Question

If the work done by stretching a wire by $1mm$ is $2J$, the work necessary for stretching another wire of the same material but with half the radius of cross section and half the length by $1mm$ is

• A. $\frac{1}{4}J$
• B. $4J$
• C. $8J$
• D. $16J$

Answer 1

The correct option is A $\frac{1}{4}J$

The stretching force is given by, $F=\frac{YA\Delta l}{l}$

where $Y=$ Young's modulus , $A=$ area of cross-section of the wire, $l=$ actual length of wire and $\Delta l=$ increase in length.

or $F=\frac{Y\left(\pi r^2\right)\Delta l}{l}$

As same material so $Y$ becomes constant.

Thus, $\frac{F_1}{F_2}=\frac{r_1^2\Delta l_1}{l_1}\times \frac{l_2}{r_2^2\Delta l_2}$

Given, $r_2=r_1/2$ , $l_2=l_1/2$ and $\Delta l_1=\Delta l_2=1mm$

So, $F_1/F_2=\left(4^2/2\right)=8$ .....(1)

The work done in stretching wire by amount $\Delta l$ is $W=\left(1/2\right)F\Delta l$

Thus, $W_1/W_2=F_1/F_2$

or $W_2=\left(F_2/F_1\right)W_1=2/8=1/4$ as $(W_1=2J)$