Young’s moduli of two wires \(A\) and \(B\) are in the ratio \(7 : 4\). Wire \(A\) is \(2\, m\) long and has radius \(R\). Wire \(A\) is \(2\, m\) long and has radius \(R\). Wire \(B\) is \(1.5\, m\) long and has radius \(2\, mm\). If the two wires stretch by the same length for a given load, then the value of \(R\) is close to ......... \(mm\)

\(\frac{{{Y_A}}}{{{Y_B}}} = \frac{7}{4}\,\,\,{L_A} = 2m\,\,\,{A_A} = \pi {R^2}\)

                     \({L_B} = 1.5\,m\,\,\,\,\,\,\,\,{A_B} = \pi {\left( {2mm} \right)^2}\)

\(\frac{F}{A} = Y\left( {\frac{\ell }{L}} \right)\)

\(given\,F\,and\,\ell \,are\,same \Rightarrow \frac{{AY}}{L}\,is\,same\)

\(\frac{{{A_A}{Y_A}}}{{{L_A}}} = \frac{{{A_B}{Y_B}}}{{{L_B}}}\)

\( \Rightarrow \frac{{\left( {\pi {R^2}} \right)\left( {\frac{7}{4}{Y_B}} \right)}}{2} = \frac{{\pi {{\left( {2\,mm} \right)}^2}.{Y_B}}}{{1.5}}\)

\(R = 1.74\,mm\)