Subcontests
(3)2008 USAPhO Quarterfinal #4: Beads-and-Hoop Contraption
Two beads, each of mass m, are free to slide on a rigid, vertical hoop of mass mh. 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 m/mh 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] 2008 USAPhO Quarterfinal #2: Impulse on a Pool Ball
A uniform pool ball of radius r and mass m begins at rest on a pool table. The ball is given a horizontal impulse J of fixed magnitude at a distance βr above its center, where −1≤β≤1. The coefficient of kinetic friction between the ball and the pool table is μ. You may assume the ball and the table are perfectly rigid. Ignore effects due to deformation. (The moment of inertia about the center of mass of a solid sphere of mass m and radius r is Icm=52mr2.)[asy]
size(250);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
filldraw(circle((0,0),1),gray(.8));
draw((-3,-1)--(3,-1));
draw((-2.4,0.1)--(-2.4,0.6),EndArrow);
draw((-2.5,0)--(2.5,0),dashed);
draw((-2.75,0.7)--(-0.8,0.7),EndArrow);
label("J",(-2.8,0.7),W);
label("βr",(-2.3,0.35),E);
draw((0,-1.5)--(0,1.5),dashed);
draw((1.7,-0.1)--(1.7,-0.9),BeginArrow,EndArrow);
label("r",(1.75,-0.5),E);
[/asy](a) Find an expression for the final speed of the ball as a function of J, m, and β.(b) For what value of β does the ball immediately begin to roll without slipping, regardless of the value of μ?