Question

A satellite moving in a circular orbit 500 km above the Earth's surface. Determine the satellite's orbital period.

Answer 1

For it to be orbiting in a circle, the gravitational force must be equal to the centripetal force otherwise required to keep it going in such a circle. Let the mass of the satellite be m, the mass of the earth be M, the radius of the earth be \(r_e\), the distance above the earth's surface be d, the gravitational constant be g, orbital period be T, and the velocity of the satellite be v. Take note that this means that the distance to the center of mass of the earth is \(r_e+d\). Solving for velocity gives the following. \[\vec F_g = \vec F_c \Rightarrow G\frac{Mm}{(r_e+d)^2}=\frac{mv^2}{r_e+d} \Rightarrow v = \sqrt{\frac{GM}{r_e+d}}\]Now, we can express the following by the definition of constant velocity if we consider one full trip around the orbit.\[v = \frac{\Delta x}{\Delta t} \Rightarrow v =\frac{2\pi (r_e+d)}{T} \Rightarrow T=\frac{2\pi (r_e+d)}{v}\]Plugging in our found value for velocity gives us the following.\[T=\frac{2\pi(r_e+d)}{\sqrt{\frac{GM}{r_e+d}}} \Rightarrow \boxed{\displaystyle 2\pi \sqrt{\frac{(r_e+d)^3}{GM}}}\]