Chapter 13.5, Problem 75E

Gravitational field due to spherical shell A point mass m is a distance d from the center of a thin spherical shell of mass M and radius R. The magnitude of the gravitational force on the point mass is given by the integral

     $F\left(d\right)=\frac{GMm}{4\pi }{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{\pi }\frac{\left(d-R\cos \phi \right)\sin \phi }{{\left(R^2+d^2-2Rd\cos \phi \right)}^{3/2}}d\phi d\theta }}$,

where G is the gravitational constant.

a.    Use the change of variable x = cos φ to evaluate the integral and show that if d > R, then $F\left(d\right)=\frac{GMm}{d^2}$, which means the force is the same as if the mass of the shell were concentrated at its center.

b.    Show that if d < R (the point mass is inside the shell), then F = 0.