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

Question

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 22.0 degree. According to knowledgeable sources, the coefficient of static friction between typical human clothing and steel ranges between 0.220 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

Component of weight parallel to the wall is mg cos

Component of weight perpendicular to the wall is mg sin

Component of Rotational force parallel to the wall is mw2u sin

Component of Rotational force perpendicular to the wall is mw2cos.

now,

Case I

balancing the forces parallel to the wall

(since we need to keep the occupants from sliding down, the frictional force will act upwards)

mgcos = mw2usin + (mgsin + mw2ucos)

here after substituting the values, we find the the value of w.

Take  = 0.22 (to be on the safe side)

now

w=2.18 rad/s

w= 2*p* f

f= 0.347 rev/s

Case II

balancing the forces parallel to the wall

(since we need to keep the occupants from sliding up, the frictional force will act downwards)

mgcos = mw2usin - (mgsin + mw2ucos)

here after substituting the values, we find the the value of w.

Take  = 0.22( to be on the safe side)

now

w= 4.3977rad/s

w= 2*pi* f

f= 0.699 rev/s

CASE 1 0.347 rev/s CASE 2 0.699 rev/s

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