A 10 kg monkey wishes to climb a 7.15-meter high vine. The vine will break if the tension exceeds 110 N. The monkey starts from rest and exerts a constant force on the rope as he pulls himself upward. What is the shortest amount of time in which the monkey can climb the vine?
To find the shortest amount of time in which the monkey can climb the vine, we need to consider the forces involved.
Let's analyze the problem step by step:
Step 1: Determine the weight of the monkey.
The weight of an object is given by the equation:
Weight = mass × gravity
where the mass of the monkey is 10 kg, and the acceleration due to gravity is 9.8 m/s².
Therefore, the weight of the monkey is:
Weight = 10 kg × 9.8 m/s² = 98 N
Step 2: Determine the maximum force that can be exerted on the vine before it breaks.
The problem states that the vine will break if the tension exceeds 110 N.
Step 3: Determine the net force needed to lift the monkey.
The net force needed to lift the monkey is the difference between the weight of the monkey and the maximum force the vine can handle before breaking.
Net force = Weight - Maximum force
Net force = 98 N - 110 N = -12 N
Step 4: Determine the acceleration of the monkey.
The net force acting on the monkey is equal to the mass of the monkey multiplied by its acceleration.
Net force = mass × acceleration
-12 N = 10 kg × acceleration
acceleration = -12 N / 10 kg
acceleration = -1.2 m/s²
Step 5: Determine the time it takes for the monkey to climb the vine.
The time it takes for an object to travel a certain distance with constant acceleration is given by the equation:
Distance = Initial velocity × time + 0.5 × acceleration × time²
In this case, the monkey starts from rest, so the initial velocity is 0 m/s.
The distance the monkey needs to climb is given as 7.15 meters.
Therefore, the equation becomes:
7.15 m = 0 × time + 0.5 × (-1.2 m/s²) × time²
Simplifying the equation:
7.15 m = -0.6 m/s² × time²
time² = 7.15 m / -0.6 m/s²
time² = -11.92 s² (taking the negative value is not logical in this scenario, so we ignore it.)
To determine the shortest amount of time in which the monkey can climb the vine, we need to use Newton's second law of motion and the principles of work and power.
First, let's determine the net force required to accelerate the monkey upward.
1. Calculate the force the monkey exerts on the rope:
The monkey's weight can be calculated using the formula: weight = mass × gravitational acceleration.
weight = 10 kg × 9.8 m/s² = 98 N
Since the vine will break if the tension exceeds 110 N, the maximum force the monkey can exert is 110 N.
2. Determine the net force:
The net force is the difference between the force the monkey exerts on the rope and its weight.
net force = force exerted by the monkey - weight
net force = 110 N - 98 N = 12 N
Next, let's calculate the acceleration of the monkey.
3. Determine the acceleration:
Using Newton's second law of motion, we can calculate the acceleration using the formula: net force = mass × acceleration.
12 N = 10 kg × acceleration
acceleration = 1.2 m/s²
Now that we know the acceleration, we can find the time required for the monkey to climb the vine.
4. Calculate the time:
To calculate the time, we can use one of the equations of motion, namely: distance = initial velocity × time + 0.5 × acceleration × time².
The monkey starts from rest (initial velocity = 0 m/s), so the equation simplifies to: distance = 0.5 × acceleration × time².
Rearranging the equation, we get: time = √(2 × distance / acceleration).
Plugging in the values, we get: time = √(2 × 7.15 m / 1.2 m/s²).
Calculating the square root of 11.91, the time required for the monkey to climb the vine is approximately 3.45 seconds.
Therefore, the shortest amount of time in which the monkey can climb the vine is approximately 3.45 seconds.