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

Answer 1

Answer: (A)

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.
$\therefore \quad$ $\Delta L$ $\propto$ $\frac{L}{r^{2}}$
$\frac{L}{r^{2}}$ is maximum for wire of length $L = 100 \,cm$ and radius $r = 0.2 \,mm$.