A tuning fork of frequency 500 Hz produces 8 beats/ second when sounded with a having sonometer wire. What must be the frequency of the sonometer wire, if a slight increases be the frequency of the sonometer wire,if a slight increases in its tension produces fewer beats per second that before ?

Options

A. a. 508 Hz
B. b. 492 Hz
C. c. 500 Hz
D. d. 504 Hz

Explanation

To solve the problem step by step, we will analyze the information given and apply the relevant concepts of stationary waves and beats.

### Step 1: Understand the Concept of Beats
When two sound waves of slightly different frequencies interfere, they produce a phenomenon called beats. The number of beats per second is equal to the absolute difference between the two frequencies. Mathematically, this can be expressed as:
\[ \text{Number of beats} = |F_1 - F_2| \]
where \( F_1 \) is the frequency of the tuning fork and \( F_2 \) is the frequency of the sonometer wire.

### Step 2: Identify the Given Values
From the problem, we know:
- Frequency of the tuning fork, \( F_1 = 500 \, \text{Hz} \)
- Number of beats produced, \( \text{Beats} = 8 \, \text{beats/second} \)

### Step 3: Set Up the Equation for Beats
Using the beats formula:
\[ |F_1 - F_2| = 8 \]
This can be expressed in two possible scenarios:
1. \( F_1 - F_2 = 8 \)
2. \( F_2 - F_1 = 8 \)

### Step 4: Solve for \( F_2 \)
#### Case 1: \( F_1 - F_2 = 8 \)
\[ F_2 = F_1 - 8 = 500 \, \text{Hz} - 8 \, \text{Hz} = 492 \, \text{Hz} \]

#### Case 2: \( F_2 - F_1 = 8 \)
\[ F_2 = F_1 + 8 = 500 \, \text{Hz} + 8 \, \text{Hz} = 508 \, \text{Hz} \]

### Step 5: Analyze the Effect of Increasing Tension
The problem states that a slight increase in the tension of the sonometer wire results in fewer beats per second. This implies that the frequency of the sonometer wire \( F_2 \) must be increasing.

If \( F_2 \) were initially 492 Hz, increasing the tension would increase \( F_2 \) and thus reduce the number of beats (since the difference between \( F_1 \) and \( F_2 \) would decrease).

If \( F_2 \) were initially 508 Hz, increasing the tension would further increase \( F_2 \), which would also increase the number of beats, contradicting the information given.

### Conclusion
Thus, the only valid frequency for the sonometer wire before the increase in tension is:
\[ F_2 = 492 \, \text{Hz} \]

### Final Answer
The frequency of the sonometer wire must be **492 Hz**.

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