Question

Define gravitational potential energy. Derive expression for gravitational potential in the gravitational field of earth at distance $r$ from the centre of the earth.

Answer 1

The work obtained in bringing a body from infinity to a point in gravitational field is called gravitational potential energy.

Force of attraction between the earth and the object when an object is at the distance $a$ from the center of the earth.

$F=\frac{GmM}{x^2}$

Consider the $dW$ is the small amount of work done in bringing the body without acceleration through the very small distance $dx$

$dW=Fdx$

And total work done to bring the body from infinity to point $p$ which is at distance $r$ from the center of the earth.

$w={\int }_{\infty }^r\frac{GmM}{x^2}dx=-\frac{GMm}{r}$

Hence, the gravitational potential energy $=-\frac{GMm}{r}$