A luggage carousel at an airport has the form of a section of a large cone, steadily rotating about its vertical axis. Its metallic surface slopes downward toward the outside, making an angle of 21.0° with the horizontal. A 37.0 kg piece of luggage is placed on the carousel, 7.10 m from the axis of rotation. The travel bag goes around once in 44.0 s. (a) Calculate the force of static friction between the bag and the carousel. b) (b) The drive motor is shifted to turn the carousel at a higher constant rate of rotation, and the piece of luggage is bumped to a position 7.88 m from the axis of rotation. The bag is on the verge of slipping as it goes around once every 28.20 s. Calculate the coefficient of static friction between the bag and the carousel.

To solve this problem, we need to analyze the forces acting on the luggage and use the principles of circular motion. Let's break it down step by step:

(a) To calculate the force of static friction between the bag and the carousel, we can start by determining the net force acting on the luggage.

1. Find the gravitational force acting on the luggage:
The gravitational force is given by the weight of the luggage, which can be calculated using the formula: weight = mass × gravitational acceleration.
Given the mass of the luggage (m = 37.0 kg) and the gravitational acceleration (g ≈ 9.8 m/s²), we can calculate:
weight = m × g.

2. Determine the radial acceleration:
Since the luggage is rotating in a circle, it experiences radial acceleration toward the center of the carousel.
Radial acceleration (aᵣ) can be calculated using the formula: aᵣ = r × ω², where r is the distance from the axis of rotation and ω is the angular velocity (in radians per second).
Given the distance from the axis of rotation (r = 7.10 m) and the time for one revolution (T = 44.0 s), we can calculate:
ω = (2π rad) / T,
and then substitute ω into the formula.

3. Calculate the net radial force:
The net radial force is given by the difference between the gravitational force and the force of static friction.
Since the luggage is not slipping, the static friction force opposes the radial force.
The net radial force (Fᵣ) can be calculated using the formula: Fᵣ = m × aᵣ.

4. Determine the force of static friction:
Given that the static friction force (F_s) opposes the radial force, we have: F_s = Fᵣ.

The force of static friction between the bag and the carousel can now be calculated.

(b) To calculate the coefficient of static friction between the bag and the carousel in the second scenario, we can follow a similar process.

1. Find the new radial acceleration:
Since the luggage is on the verge of slipping, the maximum static friction force is being achieved.
The maximum static friction force is given by the equation: F_s = μ_s × N, where μ_s is the coefficient of static friction and N is the normal force.
In this case, the normal force is equal to the weight of the luggage.
Using the given time for one revolution (T = 28.20 s) and the new distance from the axis of rotation (r = 7.88 m), we can calculate the new ω and aᵣ.

2. Calculate the net radial force:
The net radial force in this scenario is the difference between the gravitational force and the maximum static friction force.
Fᵣ = m × aᵣ.

3. Determine the maximum static friction force (F_s):
Given that the luggage is on the verge of slipping, F_s is equal to the maximum static friction force, which is μ_s × N.
F_s = μ_s × m × g.

4. Calculate the coefficient of static friction:
Using the formula for static friction, F_s = μ_s × m × g, and substituting the known values, we can solve for μ_s.

By following these steps, you should be able to calculate the force of static friction in part (a) and the coefficient of static friction in part (b) of the problem.