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The figure library — page 2

Every picture here is generated from code at build time. This page lists each family of figures in its plainest form, with every essay that draws on it.

surface-energy

15 essays

Surface at fixed volume. Four shapes of identical volume with their surface areas evaluated. The sphere's is the smallest at 4.836; the flattest shape drawn carries 1.91 times as much. Surface tension is an energy per unit area, so a free drop of liquid has an incentive to be the first of these and none at all to be any of the others.

Every branch surface-energy draws, and what checks it

vector-field

15 essays

The same field, sampled as arrows. The field at a grid of points, each arrow pointing the way a positive test charge would be pushed and scaled by the strength there.

Every branch vector-field draws, and what checks it

pressure-depth

14 essays

Three vessels, one pressure. Three vessels filled to the same depth of 3 m. The pressure on each base is 29.4 kPa — identical, because pressure is set by depth — while the weight of water each holds differs by a factor of 4.7. The base of the flaring vessel carries more force than the water standing over it weighs.

Every branch pressure-depth draws, and what checks it

redshift-tower

14 essays

A photon climbing a tower. A photon emitted at the foot of a tower 22.5 m high and received at the top. It arrives with its frequency lower by gh/c² = 2.455·10⁻¹⁵ — two and a half parts in a thousand million million. Nothing was done to the photon on the way up; the two ends of the tower disagree about how fast time passes, and the frequency is the evidence. The same fraction says a clock at the foot loses 0.21 nanoseconds a day against one at the top.

Every branch redshift-tower draws, and what checks it

stern-gerlach

14 essays

A beam through a chain of analysers. An unpolarised beam of intensity 1 entering 3 analysers oriented at 0°, 90°, 0°, selecting the + then + output each time. The intensities are 0.500 and 0.500 out of analyser 1; 0.250 and 0.250 out of analyser 2; 0.125 and 0.125 out of analyser 3. The last analyser is oriented the same way as the first, and it produces a "−" beam of 0.125 — a beam of exactly the kind the first analyser removed entirely. The middle analyser did not filter the beam; it replaced what the beam had an answer to.

Every branch stern-gerlach draws, and what checks it

velocity-addition

14 essays

Composing a boost with a speed, and never passing one. The speed one observer measures when a body moving at v is seen from a frame already moving at u, for u = 0.4, 0.6, 0.9, 1 times the speed of light. Every curve ends at one and none crosses it. The straight dashed line is the Galilean answer, u + v, which reaches 1.4c and is wrong. The flat line at the top is light: composing c with anything gives c back.

Every branch velocity-addition draws, and what checks it

wave-packet

14 essays

A packet on deep water, ω = √(gk), 1.6 s apart. A wave packet built from a Gaussian spread of wavenumbers about 1.57 per metre, drawn at two times 1.6 seconds apart, with its computed envelope ghosted around it. Between the two frames the envelope's peak moves 2.01 metres and the marked crest moves 3.95 metres, so the packet travels at 1.26 metres per second and the crests at 2.47 — a ratio of 0.51.

Every branch wave-packet draws, and what checks it

wire-frames

14 essays

The same wire, seen twice at 0.6c. Above: the wire in the laboratory. The lattice is at rest and the electrons drift, so the electrons are the contracted ones — and the wire is neutral, which means their contracted spacing is what the manufacture of a neutral wire produced. Below: the same wire seen by something moving with the electrons at 0.6c. Now the electrons are at rest and the spacing between them stretches by γ = 1.250, while the lattice moves and its spacing contracts by the same factor. The two densities no longer cancel and the wire is charged. Nothing was done to the wire; the only thing that changed is who is looking, and the magnetic force in the first frame is the electric force in the second.

Every branch wire-frames draws, and what checks it

buoyancy

13 essays

Where the upward force comes from. A block submerged with its top 1.2 m down. The pressure on the bottom face (19.6 kPa) exceeds that on the top (11.8 kPa) by exactly the weight of a column of water as tall as the block, and the sideways pressures cancel in pairs. Nothing has been added to the physics of pressure to get buoyancy out of it.

Every branch buoyancy draws, and what checks it

critical-angle

13 essays

The critical angle for n = 1.5 into n = 1. Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 41.8°, beyond which no refracted ray exists at all.

Every branch critical-angle draws, and what checks it

damped-response

13 essays

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.

Every branch damped-response draws, and what checks it

engine-efficiency

13 essays

The ceiling on a heat engine. Maximum possible efficiency against the ratio of cold to hot reservoir temperature. Reaching 100% would need a cold reservoir at absolute zero. The line is marked at ratios of 0.9, 0.7, 0.5, 0.25, where the ceiling stands at 10%, 30%, 50%, 75%.

Every branch engine-efficiency draws, and what checks it

friction-response

13 essays

What friction returns, against what it is asked for. The friction force on a block under a 50 N normal load, against the force applied to it. Below 30.0 N — the static limit μs·N — friction returns exactly what is asked for and nothing moves, so the curve is the 45° line and the coefficient never appears. At that point the surface gives way and the force drops to μk·N = 22.5 N, where it stays however hard the block is pushed. The gap above the flat line is the surplus that accelerates it: 22.5 N at the right-hand edge of the axis.

Every branch friction-response draws, and what checks it

granular-column

13 essays

The stress that stops growing with depth. Vertical stress against depth in a silo of radius 0.5 m holding grain of bulk density 1500 kg/m³, with a wall friction coefficient of 0.5 and Janssen's ratio K = 0.5. The straight line is what a liquid of the same density would do — ρgz, with no length in it anywhere. The curve is what grains do: wall friction, mobilised by the sideways stress the grains themselves exert, removes weight from the column at a rate proportional to the stress, so the stress saturates at ρgλ over a screening length λ = R/2μK = 1.00 m. Read off the drawn curve at the 8 m base, the stress is 14.7 kPa against the 117.7 kPa the liquid delivers — 88 per cent of the weight is standing on the walls. Another twenty metres of grain would move the floor's reading by less than a pascal.

Every branch granular-column draws, and what checks it

interval-hyperbola

13 essays

What puts a scale on a tilted axis. A spacetime diagram with a second observer's axes at β = 0.6. The curves are the sets of events at a fixed interval from the origin — c²t² − x² = s², one branch each for s = 0.5, s = 1, s = 1.5 — and the whole point of them is where they cross. A unit of the moving observer's time is wherever the s = 1 curve meets the tilted time axis, and on the page that point is 1.250 times as far from the origin as the stationary observer's own unit. Without the hyperbolae the tilted axes carry no scale at all, and every argument about which of two clocks is behind is unreadable off the diagram. Each drawn crossing reads back as its own interval to 0.0e+0.

Every branch interval-hyperbola draws, and what checks it

plasma-response

13 essays

Nothing propagates below 8.98 MHz. Frequency against wavenumber for a wave in a plasma of 1.0e+12 electrons per cubic metre, in units of the plasma frequency. The curve is ω² = ωₚ² + c²k², so it starts at ωₚ with zero slope and becomes the light line far above it; the whole band below ωₚ has no real k at all, which is the shaded region. At k = 3.19e-1 m⁻¹ the phase velocity read off the curve is 1.1607c and the group velocity 0.8615c, whose product is c² to a part in 10¹⁰ — the crests outrun light and the signal does not, which is the same arrangement as any other medium with a cutoff.

Every branch plasma-response draws, and what checks it