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 unstretched length of the wire is
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$F=\mathrm{k} x$
$\mathrm{T}_{1}=\mathrm{k}\left(\ell_{1}-\ell_{0}\right)$ $...(1)$
$\mathrm{T}_{2}=\mathrm{k}\left(\ell_{2}-\ell_{0}\right)$ $...(2)$
$(1) /(2)$
$\frac{\mathrm{T}_{1}}{\mathrm{T}_{2}}=\frac{\ell_{1}-\ell_{0}}{\ell_{2}-\ell_{0}}$
$\mathrm{T}_{1} \ell_{2}-\mathrm{T}_{1} \ell_{0}=\mathrm{T}_{2} \ell_{1}-\mathrm{T}_{2} \ell_{0}$
$\ell_{0}=\frac{\mathrm{T}_{1} \ell_{2}-\mathrm{T}_{2} \ell_{1}}{\mathrm{T}_{1}-\mathrm{T}_{2}}=\frac{\mathrm{T}_{2} \ell_{1}-\mathrm{T}_{1} \ell_{2}}{\mathrm{T}_{2}-\mathrm{T}_{1}}$
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