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 concepts of rotational motion.

In both parts (a) and (b), we'll start by calculating the net force acting on the luggage in the radial direction using the equation:

Net Force = Centripetal Force

Centripetal Force is given by the equation:

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

where:
- mass is the mass of the luggage
- radius is the distance between the luggage and the axis of rotation
- angular velocity is the rate at which the carousel is rotating in radians per second.

(a)
In part (a), the luggage is not slipping, so the force of static friction between the bag and the carousel provides the centripetal force.

Let's calculate the angular velocity first:
Angular velocity = (2π) / (Time for one revolution)

Plugging in the values:
Time for one revolution = 44.0 s
Angular velocity = (2π) / (44.0 s)

Next, we'll calculate the centripetal force:
Centripetal Force = (mass) x (radius) x (angular velocity)^2

Plugging in the given values:
mass = 37.0 kg
radius = 7.10 m
Centripetal Force = (37.0 kg) x (7.10 m) x [(2π) / (44.0 s)]^2

The force of static friction between the bag and the carousel is equal to the centripetal force.

(b)
In part (b), the luggage is on the verge of slipping, so the force of static friction is at its maximum value and provides the centripetal force.

We'll follow the same steps as part (a), using the given values:
mass = 37.0 kg
radius = 7.88 m
time for one revolution = 28.20 s

First, calculate the angular velocity using:
Angular velocity = (2π) / (time for one revolution)

Next, calculate the centripetal force using:
Centripetal Force = (mass) x (radius) x (angular velocity)^2

Since the luggage is on the verge of slipping, the force of static friction is at its maximum value, and we can use this to calculate the coefficient of static friction.

Coefficient of static friction = (Force of static friction) / (Normal force)

The normal force is equal to the weight of the luggage, which is given by:
Normal force = mass x gravitational acceleration

So, plug in the known values to calculate the coefficient of static friction.