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 23.0 degrees with the horizontal. A piece of luggage having mass 31.0 kg is placed on the carousel, 6.75 m from the axis of rotation. The travel bag goes around once in 48.0 s. Calculate the force of static friction between the bag and the carousel.
r=6.75
θ=23°
m=31 kg
ω=2π/48 radians/s
g=9.8 m/s/s
μ = coefficient of static friction
Assuming the bag does not slide due to friction, then
force due to gravity downwards along the incline plane, Fg
= mgsin(θ)
downward force along inclined plane due to rotation, Fr
= mrω²cos(θ)
Normal reaction due to gravity less component due to rotation, N
= mgcos(θ)-mrω²sin(θ)
μ=(Fg+Fr)/N
Can you take it from here?
Also check my logic.
To calculate the force of static friction between the bag and the carousel, we can start by finding the acceleration of the luggage.
First, we need to find the angular speed of the carousel. The angular speed (ω) is given by the formula:
ω = 2π / T
Where T is the time taken for one complete revolution. In this case, T is 48.0 s.
ω = 2π / 48.0
= 0.1309 rad/s
Next, we can find the linear velocity (v) of the luggage, which is the tangential speed at a radius of 6.75 m. The linear velocity is given by the formula:
v = ω * r
Where r is the radius. In this case, r is 6.75 m.
v = 0.1309 * 6.75
= 0.8827 m/s
Now, we can find the centripetal acceleration (a) of the luggage. The centripetal acceleration is given by the formula:
a = v^2 / r
Where v is the linear velocity and r is the radius.
a = (0.8827)^2 / 6.75
= 0.114 m/s^2
Finally, we can calculate the force of static friction (Ff) acting on the luggage. The force of static friction provides the necessary centripetal force to keep the luggage moving in a circular path.
Ff = m * a
Where m is the mass of the luggage and a is the centripetal acceleration.
Ff = 31.0 * 0.114
= 3.534 N
Therefore, the force of static friction between the bag and the carousel is 3.534 N.