The length of metallic wire is $\ell_{1}$ when tension in it is $T _{1}$. It is $\ell_{2}$ when the tension is $T _{2}$. The original length of the wire will be ...... .
JEE MAIN 2021,AIIMS 2015, Diffcult
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Assuming Hooke's law to be valid.
$T \propto(\Delta \ell)$
$T = k (\Delta \ell)$
Let, $\ell_{0}=$ natural length (original length)
$\Rightarrow T = k \left(\ell-\ell_{0}\right)$
so, $T _{1}= k \left(\ell_{1}-\ell_{0}\right)$ And $T _{2}= k \left(\ell_{2}-\ell_{0}\right)$
$\Rightarrow \frac{ T _{1}}{ T _{2}}=\frac{\ell_{1}-\ell_{0}}{\ell_{2}-\ell_{0}}$
$\Rightarrow \ell_{0}=\frac{ T _{2} \ell_{1}- T _{1} \ell_{2}}{ T _{2}- T _{1}}$
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