A tuning fork A of unknown frequency produces 5 beats/s with a fork of known frequency 340 Hz. When fork A is filed, the beat frequency decreases to 2 beats/s. What is the frequency of fork A ?

Options

A. 342 Hz
B. 345 Hz
C. 335 Hz
D. 338 Hz

Answer

To solve the problem, we need to find the frequency of tuning fork A (denoted as \( F_A \)) based on the information provided about the beat frequencies with a known tuning fork of frequency \( F_B = 340 \) Hz.

### Step-by-Step Solution:

1. **Understanding Beat Frequency**:
The beat frequency is given by the absolute difference between the frequencies of two tuning forks. Therefore, the beat frequency \( f_b \) can be expressed as:
\[ f_b = |F_A - F_B| \]

2. **Initial Condition**:
We are told that tuning fork A produces 5 beats per second with fork B (340 Hz). This gives us two possible equations:
\[ |F_A - 340| = 5 \]
This can lead to two scenarios:
- \( F_A - 340 = 5 \) → \( F_A = 345 \) Hz
- \( 340 - F_A = 5 \) → \( F_A = 335 \) Hz

3. **Filing the Fork**:
When fork A is filed, its frequency increases. We denote the new frequency of fork A after filing as \( F_A' \). Since filing increases the frequency, we have:
\[ F_A' > F_A \]

4. **New Beat Frequency**:
After filing, the beat frequency decreases to 2 beats per second. Thus, we have:
\[ |F_A' - 340| = 2 \]
This leads to two new scenarios:
- \( F_A' - 340 = 2 \) → \( F_A' = 342 \) Hz
- \( 340 - F_A' = 2 \) → \( F_A' = 338 \) Hz

5. **Analyzing the Scenarios**:
- If \( F_A = 345 \) Hz (from the first scenario), then after filing, \( F_A' \) would be greater than 345 Hz. This means \( F_A' \) cannot be 342 Hz or 338 Hz, as both are less than 345 Hz.
- If \( F_A = 335 \) Hz, then after filing, \( F_A' \) would be greater than 335 Hz. This means \( F_A' \) could be 342 Hz or 338 Hz.

6. **Finding the Correct Frequency**:
Since \( F_A' \) must be greater than 335 Hz and can be either 342 Hz or 338 Hz, the only valid frequency for \( F_A \) that satisfies all conditions is:
\[ F_A = 335 \text{ Hz} \]

### Final Answer:
The frequency of tuning fork A is \( 335 \) Hz.