Tarzan tries to cross a river by swinging from one bank to the other on a vine that is 12.0 m long. His speed at the bottom of the swing is 8.3 m/s. Tarzan does not know that the vine has a breaking strength of 1.0 ✕ 103 N. What is the largest mass that Tarzan can have and still make it safely across the river?
f = m v^2 / r ... 1.0E3 > m * 8.3^2 / 12.0
To determine the largest mass that Tarzan can have and still safely cross the river, we need to consider the forces acting on him during the swing.
1. First, let's calculate the tension force in the vine at the bottom of the swing using centripetal force.
The centripetal force required to keep Tarzan swinging in a circular path can be calculated using the formula:
F = (m * v^2) / r
Where:
F is the centripetal force
m is the mass of Tarzan
v is the velocity of Tarzan at the bottom of the swing
r is the length of the vine
From the given information:
v = 8.3 m/s
r = 12.0 m
Substituting the values into the formula:
F = (m * (8.3 m/s)^2) / 12.0 m
2. Next, let's determine the maximum tension force the vine can withstand without breaking.
The maximum tension force the vine can withstand is given as 1.0 * 10^3 N.
3. Finally, we equate the tension force in the vine to the maximum tension force to find the largest mass Tarzan can have.
(m * (8.3 m/s)^2) / 12.0 m = 1.0 * 10^3 N
Simplifying the equation:
(m * 68.89 m^2/s^2) / 12.0 m = 1.0 * 10^3 N
68.89 m^2/s^2 * m / 12.0 = 1.0 * 10^3 N
Dividing both sides of the equation by 68.89 m^2/s^2:
m / 12.0 = 1.0 * 10^3 N / 68.89 m^2/s^2
m / 12.0 ≈ 14.5 N
Multiplying both sides of the equation by 12.0:
m ≈ 14.5 N * 12.0
m ≈ 174.0 kg
Therefore, the largest mass that Tarzan can have and still safely cross the river is approximately 174.0 kilograms.