Design a carnival ride on which standing passengers are pressed against the inside curved wall of a rotating vertical cylinder. It is to turn at most at 1/3 revolution per second. Assuming a minimum coefficient of friction of 0.70 between clothing and wall, what diameter should the ride have if we can safely make the floor drop away when it reaches running speed?
Ac = v2/R =omega^2 r = g/.7 =9.81/.7 = 14
r = 14/omega^2
but omega = 2 pi (1/3) = 2.09 radians/s
omega^2 = 4.38
r = 14/4.38 = 3.19 meters
To determine the diameter of the carnival ride, we need to consider the forces acting on the standing passengers and use the given information.
First, let's analyze the forces involved:
1. Centripetal Force: The standing passengers are pressed against the inside curved wall of the rotating vertical cylinder due to the centripetal force. This force is provided by the friction between the passengers' clothing and the wall.
2. Gravity: The passengers also experience the force of gravity pulling them downward.
To find the diameter of the ride, we need to consider the maximum force of friction and the maximum speed at which the ride can turn.
Step 1: Calculate the maximum force of friction:
The maximum force of friction can be calculated using the coefficient of friction and the normal force acting on the passengers. Since the passengers are pressed against the wall, the normal force is equal to their weight.
Given the minimum coefficient of friction between clothing and wall is 0.70, we can use this value to find the maximum force of friction.
Step 2: Find the maximum speed:
The ride should not turn faster than 1/3 revolution per second. We can convert this into angular velocity using the formula:
Angular velocity (ω) = 2πf
Where f is the frequency (1/3 revolution per second) and ω is in radians per second.
Step 3: Determine the maximum acceleration:
To find the maximum acceleration, we need to relate the radius of the ride to the angular velocity.
Maximum acceleration (a) = ω^2 * r
Where ω is the angular velocity and r is the radius of the ride.
Step 4: Relate the maximum acceleration to the maximum force of friction:
The maximum acceleration should correspond to the maximum force of friction. We can relate these using the formula:
Maximum force of friction (F) = m * a
Where m is the mass of a passenger.
Step 5: Relate the maximum force of friction to the normal force:
The maximum force of friction should also correspond to the maximum normal force acting on the passengers. Since the maximum force of friction is equal to the coefficient of friction times the normal force, we can write:
F = μ * N
Since N is equal to mg, where g is the acceleration due to gravity, we can solve for N:
N = m * g
Now we can equate the maximum force of friction to the maximum normal force to find the maximum mass (m) of a passenger:
μ * m * g = m * a
Step 6: Calculate the diameter:
The diameter of the ride can be found using the maximum acceleration calculated earlier. The radius (r) is half of the diameter (d), so we can write:
a = ω^2 * r
a = ω^2 * (d/2)
Solving for d, we get:
d = 2 * a / ω^2
Using the rotational frequency (f) and the given information, we can calculate the angular velocity (ω) and then substitute it into the formula to find the diameter (d).