Two vibrating strings of the same material but lengths `L` and `2L` have radii `2r` and `r` respectively. They are stretched under the same tension. Both the string vibrate in their fundamental nodes, the one of length `L` with freuqency `v_(1)` and the other with frequency `v_(2)`. the ratio `v_(1)//v_(2)` is given by

Options

A. 1
B. `1/2`
C. `3/2`
D. 2

Answer

To solve the problem, we need to find the ratio of the frequencies \( v_1 \) and \( v_2 \) of two vibrating strings of different lengths and radii but under the same tension.

### Step-by-Step Solution:

1. **Understanding the Frequency Formula**:
The frequency \( v \) of a vibrating string is given by the formula:
\[ v = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
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.

2. **Calculating Mass per Unit Length (\( \mu \))**:
The mass per unit length \( \mu \) can be expressed as:
\[ \mu = \text{Density} \times \text{Cross-sectional Area} \]
For a cylindrical string, the cross-sectional area \( A \) is given by:
\[ A = \pi r^2 \]
Therefore, for the two strings:
- For string 1 (length \( L \), radius \( 2r \)):
\[ A_1 = \pi (2r)^2 = 4\pi r^2 \]
\[ \mu_1 = \rho \cdot A_1 = \rho \cdot 4\pi r^2 \]
- For string 2 (length \( 2L \), radius \( r \)):
\[ A_2 = \pi r^2 \]
\[ \mu_2 = \rho \cdot A_2 = \rho \cdot \pi r^2 \]

3. **Substituting into the Frequency Formula**:
- For string 1:
\[ v_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu_1}} = \frac{1}{2L} \sqrt{\frac{T}{4\rho \pi r^2}} = \frac{1}{2L} \cdot \frac{1}{2} \sqrt{\frac{T}{\rho \pi r^2}} = \frac{1}{4L} \sqrt{\frac{T}{\rho \pi r^2}} \]
- For string 2:
\[ v_2 = \frac{1}{2(2L)} \sqrt{\frac{T}{\mu_2}} = \frac{1}{4L} \sqrt{\frac{T}{\rho \pi r^2}} \]

4. **Finding the Ratio of Frequencies**:
Now, we can find the ratio \( \frac{v_1}{v_2} \):
\[ \frac{v_1}{v_2} = \frac{\frac{1}{4L} \sqrt{\frac{T}{\rho \pi r^2}}}{\frac{1}{4L} \sqrt{\frac{T}{\rho \pi r^2}}} = 1 \]

### Final Answer:
The ratio \( \frac{v_1}{v_2} \) is:
\[ \frac{v_1}{v_2} = 1 \]