A money of mass 40 kg climbs up a rope, of breaking load 600 N hanging from a ceiling. If it climbs up the rope with the maximum possible acceleration, then the time taken by monkey to climb up is [length of rope is 10 m]

Options

A. 2 s
B. 1 s
C. 4 s
D. 3 s

Answer

To solve the problem step by step, we need to analyze the forces acting on the monkey and then use kinematic equations to find the time taken to climb the rope.

### Step 1: Determine the maximum force the monkey can exert.
The breaking load of the rope is given as 600 N. This means the maximum tension (T) in the rope while the monkey is climbing cannot exceed 600 N.

### Step 2: Calculate the weight of the monkey.
The weight (W) of the monkey can be calculated using the formula:
\[ W = m \cdot g \]
where:
- \( m = 40 \, \text{kg} \) (mass of the monkey)
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)

Calculating the weight:
\[ W = 40 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 392 \, \text{N} \]

### Step 3: Determine the net force acting on the monkey.
The net force (F_net) acting on the monkey when it climbs with maximum acceleration can be calculated using:
\[ F_{\text{net}} = T - W \]
Given that the maximum tension \( T = 600 \, \text{N} \):
\[ F_{\text{net}} = 600 \, \text{N} - 392 \, \text{N} = 208 \, \text{N} \]

### Step 4: Calculate the maximum acceleration of the monkey.
Using Newton's second law:
\[ F_{\text{net}} = m \cdot a \]
We can rearrange this to find the acceleration (a):
\[ a = \frac{F_{\text{net}}}{m} = \frac{208 \, \text{N}}{40 \, \text{kg}} = 5.2 \, \text{m/s}^2 \]

### Step 5: Use kinematic equations to find the time taken to climb the rope.
We will use the equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
where:
- \( s = 10 \, \text{m} \) (length of the rope)
- \( u = 0 \, \text{m/s} \) (initial velocity)
- \( a = 5.2 \, \text{m/s}^2 \)

Substituting the known values:
\[ 10 = 0 \cdot t + \frac{1}{2} \cdot 5.2 \cdot t^2 \]
This simplifies to:
\[ 10 = 2.6 t^2 \]
Now, solving for \( t^2 \):
\[ t^2 = \frac{10}{2.6} \approx 3.846 \]
Taking the square root to find \( t \):
\[ t \approx \sqrt{3.846} \approx 1.96 \, \text{s} \]

### Final Answer:
The time taken by the monkey to climb up the rope is approximately **1.96 seconds**.

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