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Two wires of the same material (Young's modulus $Y$ ) and same length $L$ but radii $R$ and $2R$ respectively are joined end to end and a weight $W$ is suspended from the combination as shown in the figure. The elastic potential energy in the system is
$\frac{{3{W^2}L}}{{4\pi {R^2}Y}}$
$\frac{{3{W^2}L}}{{8\pi {R^2}Y}}$
$\frac{{5{W^2}L}}{{8\pi {R^2}Y}}$
$\frac{{{W^2}L}}{{\pi {R^2}Y}}$
Solution
Energy stored $\mathrm{U}=\frac{1}{2} \times \frac{(\text { stress })^{2}}{\mathrm{Y}} \times$ volume
$U_{T}=U_{1}+U_{2}$
$=\frac{1}{2 \mathrm{Y}}\left[\left(\frac{\mathrm{W}}{\mathrm{A}_{1}}\right)^{2} \times \mathrm{A}_{1} \times \mathrm{L}+\left(\frac{\mathrm{W}}{\mathrm{A}_{2}}\right)^{2} \times \mathrm{A}_{2} \times \mathrm{L}\right]$
$=\frac{W^{2} L}{2 Y}\left[\frac{1}{A_{1}^{2}}+\frac{1}{A_{2}^{2}}\right]$
$=\frac{W^{2} L}{2 Y}\left[\frac{1}{\pi R^{2}}+\frac{1}{\pi(2 R)^{2}}\right]$
$=\frac{W^{2} L}{2 \pi Y R^{2}}\left(1+\frac{1}{4}\right)=\frac{5 W^{2} L}{8 \pi Y R^{2}}$
