Aristotle's Wheel

Definition Fix two concentric circles with radii $R > r$ together as one rigid wheel on a common axle, and let the outer circle roll without slipping along a straight road through one full turn. The axle then moves forward $2\pi R$, the outer circumference. The inner circle is carried along with the axle, so it also moves forward $2\pi R$, and all the while its lowest point touches a line parallel to the road at height $R - r$. Every point of the inner circle touches this line at some moment during the turn, one after another, as if the inner circle had been unrolled along a segment of length $2\pi R$. That seems to show that the circumferences $2\pi R$ and $2\pi r$ are equal. The resolution is that only the outer circle rolls without slipping, because only it is held against the road. If the wheel turns at angular speed $\omega$, the bottom of the inner circle moves forward along its line at speed $(R - r)\omega$, so the inner circle turns and slides at once. Of the distance $2\pi R$, the turning of the inner circle accounts for $2\pi r$ and sliding for the remaining $2\pi(R - r)$. Pairing the points of two curves one to one does not show that the curves have equal lengths. If the wheel instead rests on a raised rail under the inner circle and that circle rolls without slipping, the axle moves only $2\pi r$ per turn and the outer circle slides. While the wheel moves, its two circles can never both roll without slipping on stationary tracks.

Definition Circles in one plane are concentric if they have the same center. Measure angles in radians, so that an angle of $\theta$ radians at the center of a circle of radius $\rho$ cuts off an arc of length $\rho\theta$ and a full turn is $2\pi$ radians. When concentric circles are parts of one rigid body that turns about their common center, every circle turns through the same angle, but an angle $\theta$ moves a point of the circle of radius $\rho$ along an arc of length $\rho\theta$.

Definition The angular speed $\omega$ of a turning body is the angle it turns through per unit of time, here in radians per second. All parts of a rigid body turn at the same angular speed. A point at distance $\rho$ from the axle moves around the axle at speed $\rho\omega$, as seen by an observer who travels along with the axle without turning.

Definition Let a circle of radius $\rho$ belong to a rigid wheel and touch a stationary straight line at its lowest point. Suppose the axle moves along the line at speed $v$ while the wheel turns at angular speed $\omega$ in the direction that carries its bottom backward relative to the axle, as it does when a wheel rolls forward. The point of the circle at the bottom then moves along the line with velocity $v - \rho\omega$, counted positive in the direction of travel. The circle rolls without slipping when $v = \rho\omega$, so that its bottom point is momentarily at rest. It slides forward when $v > \rho\omega$ and slides backward when $v < \rho\omega$. If $v$ and $\omega$ stay constant through one full turn, the axle moves $d = 2\pi v / \omega$, and the bottom points of the circle slide a total distance $d - 2\pi\rho$ along the line, where a negative value means sliding backward.