Question

A steel wire of length $l=1\text{m}$ and of circular cross-section $1\text{mm}$ is under tension $F$. The extension is $x$. If it is recast into a wire of square cross-section of side $1\text{mm}$ under same tension, it has extension ${\pi }^{n}x$. Where $n=$

• A. 2
• B. 2.0
• C. 2.00

Answer 1

Answer: (A), (B), (C)

Given:
$L_1=1\text{m}$
$r_1=1\times {10}^{-3}\text{m}$
$\text{Extension}=x$
Area of the cross-section of wire:
$A=\pi r^2=\pi {\text{mm}}^2$
Wire Volume:
$V=Al\pi {\text{mm}}^3$
The length of the wire $l$ becomes after melting $=\frac{V}{A}=\pi \text{mm}$
Now applying the formula:
$Y=\frac{\text{stress}}{\text{strain}}$
$\text{For stress}=\frac{F}{A}$
$\text{For strain}=\frac{\Delta l}{l}$
Now substituting the values
$Y=\frac{F\times l}{A\times \Delta l}$
Now substituting the values and solve it,
$\frac{F_1\times l_1}{A_1\times \Delta l_1}=\frac{F_2\times l_2}{A_2\times \Delta l_2}$
$\frac{1}{x\pi }=\frac{\pi }{\Delta l_2}$
$\Delta l_2={\pi }^{2}x$
Therefore, the extension is ${\pi }^{2}x$ ....(1)
But according to question it is given that
the extension under the same load is ${\pi }^{n}x$ ....(2)
By comparing equations (1) and (2),
The value of $n=2$.