A knife-edge divides a sonometer wire into two parts. The fundamental frequencies of the two parts are `n_(1)` and `n_(2)`. The fundamental frequency of the sonometer wire when the knife-edge is removed will be

Options

A. `n_(1)+n_(2)`
B. `(1)/(2)(n_(1)+n_(2))`
C. `sqrt(n_(1)n_(2))`
D. `(n_(1)n_(2))/(n_(1)+n_(2))`

Answer

To find the fundamental frequency of the sonometer wire when the knife-edge is removed, we can follow these steps:

### Step 1: Understand the Setup
The sonometer wire is divided into two parts by a knife-edge. Let the lengths of the two parts be \(L_1\) and \(L_2\). The fundamental frequencies of these two parts are \(n_1\) and \(n_2\) respectively.

### Step 2: Relate Frequency and Length
The fundamental frequency of a vibrating string is inversely proportional to its length. This relationship can be expressed as:
\[ n \propto \frac{1}{L} \]
For the two parts, we can write:
\[ n_1 \propto \frac{1}{L_1} \quad \text{and} \quad n_2 \propto \frac{1}{L_2} \]

### Step 3: Express Lengths in Terms of Frequencies
From the above relationships, we can express the lengths in terms of the frequencies:
\[ L_1 = \frac{v}{2n_1} \quad \text{and} \quad L_2 = \frac{v}{2n_2} \]
where \(v\) is the velocity of the wave in the string.

### Step 4: Total Length of the Wire
The total length of the wire when the knife-edge is removed is:
\[ L = L_1 + L_2 \]
Substituting the expressions for \(L_1\) and \(L_2\):
\[ L = \frac{v}{2n_1} + \frac{v}{2n_2} \]

### Step 5: Simplify the Equation
Factoring out \(v/2\):
\[ L = \frac{v}{2} \left(\frac{1}{n_1} + \frac{1}{n_2}\right) \]

### Step 6: Relate Total Length to Frequency
Now, we can express the fundamental frequency \(n\) of the entire wire when the knife-edge is removed:
\[ n \propto \frac{1}{L} \]
Thus,
\[ n = \frac{v}{2L} \]
Substituting the expression for \(L\):
\[ n = \frac{v}{2} \cdot \frac{1}{\frac{v}{2} \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} = \frac{1}{\frac{1}{n_1} + \frac{1}{n_2}} \]

### Step 7: Final Expression for Frequency
This leads us to the final expression for the fundamental frequency \(n\):
\[ \frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} \]
Thus, the fundamental frequency \(n\) can be expressed as:
\[ n = \frac{n_1 n_2}{n_1 + n_2} \]

### Conclusion
The fundamental frequency of the sonometer wire when the knife-edge is removed is given by:
\[ n = \frac{n_1 n_2}{n_1 + n_2} \]

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