(a)[2]
Explain how a satellite can remain in a circular orbit around a planet.
(b)[4]
Treat the Earth and the Moon as uniform spheres isolated in space. The Earth has radius $R$ and mean density $\rho$. The Moon, of mass $m$, moves in a circular orbit around the Earth at radius $nR$, as shown in Fig. 1.1. The Moon completes one orbit of the Earth in time $T$. Show that the mean density $\rho$ of the Earth is given by the expression $\rho = \frac{3\pi n^3}{G T^2}$.
(c)[3]
The Earth’s radius $R$ is $6.38 \times 10^3\ \text{km}$ and the centre-to-centre distance between the Earth and the Moon is $3.84 \times 10^5\ \text{km}$. The orbital period $T$ of the Moon about the Earth is $27.3$ days. Use the expression in (b) to calculate $\rho$.