A sonometer wire under a tension of `10 kg` weight is in unsion with a tuning fork of frequency `320 H_(Z)` . To make the wire vibrate in unsion with a tuning fork of frequency `256 h_(Z)` , the tension should be altered by

Options

A. `3.6 kg` decreased
B. `3.6 kg` increased
C. `6.4 kg` decreased
D. `6.4 kg` increased

Explanation

To solve the problem, we need to determine how the tension in the sonometer wire should be altered to change the frequency from 320 Hz to 256 Hz.

### Step-by-Step Solution:

1. **Understanding the relationship between frequency and tension**:
The frequency \( f \) of a vibrating wire is related to the tension \( T \) in the wire by the formula:
\[ f \propto \sqrt{T} \]
This means that the frequency is directly proportional to the square root of the tension.

2. **Setting up the equations**:
Let \( T_1 \) be the initial tension corresponding to the frequency \( f_1 = 320 \, \text{Hz} \), and \( T_2 \) be the new tension corresponding to the frequency \( f_2 = 256 \, \text{Hz} \).
We can express this relationship mathematically as:
\[ f_1 = k \sqrt{T_1} \]
\[ f_2 = k \sqrt{T_2} \]
where \( k \) is a constant.

3. **Dividing the equations**:
To eliminate \( k \), we can divide the second equation by the first:
\[ \frac{f_2}{f_1} = \frac{\sqrt{T_2}}{\sqrt{T_1}} \]
Squaring both sides gives:
\[ \left(\frac{f_2}{f_1}\right)^2 = \frac{T_2}{T_1} \]

4. **Substituting the known values**:
Substitute \( f_1 = 320 \, \text{Hz} \) and \( f_2 = 256 \, \text{Hz} \):
\[ \left(\frac{256}{320}\right)^2 = \frac{T_2}{T_1} \]

5. **Calculating the ratio**:
Calculate \( \frac{256}{320} \):
\[ \frac{256}{320} = 0.8 \]
Thus,
\[ (0.8)^2 = 0.64 \]
Therefore,
\[ \frac{T_2}{T_1} = 0.64 \]

6. **Finding \( T_2 \)**:
Given that \( T_1 = 10 \, \text{kg} \) (the tension under the weight), we can find \( T_2 \):
\[ T_2 = 0.64 \times T_1 = 0.64 \times 10 \, \text{kg} = 6.4 \, \text{kg} \]

7. **Calculating the change in tension**:
The change in tension required is:
\[ T_1 - T_2 = 10 \, \text{kg} - 6.4 \, \text{kg} = 3.6 \, \text{kg} \]

8. **Conclusion**:
The tension should be decreased by \( 3.6 \, \text{kg} \).

### Final Answer:
The tension should be altered by decreasing it by **3.6 kg**.