A string has tension T. For tripling the frequency. The tension is string will become

Options

A. 3T
B. `T/9`
C. `T/sqrt(3)`
D. 9T

Explanation

To solve the problem of how the tension in a string must change to triple its frequency, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship**: The frequency \( f \) of a string fixed at both ends is related to the tension \( T \) in the string by the formula:
\[ f \propto \sqrt{T} \]
This means that the frequency is proportional to the square root of the tension.

2. **Setting Initial Conditions**: Let the initial tension be \( T_1 = T \) and the initial frequency be \( f_1 \). Therefore, we can express the relationship as:
\[ f_1 \propto \sqrt{T} \]

3. **New Frequency Condition**: We want to triple the frequency, so the new frequency \( f_2 \) will be:
\[ f_2 = 3f_1 \]

4. **Expressing New Tension**: According to the proportionality, we can write:
\[ f_2 \propto \sqrt{T_2} \]
where \( T_2 \) is the new tension we need to find.

5. **Setting Up the Ratio**: Now, we can set up the ratio of the frequencies:
\[ \frac{f_1}{f_2} = \frac{\sqrt{T_1}}{\sqrt{T_2}} \]
Substituting \( f_2 = 3f_1 \):
\[ \frac{f_1}{3f_1} = \frac{\sqrt{T}}{\sqrt{T_2}} \]
This simplifies to:
\[ \frac{1}{3} = \frac{\sqrt{T}}{\sqrt{T_2}} \]

6. **Squaring Both Sides**: To eliminate the square roots, we square both sides:
\[ \left(\frac{1}{3}\right)^2 = \frac{T}{T_2} \]
This gives us:
\[ \frac{1}{9} = \frac{T}{T_2} \]

7. **Finding \( T_2 \)**: Rearranging the equation to solve for \( T_2 \):
\[ T_2 = 9T \]

### Final Answer:
Thus, to triple the frequency of the string, the tension in the string must become \( 9T \).

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