First slide

Motion of centre of mass

Question

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 $θ = 60^{0}$ .

Moderate

A

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

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

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

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

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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