A fork of unknown frequency gives 3 beats/sec. when sounded with a fork of frequency 256 Hz. On loading the fork of unknown frequency with wax, the number of beats produced per second remain 3. Calculate the unknown frequency of the fork.

Answer

To solve the problem, we need to determine the unknown frequency of a tuning fork that produces beats with a known frequency fork of 256 Hz. The steps to find the solution are as follows:

### Step 1: Understand the concept of beats
Beats occur when two sound waves of slightly different frequencies interfere with each other. The beat frequency is equal to the absolute difference between the two frequencies.

### Step 2: Set up the equation for the initial scenario
Let the frequency of the unknown fork be \( f \). The beat frequency when sounded with the 256 Hz fork is given as 3 beats per second. Therefore, we can write the following equations:
- \( |f - 256| = 3 \)

### Step 3: Solve the equation for two possible frequencies
From the equation \( |f - 256| = 3 \), we can derive two possible cases:
1. \( f - 256 = 3 \) → \( f = 259 \) Hz
2. \( f - 256 = -3 \) → \( f = 253 \) Hz

### Step 4: Analyze the effect of loading the fork with wax
When the fork is loaded with wax, its frequency decreases. The problem states that the number of beats produced per second remains 3. This means that the new frequency after loading the fork must still maintain a beat frequency of 3 with the 256 Hz fork.

### Step 5: Determine the possible frequencies after loading
If the original frequency was 259 Hz, after loading with wax, the frequency would decrease. The beat frequency would then be:
- If the frequency drops to 256 Hz, the beat frequency would be \( |256 - 256| = 0 \) (not possible).
- If it drops to 255 Hz, the beat frequency would be \( |255 - 256| = 1 \) (not possible).
- If it drops to 253 Hz, the beat frequency would be \( |253 - 256| = 3 \) (possible).

If the original frequency was 253 Hz, loading it with wax would further decrease the frequency, increasing the beat frequency, which contradicts the condition of maintaining 3 beats per second.

### Conclusion
The only consistent scenario is that the original unknown frequency must be 259 Hz. After loading with wax, it can drop to 256 Hz, maintaining the beat frequency of 3.

Thus, the unknown frequency of the fork is:
**Unknown frequency = 259 Hz**