A resonance tube is old and has jagged end. It is still used in the laboratory to determine. Velocity of sound in air. A tuning fork of frequency `512Hz` produces first resonance when the tube is filled with wtaer to a mark 11 cm below a reference mark, near the open end of the tube. the experiment is repeated with another fork of frequency `256Hz` which produces first resonance when water reaches a mark `27 cm` below the reference mark. the velocity of sound in air, obtained in the experiment, is close to :

Options

A. (a)`335ms^(-1)`
B. (b)`341ms^(-1)`
C. (c)`328ms^(-1)`
D. (d)`322ms^(-1)`

Explanation

To determine the velocity of sound in air using the resonance tube experiment, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Resonance in the Tube:**
- The first resonance occurs when the length of the air column in the tube is equal to one-fourth of the wavelength (λ) of the sound produced by the tuning fork.
- For the first tuning fork with frequency \( f_1 = 512 \, \text{Hz} \), the air column length \( L_1 = 11 \, \text{cm} = 0.11 \, \text{m} \).
- We can express this relationship as:
\[ L_1 = \frac{\lambda_1}{4} \implies \lambda_1 = 4L_1 = 4 \times 0.11 = 0.44 \, \text{m} \]

2. **Using the Wave Equation:**
- The velocity of sound \( v \) can be calculated using the formula:
\[ v = f_1 \lambda_1 \]
- Substituting the values:
\[ v = 512 \times 0.44 \]

3. **Calculating the Velocity:**
- Performing the multiplication:
\[ v = 225.28 \, \text{m/s} \]

4. **Repeating for the Second Tuning Fork:**
- For the second tuning fork with frequency \( f_2 = 256 \, \text{Hz} \), the air column length \( L_2 = 27 \, \text{cm} = 0.27 \, \text{m} \).
- Similarly, we express this as:
\[ L_2 = \frac{\lambda_2}{4} \implies \lambda_2 = 4L_2 = 4 \times 0.27 = 1.08 \, \text{m} \]
- Now, we can calculate the velocity using the second tuning fork:
\[ v = f_2 \lambda_2 \]
- Substituting the values:
\[ v = 256 \times 1.08 \]

5. **Calculating the Velocity Again:**
- Performing the multiplication:
\[ v = 276.48 \, \text{m/s} \]

6. **Finding the Average Velocity:**
- To find a more accurate value for the velocity of sound, we can average the two calculated velocities:
\[ v_{avg} = \frac{225.28 + 276.48}{2} = \frac{501.76}{2} = 250.88 \, \text{m/s} \]

7. **Final Result:**
- The velocity of sound in air obtained from the experiment is approximately \( 250.88 \, \text{m/s} \).

### Conclusion:
The closest option for the velocity of sound in air based on the calculations is option C.

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