Question

The radii and Young's modulus of two uniform wires $A$ & $B$ are in the ratio $2:1$ and $1:2$ respectively. Both the wires are subjected to the same longitudinal force. If increase in the length of wire $A$ is $1\%$ . Then the percentage increase in length of wire $B$ is :

• A. $1$
• B. $1.5$
• C. $2$
• D. $3$

Answer 1

The correct option is C $2$

$Y=\frac{F/A}{△l/l}$
$△l=\frac{Fl}{\pi r^{2}Y}$
$△l_A=\frac{Fl}{\pi r_A^2{Y}_A}.............\left(1\right)$

$△l_B=\frac{Fl}{\pi r_B^2{Y}_B}.............\left(2\right)$

Dividing 1 by 2

$\frac{△l_A}{△l_B}=\frac{r_B^2{Y}_B}{r_A^2{Y}_A}=\left(\frac{r_B}{r_A}{)}^2\right(\frac{Y_B}{Y_A})=(\frac{1}{2}{)}^2\left(\frac{2}{1}\right)=\frac{1}{2}$
$△l_B=2△l_A=2\times 1$ %
$=2$ %