## College Physics (4th Edition)

The orbital radius of a geosynchronous satellite is $42,200~km$
When a satellite orbits a planet, the gravitational force provides the centripetal force to keep the satellite moving in a circle. Let $M_p$ be the mass of the planet and let $M_s$ be the mass of the satellite. We can find an expression for the angular speed of a satellite: $\frac{G~M_p~M_s}{R^2} = \frac{M_s~v^2}{R}$ $v = \sqrt{\frac{G~M_p}{R}}$ We can find an expression for a satellite's orbital period $P$: $P = \frac{2\pi~R}{v}$ $P = \frac{2\pi~R}{\sqrt{\frac{G~M_p}{R}}}$ $P = 2\pi~\sqrt{\frac{R^3}{G~M_p}}$ We can find use the expression for a satellite's orbital period $P$ to find the orbital radius $R$: $P = 2\pi~\sqrt{\frac{R^3}{G~M_p}}$ $R = (\frac{P^2~G~M_p}{4\pi^2})^{1/3}$ $R = [~\frac{(86,400~s)^2~(6.67\times 10^{-11}~m^3/kg~s^2)~(5.97\times 10^{24}~kg)}{4\pi^2}~~]^{1/3}$ $R = 42,200,000~m$ $R = 42,200~km$ The orbital radius of a geosynchronous satellite is $42,200~km$