Question

# A solid cylinder is projected on a rough surface having coefficient of friction μ with velocity v0 and angular velocity ω0(where ω0=v02R,R is radius). The time t at which rolling without slipping occurs is :

A
v03μg
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B
v06μg
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C
v02μg
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D
v0μg
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Solution

## The correct option is B v06μgThe velocity of point of contact is vo−ωoR=vo/2. So the rigid body is slipping on the ground in forward direction, which means frictional force will act in backward direction.This will reduce the linear velocity and increase the angular velocity till there is rolling without slipping.Frictional force is μmg. Le the time after which rolling-without-slipping starts is t.So the linear velocity at time t, is vt=vo−(μmg/m)t=vo−μgtthe angular velocity after time t is, ωt=ωo+μmgRmR2/2t=ωo+2μgt/R=vo/2R+2μgt/Rfor rolling-without-slipping, vt=ωtR⟹vo−μgt=vo/2+2μgt⟹t=vo6μg

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