 # Why Is Gravitational Mass Equal To Inertial Mass?

Following is the derivation that explains why gravitational mass is equal to the inertial mass.

Consider two bodies B and B’ with gravitational mass as Mg and Mg’ respectively. These bodies are kept in the gravitational field of the Earth which has M as the gravitational mass at a distance R from the Earth.

Following is the force experienced by the body B:

$F=frac{GMM_{g}}{R^{2}}$ (equ. 1)

The force experienced by the body B’ is:

$F=frac{GMM_{g}’}{R^{2}}$ (equ. 2)

Dividing equ. 1 and equ. 2, we get

$frac{F}{F’}=frac{GMM_{g}}{R^{2}}timesfrac{R^{2}}{GMM_{g}}=frac{M_{g}}{M_{g}’}$ (equ. 3)

Consider two bodies with inertial mass Mi and Mi’ respectively. These bodies are influenced by the acceleration due to gravity g making both the bodies fall downward in the vacuum from the same height.

F = Mig

F’ = Mi’g

Again dividing F and F’, we get

$frac{F}{F’}=frac{M_{i}g}{M_{i}’g}=frac{M_{i}}{M_{i}’}$ (equ.4)

From equ. 3 and equ. 4, we get

$frac{Mg}{Mg’}=frac{M_{i}}{M_{i}’}$

∴ Mg ∝ Mi

Thus, it can be concluded that since the gravitational mass of the body is proportional to the inertial mass of the body, the gravitational mass and inertial mass are equal.