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 wire is :

• A. $\frac{L_1+L_2}{2}$
• 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

The correct option is C $\frac{T_2{L}_1-T_1{L}_2}{T_2-T_1}$

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