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Q.
A wire fixed at the upper end stretches by length $l$ by applying a force $F$. The work done in stretching is
Mechanical Properties of Solids
Solution:
Young's modulus $Y=\frac{F L}{Al}$
$\therefore F=\frac{Y A l}{L}\,\,\,\,\,\,\,\dots(i)$
or $ dW=F \,d l=\frac{Y A l(d l)}{L}$
or $\int d W=\frac{Y A}{L} \int\limits_{0}^{l} l d l=\frac{Y A l^{2}}{2 L}$
or Workdone $=\frac{Y A l^{2}}{2 L}\,\,\,\,\,\dots(ii)$
From (i) and (ii)
Workdone $=\frac{F l}{2}$