Question

A metal wire of length $L$ and radius $r$ is clamped rigidly at one end. A force $F$ is applied at another end so that its length increases by $L$. The increase in length of another metal wire (same metal) of length $2L$ and radius $2r$, when stretched by a force $2F$, will be

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

Answer 1

The correct option is B $L$

As given in the question, both wires are made up of same metals. Hence the Young's modulus of both wires must be same.

The Young's modulus is given by:
$Y=\frac{FL}{\pi r^{2}l}$ $\therefore l\propto \frac{FL}{r^2}$

Here, $l=$ change in length of the wire.

The ration of change in lengths:
$(F_2=2F,L_2=2L,r_2=2r,l_1=L)$
$\frac{l_1}{l_2}=\left(\frac{F_1}{F_2}\right)\left(\frac{L_1}{L_2}\right)\left(\frac{r_2^2}{r_1^2}\right)\frac{l_1}{l_2}=\left(\frac{F}{2F}\right)\left(\frac{L}{2L}\right)\left(\frac{4r^2}{r^2}\right)=1:1$

or $l_1=l_2$
$\Rightarrow l_2=L$