Tarzan plans to cross a gorge by swinging in an arc from a hanging vine. If his arms are capable of exerting a force of 1720 N on the rope, what is the maximum speed (in meters/second) he can tolerate at the lowest point of his swing? His mass is 81.0 kg, and the vine is 12.0 m long.
1720=mg+mv^2/r
solve for v.
To find the maximum speed Tarzan can tolerate at the lowest point of his swing, we can find the tension in the vine when Tarzan is at the lowest point of his swing.
1. First, let's find the tension in the vine when Tarzan is at the lowest point of his swing.
At the lowest point, there are two forces acting on Tarzan: his weight and the tension in the vine.
Weight (W) = mass (m) * acceleration due to gravity (g)
W = 81.0 kg * 9.8 m/s^2 = 794.8 N
The net force acting on Tarzan at the lowest point is the difference between the tension and weight.
Net Force = Tension - Weight
We know that Tarzan's arms are capable of exerting a force of 1720 N on the rope. So the tension in the vine is 1720 N when Tarzan is at the lowest point of his swing.
Net Force = 1720 N - 794.8 N = 925.2 N
2. Next, let's find the maximum speed Tarzan can tolerate at the lowest point of his swing using the concept of conservation of mechanical energy.
At the highest point of his swing, Tarzan has the maximum potential energy and zero kinetic energy.
At the lowest point of his swing, Tarzan has the maximum kinetic energy and zero potential energy.
The potential energy at the highest point is equal to the kinetic energy at the lowest point.
Potential Energy = Kinetic Energy
The potential energy at the highest point is given by:
Potential Energy = mass (m) * gravity (g) * height (h)
The height (h) is half of the length of the vine since Tarzan swings in an arc.
height (h) = 12.0 m / 2 = 6.0 m
Potential Energy = 81.0 kg * 9.8 m/s^2 * 6.0 m = 4768.4 J
The kinetic energy at the lowest point is given by:
Kinetic Energy = (1/2) * mass (m) * velocity^2
We need to find the velocity (v) at the lowest point.
Equating potential energy and kinetic energy, we have:
4768.4 J = (1/2) * 81.0 kg * velocity^2
Solving for velocity (v):
velocity^2 = (4768.4 J * 2) / (81.0 kg)
velocity^2 = 117.095
velocity = √117.095 ≈ 10.82 m/s
Therefore, the maximum speed Tarzan can tolerate at the lowest point of his swing is approximately 10.82 meters/second.