Consider a point mass \(m\) located a distance \(R\) from the origin, and a spherical shell of mass \(\Delta M\), radius \(a\), and thickness \(\Delta a\), centered on the origin. The shell has uniform mass density \(ho\).

(a) Find \(\Delta M\) in terms of the other parameters given, assuming \(\Delta a \ll a\). Show that the gravitational potential energy of the point mass \(m\) due to the shell's gravity is

(b) \(-G \Delta M m / R\) for \(R>a\);

(c) a constant for \(R

(d) Then show that if a mass distribution is spherically symmetric the gravitational field inside it is directed radially inward, and its magnitude at radius \(R\) from the center is simply \(G M(R) / R^{2}\), where \(M(R)\) is the mass within the sphere whose radius is \(R\). That is, a shell whose radius is greater than \(R\) exerts no net gravitational force on \(m\).