Question

4. Assume the earth is of uniform density (not actually true). Imagine that a hole is drilled through the earth, passing through its center. A ball of mass m is dropped into the hole. A) Show that

the mass oscillates about the center of the earth in simple harmonic motion. B) Show that the period of the motion is T = (2πRE3)/GmE where RE and mE are the radius and mass of the earth respectively. [Hint: first apply Newton’s law of gravitation to figure out the gravitational force on the ball when it is at some radius r within the earth. It can be shown (and you can assume) that only the mass of the earth within that radius r contributes to the gravitational force experienced by the ball.]