Question

Two wires of same material (Young;s modulus Y) and same length L but different radii r and 2r respectively are joined end to end and a weight W is suspended from the combinationa s shown in figure. The elastic potential energy in the system is:

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

Answer 1

The correct option is C $\frac{5W^{2}L}{8\pi r^{2}Y}$

$\begin{array}{c}\text{Two wires have same young's modulus}y\text{and same tension}\\ \text{W.}\\ \text{Elastic potential energy in topwire is}\text{.}\end{array}$

$\begin{array}{c}=\frac{1}{2}\times \frac{(\text{Stress}{)}^2}{y}\times A_1\times L\\ =\frac{1}{2y}\frac{W^2}{A_1^2}\times A_1\times L\\ =\frac{1}{2y}\times \frac{w^{2}L}{4\pi r^2}\end{array}$

$\text{Elastic potential energy in second wires.}$

$\begin{array}{c}=\frac{1}{2}\times \frac{W^2}{A_2^2}\times A_2\times L\\ =\frac{1}{2}\times \frac{W^{2}L}{y\pi r^2}\end{array}$

$\begin{array}{c}{\text{Net elastic potential energy is =}}^{}(\frac{1}{8}+\frac{1}{2})\frac{w^{2}L}{y\pi r^2}\\ =\frac{5w^{2}L}{8\pi r^{2}y}\end{array}$