The Graviton is an amusement park ride in which riders stand against the inner w

Question

The Graviton 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 Graviton allows them to remain safely inside the ride. Most Gravitons feature vertical walls, but the example shown in the figure has tapered walls of 22.4degree. According to knowledgeable sources, the coefficient of static friction between typical human clothing and steel ranges between 0.210 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? What is the maximum rotational speed at which the riders will not slide up the walls of the ride?

Explanation / Answer

Here, to not slide down -

The condition must be
mw^2rcos+mgsin>mgcos
then
w>[(g(cos-sin)/(rcos)]^1/2
w>[9.8(cos22.4-0.30*sin22.4) / (0.30*3*cos22.4)]^1/2 [avg value of = (0.210+0.390)/2 = 0.30, is considered]
w>3.05 rad /s
or
w>0.48 rev/s

So, the minimum rotational speed needed to keep the occupants sliding down the wall during ride = 0.48 rev/s.

For not sliding up must be
mw^2rsin<mgsin+mgcos
then
w<[g(sin+cos)/(rsin)]^1/2
w<[9.8(0.30*sin22.4+cos22.4) / (0.30*3*sin22.4)]^1/2
w<5.40 rad/s
or
w<0.86 rev/s

Therefore, maximum rotaational speed at which the riders will not slide up the walls of the ride = 0.86 rev/s

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