A sound source of frequency `512 Hz` is producing `6` beats with a guitar. If the string of gultar is stretched slightly then beat frequency decreases. The original frequency of guitar is

Options

A. `506 Hz`
B. `512 Hz`
C. `518 Hz`
D. `524 Hz`

Answer

To find the original frequency of the guitar, we can follow these steps:

### Step 1: Understand the concept of beats
When two sound waves of slightly different frequencies interfere, they produce a phenomenon called beats. The beat frequency is equal to the absolute difference between the two frequencies.

### Step 2: Identify the given values
- Frequency of the sound source (f_source) = 512 Hz
- Beat frequency (f_beat) = 6 beats

### Step 3: Determine the relationship between frequencies
The beat frequency is given by:
\[ f_{beat} = |f_{source} - f_{guitar}| \]
Where \( f_{guitar} \) is the frequency of the guitar.

### Step 4: Set up the equation
Since the beat frequency is 6, we can write:
\[ |512 - f_{guitar}| = 6 \]

This gives us two possible equations:
1. \( 512 - f_{guitar} = 6 \)
2. \( f_{guitar} - 512 = 6 \)

### Step 5: Solve the first equation
From the first equation:
\[ 512 - f_{guitar} = 6 \]
\[ f_{guitar} = 512 - 6 \]
\[ f_{guitar} = 506 \, \text{Hz} \]

### Step 6: Solve the second equation
From the second equation:
\[ f_{guitar} - 512 = 6 \]
\[ f_{guitar} = 512 + 6 \]
\[ f_{guitar} = 518 \, \text{Hz} \]

### Step 7: Analyze the situation
Since the problem states that the beat frequency decreases when the guitar string is stretched, it indicates that the frequency of the guitar is initially lower than the sound source frequency. Therefore, the original frequency of the guitar must be:
\[ f_{guitar} = 506 \, \text{Hz} \]

### Conclusion
The original frequency of the guitar is **506 Hz**.

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