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तनाव $T_{1}$ होने पर किसी धातु तार की लम्बाई $\ell_{1}$ और तनाव $T _{2}$ होने पर उसकी लम्बाई $\ell_{2}$ है। इस तार की प्राकृत लम्बाई है।
$\sqrt{l_{1} l_{2}}$
$\frac{l_{1} T_{2}-l_{2} T_{1}}{T_{2}-T_{1}}$
$\frac{l_{1} T_{2}+l_{2} T_{1}}{T_{2}+T_{1}}$
$\frac{l_{1}+l_{2}}{2}$
Solution
Increase in length under tension $\mathrm{T}_{1}=l_{1}-l$ Increase in length under tension $\mathrm{T}_{2}=l_{2}-l$ $\mathrm{Y}=\frac{\mathrm{T}_{1}}{\mathrm{~A}} \times \frac{l}{l_{1}-l}$ and $\mathrm{Y}=\frac{\mathrm{T}_{2}}{\mathrm{~A}} \times \frac{l}{l_{2}-l}$
Since material of wire is same hence $\mathrm{Y}$ is same.
$\therefore \frac{\mathrm{T}_{1}}{\mathrm{~A}} \times \frac{l}{l_{1}-l}=\frac{\mathrm{T}_{2}}{\mathrm{~A}} \times \frac{l}{l_{2}-l}$
$\therefore \mathrm{T}_{1}\left(l_{2}-l\right)=\mathrm{T}_{2}\left(l_{1}-l\right)$
$\therefore \mathrm{T}_{1} l_{2}-\mathrm{T}_{1} l=\mathrm{T}_{2} l_{1}-\mathrm{T}_{2} l$
$\therefore \mathrm{T}_{1} l_{2}-\mathrm{T}_{2} l_{1}=\left(\mathrm{T}_{1}-\mathrm{T}_{2}\right) l$
$\therefore l=\frac{\mathrm{T}_{1} l_{2}-\mathrm{T}_{2} l_{1}}{\mathrm{~T}_{1}-\mathrm{T}_{2}}$ or $\frac{\mathrm{T}_{2} l_{1}-\mathrm{T}_{l} l_{2}}{\mathrm{~T}_{2}-\mathrm{T}_{1}}$
