An air column in a pipe which is closed at one end will be in resonance with a vibrating tuning fork of frequency 264 Hz if the length of the air column in cm is (Speed of sound in air = 340 m/s)

Options

A. 32.19 cm
B. 64.39 cm
C. 100 cm
D. 140 cm

Explanation

To find the length of the air column in a pipe that is closed at one end and will resonate with a tuning fork of frequency 264 Hz, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Resonance Condition**:
For a pipe closed at one end, the fundamental frequency (first harmonic) has a wavelength (\(\lambda\)) related to the length of the pipe (L) by the formula:
\[ L = \frac{\lambda}{4} \]
This means that the wavelength is four times the length of the air column:
\[ \lambda = 4L \]

2. **Use the Speed of Sound Formula**:
The speed of sound in air (v) is related to frequency (f) and wavelength (\(\lambda\)) by the equation:
\[ v = f \cdot \lambda \]
Given that the speed of sound in air is 340 m/s and the frequency of the tuning fork is 264 Hz, we can substitute these values into the equation.

3. **Substitute the Wavelength**:
Replace \(\lambda\) in the speed of sound formula with \(4L\):
\[ v = f \cdot (4L) \]
Substituting the known values:
\[ 340 = 264 \cdot (4L) \]

4. **Solve for L**:
Rearranging the equation to solve for L:
\[ L = \frac{340}{264 \cdot 4} \]
Calculate the denominator:
\[ 264 \cdot 4 = 1056 \]
Now substitute back:
\[ L = \frac{340}{1056} \]
Performing the division:
\[ L \approx 0.3219 \text{ m} \]
To convert this into centimeters:
\[ L \approx 0.3219 \times 100 \approx 32.19 \text{ cm} \]

5. **Final Answer**:
The length of the air column in cm that will be in resonance with the tuning fork is approximately:
\[ L \approx 32.19 \text{ cm} \]