Tarzan plans to cross a gorge by swinging in an arc from a hanging vine. If his arms are capable of exerting 1350 N on the vine, what is his acceleration at the point where the vine makes an angle of 45 degrees from the vertical assuming maximum speed at this point. His mass is 78 kg and the vine is 5.2 m long
To find Tarzan's acceleration at the point where the vine makes an angle of 45 degrees from the vertical, we can use the concept of centripetal acceleration.
1. First, let's determine the net force acting on Tarzan as he swings on the vine. The net force is the difference between the force exerted by Tarzan on the vine and the force due to gravity:
Net force = Tarzan's force - Force due to gravity
Tarzan's force = 1350 N (given)
Force due to gravity = mass * acceleration due to gravity
= 78 kg * 9.8 m/s^2 (standard value for acceleration due to gravity)
2. Next, we need to find the tension in the vine at the point where it makes an angle of 45 degrees. This tension is responsible for providing the centripetal force to keep Tarzan moving in the circular path. The tension force in the vine can be calculated using:
Tension = (mass * velocity^2) / radius
We can find the velocity using conservation of energy. The potential energy at the highest point is equal to the kinetic energy at the lowest point:
Potential energy = Kinetic energy
m * g * h = (1/2) * m * v^2
Solving for v (velocity), we get:
v = sqrt(2 * g * h)
where h is the vertical height from the highest point of the swing to the point where the vine makes an angle of 45 degrees.
3. The radius of the circular path is given by the length of the vine, which is 5.2 m.
4. Now that we have the tension and all the forces acting on Tarzan, we can calculate his acceleration using Newton's second law:
Net force = mass * acceleration
Substituting the values into the equation, we have:
Tarzan's force - Force due to gravity = mass * acceleration
(1350 N) - (78 kg * 9.8 m/s^2) = 78 kg * acceleration
Solving for acceleration will give us the answer.
Note: It's important to clarify that this calculation assumes Tarzan achieves maximum speed at the point where the vine makes an angle of 45 degrees.