Question

Two wires having same length and material are stretched by same force. Their diameters are in the ratio 1 : 3. The ratio of strain energy per unit volume for these two wires (smaller to larger diameter) when stretched is

• A. 3 : 1
• B. 9 : 1
• C. 27 : 1
• D. 81 : 1

Answer 1

Answer: (D)

81 : 1

Strain energy per unit volume $U=\frac{1}{2}\times Stress\times Strain$

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

$⟹$ $U=\frac{1}{2Y}\times (Stress{)}^2$

where $Stress=\frac{F}{A}=\frac{F}{\pi d^2/4}$

$⟹$ $U\propto \frac{1}{d^4}$

Given : $\frac{d_s}{d_l}=\frac{1}{3}$

Thus ratio of strain energy per unit volume $\frac{U_s}{U_l}=(\frac{d_l}{d_s}{)}^4$

$⟹$ $\frac{U_s}{U_l}=(\frac{3}{1}{)}^4=81$