Illustrated Physics
physics, drawn
Concept

Cycloid — where it appears

The curve traced by a point on the rim of a circle rolling along a straight line, a series of arches with cusps where the point touches the line. It is the fastest descent curve under gravity and the path along which a pendulum's period does not depend on its amplitude.

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

Four beads, four heights, one arrival time. One arch of a cycloid of radius 1, drawn with four beads on it at heights 0.061, 0.235, 0.592, 2.000 — a range of 33.0 to one. Each bead's time to slide, from rest and without friction, to the bottom of the arch is computed as a quadrature of ds/v along the curve as drawn, and the four answers are 1.003205 s, 1.003205 s, 1.003205 s, 1.003205 s: identical to 6.9e-13 of themselves. The closed form for this curve is π√(a/g) = 1.003205 s, which the quadrature reproduces without being told it. A bead let go at the cusp travels 5.7 times as far as the lowest one and arrives with it, because the extra distance is exactly paid for by the extra speed the extra height buys.

The curve that does not ask where it started

A pendulum's period depends on how far it swings, and the dependence is small but never zero. There is exactly one curve for which it is zero, and the reason has nothing to do with pendulums: on that curve the height above the bottom is proportional to the square of the distance travelled along it, which makes the motion harmonic by construction rather than by approximation.

mechanics · Pendulum 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

Named alongside it

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

Amplitude dependenceArc lengthBrachistochroneConstraintEscapementEvoluteIsochronismLength contractionThe Lorentz factorPendulumRelativity of simultaneityRolling without slipping

All concepts