Circular Orbits
The Earth goes round the sun. There are satellites that go round the Earth. We can use our knowledge of the law of gravitation and centripetal force to understand these motions.
Consider a satellite going round the Earth in a circular orbit. There is only one force acting on the satellite - the gravitational force from the Earth. There are forces from the sun and the moon, but these are much weaker if the satellite is close to the Earth.
If the satellite is going round in a circle, there must be a centripetal force. So the gravitational force must be the centripetal force.
Therefore, GMm/r2 = mv2/r, where M is the mass of the Earth, m the mass of the satellite, r its distance from the centre of the Earth, and v its velocity.
Notice that m appears on both sides. It can be divided out. Then m does not appear in the relation. The remaining variables are v and r. E.g. this allows us to find the radius if we know M and v.
We can get some useful relations from GMm/r2 = mv2/r.
If we multiply both sides by r/2m, we get GMm/2r = mv2/2. The right hand side is the kinetic energy.
Lets write v in terms of the circumference 2πr and period T of the orbit. So v = 2πr/T. Substituting, we get GMm/r2 = m(2πr/T)2/r. Simplifying, we find T2 = 4π2r3/GM. This tells us that T2 is proportional to r3.
Copyright 2010 by Kai Hock. All rights reserved.
Last updated: 28 September 2010.