Presets
Playback
Wheel
Wheel inertia, spin rate, bearing friction, brake.
Stool
Person plus stool inertia and bearing friction.
Arms
Tilt the axle with the slider or the buttons.
Layers
Model checks
The science
What happens
Sit on a swivel stool holding a spinning bicycle wheel with its axle vertical. Tilt the axle and the stool starts to turn; flip the wheel upside down and the stool turns fastest. Brake the wheel while it is level and the stool picks up the wheel's spin. Nothing external pushes you around the vertical axis, so the total angular momentum about that axis cannot change: whatever the wheel gives up, you and the stool receive.
Equations
Vertical angular momentum of the system: L_z = I_s w + I_w Omega cos(theta), where I_s is the moment of inertia of person plus stool, w the stool spin rate, I_w and Omega the wheel's moment of inertia and spin rate, and theta the tilt of its axle from vertical.
With negligible friction L_z stays at its start value I_w Omega, so w = I_w Omega (1 - cos(theta)) / I_s. A quarter tilt (theta = 90 deg) gives w = I_w Omega / I_s and a full flip (180 deg) gives twice that. Braking the wheel while level (theta = 0) transfers I_w Omega to the stool.
Your hands feel a torque only while the axle direction changes: |tau| = |dL_w/dt| = I_w Omega (d theta / dt). Bearing friction on the stool removes angular momentum dL_z/dt = -b w; friction inside the wheel only moves angular momentum between wheel and person.
Honest simplifications
The axle is tilted in a fixed plane at a controlled rate and its centre lies on the stool axis, so the orbital angular momentum of the wheel is folded into I_s. The person, stool and wheel mount are one rigid rotor with a constant I_s, the wheel is an ideal axisymmetric rotor, and air drag is ignored. The tilting arms are drawn with a simple two-bone model and are not a biomechanical simulation.
Checks in the page
Check 1 verifies that I_s w + I_w Omega cos(theta) stays constant through a 3 s flip. Check 2 compares the simulated stool rate after a flip with L (1 - cos theta) / I_s. Check 3 compares the torque on the hands L dtheta/dt with a finite difference of the wheel's angular-momentum vector.
What to try
- Press Flip 180: the total (white) arrow stays put while the wheel arrow reverses and the stool arrow takes the difference.
- Compare Tilt 90 with Flip 180: the stool rate doubles.
- Choose a light stool or a heavy wheel to turn faster.
- Switch on the brake while the axle is vertical to hand the wheel spin to the stool.