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Question

The length of a metal wire is $$l_{1}$$ when the tension in it is $$T_{1}$$ and is $$l_{2}$$ when the tension is $$T_{2}$$. The natural length of wire is


A
l1+l22
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B
l1l2
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C
l1T2l2T1T2T1
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D
l1T2+l2T1T2+T1
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Solution

The correct option is D $$\dfrac {l_{1}T_{2} - l_{2}T_{1}}{T_{2} - T_{1}}$$
Let the natural length of wire be  $$l_o$$.
Using Hooke's law,    $$Y = \dfrac{Tl_o}{A\Delta l}$$
where  $$\Delta l = l-l_o$$
We get  $$l-l_o = \dfrac{Tl_o}{AY}$$
Case 1 :  Tension is $$T_1$$ and length of wire $$l=l_1$$
$$\therefore$$  $$l_1-l_o = \dfrac{T_1l_o}{AY}$$       .....(1)
Case 2 :  Tension is $$T_2$$ and length of wire $$l=l_2$$
$$\therefore$$  $$l_2-l_o = \dfrac{T_2l_o}{AY}$$       .....(2)
Dividing both equations   $$\dfrac{l_1 - l_o}{l_2 - l_o} = \dfrac{T_1 }{T_2}$$
$$\implies$$  $$l_o = \dfrac{l_1T_2 - l_2T_1}{T_2-T_1}$$

Physics

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