What are the orbital period of the Moon around the Earth and the rotational period of the Moon about its own axis?

Two small,equal masses are attached by a light rod. This object orbits a planet; the length of the rod is smaller than the radius of the orbit,but not negligible. The rod rotates about its axis in such a way that it remains vertical with respect to the planet. Determine the state of the rod (tension or compression) and the stability of the equilibrium with respect to a small angular perturbation.

Who decides the upward direction in an artificial satellite?

$A$ satellite of mass $m$ revolves around the earth of radius $R$ at a height $x$ from its surface. If $g$ is the acceleration due to gravity on the surface of the earth,the orbital speed of the satellite is

$A$ spaceship orbits around a planet at a height of $20 \, km$ from its surface. Assuming that only the gravitational field of the planet acts on the spaceship, what will be the number of complete revolutions made by the spaceship in $24 \, hours$ around the planet? [Given: Mass of planet $= 8 \times 10^{22} \, kg$, Radius of planet $= 2 \times 10^6 \, m$, Gravitational constant $G = 6.67 \times 10^{-11} \, N \cdot m^2/kg^2$]

Given the radius of Earth $R$ and the length of a day $T$,the height of a geostationary satellite is: [$G$ = Gravitational Constant,$M$ = Mass of Earth]