Question

A wire of length $L$ and radius $r$ is clamped at one end. On stretching the other end of the wire with a force $F$, the increase in its length is $l$. If another wire of same material but of length $2L$ and radius $2r$ is stretched with a force $2F$, the increase in its length will be:

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

Answer 1

The correct option is C $l$

Let $Y$ be Young's modulus of the material of the wire, Then for the first wire
$Y=\frac{F/\pi r^2}{l/L}=\frac{FL}{\pi r^{2}l}...\left(i\right)$
As both the wires are made of the same material, so their
Young modulus is same
Let the extension produced in second wire be $l^{\prime }$. Then
$Y=\frac{2F/\pi {\left(2r\right)}^2}{l^{\prime }/2L}=\frac{FL}{\pi r^2{l}^{\prime }}...\left(ii\right)$
Equating (i) and (ii) ,we get
$\frac{FL}{\pi r^{2}l}=\frac{FL}{\pi r^2{l}^{\prime }}or{l}^{\prime }=l$