May the Forces Be With You

Gravity and the Centrifugal Force


One way to analyze the problem is to calculate the Lagrangian, and solve for the equation of motion. If you're looking for steady-state solutions (solutions in which the angle of the bead isn't changing), set theta-double-dot to zero and the equilibrium (steady-state) points fall out - either theta equals zero, or theta equals the arc-cosine of g divided by the radius of the circle times omega-squared. So there is the possibility of steady state solutions at points other than theta=zero.

But that still doesn't tell us which points are stable. Stable points are those at which the bead will stay near, even if you nudge the bead a bit; a ball at the bottom of a hill is at a stable point, but a ball at the top of a hill is not - even though all the forces are balanced at the top of the hill, a slight nudge will result in the ball going further and further from the top.

Before we get into that question though, I have a detour to make.

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