I feel like i did this question before but, my brain is completely blink right n

Question

I feel like i did this question before but, my brain is completely blink right now and i need to finish this question before midnight.

The Gravitron is an amusement park ride in which riders stand against the inner wall of a large spinning steel cylinder. At some point, the floor of the Graviton drops out, instilling the fear in riders that they will fall a great height. However, the spinning motion of the Gravitron allows them to remain safely inside the ride. Most Gravitrons feature vertical walls, but the example shown in the figure has tapered walls of 22.8°. According to knowledgeable sources, the coefficient of static friction between typical human clothing and steel ranges between 0.240 to 0.390. In the figure, the center of mass of a 54.6 kg rider resides 3.00 m from the axis of rotation As a safety expert inspecting the safety of rides at a county fair, you want to reduce the chances of injury. What minimum rotational speed (expressed in rev/s) is needed to keep the occupants from sliding down the wall during the ride? Number rev/s What is the maximum rotational speed at which the riders will not slide up the walls of the ride? Number rev/s

Explanation / Answer

There is a component of the outward force pushing the customers up the incline = mV²/R*sin22.8
There is a component of the outward force pushing the customers against the incline = mV²/R*cos22.8
There is a component of the customers weight pushing the customers against the incline = mg*sin22.8
There is a component of the customers weight pushing the customers down the incline = mg*cos22.8
The first force plus the second times plus the third times must exceed the 4th.
9.8*sin67.2 = 0.24*(9.8cos67.2 + ²*3.00*sin67.2) + ²*3.00*cos67.2
This solves to = 2.1089 rad/s 0.34 rev/s minimum

For the maximum, friction points downslope!
9.8*sin67.2 + 0.24*(9.8cos67.2 + ²*3.00*sin67.2) = ²*3.00*cos67.2
which solves to = 4.465 rad/s 0.71 rev/s maximum

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