynamics",
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"source": "APhO_2025",
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"image_question": [
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"image_question/APhO_2025_2_c_1.png"
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]
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},
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{
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"id": "APhO_2025_2_C_2",
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"context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part C: Phase transitions in spin chains] \n\nNext we consider the same chain made of $N$ spins from Part B, except the spin vectors are now restricted to point either up or down along the $z$-axis, so that the spin component along $z$ can be written as $S_{i, z} = s_{i} S$, where $s_{i} = \\pm 1$, see Figure C.1. In addition to the nearest neighbor interactions, we could have an external magnetic field pointing along the $z$-axis so that the total energy of the system is given by \n$E = -\\tilde{J} \\sum_i s_i s_{i+1} - h \\sum_i s_i$ \nWe assume $\\tilde{J} \\geq 0$, and $h$ is a constant dependent on the magnetic field. The spin system is at equilibrium with a heat bath at temperature $T$. Ignore the edges of the chain. \n\n[figure1] \nFigure C.1. \n\n(C.1) Assume first that $\\tilde{J} = 0$, what is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_{\\uparrow}$ to being anti-aligned to the magnetic field $p_{\\downarrow}$? Express $p_{\\uparrow} / p_{\\downarrow}$ in terms of $h, T$ and fundamental constants.",
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"question": "Find the average polarization of the system $\\bar{s} \\equiv \\frac{1}{N} \\sum_{i} s_{i}$ for $N \\gg 1$ in terms of $h, T$ and fundamental constants. If the magnetic field $h$ can range from $-h_{0}$ to $h_{0}$, make a sketch of $\\bar{s}$ as a function of $h$ for the cases $h_{o} \\gg k_{B} T$, $h_{o} \\approx k_{B} T$ and $h_{o} \\ll k_{B} T$.",
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"marking": [
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[
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"Award 0.2 pt if the answer deduces the expression for the average polarization as $\\bar{s} = p_{\\uparrow} - p_{\\downarrow}$, where $p_{\\uparrow}$ and $p_{\\downarrow}$ are the probabilities of spin-up and spin-down states, respectively. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer uses the normalization condition $p_{\\uparrow} + p_{\\downarrow} = 1$. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer gives the correct final result $\\bar{s} = \\tanh\\left( \\frac{h}{k_B T} \\right)$, where $h$ is the Zeeman energy, $k_B$ is the Boltzmann constant, and $T$ is the temperature. Otherwise, award 0 pt.",
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"Award 0.2 pt if the answer provides a correct sketch of $\\bar{s}$ versus $h/h_0$ in the case $h_0 \\gg k_B T$ (sharp step-like curve approaching $\\pm 1$). Otherwise, award 0 pt.",
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"Award 0.2 pt if the answer provides a correct sketch of $\\bar{s}$ versus $h/h_0$ in the case $h_0 \\approx k_B T$ (smooth nonlinear S-shaped curve). Otherwise, award 0 pt.",
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"Award 0.2 pt if the answer provides a correct sketch of $\\bar{s}$ versus $h/h_0$ in the case $h_0 \\ll k_B T$ (almost flat line near zero). Otherwise, award 0 pt."
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]
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],
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"answer": [
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"\\boxed{$\\bar{s} = \\tanh(\\frac{h}{k_B T})$}"
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],
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"answer_type": [
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"Expression"
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],
|
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"unit": [
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null
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],
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"points": [
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1.0
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|
],
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"modality": "text+illustration figure",
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"field": "Thermodynamics",
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"source": "APhO_2025",
|
|
|
"image_question": [
|
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"image_question/APhO_2025_2_c_1.png"
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]
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},
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{
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"id": "APhO_2025_2_C_3",
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"context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part C: Phase transitions in spin chains] \n\nNext we consider the same chain made of $N$ spins from Part B, except the spin vectors are now restricted to point either up or down along the $z$-axis, so that the spin component along $z$ can be written as $S_{i, z} = s_{i} S$, where $s_{i} = \\pm 1$, see Figure C.1. In addition to the nearest neighbor interactions, we could have an external magnetic field pointing along the $z$-axis so that the total energy of the system is given by \n$E = -\\tilde{J} \\sum_i s_i s_{i+1} - h \\sum_i s_i$ \nWe assume $\\tilde{J} \\geq 0$, and $h$ is a constant dependent on the magnetic field. The spin system is at equilibrium with a heat bath at temperature $T$. Ignore the edges of the chain. \n\n[figure1] \nFigure C.1. \n\n(C.1) Assume first that $\\tilde{J} = 0$, what is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_{\\uparrow}$ to being anti-aligned to the magnetic field $p_{\\downarrow}$? Express $p_{\\uparrow} / p_{\\downarrow}$ in terms of $h, T$ and fundamental constants. \n\n(C.2) Find the average polarization of the system $\\bar{s} \\equiv \\frac{1}{N} \\sum_{i} s_{i}$ for $N \\gg 1$ in terms of $h, T$ and fundamental constants. If the magnetic field $h$ can range from $-h_{0}$ to $h_{0}$, make a sketch of $\\bar{s}$ as a function of $h$ for the cases $h_{o} \\gg k_{B} T$, $h_{o} \\approx k_{B} T$ and $h_{o} \\ll k_{B} T$. \n\nIn the remaining questions, we turn off the magnetic field, so $h = 0$, and set $\\tilde{J} > 0$.",
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"question": "What is the energy $E_{g}$ of the ground state (the lowest energy state)? Express your answer in terms of $\\tilde{J}$ and $N$.",
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"marking": [
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[
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"Award 0.1 pt if the answer recognizes that the energy of the system is minimized when all spins align in the same direction. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer gives the correct ground state energy as $E_g = -\\tilde{J}(N-1) \\approx -\\tilde{J}N$, where $\\tilde{J}$ is the effective exchange coupling and $N$ is the number of spins (both $N-1$ and $N$ are acceptable). Otherwise, award 0 pt."
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]
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],
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"answer": [
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"\\boxed{$E_g \\simeq -\\tilde{J} N$}"
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],
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"answer_type": [
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"Expression"
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],
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"unit": [
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null
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],
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"points": [
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0.2
|
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|
],
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"modality": "text+illustration figure",
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|
"field": "Modern Physics",
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|
"source": "APhO_2025",
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|
"image_question": [
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"image_question/APhO_2025_2_c_1.png"
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]
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},
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{
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|
"id": "APhO_2025_2_C_4",
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|
|
"context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part C: Phase transitions in spin chains] \n\nNext we consider the same chain made of $N$ spins from Part B, except the spin vectors are now restricted to point either up or down along the $z$-axis, so that the spin component along $z$ can be written as $S_{i, z} = s_{i} S$, where $s_{i} = \\pm 1$, see Figure C.1. In addition to the nearest neighbor interactions, we could have an external magnetic field pointing along the $z$-axis so that the total energy of the system is given by \n$E = -\\tilde{J} \\sum_i s_i s_{i+1} - h \\sum_i s_i$ \nWe assume $\\tilde{J} \\geq 0$, and $h$ is a constant dependent on the magnetic field. The spin system is at equilibrium with a heat bath at temperature $T$. Ignore the edges of the chain. \n\n[figure1] \nFigure C.1. \n\n(C.1) Assume first that $\\tilde{J} = 0$, what is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_{\\uparrow}$ to being anti-aligned to the magnetic field $p_{\\downarrow}$? Express $p_{\\uparrow} / p_{\\downarrow}$ in terms of $h, T$ and fundamental constants. \n\n(C.2) Find the average polarization of the system $\\bar{s} \\equiv \\frac{1}{N} \\sum_{i} s_{i}$ for $N \\gg 1$ in terms of $h, T$ and fundamental constants. If the magnetic field $h$ can range from $-h_{0}$ to $h_{0}$, make a sketch of $\\bar{s}$ as a function of $h$ for the cases $h_{o} \\gg k_{B} T$, $h_{o} \\approx k_{B} T$ and $h_{o} \\ll k_{B} T$. \n\nIn the remaining questions, we turn off the magnetic field, so $h = 0$, and set $\\tilde{J} > 0$. \n\n(C.3) What is the energy $E_{g}$ of the ground state (the lowest energy state)? Express your answer in terms of $\\tilde{J}$ and $N$. \n\nInstead of considering the interactions between each spin and its neighbors, we assume that each spin sees an average polarization $\\bar{s}$ from its nearest-neighbors.",
|
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|
"question": "Approximate the energy of the system as a sum over all spins $E = -\\tilde{J}_{\\text{eff}} \\sum_i s_i$ and express $\\tilde{J}_{\\text{eff}}$ in terms of $\\tilde{J}$ and $\\bar{s}$.",
|
|
|
"marking": [
|
|
|
[
|
|
|
"Award 0.1 pt if the answer realizes that $s_{i+1}$ can be replaced with the average polarization $\\bar{s}$. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer gives the correct final result $E = -\\tilde{J}_{\\text{eff}} \\sum_i s_i = -\\tilde{J} \\sum_i s_i \\bar{s}$, where $\\tilde{J}_{\\text{eff}} = \\tilde{J} \\bar{s}$ is the effective coupling constant. Otherwise, award 0 pt."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{$\\tilde{J}_{\\text{eff}} = \\tilde{J} \\bar{s}$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.2
|
|
|
],
|
|
|
"modality": "text+illustration figure",
|
|
|
"field": "Modern Physics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_2_c_1.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_2_C_5",
|
|
|
"context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part C: Phase transitions in spin chains] \n\nNext we consider the same chain made of $N$ spins from Part B, except the spin vectors are now restricted to point either up or down along the $z$-axis, so that the spin component along $z$ can be written as $S_{i, z} = s_{i} S$, where $s_{i} = \\pm 1$, see Figure C.1. In addition to the nearest neighbor interactions, we could have an external magnetic field pointing along the $z$-axis so that the total energy of the system is given by \n$E = -\\tilde{J} \\sum_i s_i s_{i+1} - h \\sum_i s_i$ \nWe assume $\\tilde{J} \\geq 0$, and $h$ is a constant dependent on the magnetic field. The spin system is at equilibrium with a heat bath at temperature $T$. Ignore the edges of the chain. \n\n[figure1] \nFigure C.1. \n\n(C.1) Assume first that $\\tilde{J} = 0$, what is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_{\\uparrow}$ to being anti-aligned to the magnetic field $p_{\\downarrow}$? Express $p_{\\uparrow} / p_{\\downarrow}$ in terms of $h, T$ and fundamental constants. \n\n(C.2) Find the average polarization of the system $\\bar{s} \\equiv \\frac{1}{N} \\sum_{i} s_{i}$ for $N \\gg 1$ in terms of $h, T$ and fundamental constants. If the magnetic field $h$ can range from $-h_{0}$ to $h_{0}$, make a sketch of $\\bar{s}$ as a function of $h$ for the cases $h_{o} \\gg k_{B} T$, $h_{o} \\approx k_{B} T$ and $h_{o} \\ll k_{B} T$. \n\nIn the remaining questions, we turn off the magnetic field, so $h = 0$, and set $\\tilde{J} > 0$. \n\n(C.3) What is the energy $E_{g}$ of the ground state (the lowest energy state)? Express your answer in terms of $\\tilde{J}$ and $N$. \n\nInstead of considering the interactions between each spin and its neighbors, we assume that each spin sees an average polarization $\\bar{s}$ from its nearest-neighbors. \n\n(C.4) Approximate the energy of the system as a sum over all spins $E = -\\tilde{J}_{\\text{eff}} \\sum_i s_i$ and express $\\tilde{J}_{\\text{eff}}$ in terms of $\\tilde{J}$ and $\\bar{s}$.",
|
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"question": "(1) Using your result from C.2, find an equation that the average polarization $\\bar{s}$ must satisfy. \n(2) The number of solutions to this equation depends on $T$. Find the critical temperature $T_{c}$ at which the number of solutions changes. Express your answer in terms of $\\tilde{J}$ and fundamental constants.",
|
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"marking": [
|
|
|
[
|
|
|
"Award 0.1 pt if the answer correctly states the self-consistent equation for the polarization as $\\bar{s} = \\tanh\\left( \\frac{\\tilde{J}_{\\text{eff}}}{k_B T} \\right)$. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer replaces $\\tilde{J}_{\\text{eff}} = \\tilde{J} \\bar{s}$ into the result from C.2, leading to $\\bar{s} = \\tanh\\left( \\frac{\\tilde{J} \\bar{s}}{k_B T} \\right)$. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer realizes that for $\\tilde{J} \\ll k_B T$, there exists only one trivial solution $\\bar{s} = 0$. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer realizes that for $\\tilde{J} \\gg k_B T$, there exist two non-trivial solutions for $\\bar{s}$. Otherwise, award 0 pt.",
|
|
|
"Award 0.3 pt if the answer clearly states the condition when the number of solutions changes, namely when the slope condition at $\\bar{s}=0$ is satisfied: $\\frac{d}{d \\bar{s}} \\tanh\\left( \\frac{\\tilde{J} \\bar{s}}{k_B T_c} \\right) \\right|_{\\bar{s}=0} = \\frac{d}{d \\bar{s}} \\bar{s} \\right|_{\\bar{s}=0}$, which simplifies to $\\frac{\\tilde{J}}{k_B T_c} = 1$. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer gives the correct final critical temperature $T_c = \\frac{\\tilde{J}}{k_B}$. Otherwise, award 0 pt."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{$\\bar{s} = \\tanh\\left(\\frac{\\tilde{J} \\bar{s}}{k_B T}\\right)$}",
|
|
|
"\\boxed{$T_c = \\frac{\\tilde{J}}{k_B}$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Equation",
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null,
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.3,
|
|
|
0.9
|
|
|
],
|
|
|
"modality": "text+illustration figure",
|
|
|
"field": "Thermodynamics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_2_c_1.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_2_C_6",
|
|
|
"context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part C: Phase transitions in spin chains] \n\nNext we consider the same chain made of $N$ spins from Part B, except the spin vectors are now restricted to point either up or down along the $z$-axis, so that the spin component along $z$ can be written as $S_{i, z} = s_{i} S$, where $s_{i} = \\pm 1$, see Figure C.1. In addition to the nearest neighbor interactions, we could have an external magnetic field pointing along the $z$-axis so that the total energy of the system is given by \n$E = -\\tilde{J} \\sum_i s_i s_{i+1} - h \\sum_i s_i$ \nWe assume $\\tilde{J} \\geq 0$, and $h$ is a constant dependent on the magnetic field. The spin system is at equilibrium with a heat bath at temperature $T$. Ignore the edges of the chain. \n\n[figure1] \nFigure C.1. \n\n(C.1) Assume first that $\\tilde{J} = 0$, what is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_{\\uparrow}$ to being anti-aligned to the magnetic field $p_{\\downarrow}$? Express $p_{\\uparrow} / p_{\\downarrow}$ in terms of $h, T$ and fundamental constants. \n\n(C.2) Find the average polarization of the system $\\bar{s} \\equiv \\frac{1}{N} \\sum_{i} s_{i}$ for $N \\gg 1$ in terms of $h, T$ and fundamental constants. If the magnetic field $h$ can range from $-h_{0}$ to $h_{0}$, make a sketch of $\\bar{s}$ as a function of $h$ for the cases $h_{o} \\gg k_{B} T$, $h_{o} \\approx k_{B} T$ and $h_{o} \\ll k_{B} T$. \n\nIn the remaining questions, we turn off the magnetic field, so $h = 0$, and set $\\tilde{J} > 0$. \n\n(C.3) What is the energy $E_{g}$ of the ground state (the lowest energy state)? Express your answer in terms of $\\tilde{J}$ and $N$. \n\nInstead of considering the interactions between each spin and its neighbors, we assume that each spin sees an average polarization $\\bar{s}$ from its nearest-neighbors. \n\n(C.4) Approximate the energy of the system as a sum over all spins $E = -\\tilde{J}_{\\text{eff}} \\sum_i s_i$ and express $\\tilde{J}_{\\text{eff}}$ in terms of $\\tilde{J}$ and $\\bar{s}$. \n\n(C.5) Using your result from C.2, find an equation that the average polarization $\\bar{s}$ must satisfy. The number of solutions to this equation depends on $T$. Find the critical temperature $T_{c}$ at which the number of solutions changes. Express your answer in terms of $\\tilde{J}$ and fundamental constants.",
|
|
|
"question": "Find all possible values of $\\bar{s}$ when $T < T_{c}$ and $T_{c} - T \\ll T_{c}$. Express your answers in terms of $T$ and $T_{c}$. Sketch all possible values of $\\bar{s}$ for the temperature $T$ in the range $0 \\leq T \\leq 2 T_{c}$.",
|
|
|
"marking": [
|
|
|
[
|
|
|
"Award 0.1 pt if the answer uses the proper approximation $\\tanh(x) \\approx x - \\frac{1}{3}x^3$ for small $x$, leading to $\\bar{s} = \\frac{T_c}{T} \\bar{s} - \\frac{1}{3}\\left( \\frac{T_c}{T} \\bar{s} \\right)^3$. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer derives a correct non-trivial solution for $\\bar{s}$, i.e., $\\bar{s} = \\sqrt{3 \\left[ \\left(\\frac{T}{T_c}\\right)^2 - \\left(\\frac{T}{T_c}\\right)^3 \\right]} = \\sqrt{3 \\left(\\frac{T}{T_c}\\right)^2 \\cdot \\left(1 - \\frac{T}{T_c}\\right) } \\approx \\sqrt{3 \\frac{T_c - T}{T_c}}$, even if not fully simplified. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer derives a correct non-trivial solution for $\\bar{s}$, i.e., $\\bar{s} = - \\sqrt{3 \\left[ \\left(\\frac{T}{T_c}\\right)^2 - \\left(\\frac{T}{T_c}\\right)^3 \\right]} = - \\sqrt{3 \\left(\\frac{T}{T_c}\\right)^2 \\cdot \\left(1 - \\frac{T}{T_c}\\right) } \\approx - \\sqrt{3 \\frac{T_c - T}{T_c}}$, even if not fully simplified. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer sketches $\\bar{s} = 0$ as the only solution for $T > T_c$. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer shows that the two non-trivial solutions $\\bar{s}$ emerge vertically at $T = T_c$. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer sketches $\\bar{s} = 0$ as a valid solution also for $T < T_c$. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer shows that the two non-trivial solutions monotonically increase in magnitude and approach $\\pm 1$ as $T \\to 0$ (award 0.1 pt each). Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer states that either of the non-trivial solutions has zero slope as $T \\to 0$. Otherwise, award 0 pt."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{$\\bar{s} = 0$}",
|
|
|
"\\boxed{$\\bar{s} = \\sqrt{3 \\cdot \\frac{T_c - T}{T_c}}$}",
|
|
|
"\\boxed{$\\bar{s} = -\\sqrt{3 \\cdot \\frac{T_c - T}{T_c}}$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Numerical Value",
|
|
|
"Expression",
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null,
|
|
|
null,
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.2,
|
|
|
0.4,
|
|
|
0.4
|
|
|
],
|
|
|
"modality": "text+illustration figure",
|
|
|
"field": "Thermodynamics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_2_c_1.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_2_C_7",
|
|
|
"context": "[Waves and Phase Transitions in Spin Systems] \n\n[Introduction] \n\nIn classical physics, angular momentum arises from the motion of an object around an axis - whether it be a spinning top, a rotating planet, or an orbiting electron in the atom. However, in quantum physics, fundamental particles possess an intrinsic and quantized form of angular momentum called spin. This property plays a crucial role in various physical phenomena, ranging from materials properties, such as magnetism, to modern applications, such as quantum computing. \n\nIn this problem we will treat spin classically, which will lead to some qualitatively correct results. You will explore the physics of spin systems through spin-spin interactions, evolution under magnetic fields, and statistical physics to understand the emergence of spin waves and phase transitions in magnets. \n\nUseful information: \n$\\cosh(x) \\equiv \\frac{e^x + e^{-x}}{2}, \\sinh(x) \\equiv \\frac{e^x - e^{-x}}{2}, $\\tanh(x) \\equiv \\frac{\\sinh(x)}{\\cosh(x)} \\approx x - \\frac{1}{3} x^3$ for $|x| \\ll 1$. \n\nThe magnetic field due to a magnetic dipole of moment $\\vec{\\mu}$ at a position $\\vec{r}$ away from it is given by ($\\mu_{0}$ is is the vacuum permeability): \n$\\vec{B} = \\frac{\\mu_0}{4\\pi} \\left( \\frac{3(\\vec{\\mu} \\cdot \\vec{r})\\vec{r}}{r^5} - \\frac{\\vec{\\mu}}{r^3} \\right)$ \n\n[Part C: Phase transitions in spin chains] \n\nNext we consider the same chain made of $N$ spins from Part B, except the spin vectors are now restricted to point either up or down along the $z$-axis, so that the spin component along $z$ can be written as $S_{i, z} = s_{i} S$, where $s_{i} = \\pm 1$, see Figure C.1. In addition to the nearest neighbor interactions, we could have an external magnetic field pointing along the $z$-axis so that the total energy of the system is given by \n$E = -\\tilde{J} \\sum_i s_i s_{i+1} - h \\sum_i s_i$ \nWe assume $\\tilde{J} \\geq 0$, and $h$ is a constant dependent on the magnetic field. The spin system is at equilibrium with a heat bath at temperature $T$. Ignore the edges of the chain. \n\n[figure1] \nFigure C.1. \n\n(C.1) Assume first that $\\tilde{J} = 0$, what is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_{\\uparrow}$ to being anti-aligned to the magnetic field $p_{\\downarrow}$? Express $p_{\\uparrow} / p_{\\downarrow}$ in terms of $h, T$ and fundamental constants. \n\n(C.2) Find the average polarization of the system $\\bar{s} \\equiv \\frac{1}{N} \\sum_{i} s_{i}$ for $N \\gg 1$ in terms of $h, T$ and fundamental constants. If the magnetic field $h$ can range from $-h_{0}$ to $h_{0}$, make a sketch of $\\bar{s}$ as a function of $h$ for the cases $h_{o} \\gg k_{B} T$, $h_{o} \\approx k_{B} T$ and $h_{o} \\ll k_{B} T$. \n\nIn the remaining questions, we turn off the magnetic field, so $h = 0$, and set $\\tilde{J} > 0$. \n\n(C.3) What is the energy $E_{g}$ of the ground state (the lowest energy state)? Express your answer in terms of $\\tilde{J}$ and $N$. \n\nInstead of considering the interactions between each spin and its neighbors, we assume that each spin sees an average polarization $\\bar{s}$ from its nearest-neighbors. \n\n(C.4) Approximate the energy of the system as a sum over all spins $E = -\\tilde{J}_{\\text{eff}} \\sum_i s_i$ and express $\\tilde{J}_{\\text{eff}}$ in terms of $\\tilde{J}$ and $\\bar{s}$. \n\n(C.5) Using your result from C.2, find an equation that the average polarization $\\bar{s}$ must satisfy. The number of solutions to this equation depends on $T$. Find the critical temperature $T_{c}$ at which the number of solutions changes. Express your answer in terms of $\\tilde{J}$ and fundamental constants. \n\n(C.6) Find all possible values of $\\bar{s}$ when $T < T_{c}$ and $T_{c} - T \\ll T_{c}$. Express your answers in terms of $T$ and $T_{c}$. Sketch all possible values of $\\bar{s}$ for the temperature $T$ in the range $0 \\leq T \\leq 2 T_{c}$.",
|
|
|
"question": "Write the option letter (A or B) in your answer. \n(1) What magnetic phase of matter does $T > T_{c}$ correspond to? (A) Paramagnetic (B) Ferromagnetic. \n(2) What magnetic phase of matter does $T < T_{c}$ correspond to? (A) Paramagnetic (B) Ferromagnetic.",
|
|
|
"marking": [
|
|
|
[
|
|
|
"Award 0.1 pt if the answer correctly classifies the phase for $T > T_c$ as paramagnetic (option A). Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer correctly classifies the phase for $T < T_c$ as ferromagnetic (option B). Otherwise, award 0 pt."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{A}",
|
|
|
"\\boxed{B}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Multiple Choice",
|
|
|
"Multiple Choice"
|
|
|
],
|
|
|
"unit": [
|
|
|
null,
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.1,
|
|
|
0.1
|
|
|
],
|
|
|
"modality": "text+illustration figure",
|
|
|
"field": "Thermodynamics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_2_c_1.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_A_1",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part A: Surface Temperature of the Earth] \n\nIn this section, we study the effect of the atmosphere on the Earth surface's temperature. Assume that Earth and its atmosphere have an albedo $a = 0.3$ for solar radiation, which is the reflected fraction of the total incident radiation. You may use this value in all parts of this problem. In addition, assume the Earth radiates as a black body.",
|
|
|
"question": "Express the average net solar power received by the Earth and atmosphere system $P_{0}$ in terms of $F_{s}$, $a$ and $R_{E}$, the radius of the Earth.",
|
|
|
"marking": [
|
|
|
[
|
|
|
"Award 0.1 pt if the answer identifies the correct effective cross-sectional area as $A = \\pi R_E^2$, where $R_E$ is the Earth's radius. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer gives the correct final absorbed power expression $P_0 = (1 - a) \\pi R_E^2 F_s$, where $a$ is the albedo and $F_s$ is the solar flux. Otherwise, award 0 pt."
|
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|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{$P_0 = (1 - a) \\pi R_E^2 F_s$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.2
|
|
|
],
|
|
|
"modality": "text-only",
|
|
|
"field": "Thermodynamics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": []
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_A_2",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part A: Surface Temperature of the Earth] \n\nIn this section, we study the effect of the atmosphere on the Earth surface's temperature. Assume that Earth and its atmosphere have an albedo $a = 0.3$ for solar radiation, which is the reflected fraction of the total incident radiation. You may use this value in all parts of this problem. In addition, assume the Earth radiates as a black body. \n\n(A.1) Express the average net solar power received by the Earth and atmosphere system $P_{0}$ in terms of $F_{s}$, $a$ and $R_{E}$, the radius of the Earth.",
|
|
|
"question": "Estimate the temperature of the Earth's surface $T_{g 0}$ assuming that it is at a steady state. Ignore the atmosphere. Express your answer in $K$.",
|
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|
"marking": [
|
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|
[
|
|
|
"Award 0.1 pt if the answer sets up the energy balance condition $P_{bd} = P_0$, where $P_{bd}$ is the blackbody radiation power and $P_0$ is the absorbed solar power. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer writes the correct explicit blackbody radiation formula $P_{bd} = \\sigma A T^4$ with $A = 4 \\pi R_E^2$, where $\\sigma$ is the Stefan–Boltzmann constant, $T$ is the temperature, and $R_E$ is the Earth's radius. Otherwise, award 0 pt.",
|
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|
"Award 0.1 pt if the answer obtains the correct numerical value for the temperature, $T_{g0} \\approx 255 \\text{K} \\approx -18^{\\circ}\\text{C}$. Otherwise, award 0 pt."
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]
|
|
|
],
|
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|
"answer": [
|
|
|
"\\boxed{255}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Numerical Value"
|
|
|
],
|
|
|
"unit": [
|
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|
"K"
|
|
|
],
|
|
|
"points": [
|
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|
0.3
|
|
|
],
|
|
|
"modality": "text-only",
|
|
|
"field": "Thermodynamics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": []
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_A_3",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part A: Surface Temperature of the Earth] \n\nIn this section, we study the effect of the atmosphere on the Earth surface's temperature. Assume that Earth and its atmosphere have an albedo $a = 0.3$ for solar radiation, which is the reflected fraction of the total incident radiation. You may use this value in all parts of this problem. In addition, assume the Earth radiates as a black body. \n\n(A.1) Express the average net solar power received by the Earth and atmosphere system $P_{0}$ in terms of $F_{s}$, $a$ and $R_{E}$, the radius of the Earth. \n\n(A.2) Estimate the temperature of the Earth's surface $T_{g 0}$ assuming that it is at a steady state. Ignore the atmosphere. \n\nYour answer for (A.2) should be lower than what you would expect. We now consider adding a thin atmospheric layer at temperature $T_{a}$, see Figure A.1. The atmospheric layer transmits a net fraction $t_{\\mathrm{sw}}$ of the incident solar radiation and a net fraction $t_{\\text{lw}}$ of the Earth's thermal radiation. Otherwise, you may treat the atmosphere as a black body. \n\n[figure1] \nFigure A.1",
|
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"question": "Assuming the system is in a steady state, calculate $T_{g}$, the temperature of the ground. Use $t_{\\mathrm{sw}} = 0.9$ and $t_{\\mathrm{lw}} = 0.2$. Express your answer in $K$.",
|
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|
"marking": [
|
|
|
[
|
|
|
"Award 0.1 pt if the answer includes a correct statement of radiation balance in the region outside the atmosphere, e.g., $t_{lw} P_E + P_{atmo} = P_0$, where $t_{lw}$ is the longwave transmission coefficient, $P_E$ is the Earth's emitted power, $P_{atmo}$ is the atmospheric radiation, and $P_0$ is the absorbed solar power. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer includes a correct statement of radiation balance in the region between the Earth's surface and the atmosphere, e.g., $P_E = P_{atmo} + t_{sw} P_0$, where $t_{sw}$ is the shortwave transmission coefficient. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer uses $t_{sw}$ correctly in the equation $P_E = P_{atmo} + t_{sw} P_0$. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer uses $t_{lw}$ correctly in the equation $t_{lw} P_E + P_{atmo} = P_0$. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer gives the correct numerical result for the ground temperature, $T_g = \\left( \\frac{1+t_{sw}}{1+t_{lw}} \\right)^{1/4} T_{g0} \\approx 286 \\text{K} \\approx 13^{\\circ}\\text{C}$, where $T_{g0}$ is the temperature of the Earth's surface. Partial points: award 0.1 pt if only the analytic form is given in the answer. Otherwise, award 0 pt."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{286}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Numerical Value"
|
|
|
],
|
|
|
"unit": [
|
|
|
"K"
|
|
|
],
|
|
|
"points": [
|
|
|
0.7
|
|
|
],
|
|
|
"modality": "text+illustration figure",
|
|
|
"field": "Thermodynamics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_3_a_1.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_B_1",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part B: The absorption spectrum of atmospheric gases] \n\nThe infrared radiation emitted by Earth has low energy, incapable of exciting electrons within the molecules, but it has the ability to excite the vibrational and rotational modes of the molecules.",
|
|
|
"question": "Consider a simple diatomic molecule modeled as two point masses $m_{A}$ and $m_{B}$ connected by a spring with spring constant $k$. What is the angular frequency of vibrations $\\omega_{d}$?",
|
|
|
"marking": [
|
|
|
[
|
|
|
"Award 0.1 pt if the answer writes the correct equation of motion for particle A: $\\frac{d^2 x_A}{dt^2} = +\\frac{k}{m_A}(\\ell - \\ell_0)$, where $x_A$ is the position of particle A, $m_A$ is its mass, $k$ is the spring constant, $\\ell$ is the instantaneous spring length, and $\\ell_0$ is the natural spring length. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer writes the correct equation of motion for particle B: $\\frac{d^2 x_B}{dt^2} = -\\frac{k}{m_B}(\\ell - \\ell_0)$, where $x_B$ is the position of particle B and $m_B$ is its mass. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer correctly derives the equation of motion for the relative coordinate $\\ell = x_B - x_A$, namely $\\frac{d^2 \\ell}{dt^2} = -k\\left( \\frac{1}{m_A} + \\frac{1}{m_B} \\right)(\\ell - \\ell_0)$. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer provides the correct final angular frequency of oscillation: $\\omega_d = \\sqrt{\\frac{k}{\\mu}} = \\sqrt{ k \\frac{m_A + m_B}{m_A m_B}}$, where $\\mu = \\frac{m_A m_B}{m_A + m_B}$ is the reduced mass. Otherwise, award 0 pt."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{$\\omega_d = \\sqrt{k \\frac{m_A + m_B}{m_A m_B}}$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.5
|
|
|
],
|
|
|
"modality": "text-only",
|
|
|
"field": "Mechanics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": []
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_B_2",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part B: The absorption spectrum of atmospheric gases] \n\nThe infrared radiation emitted by Earth has low energy, incapable of exciting electrons within the molecules, but it has the ability to excite the vibrational and rotational modes of the molecules. \n\n(B.1) Consider a simple diatomic molecule modeled as two point masses $m_{A}$ and $m_{B}$ connected by a spring with spring constant $k$. What is the angular frequency of vibrations $\\omega_{d}$?",
|
|
|
"question": "Quantum mechanics dictates that vibrational excitations due to absorbing a photon can only raise the quantum energy level by one. What is the energy of the photon $E_{p}$ that can excite the vibration in B.1? Neglect recoil effects.",
|
|
|
"marking": [
|
|
|
[
|
|
|
"Award 0.2 pt if the answer gives the correct photon energy as $E = \\hbar \\omega_d$ (where $\\hbar$ is the reduced Planck constant and $\\omega_d$ is the angular frequency). Partial points: only award 0.1 pt if $h$ is used instead of $\\hbar$; no other numerical factors receive credit."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{$E_p = \\hbar \\omega_d$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.2
|
|
|
],
|
|
|
"modality": "text-only",
|
|
|
"field": "Modern Physics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": []
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_B_3",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part B: The absorption spectrum of atmospheric gases] \n\nThe infrared radiation emitted by Earth has low energy, incapable of exciting electrons within the molecules, but it has the ability to excite the vibrational and rotational modes of the molecules. \n\n(B.1) Consider a simple diatomic molecule modeled as two point masses $m_{A}$ and $m_{B}$ connected by a spring with spring constant $k$. What is the angular frequency of vibrations $\\omega_{d}$? \n\n(B.2) Quantum mechanics dictates that vibrational excitations due to absorbing a photon can only raise the quantum energy level by one. What is the energy of the photon $E_{p}$ that can excite the vibration in B.1? Neglect recoil effects. \n\nQuantum mechanics forbids the vibrational modes of symmetric diatomic molecules, such as nitrogen and oxygen (the most abundant gasses in the Earth's atmosphere) to be excited by light. This explains why $N_{2}$ and $O_{2}$ do not contribute to the green house effect. In general, the absorption of light by molecules is governed by the allowed energy transitions in them. However, the energy of the light absorbed does not have to exactly match the energy gap in the molecule. Suppose that a molecule at rest has a spectral line (an allowed transition) at frequency $f_{0}$.",
|
|
|
"question": "What is the shift in the spectral line $f - f_{0}$ if the molecule is moving with velocity $v$ towards the emitter such that $|v| \\ll c$, where $c$ is the speed of light.",
|
|
|
"marking": [
|
|
|
[
|
|
|
"Award 0.1 pt if the answer writes down an expression for the Doppler effect, even if it is incorrect. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer gives the correct frequency shift as $f - f_0 = \\frac{v}{c} f_0$, where $f$ is the observed frequency, $f_0$ is the source frequency, $v$ is the velocity of the source, and $c$ is the speed of light. Otherwise, award 0 pt."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{$f - f_0 = \\frac{v}{c} f_0$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.2
|
|
|
],
|
|
|
"modality": "text-only",
|
|
|
"field": "Optics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": []
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_B_4",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part B: The absorption spectrum of atmospheric gases] \n\nThe infrared radiation emitted by Earth has low energy, incapable of exciting electrons within the molecules, but it has the ability to excite the vibrational and rotational modes of the molecules. \n\n(B.1) Consider a simple diatomic molecule modeled as two point masses $m_{A}$ and $m_{B}$ connected by a spring with spring constant $k$. What is the angular frequency of vibrations $\\omega_{d}$? \n\n(B.2) Quantum mechanics dictates that vibrational excitations due to absorbing a photon can only raise the quantum energy level by one. What is the energy of the photon $E_{p}$ that can excite the vibration in B.1? Neglect recoil effects. \n\nQuantum mechanics forbids the vibrational modes of symmetric diatomic molecules, such as nitrogen and oxygen (the most abundant gasses in the Earth's atmosphere) to be excited by light. This explains why $N_{2}$ and $O_{2}$ do not contribute to the green house effect. In general, the absorption of light by molecules is governed by the allowed energy transitions in them. However, the energy of the light absorbed does not have to exactly match the energy gap in the molecule. Suppose that a molecule at rest has a spectral line (an allowed transition) at frequency $f_{0}$. \n\n(B.3) What is the shift in the spectral line $f - f_{0}$ if the molecule is moving with velocity $v$ towards the emitter such that $|v| \\ll c$, where $c$ is the speed of light. \n\nFor a gas at temperature $T$, the velocity of its molecules is distributed according to Maxwell's distribution. For a molecule of mass $m$, the probability to find a molecule's velocity along one dimension to be between $v$ and $v + dv$ is $p_{1}(v) d v$, where $p_{1}(v)$ is a probability distribution function given by \n$p_{1}(v) = C \\exp\\left( -\\frac{mv^{2}}{2k_{B}T} \\right)$ \n$C$ is a normalization constant ensuring the probabilities add up to one, and $k_{B}$ is the Boltzmann constant.",
|
|
|
"question": "Find the normalization constant $C$, assuming that the velocity $v$ could range from $-\\infty$ to $\\infty$.",
|
|
|
"marking": [
|
|
|
[
|
|
|
"Award 0.1 pt if the answer writes down the normalization condition $\\int_{-\\infty}^{\\infty} p(v) dv = 1$, even if it is incorrectly applied from 0 to $\\infty$. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer gives the correct result for the normalization constant as $C = \\sqrt{ \\frac{m}{2 \\pi k_B T} }$, where $m$ is the particle mass, $k_B$ is the Boltzmann constant, and $T$ is the temperature. Otherwise, award 0 pt."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{$C = \\sqrt{\\frac{m}{2 \\pi k_B T}}$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.2
|
|
|
],
|
|
|
"modality": "text-only",
|
|
|
"field": "Thermodynamics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": []
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_B_5",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part B: The absorption spectrum of atmospheric gases] \n\nThe infrared radiation emitted by Earth has low energy, incapable of exciting electrons within the molecules, but it has the ability to excite the vibrational and rotational modes of the molecules. \n\n(B.1) Consider a simple diatomic molecule modeled as two point masses $m_{A}$ and $m_{B}$ connected by a spring with spring constant $k$. What is the angular frequency of vibrations $\\omega_{d}$? \n\n(B.2) Quantum mechanics dictates that vibrational excitations due to absorbing a photon can only raise the quantum energy level by one. What is the energy of the photon $E_{p}$ that can excite the vibration in B.1? Neglect recoil effects. \n\nQuantum mechanics forbids the vibrational modes of symmetric diatomic molecules, such as nitrogen and oxygen (the most abundant gasses in the Earth's atmosphere) to be excited by light. This explains why $N_{2}$ and $O_{2}$ do not contribute to the green house effect. In general, the absorption of light by molecules is governed by the allowed energy transitions in them. However, the energy of the light absorbed does not have to exactly match the energy gap in the molecule. Suppose that a molecule at rest has a spectral line (an allowed transition) at frequency $f_{0}$. \n\n(B.3) What is the shift in the spectral line $f - f_{0}$ if the molecule is moving with velocity $v$ towards the emitter such that $|v| \\ll c$, where $c$ is the speed of light. \n\nFor a gas at temperature $T$, the velocity of its molecules is distributed according to Maxwell's distribution. For a molecule of mass $m$, the probability to find a molecule's velocity along one dimension to be between $v$ and $v + dv$ is $p_{1}(v) d v$, where $p_{1}(v)$ is a probability distribution function given by \n$p_{1}(v) = C \\exp\\left( -\\frac{mv^{2}}{2k_{B}T} \\right)$ \n$C$ is a normalization constant ensuring the probabilities add up to one, and $k_{B}$ is the Boltzmann constant. \n\n(B.4) Find the normalization constant $C$, assuming that the velocity $v$ could range from $-\\infty$ to $\\infty$.",
|
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"question": "Find the probability distribution function $p_{2}(f)$ to find a molecule with a spectral line $f_{0}$ shifted to $f$ due to thermal motion, up to a normalization factor, in terms of $f, f_{0}, T, m$ and fundamental constants. Use $C$ to represent the normalization factor.",
|
|
|
"marking": [
|
|
|
[
|
|
|
"Award 0.1 pt if the answer correctly replaces the velocity $v$ in the distribution using the Doppler effect relation $v = \\frac{f - f_0}{f_0} c$, where $f$ is the observed frequency, $f_0$ is the source frequency, and $c$ is the speed of light. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer writes the correct exponential dependence for the probability distribution as $p(f) \\propto \\exp \\left[ - \\frac{m c^2}{2 k_B T} \\left( \\frac{f - f_0}{f_0} \\right)^2 \\right]$, where $m$ is the particle mass, $k_B$ is the Boltzmann constant, and $T$ is the temperature. Otherwise, award 0 pt."
|
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]
|
|
|
],
|
|
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"answer": [
|
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|
"\\boxed{$p_2(f) = C \\exp\\left[-\\frac{mc^2}{2k_B T} \\left(\\frac{f - f_0}{f_0}\\right)^2\\right]$}"
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],
|
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"answer_type": [
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|
"Expression"
|
|
|
],
|
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|
"unit": [
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null
|
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|
],
|
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"points": [
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0.3
|
|
|
],
|
|
|
"modality": "text-only",
|
|
|
"field": "Thermodynamics",
|
|
|
"source": "APhO_2025",
|
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|
"image_question": []
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|
},
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{
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|
|
"id": "APhO_2025_3_B_6",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part B: The absorption spectrum of atmospheric gases] \n\nThe infrared radiation emitted by Earth has low energy, incapable of exciting electrons within the molecules, but it has the ability to excite the vibrational and rotational modes of the molecules. \n\n(B.1) Consider a simple diatomic molecule modeled as two point masses $m_{A}$ and $m_{B}$ connected by a spring with spring constant $k$. What is the angular frequency of vibrations $\\omega_{d}$? \n\n(B.2) Quantum mechanics dictates that vibrational excitations due to absorbing a photon can only raise the quantum energy level by one. What is the energy of the photon $E_{p}$ that can excite the vibration in B.1? Neglect recoil effects. \n\nQuantum mechanics forbids the vibrational modes of symmetric diatomic molecules, such as nitrogen and oxygen (the most abundant gasses in the Earth's atmosphere) to be excited by light. This explains why $N_{2}$ and $O_{2}$ do not contribute to the green house effect. In general, the absorption of light by molecules is governed by the allowed energy transitions in them. However, the energy of the light absorbed does not have to exactly match the energy gap in the molecule. Suppose that a molecule at rest has a spectral line (an allowed transition) at frequency $f_{0}$. \n\n(B.3) What is the shift in the spectral line $f - f_{0}$ if the molecule is moving with velocity $v$ towards the emitter such that $|v| \\ll c$, where $c$ is the speed of light. \n\nFor a gas at temperature $T$, the velocity of its molecules is distributed according to Maxwell's distribution. For a molecule of mass $m$, the probability to find a molecule's velocity along one dimension to be between $v$ and $v + dv$ is $p_{1}(v) d v$, where $p_{1}(v)$ is a probability distribution function given by \n$p_{1}(v) = C \\exp\\left( -\\frac{mv^{2}}{2k_{B}T} \\right)$ \n$C$ is a normalization constant ensuring the probabilities add up to one, and $k_{B}$ is the Boltzmann constant. \n\n(B.4) Find the normalization constant $C$, assuming that the velocity $v$ could range from $-\\infty$ to $\\infty$. \n\n(B.5) Find the probability distribution function $p_{2}(f)$ to find a molecule with a spectral line $f_{0}$ shifted to $f$ due to thermal motion, up to a normalization factor, in terms of $f, f_{0}, T, m$ and fundamental constants. Use $C$ to represent the normalization factor.",
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"question": "Sketch $p_{2}(f)$ as a function of $f - f_{0}$, and determine the shift $f^{\\star} - f_{0}$ at which $p_{2}(f^{\\star})$ is a fraction $1 / e$ of its peak value, where $e$ is the natural number.",
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"marking": [
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|
[
|
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|
"Award 0.1 pt if the answer states that the probability distribution $p(f)$ has a single peak at zero frequency shift ($f - f_0 = 0$). Otherwise, award 0 pt.",
|
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"Award 0.1 pt if the answer correctly identifies that the distribution is symmetric about $f - f_0 = 0$. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer states that the probability distribution decays to zero as $f - f_0 \\to \\pm \\infty$. Otherwise, award 0 pt.",
|
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"Award 0.1 pt if the answer gives the correct condition for the $1/e$ point of the distribution as $f^* - f_0 = f_0 \\sqrt{ \\frac{2 k_B T}{m c^2} }$, where $m$ is the particle mass, $k_B$ is the Boltzmann constant, $T$ is the temperature, and $c$ is the speed of light. Otherwise, award 0 pt."
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]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{$f^{\\star} - f_0 = f_0 \\sqrt{\\frac{2k_B T}{mc^2}}$}"
|
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|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.4
|
|
|
],
|
|
|
"modality": "text-only",
|
|
|
"field": "Thermodynamics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": []
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_C_1",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part C: Stability of air in the atmosphere] \n\nConsider a small cylindrical mass of air at height $z$ above the ground. The pressure and mass density of air at that height are $p(z)$ and $\\rho(z)$, respectively, see Figure C.1. Assume a uniform downward gravitational field $g$ and that the pressure on the Earth's surface is $p_{o}$. \n\n[figure1] \nFigure C.1",
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"question": "Assuming that the small air mass is at hydrostatic equilibrium, derive an expression of the rate of change of pressure with respect to height, $d p / d z$ in terms of $g$ and $\\rho(z)$.",
|
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"marking": [
|
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|
[
|
|
|
"Award 0.1 pt if the answer states that the sum of forces equals zero in hydrostatic equilibrium. Otherwise, award 0 pt.",
|
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|
"Award 0.1 pt if the answer correctly identifies the pressure force contributions above and below the thin layer, i.e. $p(z)S = p(z + \\mathrm{d}z)S + \\rho(z) g S \\mathrm{d}z$. Otherwise, award 0 pt.",
|
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|
"Award 0.1 pt if the answer gives the correct final hydrostatic equilibrium equation $\\frac{dp}{dz} = - \\rho(z) g$, where $\\rho(z)$ is the density and $g$ is the gravitational acceleration. Otherwise, award 0 pt."
|
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]
|
|
|
],
|
|
|
"answer": [
|
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|
"\\boxed{$\\frac{dp}{dz} = -\\rho(z) g$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.3
|
|
|
],
|
|
|
"modality": "text+illustration figure",
|
|
|
"field": "Mechanics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_3_c_1.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_C_2",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part C: Stability of air in the atmosphere] \n\nConsider a small cylindrical mass of air at height $z$ above the ground. The pressure and mass density of air at that height are $p(z)$ and $\\rho(z)$, respectively, see Figure C.1. Assume a uniform downward gravitational field $g$ and that the pressure on the Earth's surface is $p_{o}$. \n\n[figure1] \nFigure C.1 \n\n(C.1) Assuming that the small air mass is at hydrostatic equilibrium, derive an expression of the rate of change of pressure with respect to height, $d p / d z$ in terms of $g$ and $\\rho(z)$.",
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|
"question": "Express $d p / d z$ in terms of $\\mu_{\\text{air}}, g, p(z)$ and $T(z)$, the temperature at height $z$ and fundamental constants.",
|
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|
"marking": [
|
|
|
[
|
|
|
"Award 0.1 pt if the answer correctly uses the ideal gas law $pV = nRT$ and rewrites it as $\\rho(z) = \\frac{p(z) \\mu_{air}}{R T(z)}$, where $\\mu_{air}$ is the molar mass of air and $R$ is the gas constant. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer gives the correct final hydrostatic equilibrium expression $\\frac{dp}{dz} = - \\frac{\\mu_{air} p(z)}{R T(z)} g$. Otherwise, award 0 pt."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{$\\frac{dp}{dz} = -\\frac{\\mu_{\\text{air}} p(z)}{R T(z)} g$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.2
|
|
|
],
|
|
|
"modality": "text+illustration figure",
|
|
|
"field": "Thermodynamics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_3_c_1.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_C_3",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part C: Stability of air in the atmosphere] \n\nConsider a small cylindrical mass of air at height $z$ above the ground. The pressure and mass density of air at that height are $p(z)$ and $\\rho(z)$, respectively, see Figure C.1. Assume a uniform downward gravitational field $g$ and that the pressure on the Earth's surface is $p_{o}$. \n\n[figure1] \nFigure C.1 \n\n(C.1) Assuming that the small air mass is at hydrostatic equilibrium, derive an expression of the rate of change of pressure with respect to height, $d p / d z$ in terms of $g$ and $\\rho(z)$. \n\n(C.2) Express $d p / d z$ in terms of $\\mu_{\\text{air}}, g, p(z)$ and $T(z)$, the temperature at height $z$ and fundamental constants.",
|
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"question": "Assuming an isothermal atmosphere, $T(z) = T$, find an expression for $p(z)$ in terms of $z, \\mu_{\\text{air}}, g, p_{o}, T$ and fundamental constants.",
|
|
|
"marking": [
|
|
|
[
|
|
|
"Award 0.1 pt if the answer recognizes and correctly rewrites the equation as a separable differential equation $\\frac{dp}{p} = - \\frac{\\mu_{air}}{R T} g dz$. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer gives the correct final solution $p(z) = p_0 \\exp \\left( - \\frac{\\mu_{air}}{R T} g z \\right)$, where $p_0$ is the pressure at $z=0$. Otherwise, award 0 pt."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{$p(z) = p_0 \\exp\\left(-\\frac{\\mu_{\\text{air}}}{RT} gz\\right)$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.2
|
|
|
],
|
|
|
"modality": "text+illustration figure",
|
|
|
"field": "Thermodynamics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_3_c_1.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_C_4",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part C: Stability of air in the atmosphere] \n\nConsider a small cylindrical mass of air at height $z$ above the ground. The pressure and mass density of air at that height are $p(z)$ and $\\rho(z)$, respectively, see Figure C.1. Assume a uniform downward gravitational field $g$ and that the pressure on the Earth's surface is $p_{o}$. \n\n[figure1] \nFigure C.1 \n\n(C.1) Assuming that the small air mass is at hydrostatic equilibrium, derive an expression of the rate of change of pressure with respect to height, $d p / d z$ in terms of $g$ and $\\rho(z)$. \n\n(C.2) Express $d p / d z$ in terms of $\\mu_{\\text{air}}, g, p(z)$ and $T(z)$, the temperature at height $z$ and fundamental constants. \n\n(C.3) Assuming an isothermal atmosphere, $T(z) = T$, find an expression for $p(z)$ in terms of $z, \\mu_{\\text{air}}, g, p_{o}, T$ and fundamental constants. \n\nIn a real atmosphere, the temperature is not constant but changes with height. The rate of decrease of temperature with height $\\Gamma(z) = -d T / d z$ is called the lapse rate. Consider a small mass of air rising adiabatically in the atmosphere such that it remains at mechanical equilibrium with its surrounding.",
|
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"question": "For the adiabatically rising air mass, find the adiabatic lapse rate $\\Gamma_{a}$ in terms of $c_{p}$, the molar specific heat at constant pressure, $\\mu_{\\text{air}}$ and $g$.",
|
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|
"marking": [
|
|
|
[
|
|
|
"Award 0.1 pt if the answer writes the adiabatic relation in any correct form for an ideal gas, e.g., $p V^{\\gamma} = \\text{const.}$ or equivalently $p^{1-\\gamma} T^{\\gamma} = \\text{const.}$ where $\\gamma = c_p/c_v$, $c_p$ and $c_v$ are the molar specific heats at constant pressure and volume. Otherwise, award 0 pt.",
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|
"Award 0.3 pt if the answer differentiates the adiabatic relation with respect to height $z$ to relate temperature and pressure gradients as $\\frac{dT}{dz} = -\\frac{1-\\gamma}{\\gamma} \\frac{T(z)}{p(z)} \\frac{dp}{dz}$, where $T(z)$ is temperature and $p(z)$ is pressure. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer obtains the correct adiabatic lapse rate $\\Gamma_a = \\frac{\\mu_{\\text{air}}}{c_p} g$. Otherwise, award 0 pt."
|
|
|
]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{$\\Gamma_a = \\frac{\\mu_{\\text{air}}}{c_p} g$}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.6
|
|
|
],
|
|
|
"modality": "text+illustration figure",
|
|
|
"field": "Thermodynamics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_3_c_1.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_C_5",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part C: Stability of air in the atmosphere] \n\nConsider a small cylindrical mass of air at height $z$ above the ground. The pressure and mass density of air at that height are $p(z)$ and $\\rho(z)$, respectively, see Figure C.1. Assume a uniform downward gravitational field $g$ and that the pressure on the Earth's surface is $p_{o}$. \n\n[figure1] \nFigure C.1 \n\n(C.1) Assuming that the small air mass is at hydrostatic equilibrium, derive an expression of the rate of change of pressure with respect to height, $d p / d z$ in terms of $g$ and $\\rho(z)$. \n\n(C.2) Express $d p / d z$ in terms of $\\mu_{\\text{air}}, g, p(z)$ and $T(z)$, the temperature at height $z$ and fundamental constants. \n\n(C.3) Assuming an isothermal atmosphere, $T(z) = T$, find an expression for $p(z)$ in terms of $z, \\mu_{\\text{air}}, g, p_{o}, T$ and fundamental constants. \n\nIn a real atmosphere, the temperature is not constant but changes with height. The rate of decrease of temperature with height $\\Gamma(z) = -d T / d z$ is called the lapse rate. Consider a small mass of air rising adiabatically in the atmosphere such that it remains at mechanical equilibrium with its surrounding. \n\n(C.4) For the adiabatically rising air mass, find the adiabatic lapse rate $\\Gamma_{a}$ in terms of $c_{p}$, the molar specific heat at constant pressure, $\\mu_{\\text{air}}$ and $g$. \n\nTo analyze the stability of an atmosphere, we imagine starting from an equilibrium state, and then perturbing a small mass of air and analyze its response. Consider a small air mass initially in equilibrium with the surrounding air at height $z$ and temperature $T$. It is then adiabatically displaced vertically by a displacement $\\delta z_{0}$. Assume that throughout the motion, the air parcel always has the same pressure as the surrounding air at the same height. The surrounding atmosphere is unaltered and has a different lapse rate $\\Gamma$. Neglect viscosity.",
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"question": "(1) Find the equation of motion for $\\delta z$, the instantaneous vertical displacement. \n(2) Under what condition is the equilibrium at $z$ stable? \n(3) What is the angular frequency $\\omega$ of small oscillation? Express your answers in terms of $T, \\Gamma, g, \\mu_{\\text{air}}$ and $c_{p}$.",
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"marking": [
|
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|
[
|
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|
"Award 0.2 pt if the answer obtains the gravitational force with parcel density correctly as $\\delta m g = \\rho_p \\delta V g$, where $\\delta m$ is the mass of the parcel, $\\rho_p$ is the density of the parcel, and $\\delta V$ is its volume. Otherwise, award 0 pt.",
|
|
|
"Award 0.3 pt if the answer obtains the buoyancy force with air density correctly as $\\rho_a(z) g \\delta V$, where $\\rho_a$ is the surrounding air density. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer writes the correct equation of motion as $\\delta m \\frac{d^2 z}{dt^2} = \\rho_a(z) g \\delta V - \\delta m g$, and after substitution simplifies to $\\frac{d^2 z}{dt^2} = \\frac{\\rho_a(z+\\delta z) - \\rho_p(z+\\delta z)}{\\rho_p(z+\\delta z)} g$. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer correctly relates density to inverse temperature as $\\rho \\propto \\frac{1}{T}$. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer uses the appropriate approximation $T(z+\\delta z) = T(z) + \\Gamma \\delta z$, and simplifies to $\\frac{d^2 z}{dt^2} = \\frac{T(z) + \\Gamma \\delta z - T(z) - \\Gamma_a \\delta z}{T(z) + \\Gamma_a \\delta z} g$. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer provides the correct stability requirement that motion is stable whenever $\\Gamma_a > \\Gamma$, where $\\Gamma_a = \\mu_{air} g / c_p$. Otherwise, award 0 pt.",
|
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|
"Award 0.2 pt if the answer obtains the correct angular frequency of small oscillation as $\\omega = \\sqrt{\\frac{\\Gamma_a - \\Gamma}{T} g} = \\sqrt{\\frac{\\mu_{air} g / c_p - \\Gamma}{T} g}$. Otherwise, award 0 pt."
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]
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|
],
|
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"answer": [
|
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|
"\\boxed{$\\frac{d^2 z}{d t^2} = \\frac{\\Gamma - \\mu_{\\text{air}} g/c_p}{T} g \\delta_z$}",
|
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|
"\\boxed{$\\mu_{\\text{air}} g/c_p > \\Gamma$}",
|
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|
"\\boxed{$\\omega = \\sqrt{\\frac{\\mu_{\\text{air}} g/c_p - \\Gamma}{T} g}$}"
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],
|
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"answer_type": [
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"Equation",
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|
"Inequality",
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"Expression"
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|
],
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"unit": [
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null,
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null,
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null
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|
],
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"points": [
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|
1.1,
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|
0.1,
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|
0.2
|
|
|
],
|
|
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"modality": "text+illustration figure",
|
|
|
"field": "Mechanics",
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|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_3_c_1.png"
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|
]
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|
},
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|
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{
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|
|
"id": "APhO_2025_3_D_1",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part D: Moisture] \n\nEven though water constitutes a small portion of the atmosphere, it has a significant role in climate science. It is responsible for precipitation, and it is the most significant greenhouse gas. The phase of water depends on what temperature and pressure the water system is at, depicted on a $p-T$ phase diagram, see Figure D.1. When the pressure and temperature lie on the coexistence curve, both liquid and vapor water can be present in the system. The slope of the coexistence curve is given by the ClausiusClapeyron equation: \n$\\frac{d p_s}{d T} = \\frac{\\Delta S}{\\Delta V}$ \nwhere $p_{s}$ is the saturation pressure, the pressure at the phase transition, $\\Delta S$ and $\\Delta V$ are the changes in entropy and volume across the phase transitions, respectively. Treat water vapor as an ideal gas. \n\n[figure1] \nFigure D.1",
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"question": "Express $d p_{s} / d T$ for the water liquid-vapor coexistence curve in terms of the water latent heat of evaporation $L, \\mu_{\\text{H_2O}}, p_{s}, T$ and fundamental constants.",
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"marking": [
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|
[
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|
"Award 0.2 pt if the answer obtains the correct entropy change as $\\Delta S = \\frac{L m}{T}$, where $L$ is the latent heat of evaporation, $m$ is the mass of liquid water, and $T$ is the temperature. Otherwise, award 0 pt.",
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|
"Award 0.2 pt if the answer correctly states that $V_{vapor} \\gg V_{liquid}$ and therefore approximates $\\Delta V \\approx V_{vapor}$, with $V_{vapor} = \\frac{nRT}{p_s(T)}$, where $n$ is the number of moles, $R$ is the gas constant, and $p_s(T)$ is the saturation vapor pressure. Otherwise, award 0 pt.",
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|
"Award 0.1 pt if the answer obtains the correct final relation $\\frac{dp_s}{dT} = \\frac{\\mu_{H_2O} L p_s}{R T^2}$, where $\\mu_{H_2O}$ is the molar mass of water. Otherwise, award 0 pt."
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]
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|
],
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|
"answer": [
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|
"\\boxed{$\\frac{d p_s}{d T} = \\frac{\\mu_{\\text{H_2O}} L p_s}{R T^2}$}"
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],
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|
"answer_type": [
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|
"Expression"
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|
],
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|
"unit": [
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null
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|
|
],
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|
"points": [
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|
0.5
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|
|
],
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|
"modality": "text+illustration figure",
|
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|
"field": "Thermodynamics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
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|
"image_question/APhO_2025_3_d_1.png"
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|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_D_2",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part D: Moisture] \n\nEven though water constitutes a small portion of the atmosphere, it has a significant role in climate science. It is responsible for precipitation, and it is the most significant greenhouse gas. The phase of water depends on what temperature and pressure the water system is at, depicted on a $p-T$ phase diagram, see Figure D.1. When the pressure and temperature lie on the coexistence curve, both liquid and vapor water can be present in the system. The slope of the coexistence curve is given by the ClausiusClapeyron equation: \n$\\frac{d p_s}{d T} = \\frac{\\Delta S}{\\Delta V}$ \nwhere $p_{s}$ is the saturation pressure, the pressure at the phase transition, $\\Delta S$ and $\\Delta V$ are the changes in entropy and volume across the phase transitions, respectively. Treat water vapor as an ideal gas. \n\n[figure1] \nFigure D.1 \n\n(D.1) Express $d p_{s} / d T$ for the water liquid-vapor coexistence curve in terms of the water latent heat of evaporation $L, \\mu_{\\text{H_2O}}, p_{s}, T$ and fundamental constants.",
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"question": "If for some reference temperature $T_{o}$, $p_{s} = p_{s o}$, find an expression for $p_{s}(T)$ in terms of $p_{s o}, \\mu_{\\text{H_2O}}, L, T, T_{o}$ and fundamental constants.",
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"marking": [
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|
[
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|
"Award 0.1 pt if the answer recognizes a separable differential equation and obtains $\\ln [\\frac{p_s(T)}{p_{so}}] = -\\frac{\\mu_{\\text{H_2O}} L}{R} (\\frac{1}{T} - \\frac{1}{T_o})$, where $p_s$ is the saturation vapor pressure, $\\mu_{\\text{H_2O}}$ is the molar mass of water, $L$ is latent heat, $R$ is the gas constant, and $T$ is the temperature. Otherwise, award 0 pt.",
|
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|
"Award 0.1 pt if the answer obtains the correct final solution $p_s(T) = p_{so} \\exp \\left[- \\frac{\\mu_{\\text{H_2O}} L}{R} \\left( \\frac{1}{T} - \\frac{1}{T_0} \\right) \\right]$, where $p_{so}$ is the reference saturation vapor pressure at $T_0$. Otherwise, award 0 pt."
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|
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]
|
|
|
],
|
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|
"answer": [
|
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|
"\\boxed{$p_s(T) = p_{s0} \\exp\\left[-\\frac{\\mu_{\\text{H_2O}} L}{R} \\left(\\frac{1}{T} - \\frac{1}{T_0}\\right)\\right]$}"
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|
|
],
|
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|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
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|
0.2
|
|
|
],
|
|
|
"modality": "text+illustration figure",
|
|
|
"field": "Thermodynamics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_3_d_1.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_D_3",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part D: Moisture] \n\nEven though water constitutes a small portion of the atmosphere, it has a significant role in climate science. It is responsible for precipitation, and it is the most significant greenhouse gas. The phase of water depends on what temperature and pressure the water system is at, depicted on a $p-T$ phase diagram, see Figure D.1. When the pressure and temperature lie on the coexistence curve, both liquid and vapor water can be present in the system. The slope of the coexistence curve is given by the ClausiusClapeyron equation: \n$\\frac{d p_s}{d T} = \\frac{\\Delta S}{\\Delta V}$ \nwhere $p_{s}$ is the saturation pressure, the pressure at the phase transition, $\\Delta S$ and $\\Delta V$ are the changes in entropy and volume across the phase transitions, respectively. Treat water vapor as an ideal gas. \n\n[figure1] \nFigure D.1 \n\n(D.1) Express $d p_{s} / d T$ for the water liquid-vapor coexistence curve in terms of the water latent heat of evaporation $L, \\mu_{\\text{H_2O}}, p_{s}, T$ and fundamental constants. \n\n(D.2) If for some reference temperature $T_{o}$, $p_{s} = p_{s o}$, find an expression for $p_{s}(T)$ in terms of $p_{s o}, \\mu_{\\text{H_2O}}, L, T, T_{o}$ and fundamental constants. \n\nNow we consider a `moist' air mass that rises adiabatically starting from a temperature $T_{i}$. The mass mixing ratio of water vapor (the mass of water vapor relative to the total mass) is $\\phi$. Take the air mass to have a specific molar heat at constant pressure $c_{p}$. The universal gas constant is $R = 8.31 \\mathrm{J} /(\\mathrm{mol} \\mathrm{K})$.",
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"question": "Assuming that the air mass starts at $T_{i} = 17.0^{\\circ} \\mathrm{C}$ and $p_{i} = 10^{5} \\mathrm{Pa}$. Find the temperature $T_{l}$ at which liquid water starts forming in it if $\\phi = 10^{-2}$. Assume that the water content in the air mass stays constant during the rise. Use $L = 2460 \\mathrm{kJ} / \\mathrm{kg}$ and $p_{s o} = 1.94 \\times 10^{3} \\mathrm{Pa}$ at $T_{i} = 17.0^{\\circ} \\mathrm{C}$. Express your answer in $K$.",
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"marking": [
|
|
|
[
|
|
|
"Award 0.4 pt if the answer uses Dalton's law to correctly express the partial pressure of water vapor as $p_w = \\frac{n_{H_2O}}{n_{air}} p = \\frac{m_{H_2O}/\\mu_{H_2O}}{m_{air}/\\mu_{air}} p = \\phi \\frac{\\mu_{air}}{\\mu_{H_2O}} p$, where $n$ is number of moles, $m$ is mass, $\\mu$ is molar mass, $p$ is total pressure, and $\\phi$ is mixing ratio. Otherwise, award 0 pt.",
|
|
|
"Award 0.2 pt if the answer correctly relates the mole ratio to the mass ratio via $\\frac{n_{H_2O}}{n_{air}} = \\frac{m_{H_2O}/\\mu_{H_2O}}{m_{air}/\\mu_{air}}$. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer states the correct adiabatic process relation for pressure: $p(T) = p_i \\left( \\frac{T}{T_i} \\right)^{c_p/R}$, where $p_i$ is the initial pressure, $T_i$ is the initial temperature, $c_p$ is the specific heat at constant pressure, and $R$ is the gas constant. Otherwise, award 0 pt.",
|
|
|
"Award 0.5 pt if the answer shows that partial pressure of water needs to reach saturation for condensation to start. Otherwise, award 0 pt.",
|
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|
"Award 0.4 pt if the answer attempts to solve the transcendental equation iteratively by isolating $T$ on one side, such as rearranging to $T_l = \\frac{1}{\\frac{1}{T_i} - \\frac{R}{\\mu_{H_2O} L} \\ln \\left[ \\phi \\frac{\\mu_{air}}{\\mu_{H_2O}} \\frac{p_i}{p_{so}} \\left( \\frac{T_l}{T_i} \\right)^{c_p/R} \\right]}$. Otherwise, award 0 pt.",
|
|
|
"Award 0.4 pt if the answer provides the correct numerical solution $T \\approx 286.8 \\text{K} \\approx 13.7^{\\circ}\\text{C}$. Otherwise, award 0 pt."
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]
|
|
|
],
|
|
|
"answer": [
|
|
|
"\\boxed{286.8}"
|
|
|
],
|
|
|
"answer_type": [
|
|
|
"Numerical Value"
|
|
|
],
|
|
|
"unit": [
|
|
|
"K"
|
|
|
],
|
|
|
"points": [
|
|
|
2.0
|
|
|
],
|
|
|
"modality": "text+illustration figure",
|
|
|
"field": "Thermodynamics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_3_d_1.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_E_1",
|
|
|
"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part E: Sun halo] \n\nUnder suitable atmospheric conditions, a bright ring appears around the Sun which is called a halo. Halos are caused by ice crystals present in the upper troposphere. One interesting feature about halos is that they always appear at a specific angle relative to the direction of the Sun. \n\n[figure1] \nFigure E.1. On the left: A photograph showing a halo around the Sun. On the right: The path of a light ray passing through the prism.",
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"question": "Consider a simple prism with an apex angle of $\\varphi$ and direct a light ray onto it at an incidence angle $\\alpha$, as shown in Figure E.1. Let the refractive index of the prism be $n$. Express the angle of deviation $\\delta$ of the light ray after passing through the prism in terms of $\\alpha, n$ and $\\varphi$.",
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"marking": [
|
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|
[
|
|
|
"Award 0.1 pt if the answer writes Snell's law correctly for the first refraction as $\\frac{\\sin \\alpha}{\\sin \\alpha'} = n$, where $\\alpha$ is the angle of incidence, $\\alpha'$ is the refracted angle, and $n$ is the refractive index. Otherwise, award 0 pt.",
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|
"Award 0.1 pt if the answer writes Snell's law correctly for the second refraction as $\\frac{\\sin \\beta}{\\sin \\beta'} = n$, where $\\beta$ is the angle of incidence inside the prism, $\\beta'$ is the refracted angle on exit, and $n$ is the refractive index. Otherwise, award 0 pt.",
|
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|
"Award 0.2 pt if the answer expresses the deviation angle as $\\delta = \\alpha + \\beta - \\varphi$, where $\\varphi$ is the prism angle. Otherwise, award 0 pt.",
|
|
|
"Award 0.1 pt if the answer uses the geometric relation $\\alpha' + \\beta' = \\varphi$. Otherwise, award 0 pt.",
|
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|
"Award 0.2 pt if the answer performs the correct calculation steps, including substituting $\\alpha' = \\arcsin \\left( \\frac{\\sin \\alpha}{n} \\right)$ and $\\beta = \\arcsin \\left\\{ n \\sin \\left[ \\varphi - \\arcsin \\left( \\frac{\\sin \\alpha}{n} \\right) \\right] \\right\\}$. Otherwise, award 0 pt.",
|
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|
"Award 0.1 pt if the answer provides the correct final formula for $\\delta$, such as $\\delta = \\alpha + \\arcsin \\left\\{ n \\sin \\left[ \\varphi - \\arcsin \\left( \\frac{\\sin \\alpha}{n} \\right) \\right] \\right\\} - \\varphi$, or any other equivalent expression. Otherwise, award 0 pt."
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|
|
]
|
|
|
],
|
|
|
"answer": [
|
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|
"\\boxed{$\\delta = \\alpha + \\arcsin\\{n \\sin[\\varphi - \\arcsin(\\frac{\\sin\\alpha}{n})]\\} - \\varphi$}"
|
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|
],
|
|
|
"answer_type": [
|
|
|
"Expression"
|
|
|
],
|
|
|
"unit": [
|
|
|
null
|
|
|
],
|
|
|
"points": [
|
|
|
0.8
|
|
|
],
|
|
|
"modality": "text+variable figure",
|
|
|
"field": "Optics",
|
|
|
"source": "APhO_2025",
|
|
|
"image_question": [
|
|
|
"image_question/APhO_2025_3_e_1.png"
|
|
|
]
|
|
|
},
|
|
|
{
|
|
|
"id": "APhO_2025_3_E_2",
|
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"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part E: Sun halo] \n\nUnder suitable atmospheric conditions, a bright ring appears around the Sun which is called a halo. Halos are caused by ice crystals present in the upper troposphere. One interesting feature about halos is that they always appear at a specific angle relative to the direction of the Sun. \n\n[figure1] \nFigure E.1. On the left: A photograph showing a halo around the Sun. On the right: The path of a light ray passing through the prism. \n\n(E.1) Consider a simple prism with an apex angle of $\\varphi$ and direct a light ray onto it at an incidence angle $\\alpha$, as shown in Figure E.1. Let the refractive index of the prism be $n$. Express the angle of deviation $\\delta$ of the light ray after passing through the prism in terms of $\\alpha, n$ and $\\varphi$. \n\nThe most common type of halo forms when tiny ice crystals take the shape of regular hexagonal prisms. Light from the Sun falls onto randomly oriented ice crystals drifting in the atmosphere and scatters into various directions. However, in certain specific directions, the intensity of the refracted light is maximal, and this determines the angle at which the bright ring appears. \n\n[figure2] \nFigure E.2. \n\nConsider a hexagonal ice prism whose six-fold symmetry axis is perpendicular to the direction of the Sun's rays. Investigate a light ray that refracts through two rectangular faces of the prism indicated in Figure E.2. Due to the random orientation of the ice crystals, the light strikes the crystal faces at varying incidence angles $\\alpha$.",
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"question": "Calculate the deviation angle $\\delta$ for incidence angles $\\alpha = 20^{\\circ}, 30^{\\circ}, 40^{\\circ}, 50^{\\circ}, 60^{\\circ}, 70^{\\circ}$, in that order. Output the six values in degrees ($^{\\circ}$), each with three significant figures, listed sequentially and separately. The refractive index of ice is $n = 1.31$.",
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"marking": [
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[
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"Award 0.2 pt if the answer gives all six correct values (when \\alpha = 20^{\\circ}, \\delta = 27.5^{\\circ}; when \\alpha = 30^{\\circ}, \\delta = 23.0^{\\circ}; when \\alpha = 40^{\\circ}, \\delta = 21.8^{\\circ}; when \\alpha = 50^{\\circ}, \\delta = 22.5^{\\circ}; when \\alpha = 60^{\\circ}, \\delta = 24.7^{\\circ}; when \\alpha = 70^{\\circ}, \\delta = 28.7^{\\circ}). Partial points: award 0.1 pt if 3-5 values are correct; otherwise, award 0 pt.",
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"Award 0.2 pt if the calculated data points for $\\delta$ are correctly plotted against $\\alpha$ on the graph, with $\\alpha$ on the horizontal axis and $\\delta$ on the vertical axis. Otherwise, award 0 pt.",
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"Award 0.2 pt if the answer correctly identifies and shows that $\\delta$ has a local minimum (around $\\alpha \\approx 40^\\circ$). Otherwise, award 0 pt."
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]
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],
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"answer": [
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"\\boxed{$\\delta = 27.5^{\\circ}$ when $\\alpha = 20^{\\circ}$}",
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"\\boxed{$\\delta = 23.0^{\\circ}$ when $\\alpha = 30^{\\circ}$}",
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"\\boxed{$\\delta = 21.8^{\\circ}$ when $\\alpha = 40^{\\circ}$}",
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"\\boxed{$\\delta = 22.5^{\\circ}$ when $\\alpha = 50^{\\circ}$}",
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"\\boxed{$\\delta = 24.7^{\\circ}$ when $\\alpha = 60^{\\circ}$}",
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"\\boxed{$\\delta = 28.7^{\\circ}$ when $\\alpha = 70^{\\circ}$}"
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],
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"answer_type": [
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"Numerical Value",
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"Numerical Value",
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"Numerical Value",
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"Numerical Value",
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"Numerical Value",
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"Numerical Value"
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],
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"unit": [
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"degrees",
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"degrees",
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"degrees",
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"degrees",
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"degrees",
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"degrees"
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],
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"points": [
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0.1,
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0.1,
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0.1,
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0.1,
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0.1,
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0.1
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],
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"modality": "text+variable figure",
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"field": "Optics",
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"source": "APhO_2025",
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"image_question": [
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"image_question/APhO_2025_3_e_1.png",
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"image_question/APhO_2025_3_e_2.png"
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]
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},
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{
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"id": "APhO_2025_3_E_3",
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"context": "[Atmospheric Physics] \n\nThe Earth's atmosphere is a complex physical system, and predicting its behavior is crucial for environmental and meteorological purposes. However, even the best theoretical models run on modern computers are insufficient to make precise predictions. In this problem, we will attempt to understand some of the basic atmospheric phenomena based on simple models. You might need the following constants: the mean solar power per unit area at Earth, the total solar irradiance $F_{s} = 1370 \\mathrm{W} / \\mathrm{m}^{2}$, molar mass of water $\\mu_{\\text{H_2O}} \\approx 18 \\mathrm{g} / \\mathrm{mol}$ and average molar mass of air $\\mu_{\\text {air }} \\approx 29 \\mathrm{g} / \\mathrm{mol}$, The Stefan-Boltzmann constant $\\sigma=5.67 \\times 10^{-8} \\mathrm{W} /\\left(\\mathrm{m}^{2} \\mathrm{K}^{4}\\right)$. All gases in this problem can be treated as ideal gases. Assume that all air molecules have 5 degrees of freedom. You may need the following integral: \n$\\int_{-\\infty}^{\\infty} e^{-a x^2/2} dx = \\sqrt{2 \\pi/a}$, $a > 0$. \n\n[Part E: Sun halo] \n\nUnder suitable atmospheric conditions, a bright ring appears around the Sun which is called a halo. Halos are caused by ice crystals present in the upper troposphere. One interesting feature about halos is that they always appear at a specific angle relative to the direction of the Sun. \n\n[figure1] \nFigure E.1. On the left: A photograph showing a halo around the Sun. On the right: The path of a light ray passing through the prism. \n\n(E.1) Consider a simple prism with an apex angle of $\\varphi$ and direct a light ray onto it at an incidence angle $\\alpha$, as shown in Figure E.1. Let the refractive index of the prism be $n$. Express the angle of deviation $\\delta$ of the light ray after passing through the prism in terms of $\\alpha, n$ and $\\varphi$. \n\nThe most common type of halo forms when tiny ice crystals take the shape of regular hexagonal prisms. Light from the Sun falls onto randomly oriented ice crystals drifting in the atmosphere and scatters into various directions. However, in certain specific directions, the intensity of the refracted light is maximal, and this determines the angle at which the bright ring appears. \n\n[figure2] \nFigure E.2. \n\nConsider a hexagonal ice prism whose six-fold symmetry axis is perpendicular to the direction of the Sun's rays. Investigate a light ray that refracts through two rectangular faces of the prism indicated in Figure E.2. Due to the random orientation of the ice crystals, the light strikes the crystal faces at varying incidence angles $\\alpha$. \n\n(E.2) Calculate the deviation angle $\\delta$ for incidence angles $\\alpha = 20^{\\circ}, 30^{\\circ}, 40^{\\circ}, 50^{\\circ}, 60^{\\circ}, 70^{\\circ}$, in that order. Output the six values in degrees ($^{\\circ}$), each with three significant figures, listed sequentially and separately. The refractive index of ice is $n = 1.31$.",
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"question": "Using the numerical results from the previous question (E.2), determine at what angle the halo appears the brightest relative to the direction of the Sun. Express your answer in degrees ($^{\\circ}$) with three significant figures.",
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"marking": [
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[
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"Award 0.1 pt if the answer correctly reads and states the minimal value of $\\delta$ as approximately $21.8^\\circ$. Otherwise, award 0 pt.",
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"Award 0.1 pt if the answer concludes that the angular size of the halo corresponds to this minimal value of $\\delta$, i.e. about $21.8^\\circ$. Otherwise, award 0 pt."
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]
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],
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"answer": [
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"\\boxed{21.8}"
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],
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"answer_type": [
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"Numerical Value"
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],
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"unit": [
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"degrees"
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],
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"points": [
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0.2
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],
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"modality": "text+variable figure",
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"field": "Optics",
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"source": "APhO_2025",
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"image_question": [
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"image_question/APhO_2025_3_e_1.png",
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"image_question/APhO_2025_3_e_2.png"
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]
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}
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] |
