If the tension of a string is doubled, the fundamental frequency changes will be

Options

A. Twice
B. `sqrt(2)` times
C. 4 times
D. 5 times

Explanation

To solve the problem of how the fundamental frequency of a string changes when the tension is doubled, we can follow these steps:

### Step 1: Understand the formula for fundamental frequency
The fundamental 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,
- \( T \) is the tension in the string,
- \( \mu \) is the mass per unit length of the string.

### Step 2: Identify the initial conditions
Let the initial tension in the string be \( T \). Therefore, the initial fundamental frequency \( f_1 \) can be written as:
\[ f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]

### Step 3: Determine the new tension
If the tension is doubled, the new tension \( T' \) becomes:
\[ T' = 2T \]

### Step 4: Calculate the new fundamental frequency
Now, we can find the new fundamental frequency \( f_2 \) with the new tension:
\[ f_2 = \frac{1}{2L} \sqrt{\frac{T'}{\mu}} = \frac{1}{2L} \sqrt{\frac{2T}{\mu}} \]

### Step 5: Simplify the new frequency
We can simplify \( f_2 \):
\[ f_2 = \frac{1}{2L} \sqrt{2} \sqrt{\frac{T}{\mu}} = \sqrt{2} \cdot \frac{1}{2L} \sqrt{\frac{T}{\mu}} = \sqrt{2} \cdot f_1 \]

### Step 6: Conclusion
Thus, when the tension in the string is doubled, the fundamental frequency increases by a factor of \( \sqrt{2} \):
\[ f_2 = \sqrt{2} \cdot f_1 \]

### Summary
If the tension of a string is doubled, the fundamental frequency changes to \( \sqrt{2} \) times the original frequency.