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Question

A ball of mass m slides down a fixed sphere. Find the velocity of the ball when it makes an angle θ with the vertical as shown in the figure, if it is released from rest from the topmost position. The sphere has radius R and ball has radius r. Assume the sphere to be smooth.


A
2gR(1cosθ)
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B
2gr(1cosθ)
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C
2(R+r)g(1cosθ)
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D
2g(R+r)
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Solution

The correct option is C 2(R+r)g(1cosθ)
Reduction in height of centre of mass of the ball is
h=(R+r)(R+r)cosθ=(R+r)(1cosθ)



Since velocity is tangential, hence v Normal reaction (N) which means WN=0.
Also, gravity is a conservative force.

Applying mechanical energy conservation from initial position to final position:
Loss in PE=Gain in KE
mgh=12mv2
mg(R+r)(1cosθ)=12mv2
v=2(R+r)g(1cosθ)

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