What is the percentage change in the tension necessary in a somometer of fixed length to produce a note one octave lower (half of original frequency) than before?

Options

A. `25%`
B. `50%`
C. `67%`
D. `75%`

Explanation

To find the percentage change in the tension necessary in a somometer of fixed length to produce a note one octave lower (which corresponds to half of the original 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 given by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where:
- \( L \) is the length of the string (fixed in this case),
- \( T \) is the tension in the string,
- \( \mu \) is the mass per unit length of the string.

2. **Establish the relationship between frequency and tension**:
From the formula, we can see that frequency is directly proportional to the square root of the tension:
\[ f \propto \sqrt{T} \]
Therefore, if the frequency changes, the tension must change accordingly.

3. **Define the frequencies**:
Let \( f_1 \) be the original frequency and \( f_2 \) be the new frequency (one octave lower). Since one octave lower means half the frequency:
\[ f_2 = \frac{1}{2} f_1 \]

4. **Relate the tensions**:
Using the relationship between frequency and tension:
\[ \frac{f_1}{f_2} = \sqrt{\frac{T_1}{T_2}} \]
Substituting \( f_2 \):
\[ \frac{f_1}{\frac{1}{2} f_1} = \sqrt{\frac{T_1}{T_2}} \implies 2 = \sqrt{\frac{T_1}{T_2}} \]

5. **Square both sides**:
Squaring both sides gives:
\[ 4 = \frac{T_1}{T_2} \implies T_1 = 4T_2 \]

6. **Calculate the percentage change in tension**:
The percentage change in tension is given by:
\[ \text{Percentage Change} = \frac{T_1 - T_2}{T_1} \times 100 \]
Substituting \( T_1 = 4T_2 \):
\[ \text{Percentage Change} = \frac{4T_2 - T_2}{4T_2} \times 100 = \frac{3T_2}{4T_2} \times 100 = \frac{3}{4} \times 100 = 75\% \]

### Final Answer:
The percentage change in tension necessary to produce a note one octave lower is **75%**.