How much work must be done to push 1000kg communication satellite from a low orbit with h=300km, where its released by spaced shuttle to a geosynchronised orbit.

Question

anonymous · Answer

Work done is the amount of energy required to move a body from one point to another.  In this case the work is done against the gravitational field.  So to do this problem you only need to determine the difference in potential energy between the two different obits, the difference being the work done against gravity.  

The corresponding formula is then $$P.E. = \frac{GM_{E}m_{s}}{r}$$ where $M_{E}$ is the mass of the earth, $m_{s}$ the mass of the satellite, and $r$ is the distance from the centre of the earth (Recall that the earths radius is 6,370 km, so a height of h = 300 km above the surface is 6,670 km form the centre of the earth).

You will also have to calculate the distance of geostationary orbit.  this orbit occurs when the the orbital period of the satellite matches the rotation rate of the earth (84,600 seconds).  For this you will want to use the formula $$T^{2}=\frac{4\pi^{2}}{GM_{E}}R^{3}$$where $T$ is the orbital period (=84,600 seconds) and $R$ is the distance from the centre of Earth.  Therefore the work done is simply $$W=\Delta P.E. = GM_{E}m_{s}\left(\frac{1}{r}-\frac{1}{R}\right)$$