Question

The moon takes about 27.3 days to revolve round the earth in a nearly circular orbit of radius 3.84 × 10⁵ km/ Calculate the mass of the earth from these data.

Answer 1

Time period of rotation of the Moon around the Earth is given by
$T=2\mathrm{\pi }\sqrt{\frac{r^3}{\mathrm{G}M}}$,
where r is the distance between the centres of the Earth and Moon and m is the mass of the Earth.

$$\begin{gathered} \mathrm{Now},27.3=2\times 3.14\sqrt{\frac{{\left(3.84\times {10}^5\right)}^3}{6.67\times {10}^{-11}M}} \\ \Rightarrow 27.3\times 27.3=\frac{2\times 3.14\times {\left(3.84\times {10}^5\right)}^3}{6.67\times {10}^{-11}M} \\ \Rightarrow M=\frac{2\times {\left(3.14\right)}^2\times {\left(3.84\right)}^3\times {10}^{15}}{3.335\times {10}^{-11}\times {\left(27.3\right)}^2} \\ =6.02\times {10}^{24}\mathrm{kg} \end{gathered}$$
∴ The mass of the Earth is found to be 6.02 × 10²⁴ kg.