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 25.7o. According to knowledgeable sources, the coefficient of static friction between typical human clothing and steel ranges between 0.250 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?

Answer 1

For maximum rotational speed The friction force will act forward So mrw^2 cos theta lessthanorequalto mg sin theta + f mrw^2 cos theta lessthanorequalto mg sin theta + mu (mg cos theta + mrw^2 sin theta) (54.6) (3) (0.90) w^2 lessthanorequalto (54.6) (9.8) (0.433) + 0.25 ((54.6)(9.8) (.901) + (54.6) (3) w^2 (.433) 147.5 w^2 lessthanorequalto 231.68 + 0.25 (482.1 + 70.92 w^2) 129.85 w^2 lessthanorequalto 352.205 w^2 lessthanorequalto 2.71 w lessthanorequalto 1.64 So max rev/s = 1.64 times 60/2 pi almostequalto 16 rev/s \r\n For minimum rotational speed to,prevent the amn from sliding in the force of friction will act outward mrw^2 sin theta + f greaterthanorequalto mg sin theta f_max = mu N = mu (mg cos theta + mrw^2 sin theta) mrw^2 sin theta + mu(mg cos theta + mrw^2 sin theta) greaterthanorequalto mg sin theta (54.6) (3) w^2 (0.433) + 0.250 ((54.6) (9.8) (0.901) + (54.6) (3) (0.433) w^2) greaterthanorequalto (54.6) (9.8) (0.433) 70.9 w^2 + 0.250 (482.1 + 70.92 w^2) greaterthanorequalto 231.68 88.625 greaterthanorequalto 111.155