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Question
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
`x^2/(2L^2)`
`x/L`
`x^2/L`
`x^2/(2L)`
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Solution
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 `underline(x^2/(2L^2))`.
Explanation:
Consider the diagram below
Hence, change in length
ΔL = BO + OC – (BD + DC)
= 2BO – 2BD .......(∵ 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` ........[∵ x << L]
Strain = `(ΔL)/(2L)`
= `(x^2/L)/(2L)`
= `x^2/(2L^2)`
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