Question

Reduce the coupled equations of motion of a two-
body system of two equal masses into two
uncoupled equations in the centre of mass and
relative coordinates when the net external force
on the system is zero. What is the reduced mass
of the system?
​

Answer 1

Step-by-step explanation:

So, for example, if I were to calculate the angular momentum (L) of my system, I would do that by summing the angular momentum of the center of mass (CM) and the angular momentum in the CM frame. The former is easy to do, the problem is the latter. I know that while putting myself in the CM frame the only thing that gives me an angular momentum is the rotational motion of the two masses. Then I know that L⃗ =μv⃗ ×r⃗ =μr⃗ ×(r˙e^r+rθ˙e^θ), where μ is the reduced mass and v the relatice velocity. And here it is the problem. First of all, WHY does it work? I mean, not only for the angular momentum, but also for calculating the kinetic energy, the effective potential, also for calculating (if I'm not wrong) the total momentum in the frame of the CM. So my question is : why does it work? How can your problem be reduced to another problem where we have just one body with the mass equals to μ and v speed? Also, in the previous formula, I have used a θ, but I really cannot imagine what his meaning is. When considering two different bodies it's all easier to grasp