If the air column in a pipe which is closed at one end, is in resonance with a vibrating tuning fork at a frequency 260 Hz, then the length of the air column is

Options

A. 35.7 cm
B. 31.7 cm
C. 12.5 cm
D. 62.5 cm

Explanation

To find the length of the air column in a closed pipe that is in resonance with a vibrating tuning fork at a frequency of 260 Hz, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Type of Pipe**:
Since the pipe is closed at one end, it behaves like a closed organ pipe. In a closed organ pipe, the fundamental frequency (first harmonic) is produced.

2. **Understand the Relationship of Frequency and Length**:
The fundamental frequency \( f \) of a closed organ pipe is given by the formula:
\[ f = \frac{V}{4L} \]
where:
- \( f \) is the frequency (in Hz),
- \( V \) is the velocity of sound in air (in m/s),
- \( L \) is the length of the air column (in meters).

3. **Substitute the Known Values**:
We know:
- The frequency \( f = 260 \) Hz,
- The velocity of sound \( V \approx 330 \) m/s (this value can vary slightly depending on temperature and humidity, but 330 m/s is a common approximation).

Plugging these values into the formula:
\[ 260 = \frac{330}{4L} \]

4. **Rearranging the Equation to Solve for \( L \)**:
Rearranging the equation to isolate \( L \):
\[ 4L = \frac{330}{260} \]
\[ L = \frac{330}{4 \times 260} \]

5. **Calculate \( L \)**:
Now, performing the calculation:
\[ L = \frac{330}{1040} \approx 0.3173 \text{ m} \]

6. **Convert to Centimeters**:
Since the question may require the answer in centimeters:
\[ L \approx 0.3173 \text{ m} = 31.73 \text{ cm} \]

7. **Final Answer**:
The length of the air column is approximately:
\[ L \approx 31.7 \text{ cm} \]