International Physics Olympiad · Year 2025

Problems

Problem 1A uniform solid sphere of mass $m$ and radius $R$ rolls without slipping from rest down an incline at angle $\alpha$ . Its moment of inertia about its center is $I=\tfrac25mR^2$ and gravity is $g$ . Find the center-of-mass acceleration $a$ and speed $v$ after descending height $h$ . (a) Combine translation, torque, and rolling constraint to find acceleration $a$ and static-friction magnitude and direction. (b) Use energy conservation to find speed $v$ and angular speed after vertical drop $h$ , including rotational energy. (c) Check dependence on $R$ and explain why static friction does not dissipate energy in pure rolling.Solutions: 1
Problem 2A reversible Carnot engine operates between reservoirs at absolute temperatures $T_h>T_c$ . In each cycle it absorbs heat $Q_h$ from the hot reservoir. Find its efficiency $\eta$ , rejected heat $Q_c$ , and work output $W$ . (a) Apply the Clausius equality for a reversible cycle to relate $Q_h,Q_c,T_h,T_c$ . (b) Use the first law to find work $W$ and efficiency $\eta$ . (c) Examine $T_c\to T_h$ and $T_c\to0$ , explain their physical meaning, and state why absolute temperatures are required.Solutions: 1
Problem 3A photon of wavelength $\lambda$ scatters from an initially stationary electron; the scattered photon makes angle $\theta$ with the incident direction. Find the wavelength shift $\Delta\lambda$ and recoil-electron kinetic energy. Use electron mass $m_e$ , speed of light $c$ , and Planck constant $h$ . (a) Combine energy and momentum conservation with the electron’s relativistic relation to derive Compton shift $\Delta\lambda$ . (b) Find the photon energy loss and identify it with recoil-electron kinetic energy. (c) Evaluate forward scattering $\theta=0$ and backscattering $\theta=\pi$ , and state which gives the greatest recoil.Solutions: 1