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) 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 use the concepts of circular motion and the forces involved.
(a) To calculate the force of static friction:
First, let's find the net force acting on the luggage. The net force is directed towards the center of the carousel and is equal to the centripetal force required for circular motion.
The centripetal force (Fc) is given by the equation:
Fc = (m * v^2) / r
Where:
m = mass of the luggage = 37.0 kg
v = velocity of the luggage = circumference of the carousel / time taken to complete one rotation
r = radius or distance of the luggage from the axis of rotation = 7.10 m
To find the velocity, we need to calculate the circumference of the carousel. The circumference (C) is given by the equation:
C = 2 * π * r
Using the given values, we can calculate the velocity and circumference:
C = 2 * π * 7.10 m
C = 44.65 m
v = C / time taken to complete one rotation
v = 44.65 m / 44.0 s
v = 1.014 m/s
Now, we can calculate the centripetal force:
Fc = (37.0 kg * (1.014 m/s)^2) / 7.10 m
Fc = 5.349 N
The force of static friction (fs) opposes the motion and provides the centripetal force. So, fs = Fc.
Therefore, the force of static friction between the bag and the carousel is 5.349 N.
(b) To calculate the coefficient of static friction:
In this case, the luggage is on the verge of slipping, so the force of static friction is at its maximum value (fsmax), given by:
fsmax = μs * N
Where:
μs = coefficient of static friction (we need to find this)
N = normal force on the luggage
The normal force (N) is the force exerted by the carousel surface perpendicular to the luggage's weight. It can be calculated using the equation:
N = mg * cos(θ)
Where:
m = mass of the luggage = 37.0 kg
g = acceleration due to gravity = 9.8 m/s²
θ = angle the surface makes with the horizontal = 21.0°
Now, let's calculate N:
N = (37.0 kg) * (9.8 m/s²) * cos(21.0°)
N = 353.41 N
Using this value and the previously calculated force of static friction (5.349 N), we can find the coefficient of static friction:
fsmax = μs * N
5.349 N = μs * 353.41 N
μs = 5.349 N / 353.41 N
μs ≈ 0.015
Therefore, the coefficient of static friction between the bag and the carousel is approximately 0.015.