1.A straight smooth tunnel is dug through the center of Earth. A particle of mass $m$ is dropped into the tunnel. Prove that its motion is SHM and find its time period.
Sol: Restoring force $F = -\frac{GMm}{R^3} r = -m \omega^2 r \implies \omega = \sqrt{\frac{g}{R}}$. $T = 2\pi\sqrt{\frac{R}{g}} \approx 84.6\text{ minutes}$.
2.Two stars of masses $M$ and $2M$ separated by distance $d$ rotate under mutual gravitational attraction about their common center of mass. Find their orbital period $T$.
Sol: Gravitational force $F = \frac{G(M)(2M)}{d^2} = \frac{2GM^2}{d^2}$. Center of mass distance for mass $M$ is $r_1 = \frac{2}{3}d$. $M \omega^2 \left(\frac{2}{3}d\right) = \frac{2GM^2}{d^2} \implies \omega = \sqrt{\frac{3GM}{d^3}} \implies T = 2\pi \sqrt{\frac{d^3}{3GM}}$.
