A tuning fork of frequency `500 H_(Z)` is sounded on a resonance tube . The first and second resonances are obtained at `17 cm` and `52 cm` . The velocity of sound is

Options

A. `(a)170 m//s`
B. `(b)350 m//s`
C. `(c)520 m//s`
D. `(d)850 m//s`

Explanation

To solve the problem, we need to find the velocity of sound using the information provided about the resonance tube and the tuning fork.

### Step-by-Step Solution:

1. **Understanding Resonance in the Tube**:
- The first resonance occurs at a length of \( \frac{\lambda}{4} \).
- The second resonance occurs at a length of \( \frac{3\lambda}{4} \).

2. **Identifying the Lengths**:
- First resonance length \( L_1 = 17 \, \text{cm} \).
- Second resonance length \( L_2 = 52 \, \text{cm} \).

3. **Finding the Difference in Lengths**:
- The difference between the second and first resonance lengths is:
\[ L_2 - L_1 = 52 \, \text{cm} - 17 \, \text{cm} = 35 \, \text{cm} \]

4. **Relating the Length Difference to Wavelength**:
- The difference in lengths corresponds to \( \frac{3\lambda}{4} - \frac{\lambda}{4} = \frac{2\lambda}{4} = \frac{\lambda}{2} \).
- Therefore, we can set up the equation:
\[ \frac{\lambda}{2} = 35 \, \text{cm} \]

5. **Calculating the Wavelength**:
- To find \( \lambda \), we multiply both sides by 2:
\[ \lambda = 2 \times 35 \, \text{cm} = 70 \, \text{cm} \]

6. **Converting Wavelength to Meters**:
- Convert \( \lambda \) from centimeters to meters:
\[ \lambda = 70 \, \text{cm} = \frac{70}{100} \, \text{m} = 0.7 \, \text{m} \]

7. **Using the Formula for Velocity**:
- The formula for the velocity of sound is given by:
\[ v = f \times \lambda \]
- Where \( f \) is the frequency of the tuning fork, which is \( 500 \, \text{Hz} \).

8. **Calculating the Velocity**:
- Substituting the values into the formula:
\[ v = 500 \, \text{Hz} \times 0.7 \, \text{m} = 350 \, \text{m/s} \]

9. **Final Answer**:
- The velocity of sound is \( 350 \, \text{m/s} \).