What is the minimum angular speed, in rpm, for which the ride is safe?

Question

In an old-fashioned amusement park ride, passengers stand inside a 5.5-meter diameter hollow steel cylinder with their backs against the wall. The cylinder begins to rotate about a vertical axis, and 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 off. The clothing has a static coefficient of friction against steel ranging from 0.65 to 1.0 and a kinetic coefficient ranging from 0.40 to 0.70. A sign next to the entrance indicates that "No children under 30 kg allowed." What is the minimum angular speed, in revolutions per minute (rpm), for which the ride is safe?

Anonymous · Accepted Answer

Consider forces acting on a passenger in this ride

Weight acts downward

A normal force acts inward

Friction acts against the weight, and is therefore upward

In vertical direction, we expect the passenger to remain in place against the wall.

Therefore

F - W = 0

In the horizontal direction, the passenger travels in uniform circular motion once the ride is up to speed. Therefore, the centripetal acceleration is inward, which is what the horizontal forces must add to:

N = m*a_centripetal

To quantify centripetal acceleration in terms of angular velocity, use

a_centripetal = r*omega^2

By definition of radius,

r = d/2

Friction is based on the normal force:

F = mu*N

Summarize our system of equations:

N = mdomega^2/2

F = mu*N

F = m*g

Solve for omega:

N = F/mu

N = m*g/mu

mg/mu = md*omega^2/2

2g/(mud) = omega^2

omega = sqrt(2g/mud)

note: omega will be in radians/sec

Plug in data:

g:=9.8 m/sec^2; mu:=0.65; d:=5.5 m;

We accept the lowest of the static coefficient of friction, because if a passenger is sticking to the wall, this is the friction they experience.

Result:

omega = 12.878 rad/sec

In RPM:

omega = 123 RPM

Note: to do a worst case scenario, in case the passenger slides by accident, account for the lowest kinetic coefficient of friction, which is mu =0.4

The result to this version is omega = 16.42 rad/sec, or omega = 157 RPM.

Anonymous

What is the minimum angular speed, in rpm, for which the ride is safe?

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