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The length of...

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
(a) $\frac{l_{1} + l_{2}}{2}$
(b) $\sqrt{l_{1} l_{2}}$
(c) $\frac{l_{1} T_{2} - l_{2} T_{1}}{T_{2} - T_{1}}$
(d) $\frac{l_{1} T_{2} + l_{2} T_{1}}{T_{2} + T_{1}}$

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Solution

Correct option: (c)

) $Let the Young' s modulus be Y . C . S . A . = A Actual length of the wire = L For tension T_{1} : Y = \frac{\frac{T_{1}}{A}}{\frac{(L - l_{1})}{L}} . . . (1) For tension T_{2} : Y = \frac{\frac{T_{2}}{A}}{\frac{(L - l_{2})}{L}} . . . (2) From (1) and (2) : \frac{\frac{T_{1}}{A}}{\frac{(L - l_{1})}{L}} = \frac{\frac{T_{2}}{A}}{\frac{(L - l_{2})}{L}} \Rightarrow \frac{T_{1}}{(L - l_{1})} = \frac{T_{2}}{(L - l_{2})} \Rightarrow L = \frac{T_{2} l_{1} - T_{1} l_{2}}{T_{2} - T_{1}}$

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