A uniform sphere of radius $ R $ is placed on a rough horizontal surface and given a linear velocity $ v_{0} $ and angular velocity $ \omega_{0} $ as shown. The sphere comes to rest after moving some distance to the right. It follows that

Question

A uniform sphere of radius $ R $ is placed on a rough horizontal surface and given a linear velocity $ v_{0} $ and angular velocity $ \omega_{0} $ as shown. The sphere comes to rest after moving some distance to the right. It follows that, $ \mathrm{v}_{0}=\omega_{0} \mathrm{R} $, $ 2 \mathrm{v}_{0}=5 \omega_{0} \mathrm{R} $, $ 5 \mathrm{v}_{0}=2 \omega_{0} \mathrm{R} $, $ 2 \mathrm{v}_{0}=\omega_{0} \mathrm{R} $

MARKS App · Accepted Answer

friction between the sphere and the horizontal surface,f=μN, ⇒ f = μ mg, a c c e l e r a t i o n , a = μ g w h e r e , μ - c o e f f i c i e n t o f k i n e t i c f r i c t i o n , N - n o r m a l r e a c t i o n = m g m - m a s s o f t h e s p h e r e w e k n o w t h a t , t o r q u e Τ = Iα , ⇒ α = Τ I w h e r e , α - a n g u l a r a c c e l e r a t i o n , I - m o m e n t o f i n e r t i a o f t h e s p h e r e α = μ mgR 2 5 mR 2 = 5 2 g μ R Now, u sin g e q u a t i o n o f m o t i o n , v = u + at , ⇒ 0 = v 0 - at ⇒ t = v 0 a t = v 0 a = ω 0 α , w h e r e v 0 = Rω , a n d a = Rα , R - r a d i u s o f t h e s p h e r e or a α = v 0 ω 0 ∴ 2 R 5 = v 0 ω 0 ∴ 5 v 0 = 2 ω 0 R

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