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 establish the relationship between the coefficient of volume expansion of the liquid (γ) and the coefficient of linear expansion of the tube material (α) under the condition that the length of the liquid column remains constant with temperature changes.

### Step-by-Step Solution:

1. **Understand the System**:
We have a horizontal tube open at both ends containing a liquid. The length of the liquid column does not change with temperature. We denote:
- \( \gamma \): Coefficient of volume expansion of the liquid
- \( \alpha \): Coefficient of linear expansion of the tube material

2. **Define Initial Conditions**:
Let:
- \( A_0 \): Cross-sectional area of the tube at \( 0^\circ C \)
- \( L \): Length of the liquid column (constant)
- \( V_0 = L \cdot A_0 \): Initial volume of the liquid at \( 0^\circ C \)

3. **Volume at Higher Temperature**:
At a temperature \( \theta \):
- The area of the tube expands to \( A_\theta = A_0(1 + 2\alpha \Delta \theta) \), where \( \Delta \theta = \theta - 0 \).
- The volume of the liquid at this temperature becomes \( V_\theta = L \cdot A_\theta = L \cdot A_0(1 + 2\alpha \Delta \theta) \).

4. **Volume Expansion of the Liquid**:
The volume of the liquid also expands with temperature, given by:
- \( V_\theta = V_0(1 + \gamma \Delta \theta) = L \cdot A_0(1 + \gamma \Delta \theta) \).

5. **Set the Volumes Equal**:
Since the length of the liquid column does not change, we equate the two expressions for \( V_\theta \):
\[ L \cdot A_0(1 + 2\alpha \Delta \theta) = L \cdot A_0(1 + \gamma \Delta \theta) \]

6. **Cancel Common Terms**:
We can cancel \( L \) and \( A_0 \) from both sides (assuming they are not zero):
\[ 1 + 2\alpha \Delta \theta = 1 + \gamma \Delta \theta \]

7. **Simplify the Equation**:
Subtract 1 from both sides:
\[ 2\alpha \Delta \theta = \gamma \Delta \theta \]

8. **Divide by \( \Delta \theta \)** (assuming \( \Delta \theta \neq 0 \)):
\[ 2\alpha = \gamma \]

9. **Final Relationship**:
Thus, we find the relationship between the coefficients:
\[ \gamma = 2\alpha \]

### Conclusion:
The relationship between the coefficient of volume expansion of the liquid and the coefficient of linear expansion of the tube material is given by:
\[ \gamma = 2\alpha \]
The correct option is **gamma is equal to 2 alpha**.