The third ovetone produced by vibrating string of length 0.5m, is 1200 Hz. Then the velocity of the wave is

Options

A. 400 m/s
B. 600 m/s
C. 300m/s
D. 1200 m/s

Explanation

To find the velocity of the wave produced by a vibrating string of length 0.5 m when the third overtone frequency is 1200 Hz, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Overtones**:
- The third overtone is the same as the fourth harmonic (since the first overtone is the second harmonic, the second overtone is the third harmonic, and so on).

2. **Formula for Frequency of nth Harmonic**:
- The frequency of the nth harmonic (or overtone) is given by:
\[ f_n = \frac{nV}{2L} \]
- Where:
- \( f_n \) is the frequency of the nth harmonic,
- \( n \) is the harmonic number,
- \( V \) is the velocity of the wave,
- \( L \) is the length of the string.

3. **Identify the Variables**:
- For the third overtone (fourth harmonic), \( n = 4 \).
- The length of the string \( L = 0.5 \, \text{m} \).
- The frequency \( f_4 = 1200 \, \text{Hz} \).

4. **Substitute Values into the Formula**:
- Substitute \( n = 4 \), \( L = 0.5 \, \text{m} \), and \( f_4 = 1200 \, \text{Hz} \) into the frequency formula:
\[ 1200 = \frac{4V}{2 \times 0.5} \]

5. **Simplify the Equation**:
- The equation simplifies to:
\[ 1200 = \frac{4V}{1} \]
- Thus, we have:
\[ 1200 = 4V \]

6. **Solve for Velocity \( V \)**:
- Rearranging gives:
\[ V = \frac{1200}{4} = 300 \, \text{m/s} \]

### Final Answer:
The velocity of the wave is \( 300 \, \text{m/s} \).

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