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

Answer 1

Answer: (A)

Given, $r_1:r_2=1:2$ and $l_1=8$
In the first condition (cross-section area)
$A_1=\pi r_1^2$ ...(i)
In the second condition,
(cross-section area) $A_2=\pi r_2^2$ ...(ii)
On dividing Eq. (i) by Eq. (ii), we get
$\frac{A_1}{A_2}=\frac{r_1^2}{I_2^2}$ or $\frac{A_1}{A_2}={\left(\frac{1}{2}\right)}^2=\frac{1}{4}$
Stress ${}_1:$ Stress ${}_2=4:1$
Strain ${}_1:$ Strain ${}_2=4:1$
$\frac{l_1}{l_2}=\frac{4}{1}$
$\Rightarrow l_1=4l_2$
Increase in length of $B$,
$l_2=\frac{l_1}{n}=\frac{8}{2}=2mm$