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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:
1243897_504a96f4d35b4e72a1a80c7ac047c2ad.png


A
3W2L8πr2Y
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B
3W2L4πr2Y
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C
5W2L8πr2Y
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D
5W2L4πr2Y
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Solution

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

$$\begin{array}{l}\text { Two wires have same young's modulus } y \text { and same tension } \\\text { W. }\\\text { Elastic potential energy in topwire is}\text { . }\end{array}$$
$$\begin{array}{l}=\frac{1}{2} \times \frac{(\text { Stress })^{2}}{y} \times A_{1} \times L \\=\frac{1}{2 y} \frac{W^{2}}{A_{1}^{2}} \times A_{1} \times L\\=\frac{1}{2y} \times \frac{w^{2} L}{4 \pi r^{2}}\end{array}$$

$$\text { Elastic potential energy in second wires. }$$
$$\begin{array}{l}=\frac{1}{2} \times \frac{W^{2}}{A_{2}^{2}} \times A_{2} \times L \\=\frac{1}{2}\times \frac{W^{2} L}{y \pi r^{2}}\end{array}$$

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

Physics

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