Question

# Why is gravitational mass equal to inertial mass?

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## Step 1: Let us supposeCalculating the ratio of gravitational forceConsider 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 forceSimilarly 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.

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