To increase the frequency by 20 %, the tension in the string vibrating on a sonometer has to be increased by

Options

A. 0.44
B. 0.33
C. 0.22
D. 0.11

Answer

To solve the problem of how much the tension in the string must be increased to achieve a 20% increase in frequency, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between frequency and tension**:
The frequency \( F \) of a vibrating string is related to the tension \( T \) in the string by the formula:
\[ F \propto \sqrt{T} \]
This means that if the tension increases, the frequency also increases.

2. **Define initial conditions**:
Let's assume the initial frequency \( F_1 \) is 100 Hz. To find the new frequency \( F_2 \) after a 20% increase:
\[ F_2 = F_1 + 0.2 \times F_1 = 100 + 20 = 120 \text{ Hz} \]

3. **Set up the ratio of frequencies**:
Using the relationship between frequency and tension, we can write:
\[ \frac{F_2}{F_1} = \frac{\sqrt{T_2}}{\sqrt{T_1}} \]
Substituting the known frequencies:
\[ \frac{120}{100} = \frac{\sqrt{T_2}}{\sqrt{T_1}} \]
This simplifies to:
\[ \frac{6}{5} = \frac{\sqrt{T_2}}{\sqrt{T_1}} \]

4. **Square both sides**:
Squaring both sides to eliminate the square root gives:
\[ \left(\frac{6}{5}\right)^2 = \frac{T_2}{T_1} \]
This results in:
\[ \frac{36}{25} = \frac{T_2}{T_1} \]

5. **Express \( T_2 \) in terms of \( T_1 \)**:
If we let \( T_1 = 100 \) (for simplicity), we can express \( T_2 \):
\[ T_2 = T_1 \times \frac{36}{25} = 100 \times \frac{36}{25} = 144 \]

6. **Calculate the increase in tension**:
The increase in tension \( x \) is:
\[ x = T_2 - T_1 = 144 - 100 = 44 \]

7. **Calculate the percentage increase in tension**:
To find the percentage increase in tension:
\[ \text{Percentage increase} = \frac{x}{T_1} \times 100 = \frac{44}{100} \times 100 = 44\% \]
In decimal form, this is:
\[ 0.44 \]

### Conclusion:
The tension in the string must be increased by **0.44** (or 44%) to achieve a 20% increase in frequency.