A 13.5 kg monkey starts to climb a long rope to reach a banana located at a height of 6.70 m. The rope will snap if the tension exceeds 148.50 N. Calculate the least amount of time the monkey could take to reach the banana without breaking the rope.

To solve this problem, we need to calculate the minimum time the monkey can take to climb the rope without exceeding the tension limit of 148.50 N. We can do this by determining the maximum acceleration the monkey can have without breaking the rope.

### Step 1: Forces acting on the monkey
Two forces act on the monkey as it climbs:
1. **Weight of the monkey**: \( F_g = m \cdot g \), where \( m = 13.5 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \).
2. **Tension in the rope**: The rope has a maximum tension limit of 148.50 N.

We need to make sure the total force exerted by the monkey (weight plus any force due to acceleration) doesn't exceed the maximum tension in the rope.

### Step 2: Maximum allowable net force
The maximum allowable tension is given as 148.50 N. The monkey's weight is:

\[
F_g = 13.5 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 132.3 \, \text{N}
\]

Now, the net force (\( F_{net} \)) is what remains after subtracting the monkey's weight from the maximum tension the rope can handle:

\[
F_{net} = 148.50 \, \text{N} - 132.3 \, \text{N} = 16.2 \, \text{N}
\]

This net force is what accelerates the monkey upwards.

### Step 3: Maximum acceleration
Using Newton's second law, \( F = m \cdot a \), we can find the maximum acceleration (\( a_{max} \)) the monkey can have without breaking the rope:

\[
a_{max} = \frac{F_{net}}{m} = \frac{16.2 \, \text{N}}{13.5 \, \text{kg}} \approx 1.2 \, \text{m/s}^2
\]

### Step 4: Minimum time to reach the banana
The monkey needs to climb a height of 6.70 m. We can use the kinematic equation to find the minimum time \( t \) required:

\[
d = \frac{1}{2} a t^2
\]

Solving for \( t \), we get:

\[
t = \sqrt{\frac{2d}{a_{max}}}
\]

Substitute the values:

\[
t = \sqrt{\frac{2 \times 6.70 \, \text{m}}{1.2 \, \text{m/s}^2}} = \sqrt{\frac{13.4}{1.2}} \approx \sqrt{11.17} \approx 3.34 \, \text{seconds}
\]

### Final Answer:
The least amount of time the monkey could take to reach the banana without breaking the rope is approximately **3.34 seconds**.

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Would you like further details or clarifications on any part of the solution?

Here are 5 related questions to deepen understanding:
1. What happens if the monkey climbs faster than 3.34 seconds?
2. How would the problem change if the monkey's mass were different?
3. What if the rope's tension limit was higher or lower?
4. How does the monkey’s acceleration relate to its final velocity at the top?
5. What would happen if there was no tension limit on the rope?

**Tip:** When dealing with forces, always account for all forces acting on an object, including gravity and tension, to avoid exceeding limits.

Learn how to calculate the least amount of time a 13.5 kg monkey can take to climb 6.70 m on a rope without exceeding a tension limit of 148.50 N. This detailed solution uses concepts of mechanics, forces, and kinematic equations, suitable for high school physics students.

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