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I was thinking about this conceptual problem. Consider a thin ring of radius R, which is rotating about the axis passing through its center of mass. Now lets says there is no gravity and the ring is rotating at some constant angular velocity [tex]\omega[/tex] , so the angular momentum is conserved. Now if we look at some infinitesimal element of mass dm, on the circumference

then it is rotating about the axis too. So there must be a centripetal force acting on this mass element. First I couldn't think of any force which can play the role of centripetal force here.

Then I thought about a model. Lets say that some 10 beads are connected in circle with springs. So we have a polygon with 10 vertices (which are beads)and 10 sides (which are springs). Now lets set this in circular motion with some constant angular velocity. Now if we look at one of the beads, then we can imagine that two springs which connect it to the neighboring beads will get stretched. If we increase the angular velocity, then the springs will stretch even more. So we can model the thin ring as the beads connected by the springs and then its easy to see which forces act as the centripetal force. As we have seen here, its the force exerted by the two springs on a bead acts as the centripetal force. So the net force acting on a bead is directed towards the center of mass of the whole system. So this is how I resolved the question. I want to know if the reasoning is correct.

Thanks