Chapter 17, Problem 21RE

Assume that the earth is a solid sphere of uniform density with mass M and radius R = 3960 mi. For a particle of mass m within the earth at a distance r from the earth’s center, the gravitational force attracting the particle to the center is $F_r=\frac{-G{M}_{r}m}{r^2}$ where G is the gravitational constant and M_r is the mass of the earth within the sphere of radius r.

(a) Show that $F_r=\frac{-GMm}{R^3}r.$

(b) Suppose a hole is drilled through the earth along a diameter. Show that if a particle of mass m is dropped from rest at the surface, into the hole, then the distance y = y(t) of the particle from the center of the earth at time t is given by y"(t) = -k²y(t)

where k² = GM/R³ = g/R.

(c) Conclude from part (b) that the particle undergoes simple harmonic motion. Find the period T.

(d) With what speed does the particle pass through the center of the earth?