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Question

A mass m is attached to a pulley through a cord as shown in the figure. The pulley is a solid disk with radius R. The cord does not slip on the disk. The mass is released from rest at a height h from the ground and at the instant the mass reaches the ground, the disk is rotating with angular velocity ω. The mass of the disk is M=xm(2ghR2ω21). Find x.
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Solution

Let the tension be T
now mgT=ma..(1)
TR=MR2α2....(2)
a=αR...(3)
from above equations we get a=mgM2+m
now final velocity V2=2ah=2mghM2+m
Also V2=ω2R2
so M=2m(2ghR2ω21)

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