A thin light string is wrapped around the outer rim of a uniform hollow cylinder

Question

A thin light string is wrapped around the outer rim of a uniform hollow cylinder of mass 5.00 kg having inner and outer radii as shown in the figure. The cylinder is then released from rest. Part A How far must the cylinder fall before its center is moving at 6.73 m/s? Part B If you just dropped this cylinder without any string, how fast would its center be moving when it had fallen the distance in part A? Part C Why do you get two different answers? The cylinder falls the same distance in both cases. Essay answers are limited to about 500 words(3800 characters maximum, including spaces).

Explanation / Answer

Let tension in the string be T
Force Balance : mg - T = ma
and torque balance : Tr = I(alpha)
but alpha = a/r
so, Tr = Ia/r
hence, mg - Ia/r*r = ma
now, I = m(0.35^2 - 0.2^2)/2 = 0.04125m

hence, mg - 0.04125ma/0.35^2 = ma
mg - 0.336ma = ma
mg = 1.33ma
a = 7.3312 m/s/s

1. 2*7.3312*s = 6.73^2
s = 6.17m
2. 2*9.8*6.17 = v^2
v = 10.996 m/s
3. Different answers because of different accelerations in the two problems
Also, some energy is utilised for rotating the cylinder in the first case, none in second

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