The fundamental frequency of a closed pipe is 220 Hz. If `(1)/(4)` of the pipe is filled with water, the frequency of the first overtone of the pipe now is

Options

A. `220 Hz`
B. `440 Hz.`
C. `880 Hz.`
D. `1760 Hz.`

Answer

To solve the problem step by step, we need to understand the relationship between the length of the pipe, the frequency of the sound produced, and how the filling of the pipe with water affects the frequency.

### Step-by-Step Solution:

1. **Identify the Length of the Pipe:**
Let the total length of the closed pipe be \( L \).

2. **Determine the Length of the Air Column:**
Since \( \frac{1}{4} \) of the pipe is filled with water, the length of the air column is:
\[ L_{air} = L - \frac{L}{4} = \frac{3L}{4} \]

3. **Use the Fundamental Frequency Formula:**
The fundamental frequency \( f_0 \) of a closed pipe is given by the formula:
\[ f_0 = \frac{V}{4L} \]
where \( V \) is the speed of sound in air.

4. **Substitute the Given Frequency:**
We know that the fundamental frequency \( f_0 \) is 220 Hz. Therefore:
\[ 220 = \frac{V}{4L} \]

5. **Rearranging to Find \( \frac{V}{L} \):**
From the equation above, we can express \( \frac{V}{L} \):
\[ \frac{V}{L} = 220 \times 4 = 880 \text{ Hz} \]

6. **Calculate the Fundamental Frequency of the New Air Column:**
For the new air column of length \( \frac{3L}{4} \), the fundamental frequency \( f' \) is:
\[ f' = \frac{V}{4 \times \frac{3L}{4}} = \frac{V}{3L} \]

7. **Substituting \( \frac{V}{L} \):**
We can substitute \( \frac{V}{L} = 880 \) Hz into the equation for \( f' \):
\[ f' = \frac{880}{3} \]

8. **Calculate the First Overtone:**
The first overtone for a closed pipe is the third harmonic, which is three times the fundamental frequency:
\[ f_{overtone} = 3 \times f' = 3 \times \frac{880}{3} = 880 \text{ Hz} \]

9. **Final Answer:**
Therefore, the frequency of the first overtone of the pipe now is:
\[ \boxed{880 \text{ Hz}} \]