A uniform disk has a mass of 2.1 kg and a radius of 0.45 m. The disk is mounted

Question

A uniform disk has a mass of 2.1 kg and a radius of 0.45 m. The disk is mounted on
frictionless bearings and is used as a turntable. The turntable is initially rotating at 50
rpm. A thin-walled hollow cylinder has twice the mass and half the radius as the disk. It
is released from rest, just above the turntable, and on the same vertical axis. The hollow
cylinder slips on the turntable for 0.20 s until it acquires the same final angular velocity
as the turntable. (I_disk=0.5MR2
, I_cylinder=MR2
)
What is the final angular velocity of the disk in rpm?

Explanation / Answer

by conservation of momentum

initial momentum = final momentum

iniital momentum = I_disk * initial angular velocity

iniital momentum = 0.5 * mass * radius^2 * initial angular velocity

50 rpm = 5.2359 rad/sec

iniital momentum = 0.5 * 2.1 * 0.45^2 * 5.2359

final momentum = (I_disk + I_cylinder) * final angular velocity

final momentum = (0.5 * 2.1 * 0.45^2 + 2 * 2.1 * (0.45 / 2)^2) * final angular velocity

0.5 * 2.1 * 0.45^2 * 5.2359 = (0.5 * 2.1 * 0.45^2 + 2 * 2.1 * (0.45 / 2)^2) * final angular velocity

final angular velocity = 2.61795 rad/sec or 24.99 rpm

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