Question

What is the difference between the gravitational potential energy and gravitational potential? Derive an expression for gravitational potential energy.

Answer 1

Hint: Gravitational potential at any point is defined as the work done in moving unit mass from infinity to that point. It is a vector quantity that represents the direction of motion. Mathematically it is expressed as $V = - \dfrac{{GM}}{r}$
Gravitational potential energy on the other hand is the energy required by a body due to a shift in its position in a gravitational field
So, we shall integrate the work done on a infinitesimally small distance to derive the complete expression for the gravitational potential energy.

Complete step by step solution:
Gravitational potential at any point is defined as the work done in moving unit mass from infinity to that point. It is a vector quantity that represents the direction of motion.
Gravitational potential energy on the other hand is the energy required by a body due to a shift in its position in a gravitational field
Derivation of the gravitational potential energy
Let us consider an object of mass M and a unit mass m placed at infinity. Work done in moving it a small distance dx without acceleration is $dW = Fdx$ .
The force between both the masses is given by $F = \dfrac{{GMm}}{{{x^2}}}$ .
Substituting in the work equation,
$dW = \dfrac{{GMm}}{{{x^2}}}dx$
Integrating this equation from r to $\infty $ , we get
$\int\limits_r^\infty {dW = \int\limits_r^\infty {\dfrac{{GMm}}{{{x^2}}}dx} } $
$W = - [\dfrac{{GMm}}{r}]_r^\infty $
Further solving this,
$W = - [\dfrac{{GMm}}{\infty }] + [\dfrac{{GMm}}{r}]$
$ \Rightarrow W = \dfrac{{GMm}}{r}$
This is the desired expression of the gravitational potential energy.

Note:
The distance is taken from the centers of the two bodies along the line of action of force. When the radius of the bodies is appreciably large, we take into account their radii as well while calculating the force. Else we take point masses and neglect their radii. The value of G is universal and does not depend on any factors like temperature, pressure and hence it is also called the universal gravitational constant.