A metal rod of length l, cross-sectional area A, Young's modulus Y and coefficient of linear expansion `alpha` is heated to `t^(@)C`. The work that can be performed by the rod when heated is

Options

A. `(YAalphat)xx(lalphat)`
B. `1/2(YAalphat)xx(lalphat)`
C. `1/2(YAalphat)xx1/2(lalphat)`
D. `2(YAalphat)xx(lalphat)`

Answer

To find the work that can be performed by a metal rod when heated, we can follow these steps:

### Step 1: Understand the relationship between work, stress, and strain
The work done by the rod when it is heated is equal to the change in elastic potential energy. The elastic potential energy density (energy per unit volume) is given by the formula:
\[ \text{Elastic Potential Energy Density} = \frac{1}{2} \times \text{Stress} \times \text{Strain} \]

### Step 2: Define the volume of the rod
The volume \( V \) of the rod can be expressed in terms of its length \( L \) and cross-sectional area \( A \):
\[ V = A \times L \]

### Step 3: Relate stress and strain using Young's modulus
Stress \( \sigma \) is defined as:
\[ \sigma = Y \times \epsilon \]
where \( Y \) is Young's modulus and \( \epsilon \) is the strain.

### Step 4: Define thermal strain
When the rod is heated, it expands. The thermal strain \( \epsilon \) can be expressed as:
\[ \epsilon = \frac{\Delta L}{L_0} \]
where \( \Delta L \) is the change in length and \( L_0 \) is the original length. The change in length due to heating is given by:
\[ \Delta L = L_0 \alpha \Delta T \]
where \( \alpha \) is the coefficient of linear expansion and \( \Delta T \) is the change in temperature.

### Step 5: Substitute the change in length into the strain formula
Substituting \( \Delta L \) into the strain formula, we get:
\[ \epsilon = \frac{L_0 \alpha \Delta T}{L_0} = \alpha \Delta T \]

### Step 6: Substitute stress and strain into the energy density formula
Now, substituting the expressions for stress and strain into the energy density formula:
\[ \text{Elastic Potential Energy Density} = \frac{1}{2} \times (Y \times \epsilon) \times \epsilon = \frac{1}{2} Y \epsilon^2 \]
Substituting \( \epsilon = \alpha T \):
\[ \text{Elastic Potential Energy Density} = \frac{1}{2} Y (\alpha T)^2 \]

### Step 7: Calculate the total elastic potential energy
The total elastic potential energy \( U \) is then:
\[ U = \text{Elastic Potential Energy Density} \times V = \frac{1}{2} Y (\alpha T)^2 \times (A \times L) \]

### Step 8: Final expression for work done
Thus, the work that can be performed by the rod when heated is:
\[ W = \frac{1}{2} Y A \alpha^2 T^2 L \]

### Conclusion
The work done by the rod when heated is given by the formula:
\[ W = \frac{1}{2} Y A \alpha^2 T^2 L \]