Tarzan (m = 75 kg) tries to cross a river by swinging from a 15.0 m long vine. His speed at the bottom of the swing, just as he clears the water, is 9.0 m/s. What is the total force Tarzan exerts on the vine at that point?
To find the total force Tarzan exerts on the vine at the bottom of his swing, we need to consider the forces acting on him.
1. Gravitational Force: Tarzan's weight is given by the formula W = mg, where m is his mass (75 kg) and g is the acceleration due to gravity (approximately 9.8 m/s²). So, the gravitational force acting on Tarzan is Fg = 75 kg * 9.8 m/s².
2. Centripetal Force: As Tarzan swings, there is a centripetal force acting towards the center of the circular motion. This force is provided by the tension in the vine. We can calculate the centripetal force using the formula Fc = (mv²) / r, where m is Tarzan's mass, v is his speed (9.0 m/s), and r is the radius of the circular motion (half the length of the vine, 15.0 m/2).
Now, to calculate the total force exerted by Tarzan on the vine, we need to find the vector sum of these forces.
Ftotal = Fg + Fc
Substituting in the given values:
Fg = 75 kg * 9.8 m/s²
Fc = (75 kg * (9.0 m/s)²) / (15.0 m/2)
Calculating Fg and Fc, then adding them together will give the total force exerted by Tarzan on the vine at the bottom of his swing.