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Q.

A solid sphere of radius R starts rotating on rough horizontal surface with translational velocity v0 and initial angular velocity ω0=2v0/3R . The sphere starts purerolling after some time t. Find the angle by which sphere rotates upto the instant at which pure rolling starts, if v is the translational velocity at pure rolling. Assume uniformly accelerated motion up to start of pure rolling.

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a

3338vtR

b

3738vtR

c

3138vtR

d

2937vtR

answer is A.

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Detailed Solution

ω0=2v03R or, v0=32Rω0Angular momentum is conserved about A.Li=Lf ⇒25mR2×2v03R+mv0R=25mR2vR+mvR⇒v0=2119v   ...........(i)Now, ω=ω0+αt⇒v/R=2v0/3R+αt⇒α=5v19Rtwhere t is the time in which pure rolling starts.Pure rolling starts when sphere rotates by an angle,θ=ω0t+12αt2=2v03Rt+12⋅5v19Rt⋅t2=3338vtR
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A solid sphere of radius R starts rotating on rough horizontal surface with translational velocity v0 and initial angular velocity ω0=2v0/3R . The sphere starts purerolling after some time t. Find the angle by which sphere rotates upto the instant at which pure rolling starts, if v is the translational velocity at pure rolling. Assume uniformly accelerated motion up to start of pure rolling.