Two uniform strings `A and B` made of steel are made to vibrate under the same tension. If the first overtone of `A` is equal to the second overtone of `B` and if the radius of `A` is twice that of `B`, the ratio of the lengths of the strings is

Options

A. `2 : 1`
B. `3 : 2`
C. `3 : 4`
D. `1 : 3`

Explanation

To solve the problem, we need to find the ratio of the lengths of two strings A and B given that the first overtone of A is equal to the second overtone of B, and the radius of A is twice that of B.

### Step-by-Step Solution:

1. **Understanding Overtones**:
- The first overtone (n1) of string A corresponds to the second harmonic, which is given by \( f_1 = \frac{2T}{L_1} \) where \( L_1 \) is the length of string A.
- The second overtone (n2) of string B corresponds to the third harmonic, which is given by \( f_2 = \frac{3T}{L_2} \) where \( L_2 \) is the length of string B.

2. **Setting Up the Equation**:
- According to the problem, the first overtone of A is equal to the second overtone of B:
\[ 2f_1 = 3f_2 \]
- Substituting the expressions for \( f_1 \) and \( f_2 \):
\[ 2 \left( \frac{2T}{L_1} \right) = 3 \left( \frac{3T}{L_2} \right) \]
- Simplifying gives:
\[ \frac{4T}{L_1} = \frac{9T}{L_2} \]

3. **Eliminating Tension**:
- Since the tension \( T \) is the same for both strings, we can cancel it out:
\[ \frac{4}{L_1} = \frac{9}{L_2} \]

4. **Finding the Ratio of Lengths**:
- Rearranging gives:
\[ \frac{L_1}{L_2} = \frac{4}{9} \]

5. **Mass per Unit Length**:
- The mass per unit length \( m \) of a string is given by:
\[ m = A \cdot \rho \]
- Where \( A \) is the cross-sectional area and \( \rho \) is the density. The area \( A \) for a circular cross-section is \( \pi r^2 \).

6. **Substituting for Mass per Unit Length**:
- For string A:
\[ m_1 = \pi r_1^2 \cdot \rho \]
- For string B:
\[ m_2 = \pi r_2^2 \cdot \rho \]
- Given that \( r_1 = 2r_2 \):
\[ m_1 = \pi (2r_2)^2 \cdot \rho = 4\pi r_2^2 \cdot \rho = 4m_2 \]

7. **Final Ratio of Lengths**:
- Now substituting \( m_1 \) and \( m_2 \) into the length ratio:
\[ \frac{L_1}{L_2} = \frac{4}{9} \sqrt{\frac{m_2}{m_1}} = \frac{4}{9} \sqrt{\frac{1}{4}} = \frac{4}{9} \cdot \frac{1}{2} = \frac{2}{9} \]

8. **Conclusion**:
- The final ratio of the lengths of the strings is:
\[ L_1 : L_2 = 1 : 3 \]

### Final Answer:
The ratio of the lengths of strings A and B is \( 1 : 3 \).