Question

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

• A. $\frac{3w^{2}L}{4\pi R^{2}Y}$
• B. $\frac{3w^{2}L}{8\pi R^{2}Y}$
• C. $\frac{5w^{2}L}{8\pi R^{2}Y}$
• D. $\frac{w^{2}L}{\pi R^{2}Y}$

Answer 1

Answer: (C)

$k_1=\frac{Y\pi {\left(2R\right)}^2}{L}$ , $k_2=\frac{Y\pi {\left(R\right)}^2}{L}$
Equivalent $\frac{1}{k_1}+\frac{1}{k_2}=\frac{L}{4Y\pi R^2}+\frac{L}{Y\pi R^2}$
Since, $k_1{x}_1=k_2{x}_2=w$
Elastic potential energy of the system
$U=\frac{1}{2}{k}_1{x}_1^2+\frac{1}{2}{k}_2{x}_2^2$
$U=\frac{1}{2}{k}_1{\left(\frac{w}{k_1}\right)}^2+\frac{1}{2}{k}_2{\left(\frac{w}{k_2}\right)}^2$
$=\frac{1}{2}{w}^2\left\{\frac{1}{k_1}+\frac{1}{k_2}\right\}=\frac{1}{2}{w}^2\left(\frac{5L}{4{\left(Y\pi R\right)}^2}\right)$
$U=\frac{5w^{2}L}{8\pi Y{R}^2}$