A freshman smart alek lips off onces too often! The assistant principle loses her cool and grabs the 38.0kg punk by the hair- swinging him in a vertical circle of radius 1.20m! if the freshman is moving at 6.30m/s at the bottom of the swing, calculate:
A) the tensions in the punks hair at the bottom of the path
B) the tension at the top of the arc when he is moving at 5.70 m/s
C) the minimum speed needed for the punk to remain in this vertical circle
To answer these questions, we need to use the principles of circular motion and centripetal force. The centripetal force is the force that keeps an object moving in a circle.
A) To calculate the tension in the freshman's hair at the bottom of the swing, we need to find the net force acting on him. At the bottom of the swing, the tension force in the hair points upward, and the force of gravity points downward.
The net force is given by the equation:
Net force = Tension - Weight
Weight is the force of gravity acting on the freshman and is given by the equation:
Weight = mass * gravity
where mass is the mass of the freshman (38.0 kg) and gravity is the acceleration due to gravity (9.8 m/s^2).
Since the freshman is moving in a circle, the net force is equal to the centripetal force:
Net force = Tension - Weight = mass * centripetal acceleration
The centripetal acceleration is given by the equation:
Centripetal acceleration = (velocity^2) / radius
where velocity is the speed of the freshman at the bottom of the swing (6.30 m/s) and the radius is the radius of the circle (1.20 m).
So, we can now calculate the tension in the freshman's hair at the bottom of the path:
Net force = Tension - Weight = mass * centripetal acceleration
Tension - Weight = mass * (velocity^2) / radius
Tension = mass * (velocity^2) / radius + Weight
Tension = (38.0 kg * (6.30 m/s)^2) / 1.20 m + (38.0 kg * 9.8 m/s^2)
B) To calculate the tension at the top of the arc when the freshman is moving at 5.70 m/s, we can use the same equation as in part A, but with different values.
Tension = (38.0 kg * (5.70 m/s)^2) / 1.20 m + (38.0 kg * 9.8 m/s^2)
C) To find the minimum speed needed for the freshman to remain in this vertical circle, we need to consider the maximum tension force.
At the top of the arc, the tension force is at its maximum, equal to the sum of the weight and the centripetal force:
Tension = Weight + mass * centripetal acceleration
Tension = Weight + mass * (velocity^2) / radius
To find the minimum speed, we make the assumption that the tension force cannot be negative (since the freshman cannot experience negative tension). So we can set up the equation:
Weight + mass * (minimum velocity^2) / radius = 0
Solving for the minimum velocity:
Minimum velocity = sqrt((-Weight * radius) / mass)
So, to find the minimum speed needed for the freshman to remain in this vertical circle, we need to calculate sqrt((-Weight * radius) / mass).