Elastic potential energy density of a given stretch wire is proportional

Options

A. `(Stress)^2`
B. Strain
C. Stress
D. (Strain)^-1

Answer

To determine the relationship between elastic potential energy density and stress or strain in a stretched wire, we can follow these steps:

### Step 1: Understand Elastic Potential Energy Density
The elastic potential energy density (U) is defined as the elastic potential energy per unit volume. It can be expressed mathematically as:
\[ U = \frac{1}{2} \times \text{stress} \times \text{strain} \]

### Step 2: Relate Stress and Strain Using Young's Modulus
Young's modulus (Y) is defined as the ratio of stress (σ) to strain (ε):
\[ Y = \frac{\text{stress}}{\text{strain}} \]
From this, we can express stress in terms of strain:
\[ \text{stress} = Y \times \text{strain} \]

### Step 3: Substitute Stress in the Elastic Potential Energy Density Formula
Now, substituting the expression for stress into the elastic potential energy density formula:
\[ U = \frac{1}{2} \times (Y \times \text{strain}) \times \text{strain} \]
This simplifies to:
\[ U = \frac{1}{2} Y \times \text{strain}^2 \]

### Step 4: Analyze the Proportionality
From the equation \( U = \frac{1}{2} Y \times \text{strain}^2 \), we can see that the elastic potential energy density (U) is proportional to the square of the strain (ε):
\[ U \propto \text{strain}^2 \]

### Step 5: Express Elastic Potential Energy Density in Terms of Stress
We can also express strain in terms of stress:
\[ \text{strain} = \frac{\text{stress}}{Y} \]
Substituting this into the elastic potential energy density formula:
\[ U = \frac{1}{2} \times \text{stress} \times \left(\frac{\text{stress}}{Y}\right) \]
This simplifies to:
\[ U = \frac{1}{2Y} \times \text{stress}^2 \]

### Step 6: Conclude the Proportionality
From the equation \( U = \frac{1}{2Y} \times \text{stress}^2 \), we can conclude that the elastic potential energy density (U) is also proportional to the square of the stress (σ):
\[ U \propto \text{stress}^2 \]

### Final Conclusion
Thus, we can conclude that the elastic potential energy density of a stretched wire is proportional to both the square of the strain and the square of the stress.