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Q.

In the shown figure wedge of mass M has a semicircular groove of radius R. A particle of mass m is released from A. It slides on the smooth circular track and starts climing on the right face. If M = 2m, find the velocity of the wedge at $θ = 60^{0}$ .

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a

$\sqrt{\frac{gR}{10}}$

b

$\sqrt{\frac{2 gR}{27}}$

c

$\sqrt{\frac{gR}{19}}$

d

$\sqrt{\frac{gR}{38}}$

answer is C.

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Detailed Solution

Conserving momentum in the horizontal direction

$m (ucos 60^{0} - v) - Mv = 0 \Rightarrow v = \frac{mu}{2 (M + m)} = \frac{u}{6} Conserving energy, \frac{1}{2} {mu}^{2} + \frac{1}{2} {mv}^{2} = mgRcos 60^{0} \Rightarrow \frac{1}{2} m {(6 v)}^{2} + \frac{1}{2} . 2 m . v^{2} = \frac{mgR}{2} \Rightarrow v = \sqrt{\frac{gR}{38}}$

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