Question

The radii and Young’s moduli of two uniform wires A and B are in the ratio 2 : 1 and 1 : 2 respectively. Both wires are subjected to the same longitudinal force. If the increase in length of the wire A is one percent, the percentage increase in length of the wire B is

• A. 1.0
• B. 1.5
• C. 2.0
• D. 3.0

Answer 1

Answer: (C)

2.0

Answer 2

: Young’s modulus, $Y = \frac{FL}{\pi r^{2}\Delta L}$

or $\frac{\Delta L}{L} = \frac{F}{\pi r^{2}Y}$

for the same force.

$\therefore\frac{\frac{\Delta L_{B}}{L_{B}}}{\frac{\Delta L_{A}}{L_{A}}} = \left( \frac{r_{A}}{r_{B}} \right)^{2}\left( \frac{Y_{A}}{Y_{B}} \right) = \left( \frac{2}{1} \right)^{2}\left( \frac{1}{2} \right) = 2$

$\frac{\Delta L_{B}}{L_{B}} = 2\left( \frac{\Delta L_{A}}{L_{A}} \right)$

$\frac{\Delta L_{B}}{L_{B}} \times 100 = 2\left( \frac{\Delta L_{A}}{L_{A}} \times 100 \right) = 2\%$