Tarzan tries to cross a river by swinging from one bank to the other on a vine that is 12.2 m long. His speed at the bottom of the swing is 8.6 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?

To find the largest mass that Tarzan can have and still make it safely across the river, we need to determine the tension force in the vine while Tarzan is swinging.

1. First, let's calculate the gravitational force acting on Tarzan. We can use the equation:

F_gravity = m * g

where F_gravity is the gravitational force, m is the mass, and g is the acceleration due to gravity (approximately 9.8 m/s²).

2. Since Tarzan is swinging at the bottom of the swing, the tension force in the vine (T) is equal to the centripetal force required to keep Tarzan in circular motion:

T = m * v² / r

where T is the tension force, m is the mass, v is the velocity, and r is the radius of the swing (equal to the length of the vine).

3. We know that Tarzan cannot exceed the breaking strength of the vine, which is given as 1.0 × 10³ N. Therefore, we can set up the following equation:

T ≤ Breaking strength

m * v² / r ≤ Breaking strength

4. Now we can substitute the given values into the equation:

m * (8.6 m/s)² / 12.2 m ≤ 1.0 × 10³ N

5. Simplifying the equation:

m ≤ (1.0 × 10³ N) * (12.2 m) / (8.6 m/s)²

m ≤ 223.72 kg

Therefore, the largest mass that Tarzan can have and still make it safely across the river is approximately 223.72 kg.