To analyze Tarzan's swing from the cliff, we need to break down the forces acting on him and apply some fundamental physics principles. Let's tackle this step by step, starting with the forces involved and then moving on to the calculations.
Understanding the Forces
When Tarzan steps off the cliff, two primary forces act on him:
- The gravitational force (weight) acting downward, which is equal to 820 N.
- The tension in the vine, which acts along the vine's length and has a magnitude of 760 N.
Choosing a Coordinate System
We will use a coordinate system where:
- The x-axis points horizontally away from the cliff edge.
- The y-axis points vertically upward.
(a) Force of the Vine on Tarzan
To express the tension in the vine in unit vector notation, we need to resolve it into its x and y components. The angle given is 22 degrees from the vertical, which means:
- The angle with the horizontal (x-axis) is 90° - 22° = 68°.
Now, we can calculate the components:
- Tension in the x-direction: T_x = T \cdot \sin(68°) = 760 \cdot \sin(68°)
- Tension in the y-direction: T_y = T \cdot \cos(68°) = 760 \cdot \cos(68°)
Calculating these values:
- T_x ≈ 760 * 0.927 = 704.52 N
- T_y ≈ 760 * 0.374 = 284.24 N
Thus, the force of the vine on Tarzan in unit vector notation is:
F_vine = (704.52 i + 284.24 j) N
(b) Net Force Acting on Tarzan
The net force acting on Tarzan can be found by considering both the tension and the weight. The weight acts downward, so we need to express it in our coordinate system:
Weight = (0 i - 820 j) N
Now, we can find the net force:
F_net = F_vine + F_weight
F_net = (704.52 i + 284.24 j) + (0 i - 820 j)
F_net = (704.52 i - 535.76 j) N
(c) Magnitude of the Net Force
The magnitude of the net force can be calculated using the Pythagorean theorem:
|F_net| = √(F_net_x² + F_net_y²)
|F_net| = √((704.52)² + (-535.76)²)
|F_net| ≈ √(496354.38 + 287194.66) ≈ √(783549.04) ≈ 884.67 N
(d) Direction of the Net Force
The direction of the net force can be found using the arctangent function:
θ = tan⁻¹(F_net_y / F_net_x)
θ = tan⁻¹(-535.76 / 704.52)
θ ≈ -39.5°
This angle is measured from the positive x-axis downwards, indicating that the net force is directed downwards and to the left.
(e) Magnitude of Tarzan's Acceleration
To find Tarzan's acceleration, we can use Newton's second law, F = ma. Rearranging gives us:
a = F_net / m
First, we need to find Tarzan's mass:
Weight = mg → m = Weight / g = 820 N / 9.81 m/s² ≈ 83.67 kg
Now we can calculate the acceleration:
a = 884.67 N / 83.67 kg ≈ 10.59 m/s²
(f) Direction of Tarzan's Acceleration
The direction of the acceleration is the same as the direction of the net force, which we found to be approximately -39.5° from the positive x-axis.
In summary, we have determined the forces acting on Tarzan, calculated the net force, and found both the magnitude and direction of his acceleration. This analysis illustrates the application of Newton's laws in a real-world scenario, showcasing the interplay between forces in motion.
