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.