Question

A hemisphere of radius r and mass 5m 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

Answer 1

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Given A hemisphere of radius r and mass 5m 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

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

Answer 2

Answer:

The angular velocity of the particle is

ω = Vr / R = 6v / R cosθ

Explanation:

Given

Radius=r

mass =5m

when the hemisphere moves through a distance x then the partcie moves through an angle θ.

As there are no external forces acting on the particle the CM remains same horizontally .

According to Law of conservation of momentum:

m(rsinθ-x)=Mx

now differentiate w.r.t to time

5mv = m(Vr cosθ – v)

5mv = mVr cosθ – mv

5mv + mv = mVrcosθ  [ since m=5m]

6mv = mVrcosθ

6v = Vr cosθ

 Vr = 6v / cosθ

 ω = Vr / R = 6v / R cosθ