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A body hangs form the mid-point of a ...

A body hangs form the mid-point of a light wire of length `2l` and cross-sectional area `A` such that the wire sags through a verticle distance `(y lt lt l)`. If the youngs's modulus of the wire is `Y`.

What is the elastic energy stored?

A

`(YA Delta l^(2))/(2l)`

B

`(YA Delta l^(2))/(6l)`

C

`(YA Delta l^(2))/(4l)`

D

`(YA Delta l^(2))/(l)`

Text Solution

Verified by Experts

The correct Answer is:
C
`Delta l = 2(y^(2) + l^(2))^(1//2) - 2l = 2l (1 + (y^(2))/(l^(2)))^(1//2) - 2l`
`= 2l (1 + (1)/(2) (y^(2))/(l^(2)))-2l = (y^(2))/(2l^(2)) xx 2l = (y^(2))/(l)`
energy `(1)/(2) xx (Y A Delta l^(2))/(2l) - 2l = (1)/(2) "stress" xx "strain" xx "volime"`
`= (1)/(2)y ("strain")^(2) A(2l) = (1)/(2) y((Delta l)/(2l))^(2) A.2l = (yA(Delta l)^(2))/(4l)`
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