Question

A body of mass $m$ is kept at a small height $h$ above the ground. If the radius of the earth is $R$ and its mass is $M$, the potential energy of the body and the earth system (with being the reference position) is:
A) $\dfrac{{GMm}}{R} + mgh$
B) $ - \dfrac{{GMm}}{R} + mgh$
C) $\dfrac{{GMm}}{R} - mgh$
D) $ - \dfrac{{GMm}}{R} - mgh$

Answer 1

Hint: The potential energy is the energy at the rest of the body. First calculate the potential energy on the surface of the earth and then the additional potential energy of the small body at a certain height is also considered. Then the sum of the potential energy gives the total potential energy.

Formula used:
The potential energy on the surface of the earth is given by
$U = - \dfrac{{GMm}}{R}$
Where $U$ is the potential energy of the surface of the earth, $G$ is the gravitational force of the earth, $M$ is the mass of the earth and $m$ is the mass of the body.

Complete step by step solution:
It is given that the
Mass of the body is $m$
Height of the body is $h$
Radius of the earth is $R$
Mass of the earth is considered as $M$
By taking the potential energy formula,
$U = - \dfrac{{GMm}}{R}$
Since the body is placed at a small height $h$, the additional potential energy at a certain height must be considered.
The potential energy at a height $h$ is $mgh$ .
Hence the total potential energy is the sum of the potential energy of the small body and the earth’s gravity.
$\Rightarrow$ $U = - \dfrac{{GMm}}{R} + mgh$

Thus the option (B) is correct.

Note: The additional potential energy of the body is added with that of the potential energy of the earth. This is because the height of the body is very much less than the distance of the object from the centre of the earth and hence the additional force is considered.