A horizontal tube, open at both ends, contains a column of liquid. The length of this liquid column does not change with temperature. Let `gamma` : coefficient of volume expansion of the liquid and `alpha` : coefficient of linear expansion of the material of the tube

Options

A. `gamma=alpha`
B. `gamma=2alpha`
C. `gamma=3alpha`
D. `gamma=0`

Explanation

To solve the problem, we need to relate the coefficient of volumetric expansion (γ) of the liquid to the coefficient of linear expansion (α) of the tube material, given that the length of the liquid column does not change with temperature.

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a horizontal tube open at both ends containing a liquid column. The length of this liquid column remains constant despite changes in temperature.

2. **Define Initial Volume**:
- Let the initial length of the liquid column be \( L_0 \) and the cross-sectional area of the tube be \( A_0 \).
- Therefore, the initial volume of the liquid is given by:
\[ V_0 = A_0 L_0 \]

3. **Change in Area Due to Temperature**:
- When the temperature changes, the area of the tube changes due to linear expansion. The change in area \( A \) can be expressed as:
\[ A = A_0 (1 + 2\alpha \Delta T) \]
- Here, \( \Delta T \) is the change in temperature, and \( 2\alpha \) is the coefficient of areal expansion (since area expands in two dimensions).

4. **Change in Volume of the Liquid**:
- The volume of the liquid after temperature change can be expressed as:
\[ V = A L_0 \]
- Substituting the expression for \( A \):
\[ V = A_0 (1 + 2\alpha \Delta T) L_0 \]

5. **Relate the Volumes**:
- Since the volume of the liquid does not change with temperature, we set the initial volume equal to the changed volume:
\[ V_0 = V \]
- Therefore:
\[ A_0 L_0 = A_0 (1 + 2\alpha \Delta T) L_0 \]

6. **Cancel Out Common Terms**:
- We can cancel \( A_0 L_0 \) from both sides (assuming \( A_0 \) and \( L_0 \) are not zero):
\[ 1 = 1 + 2\alpha \Delta T \]

7. **Solve for Coefficient of Volumetric Expansion**:
- Rearranging gives:
\[ 0 = 2\alpha \Delta T \]
- This implies that the volumetric expansion coefficient \( \gamma \) must be related to \( \alpha \) in such a way that:
\[ \gamma = 3\alpha \]

### Final Relation:
Thus, the relationship between the coefficients of volumetric and linear expansion is:
\[ \gamma = 3\alpha \]