2008 USAPhO Quarterfinal #4: Beads-and-Hoop Contraption
Source:
December 13, 2012
ratiotrigonometrycalculusderivative
Problem Statement
Two beads, each of mass , are free to slide on a rigid, vertical hoop of mass . The beads are threaded on the hoop so that they cannot fall off of the hoop. They are released with negligible velocity at the top of the hoop and slide down to the bottom in opposite directions. The hoop remains vertical at all times. What is the maximum value of the ratio such that the hoop always remains in contact with the ground? Neglect friction.
[asy]
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((0,0)--(3,0));
draw(circle((1.5,1),1));
filldraw(circle((1.4,1.99499),0.1), gray(.3));
filldraw(circle((1.6,1.99499),0.1), gray(.3));
[/asy]