Conservation of Rotational Energy Question

Whiterice

1. The problem statement, all variables and given/known data
A string is wrapped several times around the rim of a small hoop with radius 8.00 and mass 0.180 . The free end of the string is held in place and the hoop is released from rest. After the hoop has descended 95.0 , calculate the angular speed and speed of its center.


2. Relevant equations
U = mgh
K_trans = 1/2mv^2
K_rot = 1/2Iω^2


3. The attempt at a solution
I set the starting point was at a height of .95m.
K_i + U_i = K_f + U_f
0 + mgh = 1/2mv^2+1/2Iω^2 + 0
mgh = 1/2mv^2 + 1/2m(r^2)(ω^2)
(.18)(9.8)(.95) = (.5)(.18)(v^2) + (.5)(.18)(.08^2)(ω^2)

But then I have two variables in the same equation and I'm not sure where to go. I know that the speed at the center is going to be equal to v (the linear velocity) and angular speed is ω.

ehild

It is as if the hoop rolled down on the string. There is a relation between the velocity of translation and angular speed of rotation for the case "rolling without slipping". You can figure it out if you answer the question: what distance does the hoop travel down on the string while it turns one?

ehild