In order to double the frequency of the fundamental note emitted by a stretched 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`

Explanation

To solve the problem, we need to find the factor by which the tension in the string must change in order 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 relationship between frequency, length, and tension**:
The frequency \( f \) of the fundamental note of a stretched string is given by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where:
- \( L \) is the length of the string,
- \( T \) is the tension in the string,
- \( \mu \) is the linear mass density of the string.

2. **Set up the initial and final conditions**:
Let:
- \( L_1 \) be the original length,
- \( T_1 \) be the original tension,
- \( f_1 \) be the original frequency.

The new conditions after changing the length and tension are:
- \( L_2 = \frac{3}{4} L_1 \) (length is reduced to \( \frac{3}{4} \) of the original),
- \( f_2 = 2 f_1 \) (frequency is doubled).

3. **Relate the frequencies and lengths**:
Using the frequency formula, we can write:
\[ f_1 = \frac{1}{2L_1} \sqrt{\frac{T_1}{\mu}} \]
and
\[ f_2 = \frac{1}{2L_2} \sqrt{\frac{T_2}{\mu}} \]

4. **Substitute the new length into the frequency equation**:
Since \( L_2 = \frac{3}{4} L_1 \), we have:
\[ f_2 = \frac{1}{2 \left(\frac{3}{4} L_1\right)} \sqrt{\frac{T_2}{\mu}} = \frac{2}{3 L_1} \sqrt{\frac{T_2}{\mu}} \]

5. **Set up the ratio of frequencies**:
We know \( f_2 = 2 f_1 \), so:
\[ \frac{2}{3 L_1} \sqrt{\frac{T_2}{\mu}} = 2 \left(\frac{1}{2L_1} \sqrt{\frac{T_1}{\mu}}\right) \]
Simplifying gives:
\[ \frac{2}{3} \sqrt{\frac{T_2}{\mu}} = \frac{1}{L_1} \sqrt{\frac{T_1}{\mu}} \]

6. **Eliminate \( \mu \) and rearrange**:
Cancel \( \mu \) from both sides:
\[ \frac{2}{3} \sqrt{T_2} = \sqrt{T_1} \]
Squaring both sides gives:
\[ \frac{4}{9} T_2 = T_1 \]
Therefore:
\[ T_2 = \frac{9}{4} T_1 \]

7. **Conclusion**:
The factor by which the tension must change is \( \frac{T_2}{T_1} = \frac{9}{4} \).

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