Question

The following four wires are made of the same material. Which of these will have the largest restoring force when the same elongation is produced?

• A. Length = 50 cm and diameter = 0.5 mm
• B. Length = 100 cm and diameter = 1 mm
• C. Length = 200 cm and diameter = 2 mm
• D. Length = 300 cm and diameter = 3 mm

Answer 1

The correct option is D Length = 300 cm and diameter = 3 mm

$F=KX$ where $K=\frac{YA}{L}$ for wire
Let 1, 2, 3 and 4 be the wires from options a, b, c and d respecitvely.
$\begin{array}{c}{K}_1=\frac{Y\times \frac{\pi d^2}{4}}{L}=K_1\\ K_2=\frac{Y\times \frac{\pi 4d^2}{4}}{2L}=2K_1\\ K_3=\frac{Y\times \frac{\pi 16d^2}{4}}{4L}=4K_1\\ K_4=\frac{Y\times \frac{\pi 36d^2}{4}}{6L}=6K_1\end{array}\}$ $F\propto K$
Hence $F_4$ is the largest.