Question

The length of a metal wire is $3.75\text{m}$ when the tension in it is $100\text{N}$ and is $3.50\text{m}$ when the tension is $80\text{N}$. The natural length of the wire $\left(\text{in cm}\right)$ is

Answer 1

Given,
Length of wire when the tension is $100\text{N}$ $\left(l_1\right)=3.75\text{m}$.
Length of wire when the tension is $80\text{N}$ $\left(l_2\right)=3.50\text{m}$.
Let us suppose, the natural length of the wire be $L\text{m}$.
When tension is $100\text{N}$, change in length of wire $\left(\Delta L\right)=l_1-L=3.75-L$
We know that,
Young's modulus $\left(Y\right)=\frac{F\times L}{A\times \Delta L}$
$\Rightarrow Y=\frac{100\times L}{A\times (3.75-L)}..............\left(1\right)$
Change in length of wire when tension is $80\text{N}$
$\Delta L=l_2-L=3.50-L$
Again using the formula of Young's modulus,
$Y=\frac{F\times L}{A\times \Delta L}$
$\Rightarrow Y=\frac{80\times L}{A\times (3.50-L)}...........\left(2\right)$

From (1) and (2) as the material is same, we can write that,
$\frac{100\times L}{A\times (3.75-L)}=\frac{80\times L}{A\times (3.50-L)}$
$\Rightarrow 350-100L=300-80\text{L}$
$\Rightarrow L=2.5\text{m}\text{or}250\text{cm}$