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Pure Rolling

A solid cylin...

Question

A solid cylinder of mass $M$ and radius $R$ is given a linear velocity $v_{0}$ and angular velocity $ω_{0}$ as shown and then it is released gently on a rough horizontal surface. Match the columns appropriately

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Solution

$D e p e n d i n g u p o n t h e v a l u e o f ω_{0} a n d v_{0}, t h e c y l i n d e r m a y e v e n t u a l l y m o v e f o r w a r d b a c k w a r d o r s t o p . L e t e v e n t u a l l y t h e v e l o c i t y o f t h e c y l i n d e r b e v a n d i t s a n g u l a r v e o c i t y b e ω t h e n b y c o n s e r v a t i o n o f a n g u l a r m o m e n t u m m v_{0} r - \frac{1}{2} m r^{2} ω_{0} = m v r + \frac{1}{2} m r^{2} ω a n d e v e n t u a l l y t h e c y l i n d e r i s r o l l i n g s o v = ω r \Rightarrow m v_{0} r - \frac{1}{2} m r^{2} ω_{0} = \frac{3}{2} m v r, v c a n b e + v e, - v e o r z e r o f r o m t h i s e q u a t i o n .$

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