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 $8\, mm$, 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 $_{2}=4: 1$
$\frac{l_{1}}{l_{2}}=\frac{4}{1}$
$\Rightarrow l_{1}=4 l_{2}$
Increase in length of $B$,
$l_{2}=\frac{l_{1}}{n}=\frac{8}{2}=2\, mm$