8.16 Inertial and Gravitational Mass

Given two objects \(A\) and \(B\), how can we determine the ratio of the mass of \(A\) to the mass of \(B\). One way is to use Newton’s second law of motion. If we apply equal forces \(F\) on each of the two objects,
\(
F=m_A a_A \quad \text { and also } \quad F=m_B a_B \text {. }
\)
Thus, \(\quad \frac{m_A}{m_B}=\frac{a_B}{a_A}\)
or, \(\quad m_A=\frac{a_B}{a_A} m_B\).
This equation may be used to “define the mass” of an object. Taking the object \(B\) to be the standard kilogram \(\left(m_B=1 \mathrm{~kg}\right)\), mass of any object may be obtained by measuring their accelerations under equal force and using (i). The mass so defined is called inertial mass.

Another way to compare the masses of two objects is based on the law of gravitation. The gravitational force exerted by a massive body on an object is proportional to the mass of the object. If \(F_A\) and \(F_B\) be the forces of attraction on the two objects due to the earth,
\(
F_A=\frac{G m_A M}{R^2} \text { and } F_B=\frac{G m_B M}{R^2} .
\)
Thus, \(\frac{m_A}{m_B}=\frac{F_A}{F_B}\)
or, \(\quad m_A=\frac{F_A}{F_B} m_B\).
We can use this equation to “define the mass” of an object. If \(B\) is a standard unit mass, by measuring the gravitational forces \(F_A\) and \(F_B\) we can obtain the mass of the object A. The mass so defined is called gravitational mass. When we measure the mass using a spring balance, we actually measure the gravitational mass.