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Question

A spinning cylinder of mass M and radius R is lowered on rough inclined plane of angle 30o with the horizontal and μ=1/3. The cylinder is released at a height of 3R from horizontal. If the total time taken by the cylinder to reach the bottom of the incline is ω0Rg+xRg, find the value of x.
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Solution

From t=0 till the moment when pure rolling starts friction acting is the maximum possible friction-f=μmg
Angular acceleration-
μmgcosθR=mR22α
At the moment at which pure rolling starts,
(mgsinθμmgcosθ)tm=ω0R+αtt=ω0Rgmgsinθμmgcosθ=0
So distance moved along the plane during this time is 0.
32MR2α=Mgsinθta=Rα=2gsinθ3s=3Rsinθt=2sa=6Rg

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