# Gravitational Mass

## Gravitational Mass

Gravitational mass is defined as the mass of the body by virtue of the gravitational force exerted by the earth. The gravitational mass of a body is related to gravitational pull on the body. It is the mass of the material of body, which determines the gravitational pull acting upon it. If M is the mass of the earth and R is the radius, then gravitational pull on a body of mass $${{m}_{g}}$$ is given by:

$$F=\frac{GM{{m}_{g}}}{{{R}^{2}}}\Rightarrow {{m}_{g}}=\frac{F}{GM/{{R}^{2}}}=\frac{F}{I}$$,

Here, $${{m}_{g}}$$ is the gravitational mass of the body, if $$I=1$$ then$${{m}_{g}}=F$$.

Thus the gravitational mass of a body is defined as the gravitational pull experienced by the body in a gravitational field of unit intensity.

Example:

Mean distance between the sun and earth is $$1.5\times {{10}^{11}}m$$ and the time taken by earth to complete one orbit around the sun is 1 year. Then find the mass of the sun?

Explanation:

Given,

Radius$$\left( r \right)=1.5\times {{10}^{11}}m$$,

$$T=1\,year\,=\,365\,days\,=\,\left( 365\times 24\times 60\times 60 \right)\sec$$,

$$F=\frac{{{M}_{e}}{{v}^{2}}}{r}$$  ;  $$F={{F}_{c}}$$,

Now,

$$\frac{G{{M}_{e}}{{M}_{s}}}{{{r}^{2}}}=\frac{{{M}_{e}}{{v}^{2}}}{r}=\frac{{{M}_{e}}{{\left( 2\pi r \right)}^{2}}}{r{{T}^{2}}}$$,

$$\Rightarrow {{M}_{s}}=\frac{4{{n}^{2}}{{r}^{3}}}{G{{T}^{2}}}=\frac{4\times {{\left( 1 \right)}^{2}}\times {{\left( 1.5\times {{10}^{3}} \right)}^{3}}}{6.67\times {{10}^{-11}}\times {{\left( 31536000 \right)}^{2}}}=2\times {{10}^{30}}kg$$,

$$\therefore \,\,{{M}_{s}}=2\times {{10}^{30}}kg.$$