Problem 3. Consider a hollow sphere, $a^2 < x^2 + y^2 + z^2 < b^2$, of uniform mass density $\sigma$.
Show that the gravitational force of attraction induced by the hollow sphere on a unit mass located at
$(0,0,z_0)$ is
$F(0,0,z_0) = \sigma G \hat{k} \iiint \frac{(z - z_0)dV}{[x^2 + y^2 + (z - z_0)^2]^{3/2}}$
Carrying out the integration, show that
$F(0,0,z_0) = \begin{cases} \frac{-MG}{z_0^2} \hat{k} & \text{if } |z_0| > b\\ 0 & \text{if } |z_0| < a \end{cases}$
In the above equation, M is the total mass of the hollow sphere.
