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 :
Solution:
$\frac{Energy \,stored}{Volume} = \frac{1}{2} \frac{\left(Stress\right)^{2}}{Y}$
$\frac{u_{1}}{u_{2}} = \frac{1}{4} \Rightarrow 4u_{1} = u_{2}$
$4 \frac{1}{2Y} \left[\frac{W\cdot4}{\pi d^{2}_{1}}\right]^{2} = \frac{1}{2Y} \left[\frac{W\cdot 4}{\pi d^{2}_{2}}\right]^{2}$
$4 = \left(\frac{d_{1}}{d_{2}}\right)^{4}$
$\Rightarrow \frac{d_{1}}{d_{2}} = \sqrt{2} : 1$
