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 `2x` then speed of sound will be

Options

A. `1.5V`
B. `4.14V`
C. `1.414V`
D. `2V`

Explanation

To solve the problem step by step, we will analyze the relationship between the extension in the string, the tension, and the speed of sound in the string.

### Step 1: Understand Hooke's Law
Hooke's Law states that the extension (or elongation) in a material is directly proportional to the applied force (or tension) within the elastic limit of that material. Mathematically, it can be expressed as:
\[ T \propto \Delta L \]
where \( T \) is the tension and \( \Delta L \) is the extension.

### Step 2: Establish Initial Conditions
Initially, we have:
- Extension \( \Delta L = x \)
- Speed of sound in the string \( V = \sqrt{\frac{T}{\mu}} \)
where \( T \) is the tension and \( \mu \) is the mass per unit length of the string.

### Step 3: Relate Tension to Extension
From Hooke's Law, we can express the tension in terms of the extension:
\[ T = k \Delta L \]
where \( k \) is a constant of proportionality.

For the initial condition:
\[ T_1 = k \cdot x \]

### Step 4: Change the Extension
Now, the extension is increased to \( 2x \):
\[ \Delta L = 2x \]
Using Hooke's Law again, the new tension becomes:
\[ T_2 = k \cdot (2x) = 2k \cdot x = 2T_1 \]

### Step 5: Calculate New Speed of Sound
Now we can find the new speed of sound \( V' \) in the string with the new tension:
\[ V' = \sqrt{\frac{T_2}{\mu}} = \sqrt{\frac{2T_1}{\mu}} = \sqrt{2} \cdot \sqrt{\frac{T_1}{\mu}} = \sqrt{2} \cdot V \]

### Step 6: Final Result
Thus, the new speed of sound in the stretched string when the extension is increased to \( 2x \) is:
\[ V' = \sqrt{2} \cdot V \approx 1.414 \cdot V \]

### Conclusion
Therefore, the speed of sound in the string when the extension is increased to \( 2x \) is approximately \( 1.414 \cdot V \).

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