A student is performing the experiment of resonance column. The diameter of the column tube is 6 cm. The frequency of the tuning fork is 504 Hz. Speed of the sound at the given temperature is 336 m/s. The zero of the meter scale coincides with the top end of the resonance column tube. The reading of the water level in the column when the first resonance occurs is:

Options

A. 13 cm
B. 16.6 cm
C. 18.4 cm
D. 14.8 cm

Answer

To solve the problem step by step, we need to find the reading of the water level in the resonance column when the first resonance occurs.

### Step 1: Understand the relationship between frequency, speed of sound, and wavelength.
The formula relating the speed of sound (v), frequency (f), and wavelength (λ) is given by:
\[ v = f \cdot \lambda \]

### Step 2: Calculate the wavelength for the first resonance.
In a closed tube (like the resonance column), the first resonance occurs at a length that is one-fourth of the wavelength:
\[ L + E = \frac{\lambda}{4} \]
Where \( L \) is the length of the air column and \( E \) is the end correction.

### Step 3: Calculate the wavelength using the given frequency and speed of sound.
Given:
- Speed of sound, \( v = 336 \, \text{m/s} \)
- Frequency, \( f = 504 \, \text{Hz} \)

Using the formula:
\[ \lambda = \frac{v}{f} \]
\[ \lambda = \frac{336}{504} \]

### Step 4: Simplify the wavelength calculation.
Calculating the wavelength:
\[ \lambda = \frac{336}{504} = \frac{2}{3} \, \text{m} = 0.6667 \, \text{m} \]

### Step 5: Relate the wavelength to the length of the air column.
From the relationship established earlier:
\[ L + E = \frac{\lambda}{4} \]
Substituting the value of \( \lambda \):
\[ L + E = \frac{0.6667}{4} = 0.1667 \, \text{m} = 16.67 \, \text{cm} \]

### Step 6: Calculate the end correction.
The end correction \( E \) is given by:
\[ E = 0.3 \cdot D \]
Where \( D \) is the diameter of the tube. Given \( D = 6 \, \text{cm} \):
\[ E = 0.3 \cdot 6 = 1.8 \, \text{cm} \]

### Step 7: Substitute the end correction back into the equation.
Now substituting \( E \) back into the equation:
\[ L + 1.8 = 16.67 \]
\[ L = 16.67 - 1.8 = 14.87 \, \text{cm} \]

### Step 8: Conclusion
Thus, the reading of the water level in the column when the first resonance occurs is approximately:
\[ \boxed{14.87 \, \text{cm}} \]