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Question
The length of a metal wire is l1 when the tension in it T1 and is l2 when the tension is T2. The natural length of the wire is
Options
\[\frac{\text{ l}_1 + \text{l}_2}{2}\]
\[\sqrt{\text{ l}_1 \text{l}_2}\]
\[\frac{\text{l}_1 \text{T}_2 - \text{l}_2 \text{T}_1}{\text{T}_2 - \text{T}_1}\]
\[\frac{\text{l}_1 \text{T}_2 + \text{l}_2 \text{T}_1}{\text{T}_2 + \text{T}_1}\]
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Solution
\[\text{ Let the Young's modulus be Y }. \]
\[\text{C . S . A . = A}\]
\[\text{Actual length of the wire = L}\]
\[\text{For tension T}_1 : \]
\[Y = \frac{\frac{T_1}{A}}{\frac{\left( \text{L - l}_1 \right)}{L}} . . . (1)\]
\[\text{ For tension T}_2 : \]
\[Y = \frac{\frac{T_2}{A}}{\frac{\left( \text{L - l}_2 \right)}{L}} . . . (2)\]
\[\text{ From (1) and (2): }\]
\[\frac{\frac{T_1}{A}}{\frac{\left( L - l_1 \right)}{L}} = \frac{\frac{T_2}{A}}{\frac{\left( L - l_2 \right)}{L}}\]
\[ \Rightarrow \frac{T_1}{\left( L - l_1 \right)} = \frac{T_2}{\left( L - l_2 \right)}\]
\[ \Rightarrow L = \frac{T_2 l_1 - T_1 l_2}{T_2 - T_1}\]
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