If s is stress and Y is Young's modulus of material, the energy stored per unit volum is

Options

A. `(2Y)/(s)`
B. `(s)/(2Y)`
C. `2s^(2)Y`
D. `(s^(2))/(2Y)`

Answer

To find the energy stored per unit volume in a material when given stress (s) and Young's modulus (Y), we can follow these steps:

### Step 1: Understand the relationship between stress, strain, and Young's modulus
From Hooke's Law, we know that:
\[ \text{Stress} (s) = Y \times \text{Strain} (\epsilon) \]
This can be rearranged to express strain in terms of stress and Young's modulus:
\[ \text{Strain} (\epsilon) = \frac{s}{Y} \]

### Step 2: Write the formula for energy stored per unit volume
The energy stored per unit volume (U) in a material is given by the formula:
\[ U = \frac{1}{2} \times \text{Stress} \times \text{Strain} \]
Substituting the expression for strain from Step 1, we have:
\[ U = \frac{1}{2} \times s \times \epsilon \]

### Step 3: Substitute strain into the energy formula
Substituting the expression for strain into the energy formula:
\[ U = \frac{1}{2} \times s \times \left(\frac{s}{Y}\right) \]

### Step 4: Simplify the expression
Now simplify the expression:
\[ U = \frac{1}{2} \times \frac{s^2}{Y} \]

### Step 5: Final expression for energy stored per unit volume
Thus, the energy stored per unit volume in the material can be expressed as:
\[ U = \frac{s^2}{2Y} \]

### Conclusion
The final answer for the energy stored per unit volume is:
\[ U = \frac{s^2}{2Y} \]

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