d
Now, increave in tength \(=2 P S-P R\)

\(=2 \sqrt{l^2+x^2}-2 l\).

\(=2\left(\sqrt{\ell^2+x^2}-l\right)\).

\(=2 l\left(\sqrt{1+x^2 /l^2}-1\right)\)

\(\therefore\) strain \(=\frac{2 l\left[\sqrt{1+x^2/l^2}-1\right]}{2 l}\)

\(=\left(1+x^2 / l^2\right)^{1 / 2}-1\)

\(=1+\frac{1}{2} x^2 / L^2-1=\frac{x^2}{2l^2}\)

Now, tension, \(T=\frac{m g}{2 \cos \theta}\)

\(\cos \theta=\frac{x}{\left(1+\frac{x^2}{l^2}\right)^{1 / 2} l}\)

\(=\frac{x}{\left(1+\frac{1}{2} \frac{x^2}{l^2}\right) l}\)

\(=x / l(\because l>>x)\)

\(T=\frac{m g}{2 \cos \theta}\)

\(=\frac{m g l}{2 x}\)

Stress \(=T/ A=\frac{2 m g l}{2 A x}\)

\(y=\frac{m g l / 2 A x}{x^2 / 2 l^2}=\frac{m g l^3}{A x^3}\).

\(\therefore x=l\left(\frac{m g}{\text { YA }}\right)^{1 / 3}\)