Stress and strain

Question

Two wires are made of the same material and have the same volume. The first wire has cross-sectional area *A *and the second wire has cross-sectional area *3A*. If the length of the first wire is increased by $\u2206$*l *on applying a force *F*, how much force is needed to stretch the second wire by the same amount?

Moderate

Solution

$\text{Young's modulus,}Y=\frac{Fl}{A\Delta l}$

Since initial volume of wires are same

Their areas of cross sections are A and 3 A and lengths are 3*l* and *l* respectively.

For wire 1,

$\Delta l=\left(\frac{F}{AY}\right)3l...........\left(i\right)$

$\text{For wire}2\text{, let}{F}^{\text{'}}\text{force is applied}$

$\frac{{F}^{\text{'}}}{3A}=Y\frac{\Delta l}{l}$

$\Rightarrow \Delta l=\left(\frac{{F}^{\text{'}}}{3AY}\right)l..........\left(ii\right)$

From the equations *(i)* and *(ii)*

$\Delta l=\left(\frac{F}{AY}\right)3l=\left(\frac{{F}^{\text{'}}}{3AY}\right)l\Rightarrow {F}^{\text{'}}:$

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