In reasonace tube experiment with a tuning fork of frequency `480Hz` the consecutive reasonance lengths are `30cm` and `70cm` then determine speed of sound is

Options

A. `200 m//s`
B. `256 m//s`
C. `384 m//s`
D. `240 m//s`

Answer

To determine the speed of sound using the given resonance lengths and frequency, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Frequency (f) = 480 Hz
- First resonance length (L1) = 30 cm = 0.30 m
- Second resonance length (L2) = 70 cm = 0.70 m

2. **Understand Resonance in the Tube**:
- In a resonance tube experiment, the lengths at which resonance occurs correspond to the wavelengths of the sound waves. The lengths can be expressed in terms of the wavelength (λ) as:
- For the first resonance: \( L_1 = \frac{n \lambda}{2} \)
- For the second resonance: \( L_2 = \frac{(n + 1) \lambda}{2} \)

3. **Set Up the Equations**:
- From the first resonance length:
\[ L_1 = \frac{n \lambda}{2} \implies 0.30 = \frac{n \lambda}{2} \quad \text{(1)} \]
- From the second resonance length:
\[ L_2 = \frac{(n + 1) \lambda}{2} \implies 0.70 = \frac{(n + 1) \lambda}{2} \quad \text{(2)} \]

4. **Subtract the Two Equations**:
- From (2) - (1):
\[ 0.70 - 0.30 = \frac{(n + 1) \lambda}{2} - \frac{n \lambda}{2} \]
- This simplifies to:
\[ 0.40 = \frac{\lambda}{2} \]
- Therefore, we can find λ:
\[ \lambda = 0.80 \, \text{m} \quad \text{(3)} \]

5. **Calculate the Speed of Sound**:
- The speed of sound (v) can be calculated using the formula:
\[ v = f \cdot \lambda \]
- Substituting the values:
\[ v = 480 \, \text{Hz} \times 0.80 \, \text{m} = 384 \, \text{m/s} \]

6. **Final Answer**:
- The speed of sound is **384 m/s**.