Question

Why is gravitational mass equal to inertial mass?

Answer 1

Step 1: Let us suppose

Calculating the ratio of gravitational force

Consider bodies $A,A^{*}$with gravitational mass $M_g,M_{g^{*}}$.

Let the bodies be kept in the gravitational field of the earth of Mass M and distance R.

The gravitational force on both the bodies is:

$F=\frac{GM{M}_g}{R^2},F^{*}=\frac{GM{M}_{g^{*}}}{R^2}$

The ratio of force is, $\frac{F}{F^{*}}=\frac{M_g}{M_{g^{*}}}$

Step 2: Calculating the ratio of inertial force

Similarly consider bodies $A,A^{*}$ with inertial mass $M_i,M_{i^{*}}$

The force on these bodies in influence to acceleration due to gravity is

$F=M_{i}g,F^{*}=M_{i^{*}}g$

The ratio of force is, $\frac{F}{F^{*}}=\frac{M_i}{M_{i^{*}}}$

Step 3: Calculating the relation between inertial and gravitational mass

$\frac{M_g}{M_{g^{*}}}=\frac{M_i}{M_{i^{*}}}$

Therefore, this implies that gravitational mass is proportional to inertial mass ie $M_g\alpha M_i$.

Hence both gravitational and inertial mass are the same.