In an old-fashioned amusement park ride, passengers stand inside a 3.0-m-tall, 5.0-m-diameter hollow steel cylinder with their backs against the wall. The cylinder begins to rotate about a vertical axis. Then the floor on which the passengers are standing suddenly drops away! If all goes well, the passengers will “stick” to the wall and not slide. Clothing has a static coefficient of friction against steel in the range 0.60 to 1.0 and a kinetic coefficient in the range 0.40 to 0.70.
What is the minimum rotational frequency, in rpm, for which the ride is safe?
24.4 rpm
.32
To determine the minimum rotational frequency for the ride to be safe, we need to consider the centrifugal force acting on the passengers when the floor drops away.
First, let's calculate the magnitude of the centrifugal force acting on a passenger.
The formula for centrifugal force is given by F = m * ω^2 * r, where:
- F is the centrifugal force,
- m is the mass of the passenger,
- ω is the angular velocity (in radians per second),
- r is the radius of rotation.
In this case, the radius of rotation is the distance from the center of the cylinder to the person's back, which is half the diameter of the cylinder, r = 5.0 m / 2 = 2.5 m.
We can rewrite the formula as F = m * (2πf)^2 * r, where:
- f is the frequency of rotation (in hertz), and
- ω = 2πf (convert radians per second to hertz).
To ensure the passengers stick to the wall, the centrifugal force must be greater than or equal to the maximum static friction force. The maximum static friction force is given by Fs_max = μ_static * mg, where:
- μ_static is the static coefficient of friction,
- m is the mass of the passenger, and
- g is the acceleration due to gravity (approximately 9.8 m/s^2).
Since we want to find the minimum rotational frequency, we will use the maximum static coefficient of friction (μ_static = 1.0).
Setting F = Fs_max, we have:
m * (2πf)^2 * r = μ_static * m * g.
The mass cancels out, and we can solve for the frequency (f):
(2πf)^2 = μ_static * g / r,
Simplifying further:
f^2 = (μ_static * g) / (4π^2 * r).
Taking the square root of both sides gives:
f = √[(μ_static * g) / (4π^2 * r)].
Now we can substitute the known values:
μ_static = 1.0,
g = 9.8 m/s^2,
r = 2.5 m.
Calculating the frequency, we have:
f = √[(1.0 * 9.8) / (4 * 3.14^2 * 2.5)].
f ≈ √(9.8 / 98.2),
f ≈ √0.1,
f ≈ 0.316.
Finally, since the rotational frequency is in hertz, we need to convert it to rpm (revolutions per minute):
f_rpm = f * 60.
Plugging in the value:
f_rpm ≈ 0.316 * 60,
f_rpm ≈ 18.96.
Therefore, the minimum rotational frequency for the amusement park ride to be safe is approximately 18.96 rpm.
To determine the minimum rotational frequency needed for the ride to be safe, we need to consider the maximum centrifugal force that the passengers can experience without sliding.
The centrifugal force exerted on each passenger is given by the equation:
Fc = m * r * ω^2
where Fc is the centrifugal force, m is the mass of the passenger, r is the distance from the rotational axis (in this case, the radius of the cylinder), and ω is the angular frequency, which is equal to 2π times the rotational frequency.
In this case, we want to find the minimum rotational frequency, so we need to consider the maximum value for the coefficient of friction (μ), which is 1.0.
The frictional force that prevents sliding is given by:
Ff = μ * N
where Ff is the frictional force and N is the normal force.
The normal force is equal to the weight of the passenger, which is given by:
N = m * g
where g is the acceleration due to gravity.
Since the passenger is standing against the wall, the frictional force must balance the centrifugal force in the horizontal direction:
Ff = Fc
Substituting the expressions for Ff and Fc:
μ * N = m * r * ω^2
μ * m * g = m * r * ω^2
Taking m out of the equation:
μ * g = r * ω^2
Solving for ω, we get:
ω = sqrt(μ * g / r)
Since we are given the diameter of the cylinder (5.0 m), we can calculate the radius (r) by dividing it by 2:
r = 5.0 m / 2 = 2.5 m
Substituting the values for μ (1.0) and g (9.8 m/s^2), we can calculate ω:
ω = sqrt(1.0 * 9.8 m/s^2 / 2.5 m) = sqrt(3.92) ≈ 1.98 rad/s
Finally, we can convert the angular frequency (ω) to rotational frequency (f) in revolutions per minute (rpm) using the conversion factor:
f = ω / (2π) * (1 min / 60 s) = 1.98 / (2π) * (1/60) ≈ 0.053 rpm
Therefore, the minimum rotational frequency for the ride to be safe is approximately 0.053 rpm.