The gravitational potential energy of a body at a point is defined as the amount of work done in bringing the body from infinity to that point against the field.
To calculate the gravitational potential energy of a mass m placed on the surface of a sphere. Assume uniform sphere of radius R and mass M. let a the body of mass m is placed at P, at a distance x from the centre of sphere. The force acting on this body, when it is at P is
The small work done in moving this mass through a small distance PQ = dx is
Thus the total work done in moving the mass ‘m’ from infinity to a point A, distance r from the centre of earth is
Thus, the potential energy of mass m at a distance r from the centre of the sphere of mass M is given by
and on the surface of earth it is
where hence U = Vm. So potential energy of a mass m = potential at that point × mass (m)
6.3.1 Gravitational field intensity and gravitational potential for solid and hollow sphere
(on the surface)
(on the surface)
Gravitational field intensity:
(At centre r = 0)
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