Q: 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) 2:1

(d) 1:2

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Ans: (a)

Sol: Energy stored per unit volume, $\displaystyle u = \frac{U}{V} = \frac{1}{2}Stress \times Strain $

$\displaystyle u = \frac{1}{2} Stress \times \frac{Stress}{Y} $

$\displaystyle u = \frac{1}{2} \frac{F}{A} \times \frac{F}{A Y} $

$\displaystyle u = \frac{1}{2} \frac{F^2}{A^2 Y} $

$\displaystyle u \propto \frac{1}{D^4 } $ (Since A ∝ D^{2})

$\displaystyle \frac{u_1}{u_2} = \frac{1}{4} $

$\displaystyle \frac{D_2^4}{D_1^4} = \frac{1}{4} $

$\displaystyle \frac{D_2}{D_1} = \frac{1}{\sqrt{2}} $

$\displaystyle \frac{D_1}{D_2} = \frac{\sqrt{2}}{1} $