Imagine that a hole is drilled through the center of the Earth to the other side. An object of mass m at a distance r from the center of the Earth is pulled toward the center of the Earth only by the mass within the sphere of radius r (the reddish region in Fig. P15.75)
(a) Write Newton€™s law of gravitation for an object at the distance r from the center of the Earth, and show that the force on it is of Hooke€™s law form, F = -kr, where the effective force constant is k = (4/3)) πpGm. Here P is the density of the Earth, assumed uniform, and G is the gravitational constant.
(b) Show that a sack of mail dropped into the hole will execute simple harmonic motion if it moves without friction. When will it arrive at the other side of the Earth?

Earth T2 Figure P15.75