Chapter 6.8, Problem 21P

Consider a uniform distribution of total mass $m^{\prime }$ over a spherical shell of radius $r^{\prime }$ The potential energy $\varphi$ of a mass $m$ in the gravitational field of the spherical shell is

Assuming that the earth is a spherical ball of radius $R$ and constant density, find the potential and the force on a mass $m$ outside and inside the earth. Evaluate the constants in terms of the acceleration of gravity $g$, to get $F=-\frac{mg{R}^2}{r^2}{e}_r,\text{ and }\varphi =-\frac{mg{R}^2}{r}$ $m$ outside the earth; $F=-\frac{mgr}{R}{e}_r,\text{ and }\varphi =\frac{mg}{2R}\left(r^2-3R^2\right),$ $m$ inside the earth.

Hint: To find the constants, recall that at the surface of the earth the magnitude of the force on $m$ is m g.