In order to raise a mass of 100kg `a` of mass 60kg fastens `a` rope to it and passes the rope over a smooth pulley. He climbs the rope with acceleration `5g//4` relative to the rope. The tension in the rope is: Take `g=10m//s^(2)`

Options

A. 1432N
B. 928N
C. 1219N
D. 642N

Explanation

To solve the problem, we will analyze the forces acting on both the man and the mass he is trying to lift, and use Newton's second law of motion to derive the tension in the rope.

### Step 1: Identify the given values
- Mass of the man (m1) = 60 kg
- Mass to be lifted (m2) = 100 kg
- Acceleration of the man relative to the rope (a_rel) = \( \frac{5g}{4} \)
- Acceleration due to gravity (g) = 10 m/s²

### Step 2: Calculate the absolute acceleration of the man
Let the acceleration of the rope (and the mass being lifted) be \( A \). The absolute acceleration of the man (a_m) can be expressed as:
\[ a_m = a_{rel} + A \]
Substituting the values:
\[ a_m = \frac{5g}{4} + A \]

### Step 3: Write the equations of motion for the man and the mass
For the man climbing the rope:
Using Newton's second law,
\[ T - m_1g = m_1a_m \]
Substituting \( a_m \):
\[ T - 60g = 60\left(\frac{5g}{4} + A\right) \]
\[ T - 600 = 75g + 60A \]
(1) \( T = 600 + 75g + 60A \)

For the mass being lifted:
Using Newton's second law,
\[ T - m_2g = m_2A \]
Substituting the values:
\[ T - 100g = 100A \]
(2) \( T = 100g + 100A \)

### Step 4: Set the equations for T equal to each other
From equations (1) and (2):
\[ 600 + 75g + 60A = 100g + 100A \]

### Step 5: Rearrange to solve for A
Rearranging gives:
\[ 600 + 75g - 100g = 100A - 60A \]
\[ 600 - 25g = 40A \]
Substituting \( g = 10 \):
\[ 600 - 250 = 40A \]
\[ 350 = 40A \]
\[ A = \frac{350}{40} = 8.75 \, \text{m/s}^2 \]

### Step 6: Substitute A back to find T
Now substitute \( A \) back into equation (2) to find T:
\[ T = 100g + 100A \]
\[ T = 100(10) + 100(8.75) \]
\[ T = 1000 + 875 \]
\[ T = 1875 \, \text{N} \]

### Conclusion
The tension in the rope is \( T = 1875 \, \text{N} \).

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