A mild steel wire of length 2L and cross-sectional area A is stretched, well within elastic limit, horizontally between two pillars (figure). A mass m is suspended from the mid-point of the wire. Strain in the wire is

Options

A. `(x^(2))/(2L^(2))`
B. `(x)/(L)`
C. `x^(2)//L`
D. `x^(2)//2L`

Answer

A

Explanation

(a) Consider the diagram below

Hence, change in length
`DeltaL=BO+OC-(BD+DC)`
`=2BO-2BD(because BO=OC,BD=DC)`
`=2[BO-BD]=2[(x^(2)+L^(2))^(1//2)-L]`
`=2L[(1+(x^(2))/(L^(2)))^(1//2)-1]~~2L[1+(1)/(2)(x^(2))/(L^(2))-1]=(x^(2))/(L)[because x ltlt L]`
`therefore` Strain `=(DeltaL)/(2L)=(x^(2)//L)/(2L)=(x^(2))/(2L^(2))`