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:
Correct Option: 1
(1) If force $F$ acts along the length $\mathrm{L}$ of the wire of crosssection $A$, then energy stored in unit volume of wire is given by
Energy density $=\frac{1}{2}$ stress $\times$ strain
$=\frac{1}{2} \times \frac{F}{A} \times \frac{F}{A Y} \quad\left(\because\right.$ stress $=\frac{F}{A}$ and strain $\left.=\frac{X}{A Y}\right)$
$=\frac{1}{2} \frac{F^{2}}{A^{2} Y}=\frac{1}{2} \frac{F^{2} \times 16}{\left(\pi d^{2}\right)^{2} Y}=\frac{1}{2} \frac{F^{2} \times 16}{\pi d^{4} Y}$
If $u_{1}$ and $u_{2}$ are the densities of two wires, then
$\frac{u_{1}}{u_{2}}=\left(\frac{d_{2}}{d_{1}}\right)^{4} \Rightarrow \frac{d_{1}}{d_{2}}=(4)^{1 / 4} \Rightarrow \frac{d_{1}}{d_{2}}=\sqrt{2}: 1$
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