Question

A wire of initial length $L$ and radius $r$ is stretched by a length $l$. Another wire of same material but with initial length $2L$ and radius $2r$ is stretched by a length $2l$. The ratio of the stored elastic energy per unit volume in the first and second wire is,

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

Answer 1

Answer: (D)

1 : 1

Elastic energy stored per unit volume $E=\frac{1}{2}\times Stress\times Strain$

Using Hooke's law, $Stress=Y\times Strain$

We get $E=\frac{1}{2}Y\times (Strain{)}^2$ where Young's modulus $Y$ is a material property

$⟹$ $E\propto (Strain{)}^2$

Strain in first wire, $Strai{n}_1=\frac{l}{L}$

Strain in second wire, $Strai{n}_2=\frac{2l}{2L}=\frac{l}{L}$

$\therefore$ Ratio of energy stored per unit volume $\frac{E_1}{E_2}=\frac{(l/L{)}^2}{(l/L{)}^2}=1$

$⟹$ $E_1:E_2=1:1$