Energy in a Circular Orbit

Imagine that we have an object of mass m in a circular orbit around an object of mass M. An example could be a satellite orbiting the Earth. What is the total energy associated with this object in its circular orbit?

As usual, E = U + K.
U =
- G m M
R
        and         K = ½ mv2

The only force acting on the object is the force of gravity. Applying Newton's Second Law gives:

ΣF = ma
G m M
r2
=
mv2
r
Therefore:   mv2 =
G m M
r
  and   K = ½ mv2 =
G m M
2r

The kinetic energy is positive, and half the size of the potential energy.
E =
-G m M
r
+
G m M
2r
=
-G m M
2r

A negative total energy tells us that this is a bound system. Much like an electron is bound to a proton in a hydrogen atom with a negative binding energy, the satellite is bound to the Earth - energy would have to be added to each system to remove the electron or the satellite.