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(C) Let the natural length of the wire be $\ell_0$ and the force constant be $k$.According to Hooke's Law,the tension $T$ in a wire is given by $T = k

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$\frac{l_{1} T_{2}-l_{2} T_{1}}{T_{2}-T_{1}}$

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(C) Let the natural length of the wire be $\ell_0$ and the force constant be $k$.According to Hooke's Law,the tension $T$ in a wire is given by $T = k(\ell - \ell_0)$,where $\ell$ is the stretched length.For the first case: $T_{1} = k(l_{1} - \ell_0)$  --- $(1)$For the second case: $T_{2} = k(l_{2} - \ell_0)$  --- $(2)$Dividing equation $(1)$ by $(2)$:$\frac{T_{1}}{T_{2}} = \frac{l_{1} - \ell_0}{l_{2} - \ell_0}$Cross-multiplying gives:$T_{1}(l_{2} - \ell_0) = T_{2}(l_{1} - \ell_0)$$T_{1}l_{2} - T_{1}\ell_0 = T_{2}l_{1} - T_{2}\ell_0$Rearranging to solve for $\ell_0$:$T_{2}\ell_0 - T_{1}\ell_0 = T_{2}l_{1} - T_{1}l_{2}$$\ell_0(T_{2} - T_{1}) = T_{2}l_{1} - T_{1}l_{2}$$\ell_0 = \frac{l_{1} T_{2} - l_{2} T_{1}}{T_{2} - T_{1}}$

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

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