A liquid with coefficient of volume expansion `gamma` is filled in a container of a material having coefficient of linear expansion `alpha` . If the liquid overflows on heating, then

Options

A. `gamma gt 3alpha`
B. `gamma lt 3alpha`
C. `gamma=3alpha`
D. none of these

Explanation

To solve the problem, we need to analyze the expansion of both the liquid and the container when the temperature increases. Here's the step-by-step solution:

### Step 1: Understand the coefficients of expansion
- The coefficient of volume expansion of the liquid is denoted as \( \gamma \).
- The coefficient of linear expansion of the container material is denoted as \( \alpha \).

### Step 2: Relate the coefficients of expansion
- The relationship between the coefficient of volume expansion \( \gamma \) and the coefficient of linear expansion \( \alpha \) is given by:
\[ \gamma = 3\alpha \]
This means that for every unit increase in temperature, the volume of the liquid expands more than the volume of the container.

### Step 3: Calculate the change in volume for the liquid
- When the temperature increases by \( \Delta T \), the change in volume of the liquid can be expressed as:
\[ \Delta V_L = \gamma \Delta T \]

### Step 4: Calculate the change in volume for the container
- The change in volume of the container, which has a linear expansion coefficient \( \alpha \), can be expressed as:
\[ \Delta V_C = 3\alpha \Delta T \]

### Step 5: Set up the inequality for overflow
- For the liquid to overflow, the change in volume of the liquid must be greater than the change in volume of the container:
\[ \Delta V_L > \Delta V_C \]
Substituting the expressions for \( \Delta V_L \) and \( \Delta V_C \):
\[ \gamma \Delta T > 3\alpha \Delta T \]

### Step 6: Simplify the inequality
- Since \( \Delta T \) is common on both sides and is positive, we can divide both sides by \( \Delta T \):
\[ \gamma > 3\alpha \]

### Conclusion
- This indicates that for the liquid to overflow, the coefficient of volume expansion of the liquid must be greater than three times the coefficient of linear expansion of the container material.

### Final Result
- Therefore, the condition for the liquid to overflow when heated is:
\[ \gamma > 3\alpha \]