When the length of the vibrating segment of a
sonometer wire is increased by `1%` the percentage
change in its frequency is

Options

A. `100/101`
B. `99/100`
C. 1
D. 2

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between frequency and length
The frequency \( n \) of a vibrating sonometer wire is given by the formula:
\[ n = \frac{1}{2L} \sqrt{\frac{T}{m}} \]
where:
- \( L \) is the length of the vibrating segment,
- \( T \) is the tension in the wire,
- \( m \) is the mass per unit length of the wire.

From this formula, we can see that frequency \( n \) is inversely proportional to the length \( L \):
\[ n \propto \frac{1}{L} \]

### Step 2: Define the initial and final lengths
Let the initial length of the wire be \( L_1 = L \). When the length is increased by 1%, the new length \( L_2 \) is:
\[ L_2 = L_1 + 0.01L_1 = 1.01L \]

### Step 3: Calculate the initial and final frequencies
Using the relationship \( n \propto \frac{1}{L} \):
- The initial frequency \( n_1 \) is:
\[ n_1 = \frac{1}{2L} \sqrt{\frac{T}{m}} \]
- The final frequency \( n_2 \) when the length is \( L_2 \) is:
\[ n_2 = \frac{1}{2L_2} \sqrt{\frac{T}{m}} = \frac{1}{2(1.01L)} \sqrt{\frac{T}{m}} = \frac{1}{2L} \cdot \frac{1}{1.01} \sqrt{\frac{T}{m}} = \frac{n_1}{1.01} \]

### Step 4: Calculate the percentage change in frequency
The percentage change in frequency can be calculated using the formula:
\[ \text{Percentage Change} = \frac{n_2 - n_1}{n_1} \times 100 \]
Substituting \( n_2 \):
\[ \text{Percentage Change} = \frac{\frac{n_1}{1.01} - n_1}{n_1} \times 100 \]
\[ = \frac{n_1 \left( \frac{1}{1.01} - 1 \right)}{n_1} \times 100 \]
\[ = \left( \frac{1}{1.01} - 1 \right) \times 100 \]
Calculating \( \frac{1}{1.01} \):
\[ \frac{1}{1.01} \approx 0.9901 \]
Thus:
\[ \frac{1}{1.01} - 1 \approx -0.0099 \]
Now, multiplying by 100:
\[ \text{Percentage Change} \approx -0.99\% \]

### Step 5: Conclusion
The percentage change in frequency when the length of the vibrating segment of a sonometer wire is increased by 1% is approximately:
\[ \text{Percentage Change} \approx -0.99\% \]