Unsubstantiated assumption in a problem about two particles rotating around each other

Question

In solving the following problem from Kleppner and Kolenkow

the solution assumes there is a point $C$ along the line joining $m$ and $M$ such that

1. Both $m$ and $M$ execute uniform circular motion with the same angular velocity $\omega$ about $C$

2. $C$ is inertial and thus $F=mr\omega^2=M(R-r)\omega^2$ applies

I don't understand how those two statements follow from the fact that each particle executes UCM about the other with angular velocity $\omega$.

The fact that there exists a point $C$ (namely, the center of mass) along the line joining the masses that is inertial does follow from conservation of linear momentum, but this problem is presented before the introduction of the concept of momentum, so I think there should be a way to do it without this. Moreover, I still can't deduce (1) from it.

Any hint would be appreciated.

Answer 1

You can use kinematics of circular motion to solve this.

As the bodies perform uniform circular motion about each other, it means that with respect to one of the bodies, the velocity of other body is perpendicular (condition for circular motion). As the circular motion is uniform, there is not tangential acceleration, which means the Force must be in the along the line joining the two masses(lets call the line joining the masses "AB"). As force is constant and always perpendicular to the masses' velocity, we can observe from the ground frame and say that each mass is doing uniform circular motion about a fixed point(or intertial) lying on the line joining the two masses. (As magnitude of force is constant and always perpendicular to velocity)

Now here's the tricky part, which is proving that the two centers of these moving masses are at the same point. Assume that the centers do not coincide. If they do not coincide, then the distance between the masses will continuously change, but the distance between the bodies is constant, thus, the centers coincide. Therefore we proved that the bodies perform uniform circular motion about a fixed common point on the line AB.

There is another method which is of center of mass.

F=$\frac{dp}{dt}$. Take both bodies as a system. There is no external force, so the Center of mass of the system remains at rest, therefore directly proving that the system rotates about the COM.