The extension in a string obeying Hooke's law is x. The speed of sound in the stretched string is `v`. If the extension in the string is increased to `1.5 x`, the speed of sound will be

Options

A. `1.22 v`
B. `0.61 v`
C. `1.50 v`
D. `0.75 v`

Answer

To solve the problem, we need to understand how the speed of sound in a stretched string is related to the tension in the string and the extension according to Hooke's law.

### Step-by-Step Solution:

1. **Understand the relationship between tension and extension**:
According to Hooke's law, the tension \( T \) in a string is directly proportional to the extension \( x \):
\[ T \propto x \]
This can be expressed as:
\[ T = kx \]
where \( k \) is the spring constant of the string.

2. **Speed of sound in a stretched string**:
The speed of sound \( v \) in a stretched string is given by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( \mu \) is the mass per unit length of the string.

3. **Substituting tension in the speed formula**:
Since \( T \) is proportional to \( x \), we can substitute \( T \) in the speed formula:
\[ v = \sqrt{\frac{kx}{\mu}} \]

4. **Increasing the extension**:
If the extension is increased to \( 1.5x \), the new tension \( T' \) becomes:
\[ T' = k(1.5x) = 1.5kx \]

5. **Calculating the new speed of sound**:
The new speed of sound \( v' \) in the string with the new tension is:
\[ v' = \sqrt{\frac{T'}{\mu}} = \sqrt{\frac{1.5kx}{\mu}} \]

6. **Relating the new speed to the original speed**:
We can express \( v' \) in terms of the original speed \( v \):
\[ v' = \sqrt{1.5} \cdot \sqrt{\frac{kx}{\mu}} = \sqrt{1.5} \cdot v \]

7. **Final calculation**:
The value of \( \sqrt{1.5} \) is approximately \( 1.22 \). Therefore:
\[ v' \approx 1.22v \]

### Conclusion:
The new speed of sound in the stretched string when the extension is increased to \( 1.5x \) is approximately \( 1.22v \).