Two wires A and B having same length and mate | vedclass

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(C) The energy density (strain energy per unit volume) $U$ is given by $U = \frac{1}{2} 	imes 	ext{Stress} 	imes 	ext{Strain}$.From Hooke's law,$\

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$81$

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(C) The energy density (strain energy per unit volume) $U$ is given by $U = \frac{1}{2} 	imes 	ext{Stress} 	imes 	ext{Strain}$.From Hooke's law,$	ext{Stress} = Y 	imes 	ext{Strain}$,so $U = \frac{(	ext{Stress})^2}{2Y}$.Since $	ext{Stress} = \frac{F}{A} = \frac{F}{\pi d^2 / 4}$,we have $	ext{Stress} \propto \frac{1}{d^2}$.Therefore,$U \propto \frac{1}{d^4}$.Given the ratio of diameters $\frac{d_A}{d_B} = \frac{1}{3}$,the ratio of energy densities is $\frac{U_A}{U_B} = \left(\frac{d_B}{d_A}ight)^4$.Substituting the values,$\frac{U_A}{U_B} = \left(\frac{3}{1}ight)^4 = 81:1$.

Two wires $A$ and $B$ having same length and material are stretched by the same force. Their diameters are in the ratio $1: 3$. The ratio of energy density of wire $A$ to that of wire $B$ when stretched,is (in $: 1$)

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