When a tuning fork A of unknown frequency is sounded with another tuning fork B of frequency 256Hz, then 3 beats per second are observed. After that A is loaded with wax and sounded, the again 3 beats per second are observed. The frequency of the tuning fork A is

Options

A. 250 Hz
B. 253 Hz
C. 259 Hz
D. 262 Hz

Answer

To find the frequency of tuning fork A, we can follow these steps:

### Step 1: Understand the concept of beats
When two tuning forks of slightly different frequencies are sounded together, they produce a phenomenon called beats. The beat frequency is equal to the absolute difference between the two frequencies.

### Step 2: Identify the known frequency
We know the frequency of tuning fork B is 256 Hz.

### Step 3: Use the beat frequency information
When tuning fork A is sounded with tuning fork B, 3 beats per second are observed. This means the frequency of tuning fork A (let's denote it as \( f_A \)) can be either:
- \( f_A = 256 + 3 = 259 \) Hz (if \( f_A \) is higher than \( f_B \))
- \( f_A = 256 - 3 = 253 \) Hz (if \( f_A \) is lower than \( f_B \))

### Step 4: Consider the effect of loading with wax
When tuning fork A is loaded with wax, its frequency decreases. After loading with wax, the beat frequency remains the same at 3 beats per second. This means that the new frequency of tuning fork A (after loading with wax, denoted as \( f_{A'} \)) must also produce 3 beats per second with tuning fork B.

### Step 5: Analyze the new frequency
Since loading with wax decreases the frequency, we have:
- \( f_{A'} < f_A \)

Given that the beat frequency remains 3 Hz, we can conclude that:
- \( f_{A'} = 256 + 3 = 259 \) Hz (if \( f_{A'} \) is higher than \( f_B \))
- \( f_{A'} = 256 - 3 = 253 \) Hz (if \( f_{A'} \) is lower than \( f_B \))

### Step 6: Determine the correct frequency for \( f_A \)
Since \( f_{A'} < f_A \) and both scenarios yield a beat frequency of 3 Hz, we can conclude:
- If \( f_A = 259 \) Hz, then \( f_{A'} \) would be less than 259 Hz, which could be 256 Hz or lower, resulting in 3 beats with 256 Hz.
- If \( f_A = 253 \) Hz, then \( f_{A'} \) would be less than 253 Hz, which would not produce a beat frequency of 3 Hz with 256 Hz.

Thus, the only consistent solution is:
- \( f_A = 259 \) Hz

### Final Answer
The frequency of tuning fork A is **259 Hz**.

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