String A has a length L, radius of cross-section r, density of material p and is under tension T. String B has all these quantities double those of string A. If `v_(a)` and `v_(b)` are the corresponding fundamentals frequencies of the vibrating string,then

Options

A. 1. `upsilon_(A)=2upsilon_(B)`
B. 2. `upsilon_(A)=4upsilon_(B)`
C. 3. `upsilon_(B)=4upsilon_(A)`
D. 4. `upsilon_(A)=upsilon_(B)`

Explanation

To solve the problem, we need to find the relationship between the fundamental frequencies of two strings, A and B, given their physical parameters. Let's denote the fundamental frequency of string A as \( v_A \) and that of string B as \( v_B \).

### Step-by-Step Solution:

1. **Identify the parameters for both strings:**
- For string A:
- Length: \( L \)
- Radius: \( r \)
- Density: \( \rho \)
- Tension: \( T \)
- For string B (all parameters are doubled):
- Length: \( L_B = 2L \)
- Radius: \( r_B = 2r \)
- Density: \( \rho_B = 2\rho \)
- Tension: \( T_B = 2T \)

2. **Write the formula for the fundamental frequency of a vibrating string:**
The fundamental frequency \( v \) of a string is given by:
\[ v = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where \( \mu \) (linear mass density) is given by \( \mu = \rho \cdot A \) and \( A \) is the cross-sectional area. For a circular cross-section:
\[ A = \pi r^2 \]
Therefore,
\[ \mu = \rho \cdot \pi r^2 \]

3. **Substitute \( \mu \) into the frequency formula:**
For string A:
\[ v_A = \frac{1}{2L} \sqrt{\frac{T}{\rho \cdot \pi r^2}} \]
For string B:
\[ v_B = \frac{1}{2L_B} \sqrt{\frac{T_B}{\rho_B \cdot \pi r_B^2}} \]

4. **Substituting the parameters for string B:**
\[ v_B = \frac{1}{2(2L)} \sqrt{\frac{2T}{2\rho \cdot \pi (2r)^2}} \]
Simplifying this:
\[ v_B = \frac{1}{4L} \sqrt{\frac{2T}{2\rho \cdot \pi \cdot 4r^2}} = \frac{1}{4L} \sqrt{\frac{T}{\rho \cdot \pi r^2}} \]

5. **Relate \( v_A \) and \( v_B \):**
Now we can express \( v_A \) and \( v_B \) in terms of each other:
\[ v_A = \frac{1}{2L} \sqrt{\frac{T}{\rho \cdot \pi r^2}} \]
\[ v_B = \frac{1}{4L} \sqrt{\frac{T}{\rho \cdot \pi r^2}} \]

6. **Finding the ratio \( \frac{v_A}{v_B} \):**
\[ \frac{v_A}{v_B} = \frac{\frac{1}{2L} \sqrt{\frac{T}{\rho \cdot \pi r^2}}}{\frac{1}{4L} \sqrt{\frac{T}{\rho \cdot \pi r^2}}} \]
This simplifies to:
\[ \frac{v_A}{v_B} = \frac{4L}{2L} = 4 \]
Thus, we have:
\[ v_A = 4v_B \]

### Conclusion:
The relationship between the fundamental frequencies of strings A and B is:
\[ v_A = 4v_B \]