Suppose I have a ring of radius R. And at the COM, two point masses of equal mass are placed. They're connected to the circumference with 2 massless rods such that they can move freely on those rods. Assuming no gravity or friction, if I spin the ring, will both the masses move with the same velocity at a given instant covering equal distances? Both masses are equal and centrifugal forces should be equal as well. So I think they should move equal distances as well.
At r=0, the speed of each mass is zero. It is best to assume these masses are both a very small distance from the center to eliminate the initial unstable state of zero force. What does happen depends on whether the center of the ring is held as a fixed axis. If the ring axis is fixed, then each mass experiences exactly the same effect. The symmetry of one versus the other is maintained. As the masses move away from the center, angular velocity does decrease due to increased rotational inertia. But note that this change of angular velocity is the same for both masses. This will happen even if the masses are not equal. Notice that this balance is unstable. In terms of angular motion, the artificial centrifugal force is $F=r\omega^2$. If one mass gets to a slightly larger radius than the other, then outward force and acceleration increases for that mass. This amplifies the effect.
