A particle of mass m is located outside a uniform sphere of mass M at a distance r from its centre. Find:
(a) the potential energy of gravitational interaction of the particle and the sphere,
(b) the gravitational force which the sphere exerts on the particle.

Explanation

Suppose that the sphere has a radius equal to a. We may imagine that the sphere is made up of concentric thin spherical shells (layers) with radii ranging from 0 to a, and each spherical layer is made up of elementary bands (rings). Let us first calculate potential due to an elementary band of a spherical layer at the point of location of the point mass m (say point P) (figure). As all the points of the band are located at the distance l from the point P, so,
`delta varphi=-(gammadeltaM)/(l)` (where mass of the band) (1)
`deltaM=((dM)/(4pia^2))(2pi a sin theta)(a dtheta)`
`=((dM)/(2))sin theta d theta` (2)
And `l^2=a^2+r^2-2arcos theta` (3)
Differenciating Eq. (3), we get
`ldl=arsin theta d theta` (4)
Hence using above equations
`delta varphi=-((lambdadM)/(2ar))dl` (5)
Now integrating this Eq. over the whole spherical layer
`d varphi=int delta varphi=-(gammadM)/(2ar)underset(r=-a)overset(barr+a)int`
So `d varphi=-(gammadM)/(r)` (6)
Equation (6) demonstrates that the potential produced by a thin uniform spherical layer outside the layer is such as if the whole mass of the layer were concentrated at it's centre,
Hence the potential due to the sphere at point P,
`varphi=intdvarphi=-gamma/rintdM-(gammaM)/(r)` (7)
This expression is similar to that of Eq. (6)
Hence the sought potential energy of gravitational interaction of the particle m and the sphere,
`U=mvarphi=-(gammaMm)/(r)`
(b) Using the Eq. `G_r=-(deltavarphi)/(deltar)`
`G_r=-(gammaM)/(r^2)` (using Eq. 7)
So `vecG=-(gammaM)/(r^3)vecr` and `vecF=mvecG=-(gammamM)/(r^3)vecr` (8)