Gravitational potential energy: 1. Whenever two material bodies or two electrical charges are situated at a finite distance apart, the system possesses potential energy. The potential energy is positive if the bodies or charges are present against force of repulsion and negative if force between them is attractive. In case of two material bodies, the potential energy is as gravitational potential energy, while in case of electrical charges, the potential energy is called as electrical potential energy. Hence we shall calculate the gravitational potential energy of a body of mass m situated at a distance r from earth.
2. The gravitational potential energy of a body at a point is defined as the work done in bringing the body from infinity to that point.
3. When we bring a body from outside into a gravitational field, the field will itself do work on the body. Because this work is obtained (and not done by the agent in bringing the mass), hence the gravitational potential energy is always negative.
4. When the body is at infinity, its gravitational potential energy is zero. It is called as zero level of potential energy.
5. Let us assume earth to be a uniform sphere of radius Re and mass Me. Our aim is to calculate the G.P.E of the body of mass m lying at point A, situated at a distance r from the centre O of the earth (shown in following figure). According to definition, the gravitational potential energy U of the body at point A = work done by the field in bringing the body from ∞ to point A = W.
6. Let the body be at some point P at any instant such that OP = x. Then gravitational force on the body at P is given by:
Now, small work done in moving the body through a very small distance PQ = dx, is given by
dW = Fdx = (GMem/x2)dx ……………….(1)
Therefore, total work done by the field in moving the body from ∞ to r, is:
W= ∫∞r(GMem/x2)dx = GMem ∫∞r(x-2)dx
=GMem[-1/x]∞r = -GMem[(1/r)-(1/∞]=-GMem/r …………(2)
Hence gravitational potential energy U of the body of the mass m distance r from the centre of earth is: