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 4.95kg having inner and outer radii as shown in the figure (Figure 1) . The cylinder is then released from rest.

Part A

How far must the cylinder fall before its center is moving at 6.27m/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

As the string is wrapped around the cylinder, so the cylinder will also rotate as it falls...
Let the angular speed be w ...
given .. linear speed = v = 6.27 m/sec

now.. w = v / outer radius
so.. w = 6.27 / ( 0.35 ) = 17.9143 rad/sec

let the distance by whcih the cylinder falls be x ...

so.. potential energy change = mass * g * x = 4.95 * 9.81 * x = 48.5595 * x

mass density of cylinder = 4.95 / ( pi * 0.35^2 - pi * 0.2^2 )

inertia of cylinder = 4.95 *0.35 / (0.35^2 - 0.2^2 ) * 0.35^2 / 2 - 4.95 *0.2 / (0.35^2 - 0.2^2 ) * 0.2^2 / 2
so.. I = 1.04625 kg m^2

kinetic energy change = 0.5 * mass * v^2 + 0.5 * inertia * w^2
so.. k.e. change = 0.5 * 4.95 * 6.27^2 + 0.5 * 1.04625 * 17.9143 ^2
so.. k.e. change = 265.18182433633125

for energy conservation ...
k.e. change = p.e. chang e
so.. 48.5595*x = 265.18182433633125

so.. distance by whcih it should fall = x = 5.461 m

part B

without the string .. the cylinder will only have linear speed... no rotation ...
so.. change in potential energy = same as previous = 265.18182433633125 J

cange in kinetic enegy = 0.5 * m * v^2 = 0.5 * 4.95 * v^2 = 265.18182433633125
so... v = 10.351 m/sec

part C
'We get two different answers because in the first case the string was wrapped , due to whcich the cylinder
also got an angular speed. whcile in the second case it was free... so only translational velocity was imparted

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