Question

A steel wire of length $1\text{m}$ and cross sectional area $A$ is stretched by $1\text{cm}$ under a given load. When the same load is applied to another steel wire of double length and half the cross-sectional area of the first one, the amount of stretching (extension) is

• A. $0.5\text{cm}$
• B. $2\text{cm}$
• C. $4\text{cm}$
• D. $1.5\text{cm}$

Answer 1

The correct option is C $4\text{cm}$

Young's modulus of elasticity is given as,
$Y=\frac{\text{stress}}{\text{strain}}$
i.e $Y=\frac{\left(\frac{F}{A}\right)}{\left(\frac{\Delta L}{L}\right)}$
$\Rightarrow \Delta L=\frac{FL}{AY}\text{Hence},\Delta L\propto \frac{L}{A}$
[since $F$ and $Y$ are constant here]
$\Rightarrow \frac{\Delta L_2}{\Delta L_1}=\frac{L_2}{L_1}\times \frac{A_1}{A_2}$
Or, $\frac{\Delta L_2}{\Delta L_1}=\frac{2L_1}{L_1}\times \frac{A_1}{\left(\frac{A_1}{2}\right)}=4$
$\therefore \Delta L_2=4\times \Delta L_1=4\times 1=4\text{cm}$
Here, $\Delta L_2$ is the extension produced in the second wire.