c
Consider a small element \(dx\) of radius \(r\),

\(r = \frac{{2R}}{L}x + R\)

At equilibrium change in length of the wire

\(\int\limits_0^l {dL = \int {\frac{{Mgdx}}{{\pi {{\left[ {\frac{{2R}}{L}x + R} \right]}^2}y}}} } \)

Taking limit from \(0\) to \(L\)

\(\Delta L = \frac{{Mg}}{{\pi y}} - \frac{1}{{\left[ {\frac{{2Rx}}{L} + R} \right]_0^L}} \times \frac{L}{{2R}} = \frac{{MgL}}{{3\pi {R^2}y}}\)

The equilibrium extended length of wire

\( = L + \Delta L\)

\( = L + \frac{{MgL}}{{3\pi {R^2}Y}} = L\left( {1 + \frac{1}{3}\frac{{Mg}}{{\pi Y{R^2}}}} \right)\)