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

Options

A. `170 m//s`
B. `350 m//s`
C. `520 m//s`
D. `850 m//s`

Explanation

To find the velocity of sound using the given data, we will follow these steps:

### Step 1: Understand the Resonance Condition
In a resonance tube, the first resonance occurs when there is a pressure antinode at the open end and a pressure node at the closed end. The distance from the closed end to the first resonance point is \( \frac{\lambda}{4} \) (where \( \lambda \) is the wavelength).

### Step 2: Identify the Resonance Lengths
We are given:
- First resonance length, \( L_1 = 17 \, \text{cm} \)
- Second resonance length, \( L_2 = 52 \, \text{cm} \)

### Step 3: Calculate the Difference in Lengths
The difference between the first and second resonance lengths gives us the distance between two consecutive nodes and antinodes:
\[ \Delta L = L_2 - L_1 = 52 \, \text{cm} - 17 \, \text{cm} = 35 \, \text{cm} \]

### Step 4: Relate the Difference to Wavelength
The difference in lengths corresponds to \( \frac{\lambda}{2} \):
\[ \Delta L = \frac{\lambda}{2} \implies 35 \, \text{cm} = \frac{\lambda}{2} \]
Thus, we can find the wavelength \( \lambda \):
\[ \lambda = 2 \times 35 \, \text{cm} = 70 \, \text{cm} \]

### Step 5: Convert Wavelength to Meters
Convert the wavelength from centimeters to meters:
\[ \lambda = 70 \, \text{cm} = 0.70 \, \text{m} \]

### Step 6: Use the Wave Equation
The velocity of sound \( v \) can be calculated using the formula:
\[ v = f \lambda \]
where \( f \) is the frequency of the tuning fork. Given \( f = 500 \, \text{Hz} \):
\[ v = 500 \, \text{Hz} \times 0.70 \, \text{m} = 350 \, \text{m/s} \]

### Final Answer
The velocity of sound is \( 350 \, \text{m/s} \).

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