A somometer wire produces 2 beats per second with a tuning fork, when the length of the wire is either 102cm or 104 cm . The frequency of the tuning fork is

Options

A. 206 Hz
B. 204 Hz
C. 200 Hz
D. 198 Hz

Explanation

To solve the problem, we need to find the frequency of the tuning fork when a sonometer wire produces 2 beats per second at lengths of 102 cm and 104 cm. Here’s a step-by-step solution:

### Step 1: Understand the concept of beats
Beats occur when two sound waves of slightly different frequencies interfere with each other. The number of beats per second is equal to the absolute difference between the two frequencies. In this case, we have:
- Frequency of the tuning fork = \( n \)
- Frequency of the sonometer wire at 102 cm = \( n + 2 \) (since it is higher)
- Frequency of the sonometer wire at 104 cm = \( n - 2 \) (since it is lower)

### Step 2: Set up the relationship using the sonometer formula
The frequency of a vibrating string is inversely proportional to its length. Therefore, we can write:
- \( f_1 \) (frequency at 102 cm) = \( \frac{k}{102} \)
- \( f_2 \) (frequency at 104 cm) = \( \frac{k}{104} \)

Where \( k \) is a constant that depends on the tension and mass per unit length of the wire.

### Step 3: Establish the equations
Since the frequency is inversely proportional to the length, we can set up the following equations based on the lengths:
1. \( 102 \cdot f_1 = k \)
2. \( 104 \cdot f_2 = k \)

From the above, we can express the frequencies in terms of \( k \):
- \( f_1 = \frac{k}{102} \)
- \( f_2 = \frac{k}{104} \)

### Step 4: Relate the frequencies to the tuning fork frequency
From the beats information:
- \( n + 2 = f_1 \)
- \( n - 2 = f_2 \)

### Step 5: Set up the equation
Now we can equate the expressions for \( f_1 \) and \( f_2 \) to the tuning fork frequency:
1. \( n + 2 = \frac{k}{102} \)
2. \( n - 2 = \frac{k}{104} \)

### Step 6: Solve for \( k \)
From the first equation:
\[ k = (n + 2) \cdot 102 \]
From the second equation:
\[ k = (n - 2) \cdot 104 \]

### Step 7: Equate the two expressions for \( k \)
Setting the two expressions for \( k \) equal gives:
\[ (n + 2) \cdot 102 = (n - 2) \cdot 104 \]

### Step 8: Expand and simplify the equation
Expanding both sides:
\[ 102n + 204 = 104n - 208 \]

### Step 9: Rearrange the equation
Rearranging gives:
\[ 104n - 102n = 204 + 208 \]
\[ 2n = 412 \]

### Step 10: Solve for \( n \)
Dividing both sides by 2:
\[ n = 206 \, \text{Hz} \]

### Final Answer
The frequency of the tuning fork is **206 Hz**.