5.0 Gravitational potential energy

5.1 Gravitational potential energy for a system of particles
5.2 Gravitational potential energy of a body on earth's surface

The gravitational potential energy of a body at a point in a gravitational field of another body is defined as the amount of work done in bringing the given body from infinity to that point.

Consider a body of mass $M$. The force experienced by mass $m$ placed at a distance $r$ is given by, $$F = \frac{{GMm}}{{{r^2}}}$$

Work done for displacement $dr$ is given by,

$$dW = \overrightarrow F .\,d\overrightarrow r $$

Gravitational potential energy is defined as the amount of work done in bringing the given body from infinity to any point $P$ is,

$$dW = - \frac{{GMm}}{{{r^2}}}dr$$

Also, we know.$$W = - dU$$ So, $$dU = \frac{{GMm}}{{{r^2}}}dr$$

Integrating with proper limits we get,

$$\int\limits_{{U_\infty }}^U {dU} = GMm\int\limits_\infty ^r {\frac{{dr}}{{{r^2}}}} $$$$\left[ U \right]_{{U_\infty }}^U = GMm\left[ { - \frac{1}{r}} \right]_\infty ^r$$$$U - {U_\infty } = - GMm\left( {\frac{1}{r} - \frac{1}{\infty }} \right)$$ As $\left( {{U_\infty } = 0} \right)$, $$U = - \frac{{GMm}}{r}$$