Question

The length of a metal wire is $L_1$ when the tension is $T_1$ and $L_2$ when the tension is $T_2$. The unstretched length of the wire is

• A. $\frac{L_2+L_2}{2}$
• B. $\sqrt{L_1{L}_2}$
• C. $\frac{T_2{L}_1-T_1{L}_2}{T_2-T_1}$
• D. $\frac{T_2{L}_1-T_1{L}_2}{T_2+T_1}$

Answer 1

Answer: (C)

Let the initial length of the metal wire is $L$.
The strain at tension $T_1$ is $\Delta L_1=L_1-L$
The strain at tension $T_2$ is $\Delta L_2=L_2-L$
Suppose, the Young's modulus of the wire is $Y$, then
$\frac{\frac{T_1}{A}}{\frac{\Delta L_1}{L}}=\frac{\frac{T_2}{A}}{\frac{\Delta L_2}{L}}$
where, $A$ is an cross-section of the wire. assume to be same at all the situations. $\Rightarrow \frac{T_1}{A}\times \frac{L}{\Delta L_1}$
$=\frac{T_2}{A}\times \frac{L}{\Delta L_2}$
$\Rightarrow \frac{T_1}{\left(L_1-L\right)}=\frac{T_2}{\left(L_2-L\right)}$
$T_1\left(L_2-L\right)=T_2\left(L_1-L\right);$
$L=\frac{T_2{L}_1-T_1{L}_2}{T_2-T_1}$