# Equivalence of Inertial and Gravitational Masses

## Equivalence of Inertial and Gravitational Masses

The mass of a body is the quantity of matter possessed by a body. There are two different concepts about the mass of a body. Those are Inertial and Gravitational Masses.

Inertial mass of a body is related to its inertia of linear motion, and is defined by Newton’s Second Law of Motion. Gravitational mass of a body is related to gravitational pull on the body and is defined by Newton’s Laws of Gravitation.

Consider two bodies A and B of gravitational masses mG and m’G respectively. Let these be placed at equal distance R from a third body C. Let M be the gravitational mass of the body C. According to Newton’s Law of Gravitation, the gravitational force exerted by C on A is given by:

$$F=\frac{GM{{m}_{G}}}{{{R}^{2}}}$$.

And the gravitational force exerted by C on B is:

$${{F}^{‘}}=\frac{GM{{m}_{G}}^{‘}}{{{R}^{2}}}$$.

Therefore,

$$\frac{F}{{{F}^{‘}}}=\frac{GM{{m}_{G}}/{{R}^{2}}}{GM{{m}_{G}}^{‘}/{{R}^{2}}}=\frac{{{m}_{G}}}{{{m}_{G}}^{‘}}$$ … (1)

Let, mi and m’i be the inertial masses of bides A and B, respectively. If both the bodies are allowed to fall in vacuum from the same place under the action of gravity, where the acceleration due to gravity is g, then F = mig and F’ = m’ig.

Therefore,

$$\frac{F}{{{F}^{‘}}}=\frac{{{m}_{i}}g}{{{m}_{i}}^{‘}g}=\frac{{{m}_{i}}}{{{m}_{i}}^{‘}}$$ … (2)

From equations (1) and (2), we have:

$$\frac{{{m}_{G}}}{{{m}_{G}}^{‘}}=\frac{{{m}_{i}}}{{{m}_{i}}^{‘}}$$.

It means that the gravitational mass of a body is proportional to its Inertial mass. In fact, Inertial and Gravitational masses of a body are identical. The idea of equivalence of Inertial and Gravitational mass of a body led Einstein to predict the general theory of relativity which is the modern theory of gravitation.