Question

A wire of length $1m$ and radius $1mm$ is subjected to a load. The extension is $x$. The wire is melted and then drawn into a wire of square cross-section of side $1mm$. What is the extension under the same load?

• A. ${\pi }^{2}x$
• B. $\pi x^2$
• C. $\pi x$
• D. $\pi /x$

Answer 1

The correct option is A ${\pi }^{2}x$

Given that,

Length $l=1m$

Radius $r=1mm$

Extension $=x$

Cross section area of wire$A=\pi r^2=\pi m{m}^2$

The volume of the wire $V=Al=\pi m{m}^3$

Now, after melting

The length of the wire $l=\frac{V}{A}=\pi mm$

Now, we know that

$Stress=strain\ast constant$

For case first,

$\frac{F}{A}=stress$

$\frac{\Delta l}{l}=strain$

Now,

$\frac{F\times l}{A\times \Delta l}=constant$

Now, we have

$\frac{F_1\times l_1}{A_1\times \Delta l_1}=\frac{F_2\times l_2}{A_2\times \Delta l_2}$

$\frac{1}{x\times \pi }=\frac{\pi }{\Delta l_2}$

$\Delta l_2={\pi }^{2}x$

Hence, the extension under the same load is ${\pi }^{2}x$