Hey, hopefully someone sees this. I had the same problem for a homework problem and I got the answer wrong. In the solutions, my instructor says the following.
Since there is no tangential acceleration therefore we have:
$$\dot{r}\omega +2r\dot{\omega}=0$$
And then proceeds to OP's correct...
Hopefully someone sees this. I am solving this problem as well. I don't understand how the beads can exert an upwards force on the hoop. If gravity exerts a downwards force on the beads, and the beads are always accelerating downward by less than g, then the normal force (from the hoop on the...