Question

Two steel wires having same length are suspended from a ceiling under the same load. If the ratio of their energy stored per unit volume is $1:4$, the ratio of their diameters is

• A. $2:1$
• B. $1:2$
• C. $1:\sqrt{2}$
• D. $\sqrt{2}:1$

Answer 1

The correct option is D $\sqrt{2}:1$

Formula used: $U=\frac{1}{2}\frac{(stress{)}^2}{Y}$

Given, $\frac{U_1}{U_2}=\frac{1}{4}$

$U=\frac{\text{Energy stored}}{\text{Volume}}=\frac{1}{2}\frac{{\text{(stress)}}^2}{Y}$

$U=\frac{\frac{1}{2}{\left(\frac{4W}{\pi d^2}\right)}^2}{Y}$

It is given that $\frac{U_1}{U_2}=\frac{1}{4}$

$4U_1=U_2$

$4\frac{1}{2Y}{\left[\frac{4W}{\pi d_1^2}\right]}^2=\frac{1}{2Y}{\left[\frac{4W}{\pi d_2^2}\right]}^2$

$4={\left(\frac{d_1}{d_2}\right)}^4$

$\frac{d_1}{d_2}=\sqrt{2}:1$

FINAL ANSWER: (c)