A solid sphere of radius R starts rotating on rough horizontal surface with translational velocity v₀ and initial angular velocity $ω_{0} = 2 v_{0} / 3 R$ . The sphere starts pure
rolling 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.

Question

Infinity Learn · Accepted Answer

ω$$0=2v03R$$ or, $$v0=32R$$ω$$0Angular$$ momentum is conserved about A.$$Li=Lf$$ ⇒$$25mR2$$×$$2v03R+mv0R=25mR2vR+mvR$$⇒$$v0=2119v$$   ...........$$(i)Now$$, ω$$=$$ω$$0+$$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$$=$$ω$$0t+12$$α$$t2=2v03Rt+12$$⋅$$5v19Rt$$⋅$$t2=3338vtR$$

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