A uniform rope of legnth `L` and mass `m_1` hangs vertically from a rigid support. A block of mass `m_2` is attached to the free end of the rope. A transverse pulse of wavelength `lamda_1` is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is `lamda_2`. The ratio `(lamda_2)/(lamda_1)` is

Options

A. `sqrt((m_(1)+m_(2))/m_(2)`
B. `sqrt(m_(2)/m_(1))`
C. `sqrt((m_(1)+m_(2))/m_(1)`
D. `sqrt(m_(1)/m_(2))`

Answer

To solve the problem, we need to find the ratio of the wavelengths of a transverse pulse in a rope when it is at the bottom and when it reaches the top. Let's denote the wavelength at the bottom as \( \lambda_1 \) and the wavelength at the top as \( \lambda_2 \).

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a rope of length \( L \) and mass \( m_1 \) hanging vertically from a rigid support.
- A block of mass \( m_2 \) is attached to the free end of the rope.

2. **Identifying Tension in the Rope**:
- The tension at the bottom of the rope (just below the block \( m_2 \)) is given by:
\[ T_1 = m_2 \cdot g \]
- The tension at the top of the rope (where the rope is attached to the support) is given by:
\[ T_2 = (m_1 + m_2) \cdot g \]

3. **Relationship Between Wavelength and Tension**:
- The speed of a wave in a rope is related to the tension and the mass per unit length of the rope. The wavelength is also related to the speed of the wave.
- The relationship can be expressed as:
\[ v \propto \sqrt{T} \]
- Therefore, the wavelengths at the bottom and top of the rope can be expressed in terms of the tensions:
\[ \frac{\lambda_1}{\lambda_2} = \sqrt{\frac{T_1}{T_2}} \]

4. **Substituting the Tension Values**:
- Substitute the expressions for \( T_1 \) and \( T_2 \):
\[ \frac{\lambda_1}{\lambda_2} = \sqrt{\frac{m_2 \cdot g}{(m_1 + m_2) \cdot g}} \]
- The \( g \) cancels out:
\[ \frac{\lambda_1}{\lambda_2} = \sqrt{\frac{m_2}{m_1 + m_2}} \]

5. **Finding the Ratio \( \frac{\lambda_2}{\lambda_1} \)**:
- Taking the reciprocal gives:
\[ \frac{\lambda_2}{\lambda_1} = \sqrt{\frac{m_1 + m_2}{m_2}} \]

6. **Final Result**:
- Thus, the ratio \( \frac{\lambda_2}{\lambda_1} \) is:
\[ \frac{\lambda_2}{\lambda_1} = \sqrt{\frac{m_1 + m_2}{m_2}} \]

### Conclusion:
The ratio \( \frac{\lambda_2}{\lambda_1} \) is \( \sqrt{\frac{m_1 + m_2}{m_2}} \).