A copper wire of length $1.0\, m$ and a steel wire of length $0.5\, m$ having equal cross-sectional areas are joined end to end. The composite wire is stretched by a certain load which stretches the copper wire by $1\, mm$. If the Young's modulii of copper and steel are respectively $1.0\times10^{11}\, Nm^{-2}$ and $2.0\times10^{11}\, Nm^{- 2}$, the total extension of the composite wire is ........ $mm$
JEE MAIN 2013, Medium
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${Y_c} \times \left( {\Delta {L_c}/{L_c}} \right) = {Y_s} \times \left( {\Delta {L_s}/{L_s}} \right)$
$ \Rightarrow 1 \times {10^{11}} \times \left( {\frac{{1 \times {{10}^{ - 3}}}}{1}} \right) = 2 \times {10^{11}} \times \left( {\frac{{\Delta {L_s}}}{{0.5}}} \right)$
$\therefore \Delta {L_s} = \frac{{0.5 \times {{10}^{ - 3}}}}{2} = 0.25\,mm$
Therefore, total extension of the composite
$wire = \Delta {L_c} + \Delta {L_s}$
$ = 1\,mm + 0.25\,m = 1.25\,m$
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