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)

$L=100cm,r=0.2mm$

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 ususal 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 length $L=100cm$ and radius $r=0.2mm$