In an old fashioned amusement park ride, passengers stand inside a 5.0-m diameter hollow steel cylinder with their backs against the wall. The cylinder begins to rotate about the vertical axis. Then, the floor that the passengers are standing on 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 that ranges from 0.60 to 1.0 and a kinetic coefficient in the range of 0.4 to 0.7. A sign next to the ride says “No children under 30-kg allowed!” What is the minimum angular speed, in rpm, for which the ride is safe?
Not sure what speed is safe, because we do not control what clothing riders wear, and hence their coefficient of friction.
According to the data, assuming them to be accurate, you will need to have the friction to overcome the weight of the customers. Minimum coefficient of friction is μ=0.4 (kinetic), so you will need the centripetal acceleration to be g/0.4=2.5g for the poor screaming rider to be 'sustained' by the frictional force.
Centripetal acceleration is
a=2.5g=rω²
Solve for ω and convert to RPM.
Note:
1. Investigate the effect of 2.5g on the physiological functioning of the bladder.
2. A 10% slanting slope (wider near the top) to make the "cylinder" a truncated cone will help to keep the customer in his place, and gives him a chance to survive for an "encore".
To determine the minimum angular speed for which the ride is safe, let's break down the problem and calculate the necessary conditions.
1. First, we need to find the minimum force of friction required for the passengers to stick to the wall. We'll use the kinetic coefficient of friction since the floor drops away and the passengers will experience sliding.
2. The force of friction depends on the gravitational force acting on the passengers, given by the formula Fg = mg, where m represents the mass of a passenger.
3. The normal force on each passenger is equal to their weight, perpendicular to the wall. In this case, the normal force is equal to the gravitational force, Fn = Fg.
4. The force of friction is given by Ff = μk * Fn, where μk represents the kinetic coefficient of friction.
5. To stick to the wall, the friction force provided by the static coefficient of friction (μs) should be greater than or equal to the force of friction (Ff) calculated in the previous step.
6. The static coefficient of friction, μs, gives us the minimum limit for sticking to the wall, so we'll use this value in our calculations.
7. The centripetal force required to keep the passengers against the wall is provided by the radial acceleration, given by Fr = m * ar, where m represents the mass of a passenger and ar is the radial acceleration.
8. The radial acceleration, ar, is equal to v^2 / r, where v is the linear speed and r is the radius of the cylinder.
9. The linear speed, v, is related to the angular speed, ω, of the rotating cylinder by the formula v = ω * r.
10. Combining equations, we can express the minimum angular speed, ω, as ω = sqrt(μs * g / r), where g is the acceleration due to gravity and r is the radius of the cylinder.
11. We can convert the angular speed to rpm by using the formula: ω (in rpm) = ω (in rad/s) * (60s/2π rad).
Finally, we'll substitute the given values into the equation to calculate the minimum angular speed in rpm.
Given:
- Radius of the cylinder, r = 5.0 m
- Static coefficient of friction, μs = 1.0 (assuming the maximum value)
- Acceleration due to gravity, g ≈ 9.8 m/s²
Calculations:
ω (in rad/s) = sqrt(μs * g / r) = sqrt(1.0 * 9.8 / 5.0) ≈ 1.98 rad/s
ω (in rpm) = 1.98 * 60 / (2π) ≈ 18.88 rpm
Therefore, the minimum angular speed for which the ride is safe is approximately 18.88 rpm.
To determine the minimum angular speed required for the ride to be safe, we need to consider the maximum force of static friction that can be exerted on the passengers by the wall of the cylinder.
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 static coefficient of friction, and N is the normal force. In this case, the normal force is equal to the weight of the passengers.
Since the sign next to the ride says "No children under 30-kg allowed," we can assume that the minimum weight of the passengers is 30 kg.
The normal force can be calculated using the equation:
N = m * g
Where m is the mass of the passengers and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Now, let's calculate the force of static friction for the minimum coefficient of friction (0.60):
fs = 0.60 * (30 kg * 9.8 m/s^2)
Next, we need to find the minimum angular speed at which the force of static friction will equal the force required for the passengers to "stick" to the wall. This force is provided by the centripetal force, which is given by:
Fc = m * v^2 / r
Where Fc is the centripetal force, m is the mass of the passengers, v is the velocity, and r is the radius of the cylinder.
Since the passengers are standing with their backs against the wall, the radius of the cylinder will be half of its diameter, which is 2.5 m.
We can rearrange the equation for centripetal force to solve for the velocity:
v = √(Fc * r / m)
Now, let's substitute the values into the equation:
v = √((fs * r) / m)
Substituting the values, we get:
v = √((0.60 * 30 kg * 9.8 m/s^2 * 2.5 m) / 30 kg)
Simplifying further:
v = √(14.7 m^2/s^2)
v ≈ 3.83 m/s
Finally, we need to convert the velocity to angular speed in terms of revolutions per minute (rpm). The formula for converting linear speed to angular speed is:
v = ω * r
Where v is the linear speed, ω is the angular speed, and r is the radius.
Rearranging the equation, we get:
ω = v / r
Substituting the values, we get:
ω = 3.83 m/s / 2.5 m
ω ≈ 1.53 rad/s
To convert radians per second (rad/s) to revolutions per minute (rpm), we can use the conversion factor of 1 revolution = 2π radians and 1 minute = 60 seconds:
ω = (1.53 rad/s * 60 s) / (2π rad)
ω ≈ 14.62 rpm
Therefore, the minimum angular speed required for the ride to be safe is approximately 14.62 rpm.