Illustrated Physics
physics, drawn
Concept

Rolling without slipping — where it appears

Motion of a body over a surface in which the point of contact is momentarily at rest, so friction does no work there. It ties the body's spin to its travel — a wheel's speed to its radius times its rotation — and governs everything from tyres to a coin settling on a table.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

How fast each point of a fast wheel moves. The speed through the ground of each point on the rim of a rolling wheel, in units of the speed of light, against the point's angle round the rim measured from the bottom, for axles moving at 0.5c, 0.8c, 0.95c. The dashed curves are the everyday answer, 2v sin(α/2), which puts the top at twice the axle's speed and would carry it past light. Composed relativistically, the bottom is still at rest and the top moves at 2v/(1 + v²/c²): 0.800c, 0.976c, 0.999c, always below c. Every point of the rim stays slower than light, and the fast part of the wheel is flattened into a broad plateau over the top.

The wheel whose top cannot go twice as fast

The top of a rolling wheel moves at twice the speed of its axle, and the bottom is momentarily at rest — two facts every cyclist's spokes show, blurred at the top and sharp at the bottom. Roll the wheel at a large fraction of the speed of light and the second fact survives exactly while the first cannot: the top moves at 2v/(1 + v²/c²), never as fast as light. Caught at one instant of the ground's time the spokes are bent, and over one turn the wheel advances not the circumference measured from its axle but γ times it — the rim's own length, laid down on the ground by a contact point that does not move.

relativity · Velocity addition A tilt that reaches zero at a definite time. The tilt of the disk against time from 10°, when the losses are set to end the motion after 100 s, for two ways of losing energy: viscous air squeezed out of the thin wedge under the rim (red), which takes energy faster the smaller the tilt, so that α³ falls in a straight line; and a steady rolling resistance at the contact (blue), which takes energy in proportion to the contact rate, so that α to the power 3/2 does. Both fall slowly at first and steeply at the end, and both reach zero tilt at a definite time rather than approaching it for ever: at 99 s, one second before the end, the tilt is 2.15° under the air model and 0.46° under rolling resistance. The disk's energy, (3/2)Mgaα, runs out in a finite time, and with it the tilt.

The disk that spins faster as it stops

A coin spun on a table settles with a whirr that rises in pitch and then stops dead. Everything about it is losing energy, yet the rattle gets faster to the very end. The point where the rim touches the table races round faster and faster as the coin tips flatter, while the coin's face, and the head on it, turns more and more slowly. Two plausible mechanisms of loss both predict that the tilt reaches zero, and the rattle infinity, at a definite instant — a singularity a coin on a kitchen table runs into every time.

mechanics · Rotation

Named alongside it

The objects these essays reach for when they reach for this one.

CycloidDissipationFinite time singularityLength contractionThe Lorentz factorPower lawPrecessionRelativity of simultaneityRolling resistanceRotating discVelocity additionViscosity

All concepts