A tuning fork A of frequency 512 Hz sounded with another tuning fork B gives 5 beats/sec. On loading B with a piece of wax, and on sounding, again 5 beats/sec are produced. Calculate the frequency of fork B.

Options

A. 517 Hz
B. 500 Hz
C. 600 Hz
D. 567 Hz

Explanation

To solve the problem, we need to determine the frequency of tuning fork B (fB) given that tuning fork A has a frequency of 512 Hz and they produce 5 beats per second.

### Step-by-Step Solution:

1. **Understanding Beat Frequency**:
The beat frequency is the absolute difference between the frequencies of two tuning forks. The formula for beat frequency (f_beat) is:
\[ f_{beat} = |f_A - f_B| \]
Given that \( f_A = 512 \) Hz and \( f_{beat} = 5 \) Hz, we can express this as:
\[ |512 - f_B| = 5 \]

2. **Setting Up the Equations**:
This absolute value equation gives us two possible scenarios:
- Case 1: \( 512 - f_B = 5 \)
- Case 2: \( f_B - 512 = 5 \)

From Case 1:
\[ f_B = 512 - 5 = 507 \text{ Hz} \]

From Case 2:
\[ f_B = 512 + 5 = 517 \text{ Hz} \]

So, the two potential frequencies for tuning fork B are 507 Hz and 517 Hz.

3. **Analyzing the Effect of Wax**:
When tuning fork B is loaded with wax, its frequency will decrease. If we assume that the original frequency of B was 517 Hz, then after loading with wax, it will decrease further.

If \( f_B = 517 \) Hz, after loading with wax, the frequency will drop below 517 Hz, and we can check if it can still produce 5 beats per second with tuning fork A:
\[ |512 - f_B'| = 5 \]
where \( f_B' < 517 \).

4. **Verifying the Frequencies**:
- If \( f_B = 507 \) Hz, after loading with wax, \( f_B \) would decrease further, making it impossible to maintain a beat frequency of 5 Hz with 512 Hz.
- If \( f_B = 517 \) Hz, after loading with wax, \( f_B \) will decrease, and it can still maintain a beat frequency of 5 Hz as it approaches 512 Hz.

5. **Conclusion**:
Therefore, the only viable frequency for tuning fork B that satisfies both conditions (before and after loading with wax) is:
\[ f_B = 517 \text{ Hz} \]

### Final Answer:
The frequency of tuning fork B is **517 Hz**.

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