A shaft is drilled from the surface to the center of the earth (see Fig. 13.24). As in Example 13.10 (Section 13.6), make the unrealistic assumption that the density of the earth is uniform. With this approximation, the gravitational force on an object with mass m, that is inside the earth at a distance r from the center, has magnitude (as shown in Example 13.10) and points toward the center of the earth.

(a) Derive an expression for the gravitational potential energy of the object€“earth system as a function of the object€™s distance from the center of the earth. Take the potential energy to be zero when the object is at the center of the earth.

(b) If an object is released in the shaft at the earth€™s surface, what speed will it have when it reaches the center of the earth?

Figure 13.24:

A hole through the center of the earth (assumed to be uniform). When an object is a distance r from the center, only the mass inside a sphere of radius r exerts a net gravitational force on it.

In Example 13.10:

Imagine that we drill a hole through the earth along a diameter and drop a mail pouch down the hole. Derive an expression for the gravitational force F_g on the pouch as a function of its distance from the earth€™s center. Assume that the earth€™s density is uniform (not a very realistic model; see Fig. 13.9).

Cross section through earth Spherical region of radius r F. -RE mE