Illustrated Physics
physics, drawn
Concept

Momentum flux — where it appears

The rate at which momentum is transported across a surface, which is a pressure. A wave carrying an intensity I carries a momentum flux I/c, so absorbing the energy delivers a force — 675 micronewtons per acoustic watt in water, which is how ultrasonic power is measured.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

What a scale reads while a chain falls onto it. The reading of a scale, in units of the whole chain's weight, against the length of chain that has already landed, for two ways of putting the same chain down. Lowered gently, the scale reads the weight of what is resting on it and nothing else, so the reading climbs along the diagonal to one and stops. Dropped from rest with its lower end just touching, the scale reads three times that at every instant of the fall: one part is the pile's weight and two parts is the force needed to stop the links that are arriving, which is λv² with v² = 2gx and is therefore exactly twice λgx however far the fall has got. The peak, read off the drawn curve, is 3.00 chain weights. It is reached at the instant the last link lands, and the reading then falls discontinuously to one, because the momentum flux stops all at once. The discontinuity is the part a real experiment does not show — a real chain has links of a finite size and a scale has a response time — and it is the reason a chain dropped into a bucket on a kitchen scale reads high and then settles.

The pile that lands heavier than it weighs

Drop a chain onto a scale and the reading is three times the weight of the part that has landed — not approximately, exactly, all the way through the fall. The extra two parts are the force needed to stop links that are still arriving, and the same arithmetic run backwards says that picking a chain up wastes exactly half the energy it takes to get it moving.

mechanics · Momentum The pattern that stands still while the air goes through it. Streamlines of a steady 20 metre-per-second wind over a bell-shaped ridge 800 metres high and 6.0 kilometres wide, in air whose buoyancy frequency is 1.05e-2 per second. Nothing in the picture is moving: the air crosses it from left to right at 20 metres a second and the waves stay where they are, because standing still is what selects them. The vertical wavelength is 2πU/N = 11.9 kilometres, and the crests lean upstream as they rise — checked here by finding the maximum displacement a quarter of a wavelength up, which sits well upstream of the ridge. That tilt is the signature of energy travelling upward, and it is the feature every hand-drawn version of this picture gets backwards. Cloud forms at the crest of each wave where the air is highest and coldest, which is why the lens-shaped clouds sit stationary in a moving airstream.

The wave that is required to stand still

A stratified fluid supports internal waves at every wavelength there is. Put a steady wind over a ridge and the requirement that the pattern stay put picks exactly one of them — 2πU/N, and nothing about the mountain appears in it. The clouds that mark the crests sit still while the air goes through them at twenty metres a second, and the momentum the wave carries away is delivered thirty kilometres up.

fluids · Stratification A beam of sound has a weight. The force a fully absorbed acoustic beam exerts, against its power, for 3 media. It is the power divided by the speed of sound and nothing else — a watt in water gives 675 micronewtons, which is the weight of 68 milligrams, and a watt in air gives 2.9 millinewtons because the sound is slower there. This is not an analogy with light: it is the same statement, that a wave carrying energy carries momentum, with a much smaller speed in the denominator. The consequence is that acoustic power is measured by weighing. A radiation-force balance — an absorbing target on a laboratory balance, with the transducer beneath it — is the primary standard for ultrasonic output, and every therapeutic and diagnostic transducer is calibrated against one.

Where the loudness goes

An absorption coefficient removes energy from a wave, and energy removed has to appear somewhere. It appears twice, from the same coefficient: as heat, and as momentum. So a beam of sound has a weight — a watt absorbed in water weighs sixty-eight milligrams — and acoustic power is measured by putting an absorber on a balance. The ratio of the force to the heating contains no intensity at all.

waves · Attenuation A fountain in proportion to the drop. The height of the arch above the jar against the drop below it, for three values of α, the share of each link's momentum that the pile supplies. A steady chain has one tension at the top of the arch, equal to its momentum flux λv². Going down either leg the tension falls by the weight of chain below. At the jar the tension must be (1 − α)λv²; at the floor, with the floor taking all of the arriving momentum, it is zero. That gives v²α = gh₁ and h₁ = h₂ α/(1 − α), a straight line through the origin, and no arch at all if α = 0. A measured α near 0.12 gives 27 cm of fountain at a 2 m drop, and this proportion, about a seventh of the drop, is what the experiments found.

The chain that climbs out of the jar

Pull the end of a long bead chain over the rim of a jar and let it fall to the floor, and the rest follows, as a siphon does. That much is unsurprising. What a video showed in 2013 was stranger: the chain does not run over the rim. It leaps up from the jar in a standing arch, tens of centimetres above it, with nothing supporting it. The explanation took a year and turned on something a string does not have: a link is stiff, so the pile has to push back.

mechanics · Momentum

Named alongside it

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

DissipationImpulseAbsorptionAcoustic streamingAnisotropyAttenuationBoundary conditionBuoyancy frequencyCentre of massChain fountainDispersion relationEvanescence

All concepts