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 solve this problem, we can use the concepts of circular motion and centripetal force. We'll start by analyzing the forces acting on the freshman at the bottom of the swing.

A) Tensions in the hair at the bottom of the path:
At the bottom of the swing, the only force acting on the freshman is the tension in his hair. We can calculate this tension using the equation:

Tension = Centripetal Force

The centripetal force is given by the formula:

Centripetal Force = (mass x velocity^2) / radius

Given:
- Mass of the freshman (m) = 38.0 kg
- Velocity of the freshman at the bottom of the swing (v) = 6.30 m/s
- Radius of the circle (r) = 1.20 m

Substituting these values into the formula, we get:

Centripetal Force = (38.0 kg x (6.30 m/s)^2) / 1.20 m

Calculating this, we find:

Centripetal Force = 1185.45 N

Therefore, the tension in the punk's hair at the bottom of the path is 1185.45 N.

B) Tension at the top of the arc:
At the top of the swing, two forces act on the freshman: the tension in his hair and the force of gravity. We need to consider the net force to calculate the tension.

Net Force = Tension - Weight

The weight of the freshman can be calculated using:

Weight = Mass x Gravity

Given:
- Mass of the freshman (m) = 38.0 kg
- Acceleration due to gravity (g) = 9.8 m/s^2

Weight = 38.0 kg x 9.8 m/s^2 = 372.4 N

Considering upward motion at the top of the arc, the net force is:

Net Force = Tension - Weight = m x (v^2 / r) - mg

The velocity of the freshman at the top of the arc (v) is given as 5.70 m/s. Substituting the known values, we get:

Net Force = (38.0 kg x (5.70 m/s)^2) / 1.20 m - 372.4 N

Calculating this, we find:

Net Force = 904.25 N

To obtain the tension (T), we add the weight to the net force:

Tension = Net Force + Weight = 904.25 N + 372.4 N

Calculating this, we get:

Tension = 1276.65 N

Therefore, the tension in the punk's hair at the top of the arc when he is moving at 5.70 m/s is 1276.65 N.

C) Minimum speed needed to remain in the vertical circle:
For the freshman to remain in the vertical circle, the tension in his hair must be greater than or equal to his weight at the lowest point.

So, we need to set up the inequality:

Tension ≥ Weight

Using the weight equation: Weight = Mass x Gravity

Tension ≥ m x g

Substituting the given values:

Tension ≥ 38.0 kg x 9.8 m/s^2

Calculating this, we get:

Tension ≥ 372.4 N

Therefore, to remain in the vertical circle, the minimum speed needed is when the tension in the punk's hair is at least equal to his weight, which is 372.4 N.