Question

The dimensions of four wires of the same material are given below. In which wire the increase in length will be maximum when the same tension is applied?

• A. Length 100 cm, diameter 1 mm
• B. Length 200 cm, diameter 2 mm
• C. Length 300 cm, diameter 3 mm
• D. Length 50 cm, diameter 0.5 mm

Answer 1

Answer: (D)

As , $Y=\frac{\frac{F}{A}}{\frac{l}{L}}=\frac{F\times L}{\pi r^2\times l}$
$l=\frac{F\times L}{\pi r^2\times Y}$
For given $F$ and $Y,l\propto \frac{L}{r^2}\propto \frac{L}{D^2}$
$\therefore l_1:l_2:l_3:l_4=\frac{L_1}{D_1^2}:\frac{L_2}{D_2^2}:\frac{L_3}{D_3^2}:\frac{L_4}{D_4^2}$
$\frac{L_1}{D_1^2}=\frac{100cm}{(1mmm{)}^2},\frac{L_2}{D_2^2}=\frac{2000cm}{(2mmm{)}^2},\frac{L_3}{D_3^2}=\frac{300cm}{(3mmm{)}^2},\frac{L_4}{D_4^2}=\frac{50cm}{(0.5mmm{)}^2}$
So, $l_1:l_2:l_3:l_4=1:\frac{1}{2}:\frac{1}{3}:2$
Clearly $l_4$ is maximum. So correct option is (d) ..