Find the approximate radius of the circular orbit of a satellite moving with an angular speed equal

Question

Find the approximate radius of the circular orbit of a satellite moving with an angular speed equal to the angular speed of the earth's rotation. (Mass of the earth $= 6 	imes 10^{24}~kg$, $G = 6.67 	imes 10^{-11}~N~m^2~kg^{-2}$)

Professor PC Thomas Classes · Answer

Angular speed $\omega = \frac{2\pi}{24 	imes 3600}~rad~s^{-1}$. From $\frac{GMm}{r^2} = m\omega^2 r$, we get $r = (\frac{GM}{\omega^2})^{1/3} = (\frac{6.67 	imes 10^{-11} 	imes 6 	imes 10^{24}}{(7.27 	imes 10^{-5})^2})^{1/3} \approx 4.23 	imes 10^7~m = 4.23 	imes 10^4~km$.