Absolute potential energy: The absolute gravitational potential energy of an object at a certain position is the work done by the gravitational force in displacing the object from that position to infinity where the force of gravity becomes zero.
The relation for the calculation of the work done by the gravitational force of potential energy = mgh, is true only near the surface of the earth where the gravitational force is nearly constant. But if the body is displaced through a large distance in space from, let, point 1 to N ( Fig. 4.10) in the gravitational field, then the gravitational force will not remain constant, since it varies inversely to the square of the distance.
In order to overcome this difficulty, we divide the distance between points 1 and N into small steps each of length ∆r so that the value of the force remains constant for each small step. Hence, the total work done can be calculated by adding the work done during all these steps. If r1 and r2 are the distances of points 1 and 2 respectively, from the center O of the earth (fig. 4.10), the work done during the first step i.e., displacing a body from point 1 to point 2 can be calculated as below.
The distance between the centre of this step and the center of the earth will be
The gravitational force F at the centre of this step is
Where m = mass of an object , M = mass of the earth and G = gravitational constant.
Squaring Eq. 4.12
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As this force is assumed to be constant during the interval ∆r, so the work done is