User:Spiruel/sandbox

$$\begin{align} & The\text{ angular velocity of a satellite in a geostationary orbit is the same as the angular velocity of the Earth}\text{.} \\ & \text{Therefore }\tfrac{v}{r}-\tfrac{2\pi }{T}=Surface\text{ }Velocity\text{ as }\psi satellite=\tfrac{v}{r}\text{ and }\psi planet=\tfrac{2\pi }{T}\text{ } \\ & \text{v = }\sqrt{\frac{GM}{r}}\text{ when gravitational force and centripetal force are balanced (we will be assuming stable, uniform circular orbit)} \\ & Substituting\text{ v = }\sqrt{\frac{GM}{r}}\text{ and clearing up so only one variable:} \\ & \frac{\sqrt{\frac{GM}{r}}}{r}-\tfrac{2\pi }{T}=\text{Vsurface} \\ & \frac{\sqrt{\frac{GM}{r}}}{r}=\text{Vsurface +}\tfrac{2\pi }{T} \\ & GMr=\text{ }\mathop{\text{Vsurface}}^{2}\text{ +}\tfrac{\mathop{4\pi }^{2}}{\mathop{T}^{2}} \\ & GMr=\text{ }\mathop{\text{Vsurface}}^{2}\text{ +}\tfrac{GM}{\mathop{r}^{3}} \\ & GMr-GM{{r}^{-3}}=\mathop{\text{Vsurface}}^{2} \\ & \sqrt{GM(r-{{r}^{-3}})}=\text{Vsurface} \\ \end{align}$$