Two instruments having stretched strings are being played in unison . When the tension in one of the instruments is increases by ` 1% , 3` beats are produced in ` 2 s`. The initial frequency of vibration of each wire is

Options

A. `600 Hz`
B. `300 Hz`
C. `200 Hz`
D. `150 Hz`

Explanation

To solve the problem step by step, we will use the relationship between frequency and tension in a stretched string, as well as the concept of beat frequency.

### Step-by-Step Solution:

1. **Understanding the Relationship Between Frequency and Tension**:
The frequency \( f \) of a stretched string is given by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where \( T \) is the tension, \( L \) is the length of the string, and \( \mu \) is the linear mass density of the string.

2. **Change in Tension**:
When the tension in one of the instruments is increased by 1%, the new tension \( T' \) can be expressed as:
\[ T' = T + 0.01T = 1.01T \]

3. **Calculating the New Frequency**:
The new frequency \( f' \) can be calculated using the new tension:
\[ f' = \frac{1}{2L} \sqrt{\frac{T'}{\mu}} = \frac{1}{2L} \sqrt{\frac{1.01T}{\mu}} = \sqrt{1.01} \cdot f \]

4. **Beat Frequency**:
The beat frequency \( f_b \) is given by the difference between the new frequency and the original frequency:
\[ f_b = f' - f = \sqrt{1.01} \cdot f - f = (\sqrt{1.01} - 1)f \]

5. **Given Information**:
We know that 3 beats are produced in 2 seconds, which means the beat frequency is:
\[ f_b = \frac{3 \text{ beats}}{2 \text{ seconds}} = 1.5 \text{ beats/second} \]

6. **Setting Up the Equation**:
Now we can set up the equation:
\[ (\sqrt{1.01} - 1)f = 1.5 \]

7. **Solving for Frequency \( f \)**:
Rearranging the equation gives:
\[ f = \frac{1.5}{\sqrt{1.01} - 1} \]

8. **Calculating \( \sqrt{1.01} \)**:
Approximating \( \sqrt{1.01} \):
\[ \sqrt{1.01} \approx 1.005 \]
Therefore:
\[ \sqrt{1.01} - 1 \approx 0.005 \]

9. **Final Calculation**:
Substituting this back into the frequency equation:
\[ f \approx \frac{1.5}{0.005} = 300 \text{ Hz} \]

### Conclusion:
The initial frequency of vibration of each wire is approximately **300 Hz**.