A sonometer wire 64 cm long produces 8 beats per second with a tuning fork. If length of sonometer wire is increased by one centimeter it resonates with the tuning fork. The frequency of tuning fork is observed 64x . The value of x is

Answer

To solve the problem step-by-step, we will use the relationship between the frequencies of the sonometer wire and its lengths, along with the concept of beats.

### Step 1: Understand the relationship between frequency and length
The frequency of a vibrating wire is inversely proportional to its length when the tension and mass per unit length are constant. This can be expressed as:
\[ \frac{N_1}{N_2} = \frac{L_2}{L_1} \]
where \(N_1\) and \(N_2\) are the frequencies corresponding to lengths \(L_1\) and \(L_2\), respectively.

### Step 2: Define the lengths
Given:
- Initial length \(L_1 = 64 \, \text{cm}\)
- Increased length \(L_2 = L_1 + 1 \, \text{cm} = 65 \, \text{cm}\)

### Step 3: Set up the frequency ratio
Using the relationship:
\[ \frac{N_1}{N_2} = \frac{65}{64} \]

### Step 4: Understand the beats produced
The problem states that 8 beats per second are produced when the sonometer wire is at 64 cm. The number of beats is given by:
\[ |N_1 - N_2| = 8 \]
Since \(N_1\) is greater than \(N_2\) (because the length increased), we can write:
\[ N_1 - N_2 = 8 \]

### Step 5: Express \(N_1\) in terms of \(N_2\)
From the beats equation:
\[ N_1 = N_2 + 8 \]

### Step 6: Substitute \(N_1\) into the frequency ratio
Now substitute \(N_1\) into the frequency ratio:
\[ \frac{N_2 + 8}{N_2} = \frac{65}{64} \]

### Step 7: Cross-multiply to solve for \(N_2\)
Cross-multiplying gives:
\[ 64(N_2 + 8) = 65N_2 \]
Expanding this:
\[ 64N_2 + 512 = 65N_2 \]

### Step 8: Rearrange to find \(N_2\)
Rearranging gives:
\[ 65N_2 - 64N_2 = 512 \]
\[ N_2 = 512 \, \text{Hz} \]

### Step 9: Relate \(N_2\) to the tuning fork frequency
According to the problem, the frequency of the tuning fork is given by:
\[ N_2 = 64x \]
Thus, we can set up the equation:
\[ 64x = 512 \]

### Step 10: Solve for \(x\)
Dividing both sides by 64:
\[ x = \frac{512}{64} = 8 \]

### Final Answer
The value of \(x\) is \(8\).

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