A tuning fork of frequency 440 Hz resonates with a tube closed at one end of length 18 cm and diameter 5 cm in fundamental mode. The velocity of sound in air is

Options

A. 1. `336ms^(-1)`
B. 2. `343ms^(-1)`
C. 3. `300ms^(-1)`
D. 4. `350ms^(-1)`

Explanation

To solve the problem, we need to find the velocity of sound in air using the given information about the tuning fork and the tube. Let's break it down step by step.

### Step 1: Understand the fundamental frequency in a closed tube
In a tube that is closed at one end, the fundamental mode of vibration has a wavelength (λ) given by the formula:
\[ L + 0.3 \times d = \frac{\lambda}{4} \]
where \(L\) is the length of the tube and \(d\) is the diameter of the tube.

### Step 2: Convert units
The given length of the tube is 18 cm and the diameter is 5 cm. We need to convert these measurements to meters:
- Length \(L = 18 \text{ cm} = 0.18 \text{ m}\)
- Diameter \(d = 5 \text{ cm} = 0.05 \text{ m}\)

### Step 3: Substitute the values into the formula
Now, substitute \(L\) and \(d\) into the equation:
\[ 0.18 + 0.3 \times 0.05 = \frac{\lambda}{4} \]
Calculating \(0.3 \times 0.05\):
\[ 0.3 \times 0.05 = 0.015 \]
So, we have:
\[ 0.18 + 0.015 = \frac{\lambda}{4} \]
\[ 0.195 = \frac{\lambda}{4} \]

### Step 4: Solve for the wavelength (λ)
To find the wavelength, multiply both sides by 4:
\[ \lambda = 4 \times 0.195 = 0.78 \text{ m} \]

### Step 5: Use the relationship between velocity, frequency, and wavelength
We know the relationship:
\[ v = f \times \lambda \]
where \(v\) is the velocity of sound, \(f\) is the frequency, and \(\lambda\) is the wavelength. The frequency given is 440 Hz.

### Step 6: Substitute the values into the velocity formula
Now, substituting the values we have:
\[ v = 440 \text{ Hz} \times 0.78 \text{ m} \]
Calculating this gives:
\[ v = 343.2 \text{ m/s} \]

### Step 7: Round the answer
Rounding to a reasonable number of significant figures, we find:
\[ v \approx 343 \text{ m/s} \]

### Conclusion
Thus, the velocity of sound in air is approximately **343 m/s**.