A wire of radius r stretched without tension along a straight line is lightly fixed at A and B as shown in fig. What is the tension in the wire when it is

pulled into the shape A C B ? Assume Young's modulus of the material of the wire to be Y and d < l 

From figure, the change in length
$\begin{array}{l}=\mathrm{AC}+\mathrm{CB}-\mathrm{AB}\\ =2\sqrt{\left({\mathrm{l}}^2+{\mathrm{d}}^2\right)}-2\mathrm{l}\left[\because \mathrm{AC}=\mathrm{CB}=\sqrt{\left({\mathrm{l}}^2+{\mathrm{d}}^2\right)}\right]\end{array}$
Let T be the tension in the wire. Then
Longitudinal stress $=\mathrm{T}/\left({\mathrm{\pi r}}^2\right)$
and Longitudinal strain $=\frac{\text{ Change in length }}{\text{ Original length }}$
$\begin{array}{r}=\frac{2\sqrt{\left({\mathrm{l}}^2+{\mathrm{d}}^2\right)}-2\mathrm{l}}{2\mathrm{l}}\\ \therefore \mathrm{Y}=\frac{\mathrm{T}/{\mathrm{\pi r}}^2}{\left[2\sqrt{\left({\mathrm{l}}^2+{\mathrm{d}}^2\right)}-2\mathrm{l}\right]/2\mathrm{l}},\end{array}$
$=\frac{\mathrm{Tl}}{{\mathrm{\pi r}}^2\left[\sqrt{\left({\mathrm{l}}^2+{\mathrm{d}}^2\right)}-\mathrm{l}\right]}$
or $\mathrm{T}=\frac{{\mathrm{Y\pi r}}^2\left[\sqrt{\left({\mathrm{l}}^2+{\mathrm{d}}^2\right)}-\mathrm{l}\right]}{\mathrm{l}}$
$\begin{array}{l}={\mathrm{Y\pi r}}^2\left[\left(1+\frac{{\mathrm{d}}^2}{2{\mathrm{l}}^2}\dots \right)-1\right]\\ =\frac{{\mathrm{Y\pi r}}^2{\mathrm{d}}^2}{2{\mathrm{l}}^2}\end{array}$