A uniform rope of length 12 mm and mass 6 kg hangs vertically from a rigid support. A block of mass 2 kg is attached to the free end of the rope. A transverse pulse of wavelength 0.06 m is produced at the lower end of the rope. What is the wavelength of the pulse when it reaches the top of the rope?

Options

A. 0.06 m
B. 0.03 m
C. 0.12 m
D. 0.09 m

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the Linear Mass Density (μ)
The linear mass density (μ) of the rope is calculated using the formula:
\[ \mu = \frac{m}{L} \]
where:
- \( m = 6 \, \text{kg} \) (mass of the rope)
- \( L = 12 \, \text{mm} = 0.012 \, \text{m} \) (length of the rope)

Calculating:
\[ \mu = \frac{6 \, \text{kg}}{0.012 \, \text{m}} = 500 \, \text{kg/m} \]

### Step 2: Calculate the Tension at the Lower End (T1)
At the lower end of the rope, the tension (T1) is due to the weight of the block attached:
\[ T_1 = m_{block} \cdot g \]
where:
- \( m_{block} = 2 \, \text{kg} \)
- \( g = 9.81 \, \text{m/s}^2 \) (acceleration due to gravity)

Calculating:
\[ T_1 = 2 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 19.62 \, \text{N} \]

### Step 3: Calculate the Tension at the Upper End (T2)
At the upper end of the rope, the tension (T2) is the sum of the weight of the rope and the block:
\[ T_2 = (m_{rope} + m_{block}) \cdot g \]
where:
- \( m_{rope} = 6 \, \text{kg} \)

Calculating:
\[ T_2 = (6 \, \text{kg} + 2 \, \text{kg}) \cdot 9.81 \, \text{m/s}^2 = 8 \cdot 9.81 \, \text{N} = 78.48 \, \text{N} \]

### Step 4: Calculate the Velocity of the Pulse at Both Ends
The velocity (v) of a wave on a string is given by:
\[ v = \sqrt{\frac{T}{\mu}} \]
Calculating the velocity at the lower end (v1):
\[ v_1 = \sqrt{\frac{T_1}{\mu}} = \sqrt{\frac{19.62 \, \text{N}}{500 \, \text{kg/m}}} = \sqrt{0.03924} \approx 0.198 \, \text{m/s} \]

Calculating the velocity at the upper end (v2):
\[ v_2 = \sqrt{\frac{T_2}{\mu}} = \sqrt{\frac{78.48 \, \text{N}}{500 \, \text{kg/m}}} = \sqrt{0.15696} \approx 0.396 \, \text{m/s} \]

### Step 5: Calculate the Frequency of the Pulse
The frequency (f) of the wave can be calculated using the wavelength (λ) and velocity (v):
\[ f = \frac{v}{\lambda} \]
Using the wavelength at the lower end (λ1 = 0.06 m):
\[ f = \frac{v_1}{\lambda_1} = \frac{0.198 \, \text{m/s}}{0.06 \, \text{m}} \approx 3.3 \, \text{Hz} \]

### Step 6: Calculate the Wavelength at the Upper End (λ2)
Using the frequency calculated and the velocity at the upper end:
\[ \lambda_2 = \frac{v_2}{f} = \frac{0.396 \, \text{m/s}}{3.3 \, \text{Hz}} \approx 0.12 \, \text{m} \]

### Final Answer
The wavelength of the pulse when it reaches the top of the rope is approximately:
\[ \lambda_2 \approx 0.12 \, \text{m} \]

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