Transverse waves are generated in two uniform wires A and B of the same material by attaching their free ends to a vibrating source of frequency 200 Hz. The Area of cross section of A is half that of B while tension on A is twice than on B. The ratio of wavelengths of the transverse waves in A and B is

Options

A. `1: sqrt2`
B. `sqrt2 : 1 `
C. `1:2`
D. `2:1`

Answer

To solve the problem, we need to find the ratio of the wavelengths of the transverse waves in wires A and B. We will use the relationship between wave velocity, tension, and area of cross-section, as well as the relationship between velocity, frequency, and wavelength.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Frequency (f) = 200 Hz (same for both wires)
- Area of cross-section of wire A (A_A) = 1/2 Area of cross-section of wire B (A_B)
- Tension in wire A (T_A) = 2 * Tension in wire B (T_B)

2. **Write the Formula for Wave Velocity:**
The velocity (V) of a transverse wave in a wire is given by:
\[ V = \sqrt{\frac{T}{A}} \]
where T is the tension and A is the area of cross-section.

3. **Calculate the Velocity of Wires A and B:**
- For wire A:
\[ V_A = \sqrt{\frac{T_A}{A_A}} \]
- For wire B:
\[ V_B = \sqrt{\frac{T_B}{A_B}} \]

4. **Substituting the Values:**
- From the problem, we know:
- \( A_A = \frac{1}{2} A_B \)
- \( T_A = 2 T_B \)

Substituting these into the velocity equations:
\[ V_A = \sqrt{\frac{2 T_B}{\frac{1}{2} A_B}} = \sqrt{\frac{2 T_B \cdot 2}{A_B}} = \sqrt{\frac{4 T_B}{A_B}} \]
\[ V_B = \sqrt{\frac{T_B}{A_B}} \]

5. **Finding the Ratio of Velocities:**
Now, we can find the ratio of the velocities:
\[ \frac{V_A}{V_B} = \frac{\sqrt{\frac{4 T_B}{A_B}}}{\sqrt{\frac{T_B}{A_B}}} = \sqrt{\frac{4 T_B}{T_B}} = \sqrt{4} = 2 \]

6. **Relate Velocity and Wavelength:**
The relationship between velocity (V), frequency (f), and wavelength (λ) is given by:
\[ V = f \lambda \]
Therefore, the wavelength can be expressed as:
\[ \lambda = \frac{V}{f} \]

7. **Finding the Ratio of Wavelengths:**
Since both wires are connected to the same frequency source, \( f_A = f_B = 200 \text{ Hz} \). Thus, the ratio of the wavelengths is:
\[ \frac{\lambda_A}{\lambda_B} = \frac{V_A}{V_B} \]
Substituting the ratio we found:
\[ \frac{\lambda_A}{\lambda_B} = 2 \]

### Final Answer:
The ratio of the wavelengths of the transverse waves in A and B is \( 2:1 \).