A uniform rope of length `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))`
B. `sqrt((m_1+m_2)/(m_2))`
C. `sqrt((m_2)/(m_1))`
D. `sqrt((m_1+m_2)/(m_1))`

Explanation

To solve the problem, we need to find the ratio of the wavelengths of a transverse pulse at the bottom and top of a vertically hanging rope. Here’s a step-by-step breakdown of the solution:

### Step 1: Understanding the System
We have a uniform 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. When a transverse pulse is generated at the lower end of the rope, it travels upwards.

### Step 2: Analyzing Tension in the Rope
The tension in the rope varies along its length due to the weight of the rope and the block.
- At the bottom of the rope (where the pulse is generated), the tension \( T_1 \) is due only to the weight of the block:
\[ T_1 = m_2 \cdot g \]
- At the top of the rope (where the rope is attached to the support), the tension \( T_2 \) is due to the weight of both the rope and the block:
\[ T_2 = (m_1 + m_2) \cdot g \]

### Step 3: Relating Wavelength to Tension
The wavelength of a wave on a string is related to the tension in the string. Specifically, the wavelength \( \lambda \) is proportional to the square root of the tension \( T \):
\[ \lambda \propto \sqrt{T} \]

### Step 4: Setting Up the Ratio of Wavelengths
From the relationship above, we can express the ratio of the wavelengths at the top and bottom of the rope:
\[ \frac{\lambda_1}{\lambda_2} = \sqrt{\frac{T_1}{T_2}} \]
Rearranging gives us:
\[ \frac{\lambda_2}{\lambda_1} = \sqrt{\frac{T_2}{T_1}} \]

### Step 5: Substituting the Tension Values
Now we substitute the expressions for \( T_1 \) and \( T_2 \):
\[ \frac{\lambda_2}{\lambda_1} = \sqrt{\frac{(m_1 + m_2) \cdot g}{m_2 \cdot g}} \]
The \( g \) cancels out:
\[ \frac{\lambda_2}{\lambda_1} = \sqrt{\frac{m_1 + m_2}{m_2}} \]

### Final Result
Thus, the ratio of the wavelengths is:
\[ \frac{\lambda_2}{\lambda_1} = \sqrt{\frac{m_1 + m_2}{m_2}} \]