Mass:

The mass of a body is defined as the quantity of matter possessed by the body. There are two different concepts about the mass of a body.

(i) Gravitational Mass:

The gravitational mass of a body is related to the gravitational force of attraction on the body. It is defined by Newton’s law of gravitation.

Let us consider a body of mass $m$ is placed on the surface of the earth of mass $M$ and radius $R$. So the gravitational force of attraction by the earth on the body is given by,

$F=G\frac{Mm}{R^2}$,

where $G$ is the gravitational constant.

Or, $m=\frac{F}{\frac{GM}{R^2}}$

or, $m=\frac{F}{E}$

where $E=\frac{GM}{R^2}$ is the gravitational field intensity on the surface of the earth.

Now if we put $E=1$ then $m=F$.

Hence the gravitational mass of a body is equal to the gravitational force of attraction experienced by the body in a gravitational field of unit intensity.

(ii) Inertial Mass:

The inertial mass of a body is defined by Newton’s second law of motion.

Let us consider an external force $F$ is applied on a body of mass $m$, then the body is moving with an acceleration $a$. According to Newton’s second law of motion,

$F=m\cdot{a}$.

Now if $a=1$ then $F=m$.

So the inertial mass of a body is equal to the external force when unit acceleration is produced in the body.

According to the special theory of relativity, the inertial mass of a body increases with the increase of its velocity. The inertial mass of a body moving with the velocity $v$ is given by,

$\displaystyle{m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}}$

where $m_0$ is the rest mass of the body and $c$ is the velocity of light in the vacuum.

So the properties of gravitational mass and inertial mass are the same. Thus inertial mass and gravitational mass of a body are identical.