# Solve this following

Question:

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 -

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

2. $\frac{\mathrm{T}_{2} \ell_{1}+\mathrm{T}_{1} \ell_{2}}{\mathrm{~T}_{1}+\mathrm{T}_{2}}$

3. $\frac{\mathrm{T}_{2} \ell_{1}-\mathrm{T}_{1} \ell_{2}}{\mathrm{~T}_{2}-\mathrm{T}_{1}}$

4. $\frac{\mathrm{T}_{1} \ell_{1}-\mathrm{T}_{2} \ell_{2}}{\mathrm{~T}_{2}-\mathrm{T}_{1}}$

Correct Option: , 3

Solution:

Assuming Hooke's law to be valid.

$\mathrm{T} \propto(\Delta \ell)$

$\mathrm{T}=\mathrm{k}(\Delta \ell)$

Let, $\ell_{0}=$ natural length (original length)

$\Rightarrow \mathrm{T}=\mathrm{k}\left(\ell-\ell_{0}\right)$

so, $\mathrm{T}_{1}=\mathrm{k}\left(\ell_{1}-\ell_{0}\right) \& \mathrm{~T}_{2}=\mathrm{k}\left(\ell_{2}-\ell_{0}\right)$

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

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