If two tuning fork A and B are sounded together they produce 4 beats per second. A is then slightly loaded with wax, they produce 2 beats when sounded again. The frequency of A is 256. The frequency of B will be

Options

A. 250
B. 252
C. 260
D. 262

Answer

To find the frequency of tuning fork B, 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. When two tuning forks A and B are sounded together, they produce 4 beats per second. This means:
\[ |f_A - f_B| = 4 \text{ Hz} \]

### Step 2: Determine Possible Frequencies of B
We know the frequency of tuning fork A (f_A) is 256 Hz. Therefore, the frequency of tuning fork B (f_B) can be either:
1. \( f_B = f_A + 4 = 256 + 4 = 260 \text{ Hz} \)
2. \( f_B = f_A - 4 = 256 - 4 = 252 \text{ Hz} \)

So, the possible frequencies for B are 260 Hz or 252 Hz.

### Step 3: Analyze the Effect of Loading A
When tuning fork A is slightly loaded with wax, its frequency will decrease. After loading, they produce 2 beats per second:
\[ |f_A' - f_B| = 2 \text{ Hz} \]
Where \( f_A' \) is the new frequency of A after loading.

### Step 4: Determine the New Frequency of A
Since loading A with wax decreases its frequency, we can denote the new frequency of A as:
\[ f_A' < f_A \]
Assuming \( f_A' = 256 - x \) where \( x \) is a small positive number.

### Step 5: Analyze the Two Cases for B
1. **If \( f_B = 260 \text{ Hz} \)**:
- After loading, if \( f_A' = 256 - x \), then:
\[ |(256 - x) - 260| = 2 \]
This simplifies to:
\[ | -x - 4 | = 2 \]
This leads to two equations:
- \( -x - 4 = 2 \) → \( x = -6 \) (not possible)
- \( -x - 4 = -2 \) → \( x = -2 \) (not possible)

2. **If \( f_B = 252 \text{ Hz} \)**:
- After loading, if \( f_A' = 256 - x \), then:
\[ |(256 - x) - 252| = 2 \]
This simplifies to:
\[ | 4 - x | = 2 \]
This leads to two equations:
- \( 4 - x = 2 \) → \( x = 2 \) (possible)
- \( 4 - x = -2 \) → \( x = 6 \) (not possible)

### Conclusion
Since the only valid solution occurs when \( f_B = 252 \text{ Hz} \), we conclude that the frequency of tuning fork B is:
\[ f_B = 252 \text{ Hz} \]

### Final Answer
The frequency of tuning fork B is **252 Hz**.

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