A uniform cylindrical rod of length L and radius r, is made from a material whose Young’s modulus of elasticity equals Y. When this rod is heated by temperature T and simultaneously subjected to a net longitudinal compressional force F, its length remains unchanged. The coefficient of volume expansion, of the material of the rod, is (nearly) equal to:

Options

A. `3F//(pir^(2)YT)`
B. `6F//(pir^(2)YT)`
C. `F//(3pir^(2)YT)`
D. `9F//(pir^(2)YT)`

Explanation

To solve the problem, we need to analyze the situation where a cylindrical rod is subjected to both thermal expansion due to heating and a compressive force. The key is to understand how these two factors interact when the length of the rod remains unchanged.

### Step-by-Step Solution:

1. **Understand the Effects of Heating**:
When the rod is heated, it tends to expand. The change in length (ΔL) due to thermal expansion can be expressed using the formula:
\[ \Delta L = L \cdot \alpha \cdot T \]
where \( \alpha \) is the coefficient of linear expansion and \( T \) is the change in temperature.

2. **Understand the Effects of Compression**:
The compressive force \( F \) applied to the rod will cause a change in length as well, which can be expressed using Hooke's Law:
\[ \Delta L = -\frac{F L}{A Y} \]
where \( A \) is the cross-sectional area of the rod, \( Y \) is Young's modulus, and \( L \) is the original length of the rod.

3. **Set the Changes in Length Equal**:
Since the length of the rod remains unchanged, the expansion due to heating must be equal to the contraction due to the compressive force:
\[ L \cdot \alpha \cdot T = \frac{F L}{A Y} \]

4. **Simplify the Equation**:
We can cancel \( L \) from both sides (assuming \( L \neq 0 \)):
\[ \alpha \cdot T = \frac{F}{A Y} \]

5. **Express the Coefficient of Volume Expansion**:
The coefficient of volume expansion \( \beta \) is related to the coefficient of linear expansion \( \alpha \) by the relationship:
\[ \beta = 3\alpha \]
Therefore, we can express \( \alpha \) in terms of \( \beta \):
\[ \alpha = \frac{\beta}{3} \]

6. **Substitute for Alpha**:
Substitute \( \alpha \) back into the equation:
\[ \frac{\beta}{3} \cdot T = \frac{F}{A Y} \]

7. **Solve for Beta**:
Rearranging gives:
\[ \beta = \frac{3F}{A Y T} \]

### Final Expression:
The coefficient of volume expansion \( \beta \) of the material of the rod is given by:
\[ \beta = \frac{3F}{A Y T} \]