The moon takes about 27.3 days to revolve round the earth in a nearly circular orbit of radius 3.84 × 10^{5} km/ Calculate the mass of the earth from these data.

#### Solution

Time period of rotation of the Moon around the Earth is given by \[T = 2\pi\sqrt{\frac{r^3}{GM}}\]

where r is the distance between the centres of the Earth and Moon and m is the mass of the Earth.

\[Now, 27 . 3 = 2 \times 3 . 14\sqrt{\frac{\left( 3 . 84 \times {10}^5 \right)^3}{6 . 67 \times {10}^{- 11} M}}\]

\[ \Rightarrow 27 . 3 \times 27 . 3 = \frac{2 \times 3 . 14 \times \left( 3 . 84 \times {10}^5 \right)^3}{6 . 67 \times {10}^{- 11} M}\]

\[ \Rightarrow M = \frac{2 \times \left( 3 . 14 \right)^2 \times \left( 3 . 84 \right)^3 \times {10}^{15}}{3 . 335 \times {10}^{- 11} \times \left( 27 . 3 \right)^2}\]

\[ = 6 . 02 \times {10}^{24} kg\]

∴ The mass of the Earth is found to be 6.02× 10^{24} kg.