Question

The following four wires of length L and radius r are made of the same material. Which of these will have the largest extension, when the same tension is applied?

• A. $L = 100\text{ cm, }r = 0.2\text{ mm}$
• B. $L = 200\text{ cm, }r = 0.4\text{ mm}$
• C. $L = 300\text{ cm, }r = 0.6\text{ mm}$
• D. $L = 400\text{ cm, }r = 0.8\text{ mm}$

Answer 1

Answer: (A)

$L = 100\text{ cm, }r = 0.2\text{ mm}$

Answer 2

: Young’s modulus, $Y = \frac{F}{A}\frac{L}{\Delta L} = \frac{T}{\pi r^{2}}\frac{L}{\Delta L}$

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

Where the symbols have their usual meanings.

As the four wires are made of the same material therefore Young’s modulus is the same for four wires. As the T and Y are the same for the four wires.

$\therefore\Delta L \propto \frac{L}{r^{2}}$

$\frac{L}{r^{2}}$is maximum for wire of length L = 100 cm and radius r = 0.2 mm