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

Question

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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.

$F = \frac{G m M}{x ^{2}}$

Consider the $d W$ is the small amount of work done in bringing the body without acceleration through the very small distance $d x$

$d W = F d x$

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

$w = \int_{\infty}^{r} \frac{G m M}{x ^{2}} d x = - \frac{G M m}{r}$

Hence, the gravitational potential energy $= - \frac{G M m}{r}$

Answer 2

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{G m M}{x ^{2}}

Consider the

d W

is the small amount of work done in bringing the body without acceleration through the very small distance

d x

d W = F d x

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{G m M}{x ^{2}} d x = - \frac{G M m}{r}

Hence, the gravitational potential energy

= - \frac{G M m}{r}