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 1820 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 83.0 kg, and the vine is 11.0 m long
The rope must support his weight and centripetal force at the bottom of the swing. At the breaking point,
M*(g + V^2/L) = 1820 N
L = 11.0 m
M = 83 kg
g = 9.8 m/s^2
Solve for V
11.54
To find the maximum speed Tarzan can tolerate at the lowest point of his swing, we can use the principle of conservation of mechanical energy.
1. First, let's calculate the potential energy at the highest point of Tarzan's swing. The potential energy (PE) is given by PE = m * g * h, where m is the mass, g is the acceleration due to gravity, and h is the height.
Since Tarzan is at the highest point of his swing, the height is equal to the length of the vine, h = 11.0 m.
The mass of Tarzan is given as 83.0 kg, and the acceleration due to gravity is approximately 9.8 m/s^2.
Thus, PE = 83.0 kg * 9.8 m/s^2 * 11.0 m = 9,437.4 J (joules).
2. Next, we can calculate the maximum speed Tarzan can tolerate at the lowest point by equating the potential energy to the kinetic energy (KE) at that point. The kinetic energy is given by KE = 0.5 * m * v^2, where v is the velocity.
We set KE = PE and solve for v:
0.5 * 83.0 kg * v^2 = 9,437.4 J
v^2 = (2 * 9,437.4 J) / 83.0 kg
v^2 = 226.98 m^2/s^2
Taking the square root of both sides, we get:
v = √(226.98 m^2/s^2) ≈ 15.05 m/s
Therefore, the maximum speed Tarzan can tolerate at the lowest point of his swing is approximately 15.05 meters/second.