Chapter 17.R, Problem 21E

Assume that the earth is a solid sphere of uniform density with mass M and radius $R=3960\text{ mi}$. 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 centre 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 $y=y\left(t\right)$ of the particle from the center of the earth at time t is given by

$y^{″}\left(t\right)=-k^{2}y\left(t\right)$

where $k^2=GM/R^3=g/R$.

(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?