Classical Mechanics - Special Interest Problem

Consider the artificial shell planet from last week's puzzle. The shell is constructed of material with uniform density r. The inner radius is a and the outer radius is 2a. A hole is drilled directly towards the center of the planet. An object is dropped into the hole.

1. Show that, unlike the case for a uniform solid planet, the gravitational force acting on the object is not a linear force.

2. Show that the speed of the object when it reaches the inner surface at r = a is given by
   v_r=a = a ⁸/₃prG

Hardcopy entries should be submitted in class by noon on Monday of next week. Answers should be concise but complete.