Question

Four wires of the same material are stretched by the same load. Which one of them will elongate most if their dimensions are as follows

• A. $L = 100\, cm$ , $r = 1\, mm$
• B. $L = 200\, cm$ , $r = 3\, mm$
• C. $L = 300\, cm$ , $r = 3\, mm$
• D. $L = 400\, cm$ , $r = 4\, mm$

Answer 1

Answer: (A)

$L = 100\, cm$ , $r = 1\, mm$

Answer 2

$\Delta L=\frac{F L}{A Y}$
Because, wires of the same material are stretched by the same load. So, $F$ and $Y$ will be constant.
$\therefore \Delta L \propto \frac{L}{\pi r^{2}}$
$\therefore \ \Delta L_{1} =\frac{100}{\pi \times\left(1 \times 10^{-3}\right)^{2}}$ $=\frac{100}{\pi \times 10^{-6}}=\frac{100}{\pi} \times 10^{-6}$
$\therefore \Delta L_{2} =\frac{200}{\pi \times\left(3 \times 10^{-3}\right)^{2}}$ $=\frac{200}{\pi \times 9 \times 10^{-6}}=\frac{22.2}{\pi} \times 10^{6}$
$\therefore \Delta L_{3} =\frac{300}{\pi \times\left(3 \times 10^{-3}\right)^{2}}$ $=\frac{300}{\pi \times 9 \times 10^{-6}}=\frac{333}{\pi} \times 10^{6}$
$\therefore \Delta L_{4} =\frac{400}{\pi \times\left(4 \times 10^{-3}\right)^{2}}$
$=\frac{400}{\pi \times 16 \times 10^{-6}}=\frac{25}{\pi} \times 10^{6}$
We can see that, $L=100 \,cm$ and $r=1 \,mm$ will elongate most.