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

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