Question

Two aluminium wires have lengths 1 m and 2 m and their diameters are d and d if they are stretched by applying equal forces the ratio of increase in their lengths will be

Answer 1

Step 1: Given:

Lengths of the two aluminum wires are,

$L_1=1\text{m}$

$L_2=2\text{m}$

Diameters of the two aluminum wires are,

$d_1=d$

$d_2=d$

Step 2: To Find:

The ratio of increase in their lengths will be

Step 3: Calculate the ratio of increase in their lengths:

Change in length as
$∆L=\frac{4FL}{\pi Y{D}^2}$

Here, $F$ is the force, $Y$ is young's modulus and $D$ is diameter.

The other variables in the formula are constant, and thus the proportionality is written for length and diameter as

$∆L\propto \frac{L}{D^2}$

$$\begin{gathered} \frac{∆L_1}{∆L_2}=\frac{L_1}{L_2}\times {\left(\frac{d_2}{d_1}\right)}^2 \\ =\frac{1}{2}\times {\left(\frac{d}{d}\right)}^2 \\ =\frac{1}{2} \end{gathered}$$

Hence, the ratio of increase in their lengths is $\frac{1}{2}$.