ROLLING POINT
A point on a rolling wheel traces a cycloid — contact stops dead, the top races at 2v, the flange runs backward.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
Chalk a dot on a rolling wheel and watch it. The dot does not just go around — the wheel carries it forward too, and the two motions combine into a curve with a name: the cycloid. Where you paint the dot changes everything.
No slip
Because the wheel rolls without slipping, the point touching the ground is, for that instant, not moving at all. The whole wheel is pivoting about that contact point — so a dot’s speed is just its distance from the contact, times the spin rate.
Speed
That makes the rim dot stop dead each time it touches down, and race at twice the wheel’s speed when it is overhead. A dot inside the rim never quite stops; a dot outside — a train wheel’s flange — does something stranger still.
One arch
Roll the rim point from cusp to cusp: this is the ordinary cycloid.
Audit
For the rim cycloid, contact speed is zero, top speed 2v, arch length 8r, and area 3πr² exactly.