You have a summer job helping to design the opening ceremony for the next winter

Question

You have a summer job helping to design the opening ceremony for the next winter Olympics. One of the choreographer%u2019s ideas is to have skaters race out onto the ice and grab a very large ring (the symbol of the Olympics).   Each ring is held horizontal at shoulder height by a vertical pole stuck into the ice.   The pole is attached to the ring on its circumference so that the ring can rotate horizontally around the pole.  The plan is to have the skater grab the ring at a point on the opposite side from where the pole is attached and, holding on, glide around the pole in a circle.  You have been assigned the job of determining the minimum speed that the skater must have before grabbing the ring in terms of the radius of the ring, the mass of the ring, the mass of the skater, and the constant frictional force between the skates and the ice.  The choreographer wants the skater and ring to go around the pole at least five times.  The skater is to be moving tangent to the ring just before grabbing it.

Explanation / Answer

The translational KE of the skater becomes rotational KE when attached to the ring. The skater will revolve until friction bleeds off the energy
Here's a proof
.5*m*v^2=.5*m*R^2*v^2/R^2
Where R is the diameter of the ring
1=1

The distance the skater will skate is
2*pi*R*5
and the frictional work will be
m*g*uk*2*pi*R*5
this is related to the speed of the skater as


m*g*uk*2*pi*R*5=.5*m*v^2
v(R)=sqrt(g*uk*pi*R*10)

for the radius of the ring, r
r=R/2

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