A cylindrical tube of fundamental frequnecy 20 Hz in air is open at both the ends. The tube is now half dipped in a liquid. What will be the fundamenntal frequency of air column in this case ?

Options

A. 20 Hz
B. 10 Hz
C. 30 Hz
D. 40 Hz

Explanation

To solve the problem, we need to determine the fundamental frequency of a cylindrical tube that is half dipped in a liquid. The tube is initially open at both ends and has a fundamental frequency of 20 Hz in air. When half of the tube is submerged in a liquid, the effective length of the air column changes, which affects the frequency.

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
- The tube is open at both ends and has a fundamental frequency of 20 Hz.
- The formula for the fundamental frequency \( f \) of an open organ pipe is given by:
\[ f = \frac{V}{2L} \]
where \( V \) is the speed of sound in air and \( L \) is the length of the tube.

2. **Relating Frequency to Length**:
- From the initial condition, we can express the speed of sound in terms of the length of the tube:
\[ 20 = \frac{V}{2L} \]
- Rearranging gives:
\[ V = 40L \]

3. **Changing the Condition**:
- When the tube is half dipped in a liquid, the effective length of the air column is halved. The length of the air column now becomes \( L' = \frac{L}{2} \).
- Since the lower half is submerged, the upper half behaves like a closed organ pipe.

4. **Fundamental Frequency of a Closed Organ Pipe**:
- The formula for the fundamental frequency \( f_c \) of a closed organ pipe (one end closed) is given by:
\[ f_c = \frac{V}{4L'} \]
- Substituting \( L' = \frac{L}{2} \):
\[ f_c = \frac{V}{4 \cdot \frac{L}{2}} = \frac{V}{2L} \]

5. **Substituting the Value of V**:
- We already found that \( V = 40L \). Now substituting this into the frequency formula:
\[ f_c = \frac{40L}{2L} = 20 \text{ Hz} \]

6. **Conclusion**:
- The fundamental frequency of the air column when the tube is half dipped in a liquid remains 20 Hz.

### Final Answer:
The fundamental frequency of the air column when the tube is half dipped in a liquid is **20 Hz**.