Question

A wire of length Land radius r is rigidly fixed at one end. On stretching the other end of the wire with a force F the increase in length is l. If another wire of the same material but double the length and radius is stretched with a force 2F, then increase in length is ______.

Options

• ${\displaystyle \frac{l}{4}}$

• 2l

• ${\displaystyle \frac{l}{2}}$

• l

Answer 1

A wire of length Land radius r is rigidly fixed at one end. On stretching the other end of the wire with a force F the increase in length is l. If another wire of the same material but double the length and radius is stretched with a force 2F, then increase in length is l.

Explanation:

Let the length of wire = L

Radius = r

When a force Fis applied, then increase in length = (l).

Since, Young's modulus

Y = ${\displaystyle \frac{\frac{\text{F}}{\pi {\text{r}}^2}}{\frac{l}{\text{L}}}}$

${\displaystyle \Rightarrow \text{Y}=\frac{\text{Fl}}{\pi {\text{r}}^{2}l}}$    ...(i)

For another wire,

length of wire, L' = 2L

Radius of wire, r' = 2r

Applied force, F' = 2F

${\displaystyle \therefore \text{Y}=\frac{\text{F'}\text{L'}}{\pi {\text{r}}^{2}l\prime }}$ where, /'= elongation in length

${\displaystyle =\frac{2\text{F}\cdot 2\text{L}}{\pi {\left(2\text{r}\right)}^2\cdot \text{l'}}}$

${\displaystyle =\frac{4\text{FL}}{4\pi {\text{r}}^2\text{l'}}}$

Y = ${\displaystyle \frac{\text{FL}}{\pi {\text{r}}^2\text{l'}}}$   ...(ii)

From Eqs. (i) and (ii), we have l' = l