Question

If x longitudinal strain is produced in a wire of Young's modulus y then energy stored in the material of the wire per unit volume is

• A. $yx^2$
• B. $y^2x$
• C. $\frac{1}{2}y^2x$
• D. $\frac{1}{2}yx^2$

Answer 1

Answer: (D)

$\frac{1}{2}yx^2$

Answer 2

The energy stored per unit volume ($U/V$) in an elastic material under strain is given by: $\frac{U}{V} = \frac{1}{2} \times \text{stress} \times \text{strain}$ Young's modulus ($y$) is defined as the ratio of stress to strain: $y = \frac{\text{stress}}{\text{strain}}$ Therefore, the stress in the wire is: $\text{stress} = y \times \text{strain}$ Given that the longitudinal strain is $x$, the stress is: $\text{stress} = yx$ Substituting the stress and strain into the energy stored per unit volume formula: $\frac{U}{V} = \frac{1}{2} \times (yx) \times x$ $\frac{U}{V} = \frac{1}{2} yx^2$