Gravitational Potential Due to Point Mass

Sam was watching Interstellar movie.

He saw how gravity and time affect our universe.

In the end part of the movie, he watched that the Cooper get fell into a massive black hole.

In front of the black hole, Cooper looks like a point-sized.

And the gravitational force of a black hole pulling the Cooper inside it.

This happens due to the massive size of the black hole and its gravitational potential energy which pulling the point size mass of Cooper towards it.

Let's understand the effect of gravitational potential due to a large mass on a small mass object.

Suppose you have a point-sized body of mass $M$.

The gravitational field of mass $M$ is attractive in nature and work towards the centre of the body all around it.

You have the other test mass of mass $m$, which is at a distance infinity.

We have to bring that mass $m$ at the point $P$ which is at distance $R$ from the mass $M$.

To bring the mass $m$ under the gravitational influence of $M$, you have to do some work.

Let the distance between $M$ and $m$ is $r$.

Let we displace the mass $m$ by a distance $dr$ towards the mass $M$.

The work done to displace the mass $m$ by $dr$ distance is given as dot product of gravitational force and displacement.

The gravitational field is towards the $M$ and $m$ is also moving towards the mass $M$. So, the angle between them is zero.

Put the value of the gravitational force(F).

Now integrate the equation from infinity to $R$ because mass $m$ is carried from infinity to the point $P$.

Here, $G$, $M$ and $m$ are constant. So, they are taken out of the integration.

Gravitational potential(V) is the ratio of work done(W) to the mass(m) of the object.

Gravitational potential of a point mass to bring it to the point $P$ is given as

Unit of the gravitational potential is J/kg.

Revision

The gravitational potential for point mass is defined as the external work to be done to bring test mass from infinity to a particular point under the influence of point mass per unit test mass.

Gravitational potential(V) due to point mass is written as-

The End