**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,

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

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

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