Illustrated Physics
physics, drawn
Concept

Critical damping — where it appears

The amount of damping at which a displaced oscillator returns to rest as quickly as possible without overshooting. Less damping gives decaying oscillations and more gives a slow creep back, which is why door closers, instrument needles and car suspensions are designed near it.

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

One equation, and the number that decides what it does. The same oscillator released from rest at one unit of displacement, with 4 different amounts of velocity-proportional loss. Every curve is a numerical integration of a single equation with no case analysis in it; what changes between them is one dimensionless number, the damping ratio. Below one, the two exponents are a complex conjugate pair and the motion crosses zero over and over inside a decaying envelope. At one they collide into a real double root and the trace reaches the axis and stops. Above one they are two distinct real numbers, the slower of them dominates, and the return is sluggish — an overdamped system takes longer to come back than a critically damped one, which is the thing the word 'over' is doing in its name. The ratios drawn are 0.15, 0.5, 1, 2, and the first zero crossing happens at 1.75 s for ζ = 0.15, 2.42 s for ζ = 0.5, never for ζ = 1, never for ζ = 2.

The three ways of coming to rest

An oscillator with friction in it can swing and fade, arrive and stop, or crawl back so slowly it looks stuck. One equation and one number decide which. The number is not what most people would guess, and neither is the value that returns fastest.

mechanics · Pendulum The column that overshoots its own height. The height of water in a vertical glass tube, as a fraction of Jurin's equilibrium height 2γ/ρgr, against time for tubes of radius 0.2, 0.4, 1, 2 mm, integrated from the capillary pull, the weight and the Poiseuille drag with the inertia of the rising column. The 0.2 mm tube (Jurin height 74.4 mm) creeps up without overshooting; the 0.4 mm tube (Jurin height 37.2 mm) creeps up without overshooting; the 1 mm tube (Jurin height 14.9 mm) overshoots to 1.33 of it; the 2 mm tube (Jurin height 7.4 mm) overshoots to 1.46 of it. The linearised motion about the equilibrium is underdamped when the radius exceeds (32η²γ/ρ³g²)^(1/5) = 0.48 mm for water: wider tubes ring, narrower ones do not.

The pore that lifts highest fills slowest

Jurin's law says how high water climbs in a tube and nothing about how long it takes. The rise has three laws in succession — the column's inertia, then its viscous drag, then its weight — and the same narrowness that lifts water highest throttles it hardest, so for any deadline there is a best pore. A wide enough tube does something Jurin's law cannot: it overshoots and rings. And whatever the route, exactly half the energy the surface releases is lost.

fluids · Capillarity

Named alongside it

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

DissipationCapillary riseCharacteristic equationDampingEnergyExponential decayInertiaJurin lawLinearityLogarithmic decrementPhase portraitPoiseuille flow

All concepts