c
\(2T\sin \theta = mg\) \( \Rightarrow T = \frac{{mg}}{{2\sin \theta }} = \frac{{mgL}}{{2x}}\) [ \(\theta \) \(\sin \theta = \frac{x}{L}\)]

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

\( = L\left[ {1 + \frac{1}{2}\frac{{{x^2}}}{{{L^2}}} - 1} \right]\)\( = \frac{{{x^2}}}{{2L}}\)

\(Y = \frac{T}{A}\frac{L}{l}\)

\(\therefore\)  \(T = \frac{{YAl}}{L}\)

\(\frac{{mgL}}{{2x}} = \frac{{YA}}{L}.\frac{{{x^2}}}{{2L}}\)

\(\therefore m = \frac{{YA{x^3}}}{{g{L^3}}}\)