Gravitational potential energy

Gravitational potential energy:

               The  amount  of  work  done  in moving  a unit  mass  from  infinity  to  the point  under  consideration ,  in  the gravitational  field  of  the  body  is  called gravitational  potential  at  that  point.

               The  gravitational  potential  is scalar quantity and it's S.I unit is joule/kg.

Gravitational potential due to a point mass :

               Consider a point A at a distance ‘r' from a particle of mass M, as shown in figure.

               Then intensity of Gravitational field at point A is given by:

                E = - GM/r²   (along AM )

               .•.   Difference   of   Gravitational potential between two points A and B at a distance ‘dr' apart is :

                VA-VB= dV = - E dr     .......(1)

                .•. E = - dV/dr                ....... (2)

                This  is  the  relation  between gravitational  field  and  gravitational potential.

                 .•. the gravitational field is defined as  the  negative  gradient  of  gravitational potential.

                  Now, from eq. (1) , we get:

                  dV = - ( - GM/r²) dr = GM/r² dr

                  Hence, potential V at point A ( i.e work  done  in  moving  unit  mass  from infinity to point A) in the gravitational field of M is given by:

                   Thus, gravitational potential is zero at infinity. And  at all other points it is less than zero i.e it has a negative value.

Gravitational potential energy:

                    The  gravitational  potential energy  of  a  body  at  a  point  in  the gravitational field is defined as the amount of work done in bringing the body from infinity to the desired point.

                    Consider a point A at a distance ‘r' from a  body  of  mass M.  A  particle  of mass ‘m' is placed at B , at a distance (r+dr ) in the gravitational field of M.

Gravitational force of attraction acting on mass ‘m' is :

                      F = - GMm/r²  (along AM )

                     Hence, the work done (dW) by the force F in moving the mass‘m' from B to A , through a small distance ‘dr' is given by:

                     dW = - F dr = - (- GM/r²) dr

.•.  total work done in moving the mass ‘m' from ∞ to point A is given by:

                  .•. W = - GMm/r      .....(1)

                  This work done is stored in the body as it's gravitational potential energy (U) and it is given by :

                   U = - GMm/ r       .......(2)

.•. Gravitational P.E (U) =gravitational potential (V) × mass of the body (m)

                   Gravitational potential energy (U) is always negative . It is a scalar quantity and it's SI unit is joule ( J ).