Proof That Body in Circular Orbit Has Total Energy That is Half of Its Gravitational Potential Energy
That a body in orbit around another body has negative total energy is quite important. It allows that bound orbits are possible, applies to any system where the force maintaining the orbit is an inverse square law. The obvious examples are planetary motion and the orbit of electrons around the nucleus.
Suppose then that an object of mass
orbits a central body of mass
in a circular orbit of radius
The (attractive) gravitational force between them is
The centripetal force is
where
is the speed of the orbiting body.
Equating these two gives
(1)
The total energy of the body is the sum of the positive
and the negative
Substituting (1) into this expression gives
It is important to realise that this example only applies for circular orbits. If the orbit is not perfectly circular, then energy is constant being exchanged between gravitational and kinetic energy, so obviously the total energy, which is constant, cannot be half of the constantly changing gravitational potential energy.