Derive an expression for gravitational potential energy of body of mass M at distance r from centre of earth.

Explanation

Gravitational potential energy: Gravitational potential energy of a body at a point is the work done in bringing a body from infinity to that point.

Expression for Gravitational potential energy : Consider a body of mass m placed at distance x from earth of mass M. Then, gravitational force between the body and the earth, `F= (GMm)/(x^(2))`, The small amount of work done in displacing a body through very small distance dx is given by
`dW= F dx = (GM m)/(x^(2))dx`
Therefore, work done is bringing the body from infinity to the point A
`int dW= underset(x=oo)overset(x=r)int (GMm)/(x^(2))dx rARr W= GM m int_(oo)^(r )int (1)/(x^(2)) dx= GM m [(-1)/(x)]_(oo)^(r )`
`rArr W= -GM m[(1)/(r )- (1)/(oo)]= (GMm)/(r)`
This work done is stored inside the body as gravitational potential energy.
`therefore` Gravitational potential energy, `U= (-GMm)/(r )`