Ferris Wheel - 2D Side View: Swinging Gondolas, Apparent Weight and Resonance

Presets

Force arrows on the tracked cabin

Rigid seats (cabins locked to the rim)

Live readouts

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Graphs (tracked cabin, red)

Apparent weight T/(mg) vs time (g units)

Cabin angle from vertical (deg) vs time

Phase plane: angle (x, deg) vs angular velocity (y, deg/s)

Numerical checks

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How to use

Pick a preset, then change the rim speed. Cabins hang from free pivots, so they stay near vertical at gentle speeds. Press Kick cabins and watch the swing; set the speed to the resonance value shown in the readout (or use the Resonance preset) and the swing is driven up. Tick Rigid seats to see the apparent weight a rider would feel if the cabin could not swing: lighter at the top, heavier at the bottom.

The science

Model. The rim pivot of a cabin moves on a circle of radius R at wheel angle phi. The cabin is a point-mass pendulum of length L and swing angle a from the downward vertical, with viscous damping c. Newton's law perpendicular to the hanger gives:

a'' = -(g/L) sin a + (R/L) [ phi'' sin(phi-a) + phi'^2 cos(phi-a) ] - c a'

The bracket is the pivot acceleration: the centripetal part R phi'^2 and the tangential part R phi''. The hanger tension, which a rider feels as weight, is T = m (r'' + g y) . u, where u is the unit vector from the cabin to its pivot.

Rigid seat. If the seat is locked to the rim at uniform speed, the normal force on a rider at the top and bottom is:

N_top = m (g - w^2 R), N_bottom = m (g + w^2 R), w_weightless = sqrt(g/R)

Swinging cabin. For small angles the natural frequency is sqrt(g/L) and the period 2 pi sqrt(L/g) (amplitude a0 lengthens it by about a0^2/16). The rim drives the pivot at frequency w, so a wheel turning at w = sqrt(g/L) forces the cabin at resonance; damping limits the growth. Real wheels turn slowly and use dampers to stay far from this.

Simplifications. Point-mass cabins, no wind, no flexing of the wheel, no reaction of the cabins on the wheel, 2D motion only, rider fixed inside the cabin. Integration is classical RK4 with a fixed 1/240 s step.