Question

Two wires of the same material and length but diameter in the ratio 1 : 2 are stretched by the same force. The ratio of potential energy per unit volume for the two wires when stretched will be :

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

Answer 1

The correct option is D 16 : 1

Potential energy per unit volume $=\frac{1}{2}\times stress\times strain$

$\frac{1}{2}\times stress\times strain=\frac{1}{2}\times stress\times Y\times stress$ (as $strain=Y\times stress$, where $Y$ is Young's modulus)

Potential energy $\propto$ $(stress{)}^2$

As Stress $\propto (Area{)}^{-1}$

And Area $\propto r^2$

Therefore Potential Energy per unit volume(Lets say PEV) $\propto (r{)}^{-4}$

Hence,

$\frac{PE{V}_1}{PE{V}_2}=\frac{(r_2{)}^4}{(r_1{)}^4}$

Given: $\frac{r_1}{r_2}=\frac{1}{2}$

Therefore $\frac{PE{V}_1}{PE{V}_2}=\frac{(r_2{)}^4}{(r_1{)}^4}=\frac{2^4}{1}=\frac{16}{1}$

So the correct option is (D).