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Q.

A hemisphere of radius R and of mass 4m is free to slide with its base on a smooth horizontal table. A particle of mass m is placed on the top of the hemisphere. The angular velocity of the particle relative to hemisphere at an angular displacement $θ$ when velocity of hemisphere $ν$ is:

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a

$\frac{5 v}{R \sin θ}$

b

$\frac{5 v}{Rcos θ}$

c

$\frac{3 v}{R \sin θ}$

d

$\frac{2 v}{Rcos θ}$

answer is A.

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Detailed Solution

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Let $ν_{r}$ be the velocity of particle relative to hemisphere and v be the linear velocity of hemisphere at this moment. Then from conservation of linear momentum, we have
$4 mv = m (v_{r} \cos θ - v) Or 5 v = v_{r} \cos θ ∴ v_{r} = \frac{5 v}{\cos θ}$

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