Two vibrating strings of the same material but lengths L and 2L have radii 2 r and r respectively. They are strectched under the same tension. Both the strings vibrate in their fundamental modes, the one of length L with frequency `n_(1)` and the other with frequency `n_(2)`. The ratio `n_(1)//n_(2)` is given by
2
4
8
1

Options

A. 2
B. 4
C. 8
D. 1

Explanation

To solve the problem, we need to find the ratio of the frequencies \( n_1 \) and \( n_2 \) of two vibrating strings of different lengths and radii but under the same tension. Let's break it down step by step.

### Step 1: Understand the relationship between frequency, tension, and mass per unit length
The frequency of a vibrating string in its fundamental mode is given by the formula:
\[ n = \frac{v}{4L} \]
where \( v \) is the wave speed in the string and \( L \) is the length of the string.

### Step 2: Determine the wave speed in each string
The wave speed \( v \) in a string is given by:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( T \) is the tension and \( \mu \) is the mass per unit length of the string.

### Step 3: Calculate the mass per unit length \( \mu \)
The mass per unit length \( \mu \) for a cylindrical string is proportional to the square of its radius \( r \):
\[ \mu \propto r^2 \]
Thus, for the two strings:
- For string 1 (length \( L \), radius \( 2r \)):
\[ \mu_1 \propto (2r)^2 = 4r^2 \]
- For string 2 (length \( 2L \), radius \( r \)):
\[ \mu_2 \propto r^2 \]

### Step 4: Express the wave speeds in terms of the radii
Since both strings are under the same tension \( T \):
- For string 1:
\[ v_1 = \sqrt{\frac{T}{\mu_1}} = \sqrt{\frac{T}{4r^2}} = \frac{1}{2}\sqrt{\frac{T}{r^2}} = \frac{1}{2}v_0 \]
- For string 2:
\[ v_2 = \sqrt{\frac{T}{\mu_2}} = \sqrt{\frac{T}{r^2}} = v_0 \]

### Step 5: Calculate the frequencies \( n_1 \) and \( n_2 \)
Now we can calculate the frequencies:
- For string 1:
\[ n_1 = \frac{v_1}{4L} = \frac{\frac{1}{2}v_0}{4L} = \frac{v_0}{8L} \]
- For string 2:
\[ n_2 = \frac{v_2}{4(2L)} = \frac{v_0}{4(2L)} = \frac{v_0}{8L} \]

### Step 6: Find the ratio \( \frac{n_1}{n_2} \)
Now we can find the ratio of the frequencies:
\[ \frac{n_1}{n_2} = \frac{\frac{v_0}{8L}}{\frac{v_0}{8L}} = 1 \]

### Conclusion
The ratio \( \frac{n_1}{n_2} \) is equal to 1.

### Final Answer
The correct option is **1**.