Question

A wire of length $L$ and 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

• A. $\frac{l}{4}$
• B. $2l$
• C. $\frac{l}{2}$
• D. $l$

Answer 1

Answer: (D)

Let the length of wire $=L$ Radius $=r$
When a force $F$ is applied, then increase in length $=(l)$ Since, Young's modulus
$Y=\frac{\frac{F}{\pi r^{2}}}{\frac{l}{L}}$...(i)
$\Rightarrow Y=\frac{F L}{\pi r^{2} l}$
For another wire,
length of wire, $L'=2 L$
Radius of wire, $r'=2 r$
Applied force, $F'=2F$
$\therefore Y=\frac{F' L'}{\pi r'^{2} l'}$ where, $l'=$ elongation in length
$=\frac{2 F\cdot 2 L}{\pi(2 r)^{2} \cdot l'}=\frac{4 F L}{4 \pi r^{2}l'}$
$Y=\frac{F L}{\pi r^{2}l'}$...(ii)
From Eqs. (i) and (ii), we have $l'=l$