Let’s consider a system of two masses consisting of a massive body *M* and a small mass *m* separated by distance *r*. Let’s begin with them separated at infinite distance apart and both at rest. By our choice of reference point, the GPE of this system is zero at this stage.

The small *m* is now accelerating towards *M* due to , the gravitational pull *M* exerts on *m*. By the time *m* arrives at the point where , it would have gained a certain amount of KE, thanks to the work done by the gravitational pull so far. This amount of KE can be derived from the area under the *F*–*r* graph between and .

Now, this amount of KE gained is also the amount of GPE that has been lost due to the mass moving from to . Since the GPE when is 0, the GPE when must be .

Hence, the amount of GPE for two masses *M*_{1} and *M*_{2} at a distance *d* apart, is given bythe formula

The formula reminds us of the following points:

- GPE is zero when
*d*is ∞. We have chosen the reference point for GPE to be zero when the two masses are infinitely far apart. - GPE = 0 is the highest possible GPE. The closer together the two masses, the lower (the more negative) the GPE.
- GPE is always negative.

Who owns the GPE?

Technically, (where *d* is the centre-to-centre distance between *M*_{1} and *M*_{2}) belongs to both *M*_{1} and *M*_{2} as a system. Gain/loss in GPE of the system will manifest as loss/gain in KE of both masses.

But if the two masses are very unequal in size, then we usually speak of (where *r* is the distance of *m* from the centre of *M*) as if theGPE that belongs only to *m*. From Principle of Conservation of Momentum, we know that the change in *M*’s velocity is always negligible compared to the change in *m*’s velocity. So any loss/gain in the GPE of the system manifests completely as gain/loss of the KE of *m* only.

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**Video Explanation**