State Newton’s law of gravitation in words.
A satellite of mass $m$ travels in a circular orbit of radius $r$ about a planet of mass $M$. It may be assumed that the planet and the satellite are uniform spheres that are isolated in space. Show that the linear speed $v$ of the satellite is given by the expression $v = \sqrt{\frac{GM}{r}}$, where $G$ is the gravitational constant. Explain your working.
Two moons A and B travel in circular orbits around a planet, as shown in Fig. 1.1. Moon A has orbital radius $r_A$ of $1.3 \times 10^8\,\text{m}$, linear speed $v_A$ and orbital period $T_A$. Moon B has orbital radius $r_B$ of $2.2 \times 10^{10}\,\text{m}$, linear speed $v_B$ and orbital period $T_B$.
Determine the ratio $\frac{v_A}{v_B}$ between the speeds.
Determine the ratio $\frac{T_A}{T_B}$ between the periods.
The planet rotates about its own axis with angular speed $1.7 \times 10^{-4}\,\text{rad s}^{-1}$. Moon A remains directly above the same point on the planet’s surface all the time. Determine the orbital period $T_B$ of moon B.