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Question

A block is fastened at one end of a wire and is rotated in a vertical circle of radius R. Determine the ratio of change in length of the wire at the lowest point to that at the highest point of the circle. Assume that speed of the block at highest and lowest points is the same (v).

A
v+gRvgR
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B
vgRv+gR
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C
v2+gRv2gR
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D
v2gRv2+gR
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Solution

The correct option is C v2+gRv2gR
Case 1: When the block is at the lowest point of the circle.


Tlmg=mv2R
Tl=mv2R+mg

Case 2: When the block is at the highest point of the circle.


Th+mg=mv2R
Th=mv2Rmg
Now, ΔL=FLAY where Y is the Young's modulus

Change in length at the lowest point
ΔLl=(mv2R+mg)×RAY
ΔLl=(mv2+mgR)AY ......(1)
Similarly,
Change in length at highest point ΔLh
ΔLh=(mv2Rmg)RAY
ΔLh=(mv2mgR)AY ......(2)

From (1) and (2), we get
ΔLlΔLh=m(v2+gR)AYm(v2gR)AY
ΔLlΔLh=v2+gRv2gR
Hence, option (c) is the correct answer.

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