\(\frac{{{\ell _s}}}{{{\ell _b}}} = a,\frac{{{r_s}}}{{{r_b}}} = b,\frac{{{y_s}}}{{{y_b}}} = c,\frac{{\Delta \ell s}}{{\Delta {\ell _b}}} = ?\)

\(As,y = \frac{{F\ell }}{{A\Delta \ell }} \Rightarrow \Delta \ell  = \frac{{F\ell }}{{Ay}}\)

\(\Delta {\ell _s} = \frac{{3mg{\ell _s}}}{{\pi _s^2.{y_s}}}\left[ {{F_s} = \left( {M + 2M} \right)g} \right]\)

\(\Delta {\ell _b} = \frac{{2Mg{\ell _b}}}{{\pi r_b^2{y_b}}}\left[ {{F_b} = 2Mg} \right]\)

\(\therefore \frac{{\Delta {\ell _s}}}{{\Delta {\ell _b}}} = \frac{{\frac{{3Mg{\ell _s}}}{{\pi r_s^2.{y_s}}}}}{{\frac{{2Mg.{\ell _b}}}{{\pi {b^2}.{y_b}}}}} = \frac{{3a}}{{2{b^2}C}}\)