Finding minimum angular speed? in rpms

[dubsydub14]

Finding minimum angular speed?? in rpms

In an old-fashioned amusement park ride, passengers stand inside a 5.5-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 0.61 to 1.0 and a kinetic coefficient in the range 0.40 to 0.70. A sign next to the entrance says "No children under 30 kg allowed."

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

I know i need an equation that relates mass and the coefficiant frictions to the angular speed but I am having no such luck

any feedback or help would be greatly appreciated

 

[lylos]

What forces are going to be acting on the passengers?

How does the coefficient of friction play into finding the frictional force?

These questions should get you started on getting an answer.

 

[kerimek]

Thank you for your question. In order to find the minimum angular speed for the ride to be safe, we need to consider the forces acting on the passengers and the maximum friction force that can be exerted by the clothing. The equation we can use is the centripetal force equation:

F = mv^2/r

Where F is the centripetal force, m is the mass of the passenger, v is the tangential velocity, and r is the radius of the cylinder. We can rearrange this equation to solve for the tangential velocity:

v = √(Fr/m)

We also know that the maximum friction force is equal to the coefficient of static friction multiplied by the normal force, which in this case is the weight of the passenger (mg). The equation for maximum friction force is:

Ffmax = μs mg

Where μs is the static coefficient of friction. We can substitute this into our equation for tangential velocity:

v = √(μs mg r/m)

Now, we need to determine the minimum angular speed, which is equivalent to the tangential velocity divided by the radius of the cylinder. This gives us the equation:

ωmin = v/r = √(μs g)

Where g is the acceleration due to gravity (9.8 m/s^2). We can then convert this angular speed to rpm by dividing by 2π radians and multiplying by 60 seconds:

ωmin (in rpm) = (√(μs g) / 2π) * 60

Using the given range of coefficients of friction (0.61 to 1.0), we can calculate the minimum angular speed to be in the range of 7.8 to 9.6 rpm. Therefore, for the ride to be safe, the minimum angular speed should be at least 7.8 rpm. However, it is always recommended to have a higher angular speed for added safety. I hope this helps!