To determine the work done against the force of gravity as the monkey swings on the rope, we need to analyze the situation step by step. The work done against gravity can be calculated by finding the change in gravitational potential energy as the monkey swings to the highest point of his swing, where the rope makes a 60-degree angle with the vertical.
Understanding the Setup
First, let’s visualize the scenario. The monkey is hanging from a rope that is 12.0 meters long. When he swings to the side, the rope forms a 60-degree angle with the vertical. At this point, we can find the height he has risen compared to his lowest point.
Calculating the Height
To find the height (h) that the monkey rises, we can use some trigonometry. When the rope makes a 60-degree angle with the vertical, we can form a right triangle where:
- The hypotenuse is the length of the rope (12.0 m).
- The adjacent side (vertical component) is the distance from the top of the swing to the lowest point, which can be calculated using cosine.
- The opposite side (height gained) can be found using sine.
The vertical distance (adjacent side) can be calculated as:
Vertical distance = Length of rope × cos(60°)
Vertical distance = 12.0 m × cos(60°) = 12.0 m × 0.5 = 6.0 m
Now, the height (h) that the monkey rises is the difference between the length of the rope and the vertical distance:
Height gained = Length of rope - Vertical distance
Height gained = 12.0 m - 6.0 m = 6.0 m
Calculating Work Done Against Gravity
The work done against gravity is equal to the change in gravitational potential energy (PE), which can be calculated using the formula:
PE = m × g × h
Where:
- m = mass of the monkey = 8.0 kg
- g = acceleration due to gravity ≈ 9.81 m/s²
- h = height gained = 6.0 m
Now, substituting the values into the equation:
PE = 8.0 kg × 9.81 m/s² × 6.0 m
PE = 8.0 × 9.81 × 6.0 = 470.88 J
Final Thoughts
The work done against the force of gravity as the monkey swings and reaches the branch is approximately 470.88 joules. This energy represents the effort required to lift the monkey to that height against the gravitational pull. Understanding these concepts helps in grasping the principles of energy conservation and the mechanics of swinging objects.
