Question

A particle of mass m is placed at the top of a hemisphere of radius r and mass 5m, which is placed on a smooth surface.

Answer 1

The angular velocity of the particle is ω = 6v / R cosθ.

Explanation:

Correct statement:

A hemisphere of radius r and mass 5 m is free to slide with its base on a smooth horizontal surface. A particle of mass m is placed on the top of hemisphere. The angular velocity of the particle relative to hemisphere at angular displacement with vertical when velocity of hemisphere is v, is ?

Solution:

Let the relative velocity of the particle with respect to hemisphere be Vr and v be the linear velocity. By conservation of linear momentum we get

5mv = m(Vr cosθ – v)

5mv = mVr cosθ – mv

5mv + mv = mVrcosθ

6mv = mVrcosθ

6v = Vr cosθ

Vr = 6v / cosθ

ω = Vr / R = 6v / R cosθ

Hence the angular velocity of the particle is ω = 6v / R cosθ.