Define gravitational potential energy. Find the expression for gravitational potential energy at any point.

Explanation

Gravitational potential: The gravitational potential at a point in the gravitational field of a body is the amount of work done in brining the body of unit mass from infinity to that point in the field. Expression: Consider a body of unit mass placed at a distance r from mass M.

The gravitational force between unit mass and M is given by `F= (GM xx 1)/(r^(2))= (GM)/(r^(2))` ..(i)
The work done is moving unit charge from point A to point B through distance dr is given by `dW= Fdr= (GM)/(r^(2))dr` [using equation (i)] ...(ii)
Total work done in displacing unit mass from infinity to point A is given by
`W= int dW= int_(oo)^(r ) (GM)/(r^(2))dr` [using equation (ii)]
or `W= GM int_(oo)^(r ) (1)/(r^(2)) dr or W= GM [-(1)/( r)]_(oo)^(r ) = GM [(1)/(oo)- (1)/(r )]or W= -(GM)/(r )`
This work done is equal to the gravitational potential So, Gravitational potential, `V= -(GM)/(r )`