Question

The ratio of diameters of two wires of same material is $n:1$. The length of each wire is $4\text{m}$. On applying the same load, the increase in length of thin wire will be $(n>1)$ :-

• A. $n^2$ times
• B. $2n$ times
• C. $n$ times
• D. $(2n+1)$ times

Answer 1

The correct option is A $n^2$ times

Given:
$\frac{D_1}{D_2}=\frac{n}{1}$

$l_1=l_2=l=4\text{m}$

$F_1=F_2=F$

Young’s modulus is given by: $Y=\frac{\frac{F}{A}}{\frac{\Delta l}{l}}=\frac{Fl}{A\Delta l}[A=\pi r^2]$

$Y=\frac{Fl\times 4}{\pi D^2\times \Delta l}[r=\frac{D}{2}]$

or $\Delta l\propto \frac{1}{D^2}$

or $\frac{\Delta l_2}{\Delta l_1}=\frac{D_1^2}{D_2^2}=\frac{n^2}{1}$

Hence, option (a) is the correct answer.