The value of tension in a long thin metal wire has been changed

Question:

The value of tension in a long thin metal wire has been changed from $\mathrm{T}_{1}$ to $\mathrm{T}_{2}$. The lengths of the metal wire at two different values of tension $\mathrm{T}_{1}$ and $\mathrm{T}_{2}$ are $\ell_{1}$ and $\ell_{2}$ respectively. The actual length of the metal wire is :

  1. $\frac{T_{1} \ell_{2}-T_{2} \ell_{1}}{T_{1}-T_{2}}$

  2. $\frac{T_{1} \ell_{1}-T_{2} \ell_{2}}{T_{1}-T_{2}}$

  3. $\frac{\ell_{1}+\ell_{2}}{2}$

  4. $\sqrt{T_{1} T_{2} \ell_{1} \ell_{2}}$


Correct Option: 1

Solution:

$\mathrm{Y}=\frac{\mathrm{FL}}{\mathrm{A} \Delta \mathrm{L}}$

$\Rightarrow \mathrm{Y}=\frac{\mathrm{T}_{1} \ell_{0}}{\mathrm{~A}\left(\ell_{1}-\ell_{0}\right)}=\frac{\mathrm{T}_{2} \ell_{0}}{\mathrm{~A}\left(\ell_{2}-\ell_{0}\right)}$

$1=\frac{\mathrm{T}_{1}\left(\ell_{2}-\ell_{0}\right)}{\mathrm{T}_{2}\left(\ell_{1}-\ell_{0}\right)}$

$\mathrm{T}_{2} \ell_{1}-\mathrm{T}_{2} \ell_{0}=\mathrm{T}_{1} \ell_{2}-\mathrm{T}_{1} \ell_{0}$

$\left(T_{1}-T_{2}\right) \ell_{0}=T_{1} \ell_{2}-T_{2} \ell_{1}$

$\ell_{0}=\left(\frac{\mathrm{T}_{1} \ell_{2}-\mathrm{T}_{2} \ell_{1}}{\mathrm{~T}_{1}-\mathrm{T}_{2}}\right)$

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