A tuning fork of frequency 256 Hz is producing 5 beats/sec with the vibrating string of a sitar. On increasing the tension in the string, the beat frequency becomes 3 beats/sec. The initial frequency of sitar before increasing the tension is

Options

A. 256 Hz
B. 251 Hz
C. 261 Hz
D. 253 Hz

Explanation

To find the initial frequency of the sitar string before increasing the tension, we can follow these steps:

### Step 1: Understand the Beat Frequency
The beat frequency is the absolute difference between the frequencies of two sound sources. In this case, we have:
- Frequency of the tuning fork, \( f_t = 256 \, \text{Hz} \)
- Beat frequency with the sitar string, \( B_1 = 5 \, \text{beats/sec} \)

### Step 2: Determine Possible Frequencies of the Sitar String
The frequency of the sitar string, \( f_s \), can be either higher or lower than the tuning fork frequency. Therefore, we can write:
- \( f_s = f_t + B_1 \) or \( f_s = f_t - B_1 \)

Calculating these:
1. If \( f_s = 256 + 5 = 261 \, \text{Hz} \)
2. If \( f_s = 256 - 5 = 251 \, \text{Hz} \)

Thus, the possible frequencies of the sitar string before increasing the tension are \( 261 \, \text{Hz} \) or \( 251 \, \text{Hz} \).

### Step 3: Analyze the Effect of Increasing Tension
When the tension in the string is increased, the frequency of the string will increase. The new beat frequency is given as \( B_2 = 3 \, \text{beats/sec} \).

### Step 4: Determine the New Frequency of the Sitar String
If we assume the initial frequency of the sitar string was \( 261 \, \text{Hz} \) and we increase the tension, the new frequency \( f_s' \) will be greater than \( 261 \, \text{Hz} \). The beat frequency with the tuning fork will then increase, not decrease, which contradicts the given information.

On the other hand, if we assume the initial frequency of the sitar string was \( 251 \, \text{Hz} \) and we increase the tension, the new frequency \( f_s' \) will be greater than \( 251 \, \text{Hz} \). The beat frequency will then decrease because the new frequency will approach \( 256 \, \text{Hz} \).

### Step 5: Calculate the New Frequency
If we assume the new frequency after increasing tension is \( f_s' = 253 \, \text{Hz} \):
- The beat frequency with the tuning fork would then be:
\[ |f_t - f_s'| = |256 - 253| = 3 \, \text{beats/sec} \]

This matches the given condition of a beat frequency of \( 3 \, \text{beats/sec} \).

### Conclusion
Thus, the initial frequency of the sitar string before increasing the tension is:
\[ f_s = 251 \, \text{Hz} \]