Resonance tube of diameter d = 6 cm. Sounded with tuning fork of frequency t= 504 Hz. t speed of sound is equal to 336 min. than find the height of air column.

Options

A. 14.87cm
B. 10.32cm
C. 24.52cm
D. 23.32cm

Answer

To solve the problem of finding the height of the air column in a resonance tube, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Diameter of the resonance tube, \( d = 6 \, \text{cm} = 0.06 \, \text{m} \)
- Frequency of the tuning fork, \( f = 504 \, \text{Hz} \)
- Speed of sound, \( v = 336 \, \text{m/s} \)

2. **Calculate the Wavelength (\( \lambda \)):**
The relationship between speed, frequency, and wavelength is given by the formula:
\[ v = f \lambda \]
Rearranging this gives:
\[ \lambda = \frac{v}{f} \]
Substituting the known values:
\[ \lambda = \frac{336 \, \text{m/s}}{504 \, \text{Hz}} = 0.6667 \, \text{m} \]

3. **Calculate the End Correction (\( e \)):**
The end correction is given by:
\[ e = 0.3 \times d \]
Substituting the diameter:
\[ e = 0.3 \times 0.06 \, \text{m} = 0.018 \, \text{m} \]

4. **Relate the Length of the Air Column (\( L \)) to Wavelength:**
In a resonance tube, the length of the air column plus the end correction is equal to one-fourth of the wavelength:
\[ L + e = \frac{\lambda}{4} \]
Rearranging gives:
\[ L = \frac{\lambda}{4} - e \]

5. **Substituting Values to Find \( L \):**
Now substituting the values of \( \lambda \) and \( e \):
\[ L = \frac{0.6667 \, \text{m}}{4} - 0.018 \, \text{m} \]
\[ L = 0.1667 \, \text{m} - 0.018 \, \text{m} = 0.1487 \, \text{m} \]

6. **Convert \( L \) to Centimeters:**
To express \( L \) in centimeters:
\[ L = 0.1487 \, \text{m} \times 100 = 14.87 \, \text{cm} \]

### Final Answer:
The height of the air column is \( 14.87 \, \text{cm} \).

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