International Physics Olympiad · Year 2021

Problems

Problem 1Two masses $m_1,m_2$ are joined by a taut, massless inextensible string over a fixed frictionless pulley, so their accelerations have equal magnitude. Mass $m_1$ slides on a frictionless incline of angle $\alpha$ ; $m_2$ hangs freely. Take $g$ constant and assume $m_2>m_1\sin\alpha$ , so $m_2$ descends. Ignore the sizes of the bodies and all drag. (a) Draw the force diagrams, choose positive directions, and write Newton’s equation for each mass. (b) Determine the acceleration $a$ and tension $T$ . (c) Check the near-equilibrium limit when the driving forces almost balance.Solutions: 1
Problem 2A capacitor of capacitance $C$ , initially uncharged, is connected in series to a resistor $R$ and an ideal DC source of constant emf $\mathcal E$ . The switch closes at $t=0$ ; leads are ideal and source internal resistance and parasitic effects are neglected. Take positive current in the charging direction, and let $Q(t)$ be the charge on the plate connected to the positive terminal. (a) Write Kirchhoff’s loop equation and relate current to the rate of change of charge. (b) Determine $Q(t)$ , $I(t)$ , and the time constant. (c) Give the values just after switching and at long times, and check energy conservation.Solutions: 1
Problem 3Monochromatic light of wavelength $\lambda$ is incident normally on a single slit of width $b$ . A parallel screen is at distance $L$ , with $L\gg b$ so Fraunhofer diffraction applies. Intensity is observed at a small angle $\theta$ from the normal through the slit centre; ignore the finite screen size and other edge effects. (a) Divide the aperture into elements and state the condition for cancellation of their amplitudes. (b) Find the angle $\theta_1$ of the first dark minimum. (c) Derive its screen displacement $y_1$ and explain why this is a dark fringe rather than a maximum.Solutions: 1