A smooth uniform rod $A B$ of mass $M$ and length $L$ rotates freely with an angular velocity $ω_{0}$ in a horizontal plane about a stationary vertical axis passing through its end $A$ . A small sleeve of mass m starts sliding along the rod from the point $A$ . Find the velocity $v$ of the sleeve relative to the rod at the moment it reaches its other end $B$ .

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$v' = \frac{ω_{0} L M}{3 m}$

$v' = \frac{ω_{0} L}{1 + \frac{3 m}{M}}$

$v' = \frac{2 ω_{0} L}{\sqrt{1 + \frac{M}{3 m}}}$

$v' = \frac{ω_{0} L}{\sqrt{1 + \frac{3 m}{M}}}$

answer is D.

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Detailed Solution

$(\frac{M L^{2}}{3}) ω_{0} = (\frac{M L^{2}}{3}) ω + m L^{2} ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) \frac{1}{2} (\frac{M L^{2}}{3}) {ω_{0}}^{2} = \frac{1}{2} (\frac{M L^{2}}{3}) ω^{2} + [\frac{1}{2} m v'^{2} + \frac{1}{2} (m L^{2}) ω^{2}] . . . . . . . . . . . . . . . . . . . . (2) S o l v i n g, V' = \frac{ω_{0} L}{\sqrt{1 + \frac{3 m}{M}}}$