Two tuning forks, A and B, produce notes of frequencies 258 Hz and 262 Hz. An unknown note sounded with A produces certain beats. When the same note is sounded with B, the beat frequency gets doubled. The unknown frequency is

Options

A. 250 Hz
B. 252 Hz
C. 254 Hz
D. 256 Hz

Explanation

To find the unknown frequency, we can follow these steps:

### Step 1: Understand the Concept of Beats
The beat frequency is the absolute difference between the frequencies of two sound waves. If two tuning forks are sounded together, the number of beats heard per second is given by:
\[ \text{Beat Frequency} = |f_1 - f_2| \]
where \( f_1 \) and \( f_2 \) are the frequencies of the two tuning forks.

### Step 2: Identify the Given Frequencies
We are given:
- Frequency of tuning fork A, \( f_A = 258 \, \text{Hz} \)
- Frequency of tuning fork B, \( f_B = 262 \, \text{Hz} \)

Let the unknown frequency be \( f_X \).

### Step 3: Set Up the Beat Frequency with Tuning Fork A
When the unknown note \( f_X \) is sounded with tuning fork A, the beat frequency \( f_BA \) is:
\[ f_BA = |f_A - f_X| \]
Since \( f_A = 258 \, \text{Hz} \), we can write:
\[ f_BA = |258 - f_X| \]

### Step 4: Set Up the Beat Frequency with Tuning Fork B
When the same unknown note \( f_X \) is sounded with tuning fork B, the beat frequency \( f_BB \) is:
\[ f_BB = |f_B - f_X| \]
Since \( f_B = 262 \, \text{Hz} \), we can write:
\[ f_BB = |262 - f_X| \]

### Step 5: Relate the Beat Frequencies
According to the problem, the beat frequency with tuning fork B is double that with tuning fork A:
\[ f_BB = 2 \times f_BA \]

### Step 6: Substitute the Beat Frequencies
Substituting the expressions for \( f_BA \) and \( f_BB \):
\[ |262 - f_X| = 2 \times |258 - f_X| \]

### Step 7: Solve the Equation
We will consider two cases for the absolute values:

**Case 1:** \( 262 - f_X = 2(258 - f_X) \)
1. Rearranging gives:
\[ 262 - f_X = 516 - 2f_X \]
\[ 2f_X - f_X = 516 - 262 \]
\[ f_X = 254 \, \text{Hz} \]

**Case 2:** \( 262 - f_X = 2(f_X - 258) \)
1. Rearranging gives:
\[ 262 - f_X = 2f_X - 516 \]
\[ 262 + 516 = 3f_X \]
\[ 778 = 3f_X \]
\[ f_X = \frac{778}{3} \approx 259.33 \, \text{Hz} \] (not a valid solution since it does not satisfy the doubling condition)

### Step 8: Conclusion
The only valid solution from Case 1 is:
\[ f_X = 254 \, \text{Hz} \]

### Final Answer
The unknown frequency is \( 254 \, \text{Hz} \).

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