Question

When a body is strained, energy stored per unit volume is (Y-Young's modulus)

• A. $\frac{(stress)}{Y}$
• B. $\frac{ Y\times (strain)}{2}$
• C. $\frac1{2} \frac{(stress)}{Y}$
• D. $ Y\times (strain)^2$
• E. $\frac{(stress)^2}{2Y}$

Answer 1

Answer: (E)

When a body is strained, the energy stored per unit volume (also known as the strain energy density) is related to the stress and strain through Young's modulus \( Y \). 

The formula for the energy stored per unit volume is derived from the relationship between stress and strain, and it is given by: \[ \text{Energy per unit volume} = \frac{1}{2} \times \text{stress} \times \text{strain} \] Since stress (\( \sigma \)) is related to Young's modulus (\( Y \)) and strain (\( \epsilon \)) by: \[ \sigma = Y \times \epsilon \] Therefore, strain (\( \epsilon \)) can be written as: \[ \epsilon = \frac{\sigma}{Y} \] Substituting this into the energy formula: \[ \text{Energy per unit volume} = \frac{1}{2} \times \sigma \times \frac{\sigma}{Y} = \frac{1}{2} \times \frac{\sigma^2}{Y} \] Thus, the energy stored per unit volume is: \[ \text{Energy per unit volume} = \frac{1}{2} \times \frac{\text{stress}^2}{Y} \]

Correct Answer:

Correct Answer: (E) \( \frac{(stress)^2}{2Y} \)