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
| ![](../blackline.gif) |
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
| ![](../blackline.gif) |
r2
|
|
= |
mv2
| ![](../blackline.gif) |
r
|
|
Therefore: mv2 |
= |
G m M
| ![](../blackline.gif) |
r
|
|
  and K |
= |
½ mv2 |
= |
G m M
| ![](../blackline.gif) |
2r
|
|
The kinetic energy is positive, and half the size of the potential energy.
E |
= |
-G m M
| ![](../blackline.gif) |
r
|
|
+ |
G m M
| ![](../blackline.gif) |
2r
|
|
= |
-G m M
| ![](../blackline.gif) |
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.