Illustrated Physics
physics, drawn
Concept

Tautochrone — where it appears

A curve on which a body sliding under gravity reaches the bottom in the same time from whatever height it is released. Huygens showed the cycloid has the property, because its restoring force grows in proportion to the distance along it, making the motion exactly simple harmonic.

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 Three vessels emptying through the same hole. The water level against time in three vessels that each hold 25 litres to a depth of 35 cm and drain through the same 0.8 mm hole in the floor, the outflow following Torricelli's law. The cylinder empties in 16.0 hours, but its level falls fast at first and slowly at the end — a parabola in time — and it is half empty in depth after only 32% of the run. The bowl whose radius grows as the fourth root of the height lowers its level at one rate from top to bottom, a straight line, and empties in 12.9 hours. A straight-sided cone sloping outwards, the shape a potter can actually throw, comes between: half its depth goes in 49% of its run.

The vessel whose level falls evenly

Water leaves a hole in the floor of a vessel faster when the vessel is full, because the pressure at the hole is the depth of water above it. In a cylinder that makes the level fall quickly at first and slowly at the end, so hour marks on its wall crowd towards the bottom. A level that falls at one rate needs a vessel that widens as the outflow quickens: its radius must grow as the fourth root of the height. The Egyptians who made the oldest surviving water clock, about 1400 BC, gave it sloping sides, which halves the error of a cylinder. They also gave it a scale for every month, because the hours they wanted to count were not all the same length.

fluids · Hydrostatics

Named alongside it

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

IsochronismAmplitude dependenceArc lengthBrachistochroneConstraintCycloidDischarge coefficientEscapementEvoluteHydrostatic pressurePendulumPoiseuille

All concepts