If a resonance tube is sounded with a tuning fork of frequency 256 Hz, resonance occurs at 35 cm and 105 cm. The velocity of sound is about

Options

A. 360 m/s
B. 512 m/s
C. 524 m/s
D. 400 m/s

Answer

To find the velocity of sound using the resonance tube data provided, we can follow these steps:

### Step 1: Understand the resonance lengths
The resonance occurs at two lengths:
- \( L_1 = 35 \, \text{cm} \)
- \( L_2 = 105 \, \text{cm} \)

### Step 2: Relate the lengths to the wavelength
In a resonance tube, the first resonance length \( L_1 \) corresponds to \( \frac{\lambda}{4} \) (where \( \lambda \) is the wavelength) plus an end correction \( E \):
\[ L_1 + E = \frac{\lambda}{4} \tag{1} \]
For the second resonance length \( L_2 \):
\[ L_2 + E = \frac{3\lambda}{4} \tag{2} \]

### Step 3: Subtract the equations
Subtract equation (1) from equation (2):
\[ (L_2 + E) - (L_1 + E) = \frac{3\lambda}{4} - \frac{\lambda}{4} \]
This simplifies to:
\[ L_2 - L_1 = \frac{2\lambda}{4} = \frac{\lambda}{2} \]

### Step 4: Solve for the wavelength
From the above equation, we can express the wavelength \( \lambda \):
\[ \lambda = 2(L_2 - L_1) \]
Substituting the values of \( L_1 \) and \( L_2 \):
\[ \lambda = 2(105 \, \text{cm} - 35 \, \text{cm}) = 2(70 \, \text{cm}) = 140 \, \text{cm} \]

### Step 5: Convert wavelength to meters
Convert the wavelength from centimeters to meters:
\[ \lambda = 140 \, \text{cm} = 1.4 \, \text{m} \]

### Step 6: Calculate the velocity of sound
Using the formula for the velocity of sound:
\[ v = f \cdot \lambda \]
Where \( f \) is the frequency (256 Hz):
\[ v = 256 \, \text{Hz} \times 1.4 \, \text{m} = 358.4 \, \text{m/s} \]

### Step 7: Round the result
Rounding the result gives:
\[ v \approx 360 \, \text{m/s} \]

### Final Answer
The velocity of sound is approximately \( 360 \, \text{m/s} \).

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