question_answer 27)

                  A mild steel wire of length 2L and cross-sectional area A is stretched, well within elastic limit, horizontally between two pillars. A mass m is suspended from the mid point of the wire. Strain in the wire is (a) \[\frac{{{x}^{2}}}{2{{L}^{2}}}\]                             (b) \[\frac{x}{L}\]                 (c) \[\frac{{{x}^{2}}}{L}\]                            (d) \[\frac{{{x}^{2}}}{2L}\]                

Answer:

                  (a)                                 Change in length.                 \[\Delta L=AC-AO\]                 \[={{[{{L}^{2}}+{{x}^{2}}]}^{1/2}}-L=L\left[ 1+\frac{1}{2}\frac{{{x}^{2}}}{{{L}^{2}}} \right]-L=\frac{{{x}^{2}}}{2L}\]                 Longitudinal strain \[=\frac{\Delta L}{L}=\frac{{{x}^{2}}/2L}{L}=\frac{{{x}^{2}}}{2{{L}^{2}}}\]