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 determine under which condition the rope will break when the monkey climbs, we need to analyze the forces acting on the monkey and the tension in the rope.

### Step 1: Calculate 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)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)

Calculating the weight:
\[ W = 20 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 200 \, \text{N} \]

### Step 2: Determine the maximum tension the rope can withstand
The maximum tension (T_max) that the rope can withstand is given as:
\[ T_{\text{max}} = 300 \, \text{N} \]

### Step 3: Analyze the forces when the monkey climbs with acceleration
When the monkey climbs with an acceleration (a), the net force (F_net) acting on the monkey can be expressed as:
\[ F_{\text{net}} = T - W \]
Using Newton's second law, we have:
\[ F_{\text{net}} = m \cdot a \]
Therefore, we can set up the equation:
\[ T - W = m \cdot a \]
Substituting the values we have:
\[ T - 200 \, \text{N} = 20 \, \text{kg} \cdot a \]

### Step 4: Solve for tension when the rope is at its maximum limit
At the maximum tension, we set \( T = T_{\text{max}} \):
\[ 300 \, \text{N} - 200 \, \text{N} = 20 \, \text{kg} \cdot a \]
This simplifies to:
\[ 100 \, \text{N} = 20 \, \text{kg} \cdot a \]

### Step 5: Calculate the maximum allowable acceleration
Now, we can solve for acceleration (a):
\[ a = \frac{100 \, \text{N}}{20 \, \text{kg}} = 5 \, \text{m/s}^2 \]

### Step 6: Determine the conditions under which the rope will break
The rope will break if the acceleration of the monkey exceeds \( 5 \, \text{m/s}^2 \). We need to check the options provided in the question to see which one has an acceleration greater than \( 5 \, \text{m/s}^2 \).

1. **Option 1**: Climbing up with an acceleration of \( 6 \, \text{m/s}^2 \) (This will break the rope)
2. **Option 2**: Climbing up with an acceleration of \( 4 \, \text{m/s}^2 \) (This will not break the rope)
3. **Option 3**: Climbing up with uniform speed (acceleration = \( 0 \, \text{m/s}^2 \)) (This will not break the rope)
4. **Option 4**: Falling down freely under gravity (acceleration = \( 10 \, \text{m/s}^2 \), which will break the rope)

### Conclusion
The rope will break in **Option 1** (climbing with \( 6 \, \text{m/s}^2 \)) and **Option 4** (falling down freely).