Elasticity

Lengths: $\;\frac{l_A}{l_B}=\;\frac{1}{2}$Diameters: $\;\frac{d_A}{d_B}=\;\frac{3}{2}$ implies $\;\frac{r_A}{r_B}=\;\frac{3}{2}$. Thus, $\;\frac{r_A^2}{r_B^2}=\;\frac{9}{4}$.Forces: $\;\frac{F_A}{F_B}=\;\frac{3}{1}$Elastic Potential Energy Per Unit Volume:The elastic potential energy per unit volume in the wire is given by:$U = \frac{1}{2} \frac{F \cdot \Delta L}{V}$Young's Modulus Definition:From Young's modulus $Y$, we have:$Y = \frac{F \cdot L}{A \cdot \Delta L} \Rightarrow \Delta L = \frac{F \cdot L}{A \cdot Y}$Volume:$V = A \cdot L$Substituting $\Delta L$ and $V$ into the expression for $U$ :$U = \frac{1}{2} \cdot \frac{F \cdot F \cdot L}{A \cdot L \cdot A \cdot Y} = \frac{1}{2} \frac{F^2}{A^2 Y}$Ratio of Elastic Potential Energies $U_A$ and $U_B$ :$\;\frac{U_A}{U_B}=\;\frac{\;\frac{1}{2} \frac{F_A^2}{(\pi r_A^2)^2 Y}}{\frac{1}{2} \;\frac{F_B^2}{(\pi r_B^2)^2 Y}}=\;\frac{F_A^2 (r_B^2)^2}{F_B^2 (r_A^2)^2}$Now plug in the ratios:$\;\frac{U_A}{U_B}= \left(\frac{3}{1}\right)^2 \cdot \left(\frac{4}{9}\right)^2$Calculate the final ratio:$\;\;\frac{U_A}{U_B}=\;\frac{9}{1} \cdot \frac{16}{81}$$\;\;\frac{U_A}{U_B}=\;\frac{144}{81}$$\;\;\frac{U_A}{U_B}=\;\frac{16}{9}$Thus, the ratio of elastic potential energies stored per unit volume in wires $A$ and $B$ is $16: 9$.