A particle of mass m was transferred from the centre of the base of a uniform hemisphere of mass M and radius R into infinity.
What work was performed in the process by the gravitational force exerted on the particle by the hemisphere?

Explanation

One can imagine that the uniform hemisphere is made up of thin hemispherical layers of radii ranging from 0 to R. Let us consider such a layer (figure). Potential at point O, due to this layer is,
`dvarphi=-(gammadm)/(r)=-(3gammaM)/(R^3)rdr`, where `dm=(M)/((2//3)piR^2)((4pir^2)/(2))dr`
(This is because all points of each hemispherical shell are equidistant from O.)
Hence, `varphi=intdvarphi=-(3gammaM)/(R^3)underset(0)overset(R)intrdr=-(3gammamM)/(2R)`
Hence, the work done by the gravitational field force on the particle of mass `m`, to remove it to infinity is given by the formula
`A=mvarphi`, since `varphi=0` at infinity.
Hence the sought work,
`A_(0-oo)=-(3gammamM)/(2R)`

(The work done by the external agent is `-A`.)