A circular amusement-park ride consists of a vertical cylindrical chamber of radius…
PHYSICS · Multi-part · AP · Physics C: Mechanics · Force and Translational Dynamics · Force and Translational Dynamics
Problem
A circular amusement-park ride consists of a vertical cylindrical chamber of radius R that spins about its central vertical axis. Riders stand with their backs against the chamber wall. After the chamber reaches its operating angular speed, the floor drops away and the riders remain "stuck" against the wall. The coefficient of static friction between the riders’ clothing and the wall is μs. Throughout the motion, treat each rider as a particle of mass m and ignore any motion relative to the wall.
(a) Draw a clearly labeled free-body diagram for a rider after the floor has dropped away. Identify all forces acting on the rider.
(b) Derive an inequality relating the minimum angular speed ωmin of the chamber to the given quantities m, g, R, and μs that must be satisfied to prevent the rider from sliding downward.
(c) The ride operator proposes tilting the chamber inward so that the wall is banked at an angle α above the horizontal toward the axis, while keeping the same angular speed as part (b). Assuming ω=ωmin from part (b), determine an expression for the maximum bank angle αmax for which static friction can still provide enough vertical support. Express your answer in terms of μs only.
(d) Discuss qualitatively how increasing α beyond αmax would affect the magnitude and direction of the frictional force and explain why the design could become unsafe for the riders.