Question

assuming the earth to be homogenous sphere of radius R,its density in terms of G(constant of gravitation) and g(accceleration due to gravity on the surface of the earth) is

Answer 1

$$\begin{gathered} \mathrm{Using}\mathrm{the}\mathrm{relation}, \\ \mathrm{g}=\frac{\mathrm{GM}}{{\mathrm{R}}^2} \\ \mathrm{Or}\mathrm{M}=\frac{{\mathrm{gR}}^2}{\mathrm{G}}---\left(\mathrm{i}\right) \\ \mathrm{And},\mathrm{Mass}=\mathrm{Volume}\times \mathrm{density} \\ \mathrm{M}=\frac{4}{3}{\mathrm{\pi R}}^3\times \mathrm{\rho } \\ \mathrm{or}\mathrm{\rho }=\frac{3\mathrm{M}}{4{\mathrm{\pi R}}^3} \\ \mathrm{Using}\left(\mathrm{i}\right), \\ \mathrm{\rho }=\frac{3\times \frac{{\mathrm{gR}}^2}{\mathrm{G}}}{4{\mathrm{\pi R}}^3}=\frac{3\mathrm{g}}{4\mathrm{\pi RG}} \end{gathered}$$