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Question

A hemispherical bowl of radius R is rotating about its axis of symmetry which is kept vertical. A small ball kept in the bowl rotates with the bowl without slipping on its surface. If the surface of the bowl is smooth and the angle made by the radius through the ball with the vertical is α. Find the angular speed ω at which the bowl should rotate for this to happen.
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A
ω=gRcosα
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B
ω=gRsinα
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C
ω=gRsinα
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D
ω=gRcosα
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Solution

The correct option is D ω=gRcosα
We know for the maximum velocity, the block will try to move upwards as a result friction will be acting downwards.
we have given
NCosθ=FSinθ+mg.
NCosθ=μNSinθ+mg.
N(CosθμSinθ)=mg.
N=mg/(CosθμSinθ)

Second equation,
NSinθ+μNCosθ=mω²rSinθ
N(Sinθ+μCosθ)=mω²rSinθ
Substituting the above values of N in this,
We will get,
mg/(CosθμSinθ)×(Sinθ+μCosθ)=mω²rSinθ
ω²=g(Sinθ+μCosθ)/rSinθ(CosθμSinθ)
ωmax.=[g(Sinθ+μCosθ)/rSinθ(CosθμSinθ)]
Now, Similarly, ωmin. can be find. For easy calculations, Substitute the Plus sign instead of minus sign and vice versa in max. so that we can get the min. angular velocity.
ωmin.=[g(SinθμCosθ)/rSinθ(Cosθ+μSinθ)]
Hence, the option D is the correct answer.


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