 AP Free Response Question 1999 C2 A spherical, non-rotating planet has a radius R and a uniform density throughout its volume. Suppose a narrow tunnel were drilled through the planet along one of its diameters, as shown in the figure above, in which a small ball of mass m could move freely under the influence of gravity. Let r be the distance of the ball from the center of the planet.

 (a) Show that the magnitude of the force on the ball at a distance r < R from the center of the planet is given by F = —Cr, where (b) On the axes below, sketch the force F on the ball as a function of distance r from the center of the planet. The ball is dropped into the tunnel from rest at point P at the planet’s surface.

 (c) Determine the work done by gravity as the ball moves from the surface to the center of the planet.

 (d) Determine the speed of the ball when it reaches the center of the planet.

 (e) Fully describe the subsequent motion of the ball from the time it reaches the center of the planet.

 (f) Write an equation that could be used to calculate the time it takes the ball to move from point P to the center of the planet. It is not necessary to solve this equation.