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 18.0° with the horizontal. A 22.0 kg piece of luggage is placed on the carousel, 6.50 m from the axis of rotation. The travel bag goes around once in 46.0 s. (a) Calculate the force of static friction between the bag and the carousel.
Why did the luggage join the circus? Because it wanted to perform a "carousel of tricks"! Now let's get to the calculations before the luggage gets dizzy.
To calculate the force of static friction between the bag and the carousel, we need to consider the forces acting on the bag.
The weight of the bag can be calculated using the formula: weight = mass × acceleration due to gravity. Since the bag has a mass of 22.0 kg and the acceleration due to gravity is approximately 9.8 m/s^2, the weight of the bag is:
weight = 22.0 kg × 9.8 m/s^2 = 215.6 N
Now, let's determine the force of static friction. The force of static friction acts perpendicular to the metallic surface of the carousel. The angle between the metallic surface and the horizontal plane is 18.0°.
Using trigonometry, we can find the vertical component of the weight, which is responsible for the normal force:
normal force = weight × cos(angle)
normal force = 215.6 N × cos(18.0°)
To calculate the force of static friction, we use the formula: force of static friction = normal force × coefficient of static friction.
Unfortunately, the coefficient of static friction between the bag and the carousel is not provided in the question. So, we can't determine the exact force of static friction without that information.
To calculate the force of static friction between the bag and the carousel, we need to first determine the acceleration of the bag.
The acceleration of the bag can be calculated using the formula:
a = (v^2) / r
where,
a is the acceleration
v is the linear velocity
r is the radius
Since the bag goes around the carousel once in 46.0 s, the linear velocity can be calculated as:
v = (2πr) / t
where,
t is the time taken for one complete revolution (46.0 s in this case)
The radius of the carousel can be calculated using trigonometry:
r = d * sin(θ)
where,
d is the distance of the bag from the axis of rotation (6.50 m in this case)
θ is the angle of the metallic surface (18.0° in this case)
Let's perform the calculations step-by-step:
Step 1: Calculate the radius of the carousel
r = d * sin(θ)
r = 6.50 m * sin(18.0°)
r ≈ 1.837 m
Step 2: Calculate the linear velocity
v = (2πr) / t
v = (2π * 1.837 m) / 46.0 s
v ≈ 0.400 m/s
Step 3: Calculate the acceleration
a = (v^2) / r
a = (0.400 m/s)^2 / 1.837 m
a ≈ 0.087 m/s^2
Now, to calculate the force of static friction, we can use Newton's second law:
F_friction = m * a
where,
F_friction is the force of static friction
m is the mass of the bag (22.0 kg in this case)
a is the calculated acceleration
Step 4: Calculate the force of static friction
F_friction = m * a
F_friction = 22.0 kg * 0.087 m/s^2
F_friction ≈ 1.914 N
Therefore, the force of static friction between the bag and the carousel is approximately 1.914 N.
To calculate the force of static friction between the bag and the carousel, we need to consider the forces acting on the luggage.
The forces acting on the luggage are gravity (weight) and the normal force exerted by the carousel on the luggage. The normal force acts perpendicular to the surface of the carousel, and the force of static friction opposes the relative motion between the luggage and the carousel.
The force of static friction can be calculated using the equation:
fs = μs * N
Where fs is the force of static friction, μs is the coefficient of static friction, and N is the normal force.
First, let's calculate the normal force:
The weight of the luggage is given by the equation:
W = m * g
Where W is the weight, m is the mass, and g is the acceleration due to gravity.
Plugging in the values given: m = 22.0 kg and g = 9.8 m/s^2
W = 22.0 kg * 9.8 m/s^2
W = 215.6 N
Since the luggage is on an incline, we need to find the component of the weight perpendicular to the surface of the carousel. This can be calculated using:
N = W * cos(θ)
Where θ is the angle of the incline.
Plugging in the values: θ = 18.0°
N = 215.6 N * cos(18.0°)
N = 201.3 N
Now, we can calculate the force of static friction:
fs = μs * N
However, we need one more piece of information, the acceleration of the luggage. Since the luggage is going around in a circle, it experiences centripetal acceleration given by the equation:
a = (v^2) / r
Where a is the centripetal acceleration, v is the velocity, and r is the radius.
The velocity of the luggage can be calculated using:
v = (2π * r) / T
Where T is the time it takes for a complete revolution.
Plugging in the values: r = 6.50 m and T = 46.0 s
v = (2π * 6.50 m) / 46.0 s
v ≈ 2.77 m/s
Now, we can calculate the centripetal acceleration:
a = (v^2) / r
a = (2.77 m/s)^2 / 6.50 m
a ≈ 1.18 m/s^2
Finally, we can calculate the force of static friction:
fs = μs * N
We need to use the equation of motion to find the static friction force:
fs = m * a
Plugging in the values: m = 22.0 kg and a ≈ 1.18 m/s^2
fs = 22.0 kg * 1.18 m/s^2
fs ≈ 25.96 N
Therefore, the force of static friction between the bag and the carousel is approximately 25.96 N.