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.

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