A monkey of mass 20 kg climbs on a rope which can stand a maximum tension of 300 N. In which of the following case will the rope break . The monkey

Options

A. Climbs up with an acceleration of `6 m/s^2`
B. Climbs down with an acceleration of 4 `m/s^2 `
C. Climbs up with a uniform speed of 5` m/s^2 `
D. Falls down the rope nearly free under gravity

Answer

To solve the problem, we need to analyze the forces acting on the monkey when it climbs the rope. The key points to consider are the weight of the monkey, the tension in the rope, and the maximum tension the rope can withstand.

### Step-by-Step Solution:

1. **Identify the Weight of the Monkey**:
The weight (W) of the monkey can be calculated using the formula:
\[ W = m \cdot g \]
where \( m = 20 \, \text{kg} \) (mass of the monkey) and \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity).
\[ W = 20 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 200 \, \text{N} \]

2. **Determine the Maximum Tension**:
The rope can withstand a maximum tension of \( T_{max} = 300 \, \text{N} \).

3. **Analyze the Cases**:
We need to evaluate the tension in the rope for different scenarios when the monkey climbs.

- **Case 1**: The monkey accelerates upwards at \( 6 \, \text{m/s}^2 \).
- The net force acting on the monkey is given by:
\[ F_{net} = m \cdot a = 20 \, \text{kg} \cdot 6 \, \text{m/s}^2 = 120 \, \text{N} \]
- The total tension in the rope can be calculated as:
\[ T = W + F_{net} = 200 \, \text{N} + 120 \, \text{N} = 320 \, \text{N} \]
- Since \( 320 \, \text{N} > 300 \, \text{N} \), the rope will break in this case.

- **Case 2**: The monkey accelerates downwards at \( 4 \, \text{m/s}^2 \).
- The tension in the rope can be calculated as:
\[ T = W - (m \cdot a) = 200 \, \text{N} - (20 \, \text{kg} \cdot 4 \, \text{m/s}^2) = 200 \, \text{N} - 80 \, \text{N} = 120 \, \text{N} \]
- Since \( 120 \, \text{N} < 300 \, \text{N} \), the rope will not break.

- **Case 3**: The monkey moves at a uniform speed.
- When moving at a constant speed, the acceleration is zero, so:
\[ T = W = 200 \, \text{N} \]
- Since \( 200 \, \text{N} < 300 \, \text{N} \), the rope will not break.

- **Case 4**: The monkey falls freely under gravity.
- In free fall, the tension in the rope is zero:
\[ T = 0 \, \text{N} \]
- Since \( 0 \, \text{N} < 300 \, \text{N} \), the rope will not break.

4. **Conclusion**:
The only case where the rope will break is when the monkey accelerates upwards at \( 6 \, \text{m/s}^2 \).

### Final Answer:
The rope will break in **Case 1** when the monkey accelerates upwards at \( 6 \, \text{m/s}^2 \).

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