Animated Solution for Physics - Gravitation: If one wants to remove all the mass of the earth to infinity in order to break it up completely. The amount of energy that needs to be supplied will be $\frac{xG{M}^2}{5R}$, where $x$ is ………. (Round off to the nearest integer) ($M$ is the mass of earth, $R$ is the radius of earth and $G$ is the gravitational constant.)

The problem asks us to calculate the energy required to completely dismantle the Earth and send all its constituent mass to infinity. This is a classic application of the concept of self-gravitational potential energy.

Imagine the Earth not as a single solid object, but as a collection of infinitely many tiny particles. When these particles are infinitely far apart, their gravitational potential energy is zero. As gravity pulls them together to form a solid sphere, the gravitational field does positive work. Because the system does work, it loses potential energy. Thus, the assembled sphere has a negative potential energy.

This energy is known as the binding energy or self-energy of the sphere. For a uniform solid sphere of mass $M$ and radius $R$, the self-energy $U$ is given by the standard formula:

To reverse this process—that is, to break the Earth apart and send every piece back to infinity—we must do work against the attractive force of gravity. The energy we need to supply must be exactly equal in magnitude to the binding energy, but positive.

The problem states that the energy required to be supplied is $\frac{xG{M}^2}{5R}$. We simply equate our theoretical result to this given expression:

By comparing both sides, the $\frac{G{M}^2}{5R}$ terms cancel out beautifully, leaving us with:

This elegant result highlights the power of understanding the energy configurations of continuous mass distributions. Always remember to check whether you are dealing with a solid sphere or a hollow shell, as their self-energies differ!