International Physics Olympiad · Year 2024

Problems

Problem 1A small block of mass $m$ is released from rest at the top of a fixed, smooth spherical dome of radius $R$ and slides down its outside; take gravitational acceleration $g$ as constant. The angle $\theta$ is measured from the vertical through the top to the radius through the block, with $0\le\theta<\pi/2$ . Neglect the block's size and drag. (a) Use energy conservation to find its speed as a function of $\theta$ . (b) Write Newton's second law radially and state the loss-of-contact condition. (c) Find the departure angle and explain why it is independent of $m,R,g$ .Solutions: 1
Problem 2A parallel-plate capacitor of capacitance $C_0$ (air-filled) is charged to voltage $U_0$ then disconnected from the source. A dielectric slab of constant $\kappa$ is then inserted, filling the gap. Find the capacitor's energy before and after insertion, and explain the change. (a) Find plate charge, capacitance after insertion, and final voltage. (b) Calculate initial and final field energies and their ratio in terms of $\kappa$ . (c) Explain the mechanical work associated with the energy decrease as the slab is drawn in; distinguish field work from external work and state what is conserved after disconnection.Solutions: 1
Problem 3Two coherent sources of equal amplitude emit sound in phase of wavelength $\lambda$ . At point $P$ their path difference is $\Delta r$ . Find the conditions for constructive and destructive interference at $P$ , including the corresponding integer orders. (a) Derive the phase difference at $P$ from signed path difference $\Delta r$ . (b) Obtain constructive and destructive conditions with integer orders, distinguishing signed difference from its magnitude. (c) Decide whether reversing the path difference or reversing one source’s phase changes the fringe type; check zero and half-wavelength path differences.Solutions: 1