Asian Physics Olympiad · Year 2024

Problems

Problem 1Two small spheres of equal mass $m$ hang from strings of length $l$ from the same point, initially touching. Each is given an equal charge $q$ ; they repel and reach equilibrium at a small angle $\theta$ from the vertical. Find $q$ in terms of $m$ , $l$ , $\theta$ . Here $g=9.8\,\mathrm{m\,s^{-2}}$ and $k=1/(4\pi\epsilon_0)$ is Coulomb’s constant; neglect sphere size relative to the center separation, so the repulsive force is horizontal and the two strings make symmetric angles $\theta$ with the vertical. (a) Use force balance to find the electric force in terms of $m,g,\theta$ . (b) Express the center separation using $l,\theta$ and obtain the exact charge in terms of the given quantities. (c) For $\\theta\ll1$ rad, derive the small-angle formula and check its dimensions.Solutions: 1
Problem 2One mole of a monatomic ideal gas starts at $(P_0,V_0,T_0)$ . It is heated at constant pressure until its volume is $2V_0$ , then cooled at constant volume back to temperature $T_0$ . Find the work done by the gas, the heat received by it over the full process, and its change in internal energy. Heat entering the gas is positive. Treat the gas as a closed system, apply the equation of state at each state, and use the stated sign convention to distinguish work from heat. (a) Relate temperature to volume on the first leg and specify pressure and volume at the junction. (b) Find the work on each leg and the total change in internal energy. (c) Use the first law to find net heat and explain why its sign need not match the heat on the cooling leg.Solutions: 1
Problem 3A thin converging lens has focal length $f>0$ . A planar object perpendicular to the principal axis is placed a distance $3f$ from the lens. Using the thin-lens model and the convention that real images have positive image distance, find the signed image distance $v$ , lateral magnification $m$ , and describe the image. Use the stated signed-distance convention and assume a sufficiently small object so the paraxial approximation applies. (a) Determine the image position from the conjugate equation. (b) Calculate the magnification and image height if the object height is $H$ . (c) Decide whether the image is real or virtual and upright or inverted, and check the dimensional consistency.Solutions: 1