Yet Another Tunnel through the Earth

A straight, smooth (frictionless) transit tunnel is dug through a planet of radius $R$ whose mass density $\rho_0$ is constant. The tunnel passes through the center of the planet and is lined up with its axis of rotation (so that the planet's rotation is irrelevant to this problem). All the air is evacuated from the tunnel to eliminate drag forces.

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. From this you should be able to find $r(t)$ for the car, assuming that it starts at $r_0 = R$ on the North Pole at time $t = 0$.

c) How long does it take the car to get to the South Pole starting from rest at the North Pole?