A cylinder of length `l` and radius `r` is heated to temperature `T`. A longitudeinal compressive force `F` is applied on cylinder to keep its length same. Find coefficient of volume expansion.

Options

A. `(F)/(Ytpir^(2))`
B. `(3F)/(Ytpir^(2))`
C. `(2F)/(Ytpir^(2))`
D. None of these

Answer

To find the coefficient of volume expansion of a cylinder subjected to a longitudinal compressive force while being heated, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Given Parameters**:
- Length of the cylinder, \( L \)
- Radius of the cylinder, \( R \)
- Temperature, \( T \)
- Longitudinal compressive force, \( F \)

2. **Determine the Stress on the Cylinder**:
- Stress (\( \sigma \)) is defined as the force applied per unit area. The cross-sectional area \( A \) of the cylinder is given by:
\[ A = \pi R^2 \]
- Therefore, the stress can be expressed as:
\[ \sigma = \frac{F}{A} = \frac{F}{\pi R^2} \]

3. **Relate Stress to Strain Using Young's Modulus**:
- Young's modulus (\( Y \)) is defined as the ratio of stress to strain:
\[ Y = \frac{\sigma}{\epsilon} \]
- Where \( \epsilon \) (strain) is defined as the change in length (\( \Delta L \)) divided by the original length (\( L \)):
\[ \epsilon = \frac{\Delta L}{L} \]

4. **Express Change in Length Due to Temperature**:
- The change in length (\( \Delta L \)) due to thermal expansion can be expressed as:
\[ \Delta L = \alpha L T \]
- Here, \( \alpha \) is the coefficient of linear expansion.

5. **Substitute the Expression for Strain**:
- Substituting \( \Delta L \) into the strain equation gives:
\[ \epsilon = \frac{\alpha L T}{L} = \alpha T \]

6. **Combine the Equations**:
- From the definition of Young's modulus, we can write:
\[ Y = \frac{\sigma}{\epsilon} = \frac{\frac{F}{\pi R^2}}{\alpha T} \]
- Rearranging gives:
\[ \alpha = \frac{F}{Y \pi R^2} \cdot \frac{1}{T} \]

7. **Relate Linear Expansion to Volume Expansion**:
- The coefficient of volume expansion (\( \gamma \)) is related to the coefficient of linear expansion (\( \alpha \)) by the formula:
\[ \gamma = 3\alpha \]

8. **Final Expression for Coefficient of Volume Expansion**:
- Substituting the expression for \( \alpha \) into the equation for \( \gamma \):
\[ \gamma = 3 \left(\frac{F}{Y \pi R^2 T}\right) \]

### Conclusion
The coefficient of volume expansion \( \gamma \) for the cylinder is given by:
\[ \gamma = \frac{3F}{Y \pi R^2 T} \]