Question

Two wire $A$ and $B$ are of the same material. Their lengths are in the ratio of $1:2$ and the diameter are in the ratio $2:1$. If they are pulled by the same force, then increase in length will be in the ratio of

• A. $2:1$
• B. $1:4$
• C. $1:8$
• D. $8:1$

Answer 1

The correct option is C $1:8$

We know that Young's modulus $Y=\frac{F}{\pi r^2}\times \frac{L}{l}$

Since $Y$, $F$ are same for both the wires, we have,

$\frac{1}{r_1^2}\frac{L_1}{l_1}=\frac{1}{r_2^2}\frac{L_2}{l_2}$

or, $\frac{l_1}{l_2}=\frac{r_2^2\times L_1}{r_1^2\times L_2}=\frac{(D_2/2{)}^2\times L_1}{(D_1/2{)}^2\times L_2}$

or, $\frac{l_1}{l_2}=\frac{D_2^2\times L_1}{D_1^2\times L_2}=\frac{D_2^2}{(2D_2{)}^2}\times \frac{L_2}{2L_2}=\frac{1}{8}$

so, $l_1:l_2=1:8$