Question

The following four wires are made of the same material. Which of these will have the largest extension when the same tension is applied?

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

Answer 1

Answer: (A)

$Y=\frac{F/A}{\Delta \ell /\ell }\Rightarrow \Delta \ell =\frac{FL}{AY}$
$\Delta \ell \propto \frac{L}{D^2}$
$\therefore \Delta {\ell }_1:\Delta {\ell }_2:\Delta {\ell }_3:\Delta {\ell }_4$
$=\frac{50}{0.5^2}:\frac{100}{1^2}:\frac{200}{2^2}:\frac{300}{3^2}=\frac{1}{1}:\frac{2}{4}:\frac{4}{16}:\frac{6}{36}$
$=1:\frac{1}{2}:\frac{1}{4}:\frac{1}{6}$