In order to double the frequnecy of the fundamental note emitted by a stratched string the length is reduced to `3/4` th of the original length and the tension is changed. The factor by which the tension is to be changed is

Options

A. `3/8`
B. `2/3`
C. `8/9`
D. `9/4`

Answer

To solve the problem, we need to determine the factor by which the tension in the string must be changed to double the frequency of the fundamental note when the length of the string is reduced to \( \frac{3}{4} \) of its original length.

### Step-by-Step Solution:

1. **Understand the Formula for Frequency**:
The fundamental frequency \( f \) of a stretched string is given by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{m}} \]
where \( L \) is the length of the string, \( T \) is the tension, and \( m \) is the mass per unit length of the string.

2. **Set Up Initial Conditions**:
Let the original length of the string be \( L_1 \) and the original tension be \( T_1 \). The initial frequency \( f_1 \) is:
\[ f_1 = \frac{1}{2L_1} \sqrt{\frac{T_1}{m}} \]

3. **Determine New Length and Frequency**:
The new length \( L_2 \) is given as \( \frac{3}{4}L_1 \). The new frequency \( f_2 \) should be double the original frequency:
\[ f_2 = 2f_1 \]

4. **Express New Frequency**:
The new frequency \( f_2 \) can be expressed as:
\[ f_2 = \frac{1}{2L_2} \sqrt{\frac{T_2}{m}} = \frac{1}{2 \cdot \frac{3}{4}L_1} \sqrt{\frac{T_2}{m}} = \frac{2}{3L_1} \sqrt{\frac{T_2}{m}} \]

5. **Set Up the Equation**:
Since \( f_2 = 2f_1 \), we can write:
\[ \frac{2}{3L_1} \sqrt{\frac{T_2}{m}} = 2 \left( \frac{1}{2L_1} \sqrt{\frac{T_1}{m}} \right) \]
Simplifying gives:
\[ \frac{2}{3L_1} \sqrt{\frac{T_2}{m}} = \frac{1}{L_1} \sqrt{\frac{T_1}{m}} \]

6. **Cancel Common Terms**:
Multiply both sides by \( L_1 \) and \( \sqrt{m} \):
\[ \frac{2}{3} \sqrt{T_2} = \sqrt{T_1} \]

7. **Square Both Sides**:
Squaring both sides leads to:
\[ \frac{4}{9} T_2 = T_1 \]

8. **Solve for \( T_2 \)**:
Rearranging gives:
\[ T_2 = \frac{9}{4} T_1 \]

9. **Determine the Factor of Change in Tension**:
The factor by which the tension is changed is:
\[ \text{Factor} = \frac{T_2}{T_1} = \frac{9}{4} \]

### Final Answer:
The factor by which the tension is to be changed is \( \frac{9}{4} \).

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