What is the minimum angular speed, in rpm, for which the ride is safe?

Consider forces acting on a passenger in this ride Weight acts downwardA normal force acts inwardFriction acts against the weight, and is therefore upwardIn vertical direction, we expect the passenger to remain in place against the wall.Therefore F - W = 0In the horizontal direction, the passenger travels in uniform circular motion once the ride is up to speed. Therefore, the centripetal acceleration is inward, which is what the horizontal forces must add to:N = m*a_centripetalTo quantify centripetal acceleration in terms of angular velocity, usea_centripetal = r*omega^2By definition of radius,r = d/2Friction is based on the normal force:F = mu*NSummarize our system of equations:N = mdomega^2/2F = mu*NF = m*gSolve for omega:N = F/muN = m*g/mumg/mu = md*omega^2/22g/(mud) = omega^2omega = sqrt(2g/mud)note: omega will be in radians/secPlug in data:g:=9.8 m/sec^2; mu:=0.65; d:=5.5 m;We accept the lowest of the static coefficient of friction, because if a passenger is sticking to the wall, this is the friction they experience.Result:omega = 12.878 rad/sec In RPM:omega = 123 RPMNote: to do a worst case scenario, in case the passenger slides by accident, account for the lowest kinetic coefficient of friction, which is mu =0.4The result to this version is omega = 16.42 rad/sec, or omega = 157 RPM.... Show More