Question

Two wires $A$ and $B$ of same material and of equal length with the radii in the ratio $1:2$ are subjected to identical loads. If the length of $A$ increases by $8mm$, then the increase in length of $B$ is

• A. $2mm$
• B. $4mm$
• C. $8mm$
• D. $16mm$
• E. $1mm$

Answer 1

The correct option is A $2mm$

Given, $r_1:r_2=1:2$ and $l_1=8$
In the first condition (cross-section area)
$A_1=\pi r_1^2....\left(i\right)$
In the second condition,
(cross-section area) $A_2=\pi r_2^2....\left(ii\right)$
On dividing Eq. (i) by Eq. (ii), we get
$\frac{A_1}{A_2}=\frac{r_1^2}{r_2^2}$ or $\frac{A_1}{A_2}={\left(\frac{1}{2}\right)}^2=\frac{1}{4}$
$Stres{s}_1:Stres{s}_2=4:1$
$Strai{n}_1:Strai{n}_2=4:1$
$\frac{l_1}{l_2}=\frac{4}{1}\Rightarrow l_1=4l_2$
Increase in length of $B$,
$l_2=\frac{l_1}{n}=\frac{8}{2}=2mm$.