To solve the problem step by step, we will follow the reasoning provided in the video transcript.
### Step 1: Identify the Given Data
- Length of the rope, \( L = 100 \, \text{m} \)
- Breaking strength of the rope, \( T_{max} = 108 \, \text{N} \)
- Mass of the monkey, \( m = 6 \, \text{kg} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)
### Step 2: Write the Forces Acting on the Monkey
When the monkey climbs the rope, the forces acting on it are:
- The tension in the rope \( T \) acting upwards.
- The weight of the monkey \( mg \) acting downwards.
- The pseudo force \( ma \) acting downwards due to the upward acceleration \( a \).
### Step 3: Apply Newton's Second Law
According to Newton's second law, the net force acting on the monkey can be expressed as:
\[
T - mg - ma = 0
\]
Rearranging gives us:
\[
T = mg + ma
\]
### Step 4: Substitute Known Values
Substituting the known values into the equation:
\[
T = 6 \times 10 + 6a
\]
\[
T = 60 + 6a
\]
### Step 5: Set Maximum Tension Equal to Breaking Strength
Since the maximum tension \( T_{max} \) is given as \( 108 \, \text{N} \), we set up the equation:
\[
60 + 6a = 108
\]
### Step 6: Solve for Acceleration \( a \)
Now, we solve for \( a \):
\[
6a = 108 - 60
\]
\[
6a = 48
\]
\[
a = \frac{48}{6} = 8 \, \text{m/s}^2
\]
### Step 7: Use the Equation of Motion to Find Time
We use the equation of motion:
\[
s = ut + \frac{1}{2} a t^2
\]
Where:
- \( s = 100 \, \text{m} \) (distance to climb)
- \( u = 0 \, \text{m/s} \) (initial velocity)
- \( a = 8 \, \text{m/s}^2 \) (acceleration)
Substituting the values:
\[
100 = 0 \cdot t + \frac{1}{2} \cdot 8 \cdot t^2
\]
This simplifies to:
\[
100 = 4t^2
\]
\[
t^2 = \frac{100}{4} = 25
\]
\[
t = \sqrt{25} = 5 \, \text{s}
\]
### Step 8: Conclusion
The shortest time in which the monkey can climb the full length of the rope without breaking it is:
\[
\boxed{5 \, \text{s}}
\]
