Question

A 2000-kg communication satellite is placed in the geosynchronous orbit around the Earth. Find its speed and altitude. Give the mass of the earth is 5.98x1024kg, the radius of the Earth is 6.38x106 meter.

Answer 1

The force of attraction between the satellite and the Earth:

$$F=G\frac{Mm}{(R+h)^2}$$

is equal to the centripetal force acting upon the satellite:

$$F=m\frac{v^2}{R+h},\rightarrow\\ \space\\ \frac{GM}{R+h}=v^2.$$

The geosynchronous orbit implies that the satellite is always above a certain point on the surface of our planet, which means that it rotates with the same angular speed as the earth:

$$\Omega=\omega,\\ \frac{2\pi}{T}=\frac{v}{R+h}.$$

Therefore, we have two equations. The solution gives

$$v=\sqrt[3]{\frac{2\pi MG}{T}}=3076\text{ m/s},\\ \space\\ h=\sqrt[3]{MG\bigg(\frac{T}{2\pi}\bigg)^2}-R=35.80\cdot10^6\text{ m}.$$

In these equations we used $T=23^\text{h}56^\text{min}4.1^\text{s}.$