International Physics Olympiad · Year 2023

Problems

Problem 1A satellite of mass $m$ moves uniformly in a circular orbit of radius $r$ measured from the center of an isolated, spherically symmetric planet of mass $M$ . Neglect drag and the satellite's effect on the planet; $G$ is the gravitational constant. (a) Identify the force and state the condition for a circular orbit. (b) Derive the orbital speed $v$ and period $T$ . (c) Check their dependence on $m$ and their dimensions.Solutions: 1
Problem 2An ideal DC source of emf $\mathcal E$ is connected in series with a resistor $R>0$ and ideal inductor $L>0$ . Before the switch closes the inductor current is zero; at $t=0$ the switch closes, with positive current chosen along the source. Neglect internal resistance and take all elements as constant. (a) Use Kirchhoff's law to find $I(t)$ for $t\ge0$ . (b) Determine the steady current $I_\infty$ . (c) Find when the current reaches $0.90I_\infty$ and check the initial and long-time limits.Solutions: 1
Problem 3A small real object of height $h_o$ is placed in front of a thin converging lens of focal length $f>0$ , at distance $d_o>f$ from its optical center, with the principal axis perpendicular to the lens. Use the convention that a real object has $d_o>0$ , a real image on the far side has $d_i>0$ , and lateral magnification is $m=-d_i/d_o$ , with lens equation $1/f=1/d_o+1/d_i$ . (a) Derive the image position. (b) Find $m$ and image height $h_i$ . (c) Classify the image and examine the limit $d_o\to f^+.$ Solutions: 1