2015 年泛珠三角及中华名校物理奥林匹克邀请赛
Pan Pearl River Delta Physics Olympiad 2015
Pan Pearl River Delta Physics Olympiad 2015 2015 年泛珠三角及中华名校物理奥林匹克邀请赛 Sponsored by Institute for Advanced Study, HKUST
香港科技大学高等研究院赞助
Part-1 (Total 5 Problems) 卷-1（共5题） (9:00 am – 12:00 pm, 25 February, 2015)
Numerical answers should be given to 3 significant figures.数字答案请给三位有效数字。
1. Retrograde Motion of Mars (9 points) 火星的逆行运动 （9分）
In the history of astronomy, the phenomenon of the retrograde motion played an important role. Suppose we observe the position of Mars at midnight every night for many nights. Using distant stars and constellations as the background, we will find that Mars moves from West to East most of the time. However, there are periods of time that Mars is observed to move in opposite direction, as shown in the figure. The orbital period of Mars is 1.88 y. Assume that the orbits of Earth and Mars are circular, and the tilting of Earth’s axis can be ignored.
在天文史上，行星的逆行运动扮演了重要的角色。假设我们连续多个晚上在午夜观察火星的位置。若以远处的星体和星座为背景，我们会发现大部分时间火星是从西到东运动，但也有些时段是逆向运动，如图所示。火星的轨道周期是1.88年。假设地球和火星的轨道都是圆的，地轴的倾斜可略。
Distant constellation远处的星座 Star T星T East东 o
t > 0 t = 0 t < 0 West西
20
15
o
10
o
5
o
0o
-5
o
-10
o
-15
o
-20
o
(a) What is the orbital radius RM of Mars? Give your answer in AU (Astronomical Units, 1 AU is the average distance between Sun and Earth.) (1 points)
试求火星的轨道半径RM。答案请以AU为单位。(1 AU 是太阳与地球的平均距离。)（1分） (b) At t = 0, Sun, Earth and Mars lie on a straight line. Sketch a figure indicating the positions of Sun, Earth, Mars, and star T when t > 0. Label them by letters S, E, M, and T respectively. Mark the angular displacements ?E and ?M of Earth and Mars respectively (starting from t = 0), and the angle ? that gives the angular position of Mars as observed from Earth using distant stars and constellations as the background. (2 points)
在t = 0时，太阳、地球、火星成一直线。试作一草图，显示在t > 0时，太阳、地球、火星和星T的位置，以S，E，M和T标示。在图上标示地球和火星的角位移分别为?E和 ?M（自t = 0开始），和地球观察火星的角位置?（以远处的星体和星座为背景）。（2分）
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2015 年泛珠三角及中华名校物理奥林匹克邀请赛
Pan Pearl River Delta Physics Olympiad 2015
(c) Derive an expression for the angular position ? of Mars at time t. Express your answer in terms RE, RM, ?E, ?M and t, where ?E and ?M are the orbital angular velocity of Earth and Mars respectively. (3 points)
试推导火星在时间t时的角位置?。答案请以RE, RM, ?E, ?M和t表示，其中?E和?M分别为地球与火星的角速度。（3分）
(d) Calculate the angular position ? of Mars at t = 0.1 y, 0.2 y and 0.3 y. Give your answer in degrees. (3 points)
试计算火星在t = 0.1年, 0.2年和 0.3年时的角位置?。答案请以度数表示。（3分）
2. Rolling Ball on a Racket (10 points) 球拍滚球（10分）
As shown in the figure, a hollow spherical ball of mass M and radius R is placed on a racket of mass m. The racket has a flat surface with coefficient of static friction ?s and coefficient of kinetic friction ?k and is held horizontally.
如图所示，一个质量为M，半径为R的空心园球被放置在质量为m的球拍上。球拍具有一个平坦的表面，其静摩擦系数为?s，动摩擦系数为?k，并且被保持在水平位置。
M m F
(a) The racket is driven horizontally by a periodic force F(t)?F0cos?0t, with the ball remaining non-slipping. Calculate the maximum velocities of the oscillations of the racket and the ball, denoted as ux and uy respectively. (The moment of inertia of a hollow sphere of mass M and radius R is I = 2MR2/3.) (5 points)
球拍被周期性的力F(t)?F0cos?0t沿水平方向驱动，园球维持在不滑动的状态。试计算球拍与球振动时的最大速度，分别表示为ux和uy。（质量为M，半径为R的空心球体的转动惯量为I = 2MR2/3。） (5分)
(b) At the moment the racket is oscillating at its maximum velocity, its motion is brought to rest abruptly by an external force much stronger than the limiting frictional force between the racket and the ball in a very short duration of time. What is the final velocity of the ball? If the final velocity of the ball is 0, what is the displacement of the ball? (5 points)
在球拍振动至最大速度的一刻，其运动突然被外力煞停，这外力比球拍与球之间的极限摩擦力强得多，作用的时间也很短。问球的最终速度是多少？若球的最终速度为0，其位移是多少？（5分）
3. Balloon (10 points) 气球（10分）
R
The work done in stretching a spring is converted to its spring energy. Likewise, the work done in stretching a membrane is converted to its surface energy, given by E = ?S, where ? is called the surface tension of the membrane, and S is its surface area.
拉伸弹簧所做的功被转换成弹簧的内能。同样，拉伸薄膜所做的功被转换成它的表面能E = ?S，其中?称为薄膜的表面张力，而S是其表面面积。
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2015 年泛珠三角及中华名校物理奥林匹克邀请赛
Pan Pearl River Delta Physics Olympiad 2015
(a) Consider a balloon of radius R. What is the change in surface energy when the radius changes by dR? Hence derive an expression for the pressure due to surface tension. (2 points) 考虑半径为R的气球。当半径改变为dR时，表面能的变化是多少？由此推导表面张力形成的压力的表达式。（2分）
(b) The surface tension of balloon A is ?. When it is filled with a diatomic ideal gas, its radius becomes R0. The surface tension of balloon B is 2?. When it is filled with the same kind of ideal diatomic gas, its radius becomes R0. The temperature of the environment is T. The two balloons are then connected so that the gases are free to exchange between them until a steady state is reached. The final temperature is the same as that of the environment. What are the final radii of the two balloons respectively? You may neglect the atmospheric pressure in the analysis. (4 points)
气球A的表面张力为?。当它充满了一种双原子的理想气体，其半径是R0。气球B的表面张力为2?。当它被相同的双原子理想气体充满时，其半径是R0。环境的温度为T。然后两个气球被
连接，使得气体可以在它们之间自由交流，直至达到稳定状态。最终温度与环境相同。问两个气球最终的半径分别是什么？在分析中你可以忽略大气压力。（4分）
(c) What are the amounts of heat gain by the gases in balloons A and B respectively during the gas exchange process in (b)? (4 points)
在(b)部的气体交流过程中，气球A和B增加的热能分别是什么？（4分）
4. Fresnel Biprism (10 points) 菲涅耳双棱镜（10分）
Fresnel biprism was devised shortly after the famous Young’s double slit experiment to confirm the interference phenomenon. Nowadays, it is widely used in different applications. As shown in the figure, it consists of a single light source S and a pair of wedge-shaped prisms arranged back to back. We introduce the following notations:
在著名的杨氏双缝实验面世后不久，便产生了菲涅耳双棱镜的设计，用以确认干涉现象。如今，它被广泛用于不同的应用。如图所示，它由一个单一的光源S和一对背对背的楔形棱镜组成。我们引入以下符号：
n = refractive index of the biprism双棱镜的折射率 ? = apex angle of each prism双棱镜的顶角
b = distance between light source and biprism光源与双棱镜的距离 c = distance between biprism and screen双棱镜与屏幕的距离 ? = wavelength of light光的波长
? S ?
(a) Derive an expression for the angular deviation after a light beam has passed through one of the two prisms. (3 points)
试推导光束经过其中一个棱镜后偏转角的表达式。（3分）
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b c 2015 年泛珠三角及中华名校物理奥林匹克邀请赛
Pan Pearl River Delta Physics Olympiad 2015
(b) Derive an expression for the separation of the fringes on the screen. (4 points) 试推导屏幕上条纹距离的表达式。（4分）
(c) In a modern application on electron microscopes, the single light source is replaced by a parallel beam of wave incident normally to the flat surface of the biprism. Derive an expression for the separation of the fringes on the screen. (3 points)
在现代，这原理已应用到电子显微镜中。在这应用中，单个光源被替换成入射的平行波束，垂直于双棱镜的平面。试推导屏幕上条纹距离的表达式。（3分）
5. Ionic Crystals (11 points) 离子晶体（11分）
An ionic crystal can be modeled by a chain of positively and negatively charged ions. The ionic separation is a. The positive ions with atomic mass M are located at the positions x = na where n is even. The negative ions with atomic mass m (m < M) are located at the positions x = na where n is odd. The ions are coupled to their neighbors by springs, which provide restoring forces to their transverse displacements. The returning force is proportional to the displacements of the ions relative to their neighbors, and the spring constant is k.
我们可以一串带正电和带负电的离子，作为离子晶体的模型。离子间的距离为a。正离子的原子质量为M，处于位置x = na，其中n是偶数。负离子的原子质量为m（m < M），处于在位置x = na，其中n是奇数。相邻的离子有弹簧耦合，弹簧为离子的横向位移提供返回力。返回力正比于离子相对于相邻离子的位移，并且弹簧常数为k。
(a) Let un(t) be the transverse displacement of the ion at x = na and time t. Derive the equations of motion for both types of ions. Show that the solution of the equation of motion can be written as
令un(t)为处于x= na的离子在时间t的横向位移。试推导两种类型离子的运动方程。表明运动方程的解可以写成
?AMsin(qna??t)n even,un(t)??
Asin(qna??t)n odd.?m
Find the relation between q and ?. (3 points) 试找出q与?的关系。（3分）
(b) Find the solutions of ? in the limit q = 0, and the relation between AM and Am for each solution. (2 points)
在极限q = 0，求?的所有解，并且求在每个解中AM与Am间的关系。（2分） (c) In the limit q = 0, calculate the wave velocity of the low frequency mode. (1 point) 在极限q =0，试计算低频模式的波速。（1分）
(d) In the limit q = ?/2a, find the solutions of ?, and the relation between AM and Am for each solution. (2 points)
在极限q = ?/2a，求?的所有解，并且求在每个解中AM与Am间的关系。（2分）
(e) Sketch the angular frequency ? as a function of the wavenumber q from q = ??/2a to q = ?/2a. (2 points)
试绘出角频率?作为波数q的函数的草图，范围从q = ??/2a 到 q = ?/2a。（2分） (f) An electromagnetic wave is incident on the crystal. Which frequency mode will be excited? (1 point) 有电磁波入射到晶体。哪种频率模式会被激发？（1分）
《THE END完》
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2015 年泛珠三角及中华名校物理奥林匹克邀请赛
Pan Pearl River Delta Physics Olympiad 2015
Pan Pearl River Delta Physics Olympiad 2015 2015 年泛珠三角及中华名校物理奥林匹克邀请赛 Sponsored by Institute for Advanced Study, HKUST
香港科技大学高等研究院赞助
Part-2 (Total 2 Problems) 卷-2（共2 题） (2:00 pm – 5:00 pm, 25 February, 2015)
1. Exoplanet Microlensing (25 points) 系外行星的微透镜效应 （25分）
With the discovery of planets orbiting around stars in recent years, the observation of exoplanets from astronomical distances became a challenge to scientists. Gravitational microlensing is one of the detection methods. It makes use of Einstein’s discovery in general relativity that when a light ray passing near a spherically symmetric body of mass M, its direction will be deflected towards the body by a small angle given by
随着近年发现不少绕着恒星运行的行星，怎样观察相隔天文距离的系外行星便成为科学家的挑战。引力微透镜是其中一种检测方法。它利用爱因斯坦在广义相对论里发现的原理，就是当光线经过一个质量为M的球对称物体时，方向会朝向物体偏转，偏转的小角度为
??4GMrc2,
where G is the gravitational constant, c is the speed of light, and r is the distance of closest approach of the light ray to the body. In this problem, we will study the principle of detecting exoplanet by microlensing.
其中G是万有引力常数，c是光速，r是光线和物体的最短距离。在这个问题中，我们将研究通过微透镜效应探测系外行星的原理。
(a) Consider a distant star S located at a distance Ds from Earth E, acting as the light source. Another star L of mass M and located at distance Dl (< Ds) from Earth acts as the lens. The lines EL and ES make a small angle ? between them. Construct the following sketch in the answer book: (a1) the line EL, (a2) the line ES, (a3) the distances Dl and Ds, (a4) the angle ? (remark: although this angle is small in practice, it should not be drawn too small for the purpose of clarity), (a5) a line perpendicular to EL through L, acting as the gravitational lens, (a6) the light ray from S to E, assuming that each of the segments between S and the lens and that between the lens and E are straight lines, (a7) the deflection angle ?, (a8) the apparent angle ? of the star S as observed on Earth (relative to line EL). (3 marks)
考虑一个遥远的恒星S，离地球E的距离为Ds，作为光源。另一颗恒星L，质量为M，离地球的距离为Dl（< Ds），作为透镜。线EL和ES间的小角度为?。试在答题簿上绘出以下草图：（a1）线EL，（a2）线ES，（a3）距离Dl和Ds，（a4）角度?（注：虽然该角度实际上很小，但为清楚起见，不应把它绘得太小），（a5）一条垂直于EL而通过L的线，作为引力透镜，（a6）从S到E的光线，假定S和透镜之间的线段及透镜和E之间的线段各可视作直线，（a7）偏转角?，（a8）从地球观察星S的视角?（相对于线EL）。 （3分）
(b) Derive an equation for the angle ? in terms of the parameters Ds, Dl, G, M, c and ?, assuming that all angles are small. (3 points)
试推导?的方程式，以参数Ds, Dl, G, M, c 和 ?表达，可假设所有角度都很小。（3分）
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2015年泛珠三角及中华名校物理奥林匹克邀请赛试题及答案
