Question

Calculate the mass and density of the earth.
Given that.
Gravitational constant G = 6.67 x 10^-11 n-m^2/kg^2
The radius of the earth = 6.37 x 10^6 m
g = 9.8 m/s^2

Answer 1

$$\begin{gathered} \mathrm{Acceleration}\mathrm{due}\mathrm{to}\mathrm{gravity}: \\ \mathrm{g}=\frac{\mathrm{GM}}{{\mathrm{R}}^2} \\ \mathrm{Mass}\mathrm{of}\mathrm{earth}: \\ \mathrm{M}=\frac{{\mathrm{gR}}^2}{\mathrm{G}} \\ \mathrm{Mass}\mathrm{density}: \\ \frac{\mathrm{M}}{\mathrm{V}}=\frac{\frac{{\mathrm{gR}}^2}{\mathrm{G}}}{{\displaystyle \frac{4}{3}}{\mathrm{\pi R}}^3} \\ =\frac{{\mathrm{gR}}^2}{\mathrm{G}}\times \frac{3}{4{\mathrm{\pi R}}^3} \\ =\frac{3\mathrm{g}}{4\mathrm{\pi GR}} \\ =\frac{3\times 9.8}{4\mathrm{\pi }\times 6.67\times {10}^{-11}\times 6.37\times {10}^6} \\ =5506.5\mathrm{kg}/{\mathrm{m}}^3 \end{gathered}$$