Overdamped bead on a rotating hoop

In this section we analyze a classic problem from first-year physics, the bead on a rotating hoop. This problem provides an example of a bifurcation in a mechanical system. It also illustrates the subtleties involved in replacing Newton s law, which is a second-order equation, by a simpler first-order equation. 

Of course, this is a dicey business we can t just neglect terms because we feel like it But we will for now, and then at the end of this section we ll try to find a regime where our approximation is valid. 

Our concern now is with the first-order system = —mg sin0-1- mrOl sin 0 cos 0 

The fixed points of (2) correspond to equilibrium positions for the bead. What s your intuition about where such equilibria can occur We would expect the bead to remain at rest if placed at the top or the bottom of the hoop. Can other fixed points occur And what about stability Is the bottom always stable  

Equation (2) shows that there are always fixed points where sin0 = 0, namely 0 = 0 (the bottom of the hoop) and 0 (the top). The more interesting result is that there are two additional fixed points if 

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