Question

Calculate the gravitational potential at a location which is from the surface of earth at height 4 times the radius of earth

Answer 1

Gravitational potential at a point is the work done to bring a unit mass from infinity to that point.

Consider a body of unit mass located at a distance x metres from the center of the earth of mass M. Now the distance between the body and the earth is x and then the gravitational force of attraction between them is,

$F=\dfrac {GM}{x^2}$

which is acting towards the earth by the unit mass.

Let the unit mass move by a small distance 'dx' towards earth by this gravitational force. Then, work done to move the unit mass by a small distance 'dx' is,

$dW=\dfrac {GM}{x^2}\ dx$

Then, the total work done to move the unit mass from infinity (x = ∞) to the location which is at a height 4 times the radius of the earth from the surface of the earth (x = R + 4R = 5R) is,

$\displaystyle W=\int\limits_{\infty}^{5R}\dfrac {GM}{x^2}\ dx\\\\\\W=GM\int\limits_{\infty}^{5R}x^{-2}\ dx\\\\\\W=GM\left [\dfrac {x^{-1}}{-1}\right]_{\infty}^{5R}\\\\\\W=-GM\left [\dfrac {1}{5R}-\dfrac {1}{\infty}\right]\\\\\\\boxed{W=-\dfrac {GM}{5R}}$

This work done is the gravitational potential at that point.