Define gravitational potential energy.

Explanation

Gravitational potential energy: Potential energy of a body at a point in a gravitational field is the work done by an external agent in moving the body from infinity to that point.
Expression for Gravitational potential energy: The gravitational force is a conservative force and hence we can define a gravitational potential energy associated with this conservative force field.
Two masses `m_1` and `m_2` are initially separated by a distance r.. Assuming m I to be fixed in its position, work must be done on `m_2` to move the distance from r. to r.

To move the mass `m_2` through an infinitesimal displacement `dvecr` from `vecr` to ` vecr = dvecr` . work has to be done externally . This infinitesimal work is given by
` dW = vecF_(ext) . dvecr` .....(1)
The work is done against the gravitational force, therefore
` vecF_(ext) = (G m_1 m_2)/(r^2) hatr` ....(2)
Substituting equation (2) in (1) . we get
` dW = (Gm_1 m_2)/(r^2) hat r . d vecr` ....(3)
` dvecr = dr hat r rArr dW = (Gm_1m_2)/(r^2) hatr . (dr hat r)`
` hatr . hatr = 1` (since both are unit vectors )
` therefore dW = (Gm_1 m_2)/(r^2) dr` ...(4)
Thus the total work done for displacing the particle from r. to r is
` W = int_(r.)^(r) (Gm_1 m_2)/(r^2) dr ` ....(5)
` W = - ( (Gm_1m_2)/(r) )_(r.)^(r) `
` W = - (Gm_1m_2)/(r) + (Gm_1 m_2)/(r.)` .....(6)
` W = U(r ) - U (r.)`
where `U(r) = (- Gm_1m_2)/(r ) ` ......(7)

Case 1: If. r < r. : Since gravitational force is attractive, `m_2` is attracted by `m_1`. Then `m_2` can move from r to r. without any external work. Here work is done by the system spending its internal energy and hence the work done is said to be negative. Case 2: If r > r. : Work has to be done against gravity to move the object from r. to r. Therefore work is done on the body by external force and hence work done is positive.