Deriving Total Energy of a Binary System

[AnniB]

Homework Statement

Beginning with the kinetic and potential energies of two objects with masses m₁ and m₂, show that the total energy of a binary system is given by:
E=$$\frac{1}{2}$$$$\mu$$v² - G$$\frac{M\mu}{r}$$

Homework Equations

The one given
K =$$\frac{1}{2}$$mv²
U = G$$\frac{Mm}{r}$$

The Attempt at a Solution

I feel like this should be more of an explanation answer than much of an actual derivation, but I'm not sure if my logic follows correctly.

Assuming that both objects are orbiting in a circle around their common center of mass, both objects will have a radius r and a velocity v (since at any given orbit distance all objects have the same velocity - unless I'm remembering that wrong?). Because of this, we can treat the two masses as one mass revolving about the center of mass at the same r and v, and thus all you have to do is plug the reduced mass $$\mu$$ into the initial expression for total energy of a system, which would give you the equation from the problem.

 

[zhermes]

Your logic does work, but only for the case of equal masses in a circular orbit. What they are looking for is much more general.
If you start out by writing down the total energy (i.e. adding up all of the different terms) in the center of mass frame, you can simply to reach the desired equation.
(Hint: you'll need to use some properties of the center of mass frame, e.g. the net momentum is zero).