# 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

$$\frac{F}{F’}=\frac{GMM_{g}}{R^{2}}\times\frac{R^{2}}{GMM_{g}}=\frac{M_{g}}{M_{g}’}$$ (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.

Was this answer helpful?

4.5 (1)

(1)
(0)

#### Choose An Option That Best Describes Your Problem

Thank you. Your Feedback will Help us Serve you better.