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Question

The bob of pendulum at rest is given a sharp hit to impart a horizontal velocity of u=3gl, where l is the length of the pendulum. Select the correct statement.

A
Bob will complete the circle.
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B
String will become slack at top most point.
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C
String will become slack after rotating an angle cos113.
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D
String will become slack after rotating an angle πcos1(13)
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Solution

The correct option is D String will become slack after rotating an angle πcos1(13)

Let string slack at B.
Applying conservation of energy at point A and B.
ΔU+ΔK=0
mg(l+lcosθ)+[12mv212mu2]=0
mg[l+lcosθ]=12mv212mu2
v2=u22g[l+lcosθ]....(1)
When string slack, tension in the string becomes zero. The component of weight in radial direction balance the centrifugal force at this position.

mv2l=mgcosθ
v2=glcosθ......(2)
From equation (1) and (2),
3gl2gl[1+cosθ]=glcosθ
3cosθ=1θ=cos1(13)
Hence angle rotated α is
α=πθ=πcos1(13)

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