Harmonic Transit Tunnel through the Earth

figures/final51.1.5.eps

A straight, smooth (frictionless) transit tunnel is dug through an airless moon of radius $R$ whose mass density $\rho_0$ is constant. The moon does not rotate, and the tunnel is left in vacuum to eliminate drag forces. All answers should be given in terms of $\rho,R,G$ (and/or $M_{\rm moon}$ once you have evaluated/defined it).

a) Find the force acting on a car of mass $m$ a distance $r < R$ from the center of the planet.

b) Write Newton's second law for the car, and extract the differential equation of motion.

c) From this, find $r(t)$ for the car, assuming that it starts at $r_0 = R$ on one (e.g. the top) side at time $t = 0$.

d) How long does it take the car to get from one side of the moon to the other, starting from rest?

e) Extra Credit: Suppose a message capsule were fired in a circular orbit of radius $R$ at the same time a message capsule were dropped through the tunnel. Which would arrive on the other side first?