Question

Two wires of the same material and length but diameter in the ratio $1:2$ are stretched by the same load. The ratio of elastic potential energy per unit volume for the two wires is:

• A. $1:1$
• B. $2:1$
• C. $4:1$
• D. $16:1$

Answer 1

The correct option is D $16:1$

Elastic potential energy per unit volume is
$u=\frac{1}{2}\frac{{\left(stress\right)}^2}{Y}$
As both the wires are made up of same material, so their Young's modulus is same for both the wires.
$\therefore u\propto {\left(stress\right)}^2$

$\therefore \frac{u_1}{u_2}=\frac{{\left(stress\right)}_1^2}{{\left(stress\right)}_2^2}=\frac{{\left(F_1/A_1\right)}^2}{{\left(F_2/A_2\right)}^2}$

As both the wires are stretched by the same load, therefore $F_1=F_2$

$\therefore \frac{u_1}{u_2}={\left(\frac{A_2}{A_1}\right)}^2={\left(\frac{D_2^2}{D_1^2}\right)}^2={\left(\frac{D_2}{D_1}\right)}^4={\left(\frac{2}{1}\right)}^4=\frac{16}{1}$