Derive the differential equation(s) describing the given physical situation. Suppose a...

Derive the differential equation(s) describing the given physical situation.

Suppose a hole is drilled through the center of the earth. A body with mass in is dropped into the hole. Let the distance from the center of the earth to the mass at time t be denoted by r. See Figure 1.25.

(a) Let M denote the mass of the earth and denote the mass of that portion of the earth within a sphere of radius r. The gravitational force on where the minus sign indicates that the force is one of attraction. Use this fact to show that

[Hint: Assume that the earth is homogeneous that is, has a constant density . Use mass = density x volume.]

(b) Use Newton's second law and the result in part (a) to derive the differential equation