Question

The following four wires (given in options) are made of the same material. Which of these will have the largest extension when the same load is applied?

• A. Length $=50\text{cm}$, diameter $=0.5\text{mm}$
• B. Length $=100\text{cm}$, diameter $=1\text{mm}$
• C. Length $=200\text{cm}$, diameter $=2\text{mm}$
• D. Length $=300\text{cm}$, diameter $=3\text{mm}$

Answer 1

The correct option is A Length $=50\text{cm}$, diameter $=0.5\text{mm}$

Young's Modulus is given by,
$Y=\frac{\text{stress}}{\text{strain}}$
$\Rightarrow Y=\frac{\frac{F}{A}}{\frac{\Delta L}{L}}$
$\Rightarrow \Delta L=\frac{FL}{YA}$
As material is same, so $Y$ will be the same for each wire, and load applied $\left(F\right)$ is also equal for each wire.
$\Rightarrow \Delta L\propto \frac{L}{A}\propto \frac{L}{\pi \left(\frac{d^2}{4}\right)}$
$\Rightarrow \Delta L\propto \frac{L}{d^2}...\left(i\right)$
For option$\left(a\right)$,
$L=500\text{mm},d=0.5\text{mm}$
$\Rightarrow \frac{L}{d^2}=2000$
For option$\left(b\right)$,
$L=1000\text{mm},d=1\text{mm}$
$\Rightarrow \frac{L}{d^2}=1000$
For option$\left(c\right)$,
$L=2000\text{mm},d=2\text{mm}$
$\Rightarrow \frac{L}{d^2}=500$
For option$\left(d\right)$,
$L=3000\text{mm},d=3\text{mm}$
$\Rightarrow \frac{L}{d^2}=\frac{1000}{3}$
From Eq.$\left(i\right)$, the order of extension produced in the wires will be:
$\left(a\right)>\left(b\right)>\left(c\right)>\left(d\right)$
Since $\frac{L}{d^2}$ value is highest for the wire in option $\left(a\right)$, it will have the maximum extension.