Kleppner & Kolenkow 4.7: Ring & Beads Rise if m>3M/2

[TGupta]

This problem is Kleppner and Kolenkow's 4.7. A ring of mass M hangs from a thread, and two beads of mass m slide on it without friction. The beads are released simultaneously from the top of the ring and slide down opposite sides. Show that the ring will start to rise if m>3M/2, and find the angle at which this occurs.

Now is it not the case that, the beads exert a force on the ring which is radial in direction and has a vertical component which is directed downwards always? If that is true how can the ring go up?

 

[AlephZero]

Now is it not the case that, the beads exert a force on the ring which is radial in direction and has a vertical component which is directed downwards always?

No. Suppose the beads were sliding round the edge of a solid circular disk, instead of being on a wire.

At some point before they reach the bottom (in fact, before they reach the halfway point), the beads would lose contact with the disk.

So the radial force between the wire and the beads will change from "inwards" to "outwards" at some point. The question is asking for the condition that the vertical component of that force is greater than the weight of the ring.

 

[TGupta]

Yep, got it. Thank you very much.