Two steel wires having the 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

Two steel wires having the 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) $\sqrt{2}:1$     (c) $1: \sqrt{2}$     (d) 1 : 2

Concept: Energy stored = $ \frac{1}{2} Y. {Strain}^2 $

\[ \frac{\frac{1}{2} Y \varepsilon_1^2}{\frac{1}{2} Y \varepsilon_2^2} = \frac{1}{4} \] \[ \text{where } \varepsilon = \text{strain} =\frac{\Delta L}{L} \] \[ \frac{\left(\Delta L_1 / L\right)^2} {\left(\Delta L_2 / L\right)^2} = \frac{1}{4} \] \[ \frac{\Delta L_1}{\Delta L_2} = \frac{1}{2} \] \[ Y = \frac{F L}{A \Delta L} \] \[ \Delta L = \frac{F L}{A Y} \]  \[ \frac{\frac{F L}{\pi r_1^2 Y}}{\frac{F L}{\pi r_2^2 Y}} = \frac{1}{2} \] \[ \frac{r_2^2}{r_1^2} = \frac{1}{2} \] \[ \frac{r_1}{r_2} = \frac{\sqrt{2}}{1} \]