The negative sign indicates that the work has to be done on the body from point 1 to 2 because displacement is opposite to gravitational force. Putting the value of ∆r, we get

gravitational-force-equation-01

Similarly the work done during the second step in which the body is displaced from point 2 to 3 is

gravitational-force-equation-02

And the work done in the last step is

gravitational-force-equation-03

Hence, the total work done in displacing a body from point 1 to N is calculated by adding up the work done during all these steps.

gravitational-force-equation-04 On simplification, we get

gravitational-force-equation-05

Tid-bits

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If the point N is situated at an infinite distance from the earth, so

gravitational-force-equation-06

Therefore, general expression for the gravitational potential energy  of a body situated at distance r from the centre of earth is

gravitational-force-equation-07

This is also known as the absolute value of gravitational potential energy of a body at a distance r from the centre of the earth.

Note that when r increases, U becomes less negative i.e., U increases. It means when we raise a body above the surface of the earth its P.E. increases. The choice of zero point is arbitrary land only the difference of P.E. From one point to another is significant, wether we consider the surfaoe of the earth or the point at infinity as zero P.E. reference, the change in P.E. as we move a body above the surface of the earth, will always be positive.

Now the absolute potential energy on the surface of the earth is found by putting r=R (Radius of the Earth)

gravitational-force-equation-08

The negative sign shows that the Earth’s gravitational field for mass m is attractive. The above expression gives the work or the energy required to take the body out of the Earth’s gravitational field, where its potential energy with respect to Earth is zero.

gravitational-force-equation-09