True as far as it goes, and silent on what decides — part 1 of 5

“T = 2π√(L/g) is the period of a pendulum.”

The exact period divided by the small-angle period, against amplitude. The textbook formula is the horizontal line at one; the truth is a curve that leaves it immediately and has no elementary form. The site's gate checks the plotted curve against Simpson's rule on the elliptic integral, by a route the generator does not take.

Tested in The period that depends on the swing, computed exactly · the pendulum reading path

“A rainbow is sunlight split by raindrops acting as prisms.”

The splitting is right and it predicts nothing about where the bow is. The angle comes from a minimum of the deviation function, traced ray by ray through the drop: nothing about water, light or weather picks forty-two degrees, and the same computation puts the secondary bow on the other side of its own minimum.

Tested in The angle the rainbow has to be, and why nobody chose it · the dispersion reading path

“A wave travels at its wave speed.”

Two speeds, and in deep water they differ by a factor of two: the crests run forward through the group, rise in the middle and vanish off the front. The figure draws the same packet in a medium with no dispersion as a control, where the two speeds coincide and the crests hold station.

Tested in The packet that moves at another speed than its own crests · the wave packets reading path

“The lines in an atom's spectrum are the atom's energy levels.”

They are the differences between them, which is why the lines come in families crowding onto a limit rather than spreading out. The figure draws the ladder and the transitions between its rungs, and the wavelengths fall out of the subtraction.

Tested in The spectrum is a subtraction, not a list of values · the atomic spectra reading path

“The interference pattern builds up as the particles interfere with one another.”

Only one is ever in the apparatus. The arrivals are drawn as a sample from a fixed distribution, and the check is statistical rather than pictorial: at a thousand arrivals the histogram correlates with the intensity above 0.9, and at twenty a χ² test against a flat screen fails to reject at 95 per cent.

Tested in One arrival at a time, and the pattern still appears · the matter waves reading path

“Pressure pushes down.”

The pressure at a point in a still fluid is one number with no direction attached, and the figure showing a submerged block draws it acting on every face at once — inward on the sides, down on the top, up on the bottom. What has a direction is the force on a surface, which is pressure times the area and points along that surface's own normal.

Tested in The pressure that only knows depth · the hydrostatics reading path

“Buoyancy is an upward force that fluids exert.”

There is no separate buoyant force in the physics. The figure draws the ordinary pressure on each face of a block and adds them: the sides cancel in pairs, and the top and bottom differ by exactly the weight of a column of fluid as tall as the block. Nothing was added to hydrostatics to obtain it.

Tested in The weight of the water that is not there · the buoyancy reading path

“A floating object is in equilibrium when the upthrust equals its weight.”

That fixes how deep it sits and says nothing about which way up. Equilibrium of forces is not equilibrium of moments, and the second is decided by where the two lines of action lie — which the previous rung's arithmetic cannot see, because it never asked where the buoyancy acted.

Tested in Why a ship comes back upright · the buoyancy reading path

“Small insects walk on water because they are light.”

Weight scales as the cube of size and the surface force as the first power, so the ratio goes as the square — but the comparison that decides is against BUOYANCY, not against nothing. The capillary length, about 2.7 mm in water, is where the two are equal, and it is a fixed property of the liquid rather than of the insect.

Tested in The skin that is not a skin · the surface tension reading path

“The pressure inside a bubble is 2γ/R.”

That is a droplet, which has one surface. A soap bubble is a film with two, so its excess is 4γ/R — and the figure draws both laws on one pair of axes, exactly a factor of two apart at every radius, so the distinction is visible rather than remembered.

Tested in The small bubble blows up the big one · the surface tension reading path

“A jet of water breaks into drops because of air resistance, or because the tap wobbles.”

It breaks in a vacuum, and it breaks when nothing shakes it. Surface tension alone does it: a cylinder has more surface than the row of drops holding the same volume, so the surface is not resisting the break-up but causing it. The figure draws the growth rate against wavelength, and it is positive without anything external in the calculation.

Tested in The thread that cannot stay a thread · the surface tension reading path

“The drops come off at whatever spacing the disturbance happens to have.”

Every wavelength longer than the circumference grows, so a random disturbance contains many growing modes at once. What decides is which grows FASTEST, and the growth rate has a clear maximum — found here by searching the drawn curve rather than quoted — at kR = 0.697, which is 4.51 diameters. Random input, non-random output.

Tested in The thread that cannot stay a thread · the surface tension reading path

“Liquids rise in narrow tubes.”

Some fall. The direction is set by the contact angle, which is fixed by three surface energies rather than by the liquid alone: mercury in glass has an angle near 140°, its cosine is negative, and the same formula returns a DEPRESSION. The figure takes the contact angle as an input for that reason.

Tested in How high water will climb · the capillarity reading path

“Fluid slides along a solid surface.”

It does not. The fluid in contact with a wall moves with the wall — the no-slip condition — which is why the figure draws zero velocity at the fixed plate and full plate speed at the moving one. It is an experimental fact rather than a consequence of anything above it, and everything about drag and pipe flow follows from it.

Tested in Momentum going sideways · the viscosity reading path

“Blood pressure rises when arteries narrow because there is less room.”

The dependence is far steeper than 'less room' suggests. Resistance goes as the inverse fourth power, so a vessel narrowed by 20 per cent has its resistance multiplied by 2.4 — and the body's control of blood distribution works by small changes in radius for exactly that reason. The exponent is the mechanism, not a detail of it.

Tested in The fourth power in a pipe · the viscosity reading path

“Stirring something harder makes it flow more easily.”

For most complex fluids, yes; for a dense cornflour suspension the opposite, and violently so — its apparent viscosity RISES with shear rate, which is why it can be run across and not stood on. The figure draws both curves on one pair of axes, and the sign of the slope is the entire difference.

Tested in The fluid that answers back · the rheology reading path

“A wave's speed is a property of the medium.”

For sound and for light in a transparent solid, near enough. For water waves it is a property of the medium AND the wavelength, which is why a single storm produces waves that arrive over several days sorted by period. The figure plots speed against wavelength and it is a curve, not a horizontal line.

Tested in The speed that depends on the length · the wave packets reading path

“A gas is a continuous fluid.”

It is not, and the reason it can be treated as one is a RATIO: air's molecules travel about 68 nm between collisions, against apparatus measured in millimetres, so a region small enough to be a point still holds a hundred million molecules. Below a few microns that ratio fails and the gas stops obeying the equations of fluid mechanics — the same gas, and a different description.

Tested in How far a molecule gets · the kinetic theory reading path

“The particle moves because more molecules hit one side than the other.”

True, and the size of the imbalance is the part that matters: a micron-sized grain is struck about 10^20 times a second, and the fractional imbalance is roughly one part in 10^10 — far too small to see directly. What makes it visible is that the imbalance never averages away completely, so displacements ACCUMULATE as the square root of time.

Tested in The jiggle that proved atoms · the diffusion reading path

“Superfluid helium is a Bose–Einstein condensate.”

The ideal Bose gas explains why a transition must exist and gets the shape of the curve wrong: its condensate fraction goes as 1 − (T/Tc)^3/2 and helium's superfluid fraction as roughly 1 − (T/Tλ)^5.6, which the figure draws together so the disagreement is visible. Helium's atoms interact strongly and the calculation's do not, and only about ten per cent of helium atoms are in the condensate even at absolute zero.

Tested in The liquid that will not slow down · the superfluidity reading path

“The equivalence principle says acceleration and gravity are the same thing.”

It says they are indistinguishable *locally*, and locally is a measurement rather than a word. Two balls released a metre apart in a box falling ten metres approach by 1.57 microns; make the box large enough, or the measurement good enough, and the difference is always there. What is genuinely equivalent is a uniform field, and no real field is uniform.

Tested in The floor that cannot be told from gravity · the equivalence principle reading path

“A satellite's clock runs fast because it is higher up.”

Higher makes it run fast and moving makes it run slow, and the two have opposite signs and comparable size. The total, (GM/c²)(1/R − 3/2r), is zero at r = 3R/2 — an altitude of 3,186 km, solved for in the figure rather than quoted — negative below and positive above. A low-orbit satellite's clock loses.

Tested in The clock that runs slow lower down · the gravitational redshift reading path

“A mass acts as a lens.”

It acts as a lens with no focal plane. A glass lens deflects in proportion to the distance from its axis and therefore has one focus; a mass deflects as 1/b, so rays at different distances cross the axis at different places — a focal line running outward for ever. The consequence is that the image of a point source is a ring or an arc rather than a point, which no ordinary optical instrument does.

Tested in The bend Newton got half right · the light deflection reading path

“The distant observer and the falling observer disagree, so one of them must be wrong.”

Neither is. They are computing different quantities: elapsed proper time along one worldline, and a coordinate label attached to events by a family of static observers. Asking which is correct is asking which of two rulers laid along different paths gives the true length, and the answer is both.

Tested in Two clocks that disagree about the fall · the horizons reading path

“Gravitational waves are emitted by any accelerating mass, as electromagnetic waves are by any accelerating charge.”

A single accelerating mass radiates nothing. The monopole term is forbidden by conservation of mass and the dipole term by conservation of momentum, so the first surviving term is the quadrupole — which requires the mass distribution to change shape, not merely to move. There is no gravitational analogue of a dipole antenna.

Tested in The wave that stretches one way and squeezes the other · the gravitational waves reading path

“Radiation pressure is negligible.”

It is negligible compared with gravity for large objects and larger than gravity for small ones, and the crossover is computed rather than asserted: for rock of density 2000 kg/m³ near a solar-luminosity source it is at a grain radius of 287 nm. Both forces fall as the inverse square of the distance, so the comparison does not depend on how far away anything is.

Tested in Light has a pressure · the radiation pressure reading path

“An accelerating charge radiates.”

True, and silent about the direction. The power goes as sin²θ about the acceleration, so nothing at all is emitted along the acceleration itself — and once the charge is moving relativistically the whole pattern sweeps forward into a cone of half-angle about 1/γ. The figure computes the peak at 13.4° for a charge at 0.9c, where the naive estimate gives 25°.

Tested in A charge that turns must glow · the radiating charge reading path

“Neutrinos pass through everything.”

They pass through almost everything almost always, which is a statement about a product. The figure computes the free path in solid lead at 1.55 light-years, so a slab a metre thick stops about one neutrino in 10¹⁶ — and a detector compensates with the other factor, running 10¹³ neutrinos a second through a target of many tonnes for months.

Tested in How far a neutrino gets · the kinetic theory reading path

“In a perfect conductor the magnetic field is constant.”

The field is not constant; the flux through a material loop is. A collapsing region has the same flux through a smaller area, so the field rises as 1/R² — computed here as a factor of 4.9 × 10⁹ for a collapse from 700,000 km to 10 km. Constant flux and constant field are the same statement only when nothing moves.

Tested in The field that cannot get out · the flux freezing reading path

“Large bodies are round because gravity pulls everything toward the centre.”

Gravity pulls on small bodies too, and they are not round. What decides is a comparison: the stress ρgh under a feature against the strength of the material it rests on. Since a uniform body's surface gravity is itself proportional to its radius, the tallest possible mountain falls as 1/R — so the argument is about a competition between two quantities, and the size at which one wins is computed rather than asserted.

Tested in The size at which a body becomes round · the Self-gravity reading path

“The Planck length is √(ħG/c³), so it is exactly 1.616 × 10⁻³⁵ m.”

The combination is fixed only up to dimensionless factors. Bisecting the Compton wavelength against the Schwarzschild radius gives 2.29 × 10⁻³⁵ m — larger by exactly √2, the factor of two in rs coming through a square root. Any argument of this kind fixes an order of magnitude, and quoting four figures suggests a precision the derivation does not contain.

Tested in Where every model runs out at once · the planck scale reading path

“Each twin sees the other's clock running slow.”

True while both coast, and silent about the turnaround, where the traveller's lines of simultaneity swing across a large stretch of the stay-at-home's worldline. Counting light pulses avoids the question entirely: the traveller receives them at √((1+β)/(1−β)) times the emitted rate for the inbound half and the reciprocal for the outbound, and the totals do not match.

Tested in The twin who comes back younger · the time dilation reading path

“A source moving across the line of sight is not Doppler shifted, because its distance is not changing.”

It is shifted by 1/γ — 0.866 at half the speed of light, computed in the figure. Nothing classical predicts it, because it is not about distance changing at all: it is the source's own clock running slow, read as a colour. The transverse shift is the term that makes the effect a test of relativity rather than of arithmetic.

Tested in The shift that survives at right angles · the doppler reading path

“Entropy is proportional to the amount of stuff, so a bigger system has proportionally more.”

For ordinary matter entropy is extensive — double the volume at fixed density and it doubles. A horizon's goes as the area, so doubling the mass quadruples it. That difference is not a technicality: it says the maximum entropy that fits in a region is set by the region's boundary rather than its volume, which is the holographic bound.

Tested in The entropy that lives on a surface · the entropy reading path

“Friction opposes motion.”

It opposes relative sliding at the contact, which is not the same thing. The friction on a driven wheel points forward and is the only thing accelerating the car; the friction on a rolling ball does no work at all, because the contact point is instantaneously at rest. Stated as opposing motion, the rule gets the sign wrong for every vehicle ever built.

Tested in The force that takes what it needs · the friction reading path

“Anharmonicity is a small correction.”

It is small in the period and not small in what it accounts for. Thermal expansion, thermal conductivity in an insulator, the overtone spacing of a molecular vibration and the fact that a solid can melt at all are consequences of the cubic and quartic terms and are exactly zero in the harmonic approximation. A correction that is numerically tiny and qualitatively load-bearing is worse than a large one, because it is easy to drop.

Tested in Every minimum is a parabola · the harmonic approximation reading path

“The Coriolis force explains which way water goes down a plughole.”

The term is real and its size is computable, and over a basin it loses. The figure gives the deflection over a stated flight: 2.3 m at the pole for something moving 20 m/s for 40 seconds, which is 0.29% of the range. Scaled to a sink — a tenth of a metre per second, a few seconds — the Coriolis deflection is microns, and it is swamped by the residual swirl left from filling. What the term does decide is systems large enough for the deflection to accumulate for hours: cyclones, ocean gyres, and long-range gunnery.

Tested in The forces that are not there · the circular motion reading path

“The radius of a track in a bubble chamber gives the particle's mass.”

It gives the momentum, and the mass only in combination with something else. The radius is mv/qB, so a light fast particle and a heavy slow one of the same momentum curve identically. A mass spectrometer gets round this by fixing the energy first — accelerating everything through the same voltage — after which the radius goes as the square root of the mass, and the figure computes 96.0 mm against 100.7 mm for neon-20 and neon-22.

Tested in The force that does no work · the magnetism reading path

“The field inside a solenoid is μ₀nI.”

For an infinite solenoid, exactly. For a real one the result is an approximation whose error is set by the aspect ratio: the field on the axis at the mouth of a long solenoid is half the central value, and the outside field is not zero but small. Quoting the formula without the length is how a coil design ends up a factor of two out at the ends, which is where the sample usually is.

Tested in The field that wraps a current · the ampere law reading path

“The electric and magnetic fields in a light wave are a quarter cycle out of step, each one feeding the other.”

They are exactly in step in a plane wave in vacuum — both peak together and both vanish together. The intuition behind the quarter-cycle claim comes from an LC circuit, where energy really does slosh between two stores. In a travelling wave the energy is not sloshing; it is moving, half in each field, and the fields are in phase. The out-of-phase case is real but belongs to a standing wave, where it is the same fact as a node in one field sitting at an antinode of the other.

Tested in The term that made light · the maxwell equations reading path

“Snell's law follows from Fermat's principle.”

It follows for a flat boundary between two uniform media with light treated as a ray. The figure locates the crossing point by golden-section search on the drawn optical path length, with no law of optics used anywhere, and the angles that come out — 55.80° and 33.46° — satisfy n₁sin θ₁ = n₂sin θ₂ to one part in 10⁸. What the derivation cannot supply is the reflected beam, its amplitude, its phase, or the polarisation dependence, all of which need the boundary conditions on a wave.

Tested in The path that does not change · the fermat reading path

“The resolution limit is a consequence of the wave nature of light.”

It is a consequence of a wave passing through a finite aperture, and light is not the only wave available. An electron accelerated through 100 V has a de Broglie wavelength of 123 pm, five thousand times shorter than green light, and an electron microscope's resolution follows the same formula with that wavelength in it. The limit is about apertures and wavelengths, and the choice of what is doing the illuminating is a free parameter.

Tested in How far apart two things have to be · the diffraction reading path

“Huygens' principle explains diffraction.”

It predicts that a wave spreads past an aperture, which is the qualitative fact, and in its 1690 form it gets the amplitude wrong and predicts a backward wave that does not exist. The repair — Fresnel's obliquity factor and a phase advance of a quarter cycle — was found by requiring agreement with experiment and only justified in 1882, when Kirchhoff derived the whole construction from the wave equation and the correction fell out.

Tested in Every front is a source · the huygens reading path

“A pulse inverts when it reflects off a fixed end.”

True, and the fixed end is the limiting case rather than a separate rule. A fixed end is a medium of infinite impedance, so the reflection coefficient (1−Z₂/Z₁)/(1+Z₂/Z₁) goes to −1 and the inversion is total. At a finite ratio of 3 it is −0.500, measured off the marched pulse in the figure; at a ratio below one it is positive and there is no inversion at all.

Tested in What happens where the medium changes · the impedance reading path

“Intensity falls as one over distance squared, so amplitude does too.”

Amplitude falls as one over distance, because intensity goes as the square of amplitude. Confusing the two is a factor-of-two error in every decibel calculation and a factor-of-four error in every energy one. The rule worth keeping is that the conserved quantity is power, so it is intensity that the geometry acts on directly and amplitude that follows.

Tested in How a wave thins out · the wave motion reading path

“The heat capacity of a diatomic gas is 5/2 R because it has five degrees of freedom.”

It has seven, and two of them are frozen. The vibration contributes two quadratic terms — kinetic and potential — and would take the total to 7/2 R at every temperature. The figure computes what actually happens: hydrogen sits at 3/2 below 60 K, 5/2 through room temperature, and reaches 3.38 by 5,000 K, because a degree of freedom whose first energy step is large compared with kT cannot be excited at all.

Tested in Half a kT for every way of moving · the equipartition reading path

“The Boltzmann factor gives the probability of a state of energy E.”

It gives the probability of one *state*, not of one energy, and the difference is a degeneracy. An energy level with many states at it is that many times more likely, which is why an atom's excited populations follow g·exp(−E/kT) and why the entropy of the system itself — as opposed to the reservoir's — appears in every free-energy argument. Forgetting the degeneracy is how a spectrum comes out with the wrong line ratios.

Tested in The exponential that decides everything · the entropy reading path

“The twin who travels comes back younger because of time dilation in the outbound and return legs.”

That gets the answer and hides the geometry. Proper time is the length of a worldline in this metric, and the figure sums √(dt² − dx²) leg by leg along each drawn path: 2.000 for the twin who stays and 1.600 for the one who leaves, a ratio of 0.8000 against √(1−β²) = 0.8000. The bent path is longer on the page and shorter in time, because the minus sign makes the straight path the longest — which is the whole answer and needs no acceleration in it.

Tested in The quantity nobody argues about · the spacetime diagram reading path

“The wire is neutral, so it has no electric field.”

It is neutral in the laboratory frame, and neutrality is not frame-independent. In the frame moving with the drifting electrons the lattice is contracted and the electrons are not, so the two densities no longer cancel and the wire carries a net charge of 1.113×10⁻²⁰ C/m — small, and exactly enough to produce the force that the laboratory frame calls magnetic.

Tested in Magnetism is electricity seen sideways · the field transformation reading path

“The kinetic energy of a rolling ball is ½mv².”

That is the translation only. The energy figure prints 71 per cent translation and 29 per cent rotation for a sphere and 67/33 for a solid cylinder, so after a drop of 1.00 m the sphere reaches 3.74 m/s rather than the 4.43 m/s that ½mv² alone predicts. The two shares add with no cross term between them, which is König's theorem and is why the split can be quoted as two numbers at all.

Tested in The point that keeps moving as if nothing had happened · the momentum reading path

“A chain of masses and springs is an approximation to the wave equation.”

The relation runs the other way, and the difference is measurable. A chain of N masses has frequencies 2√(k/m)·sin(nπ/2(N+1)), proportional to n only at the bottom — within a tenth of the straight line up to mode 12 of 24 — and stopping dead at 2√(k/m). The wave equation is the limit of the chain as the spacing goes to zero, and it has no ceiling, so it permits a copper crystal vibrations of any frequency whatever when the measured spectrum ends near 7 THz.

Tested in The two pendulums that will not stop swapping · the harmonic approximation reading path

“The Poynting vector gives the energy flow at every point of space.”

Only its flux through a closed surface is determined: adding the curl of any vector field to S leaves ∇·S untouched, so the theorem fixes the total and not the route. A capacitor gap at 84.7 kV/m sitting in a 1 T field carries S = EB/µ₀ = 67.4 GW/m², fifty million times the 1.36 kW/m² of sunlight, and delivers nothing anywhere, because the divergence is zero.

Tested in Where the energy of a field actually is · the field energy reading path

“Water has a large dielectric constant because its molecules turn to line up with the field.”

They barely turn. At 300 K in a field of 10⁶ V/m the alignment energy pE is 6.2 × 10⁻²⁴ J against a thermal 4.14 × 10⁻²¹ J, so the mean alignment ⟨cos θ⟩ is 5 × 10⁻⁴ — five parts in ten thousand. Counting only that gives a relative permittivity of 12.7 where the measurement is 80, and the missing factor of 6.3 comes from neighbouring molecules being correlated rather than independent.

Tested in The attraction that needs no charge · the the field concept reading path

“The sodium doublet is resolved once the grating has more than a thousand lines.”

Two things are missing: the order, and what counts as resolved. The requirement is R = mN ≥ λ/Δλ = 982, so 600 lines in second order do it while 1200 lines in first order do no better. And the criterion is a convention — the dip measured at exactly R = 982 is 18.9 per cent, a dip first appears at 0.83 of the Rayleigh separation, and the figures report 0 per cent at R = 300 and 53.4 per cent at R = 1200.

Tested in What a thousand slits buy that two cannot · the diffraction reading path

“Coherence means having a single wavelength.”

That is half of it. Temporal coherence follows from bandwidth, spatial coherence from the source's angular size, and the two are independent. Sunlight spans 300 nm of spectrum yet is spatially coherent across 1.22λ/θ = 73 µm, because the Sun subtends 0.53° — which is why a pinhole was needed before a wavelength filter was.

Tested in Why two lamps never interfere · the coherence reading path

“A wave on a real string obeys the wave equation.”

It obeys it to order slope-squared. Replacing sin θ by tan θ costs a factor of cos θ, which is 1.5 per cent at a 10° slope and 13 per cent at 30°, and a real string's bending stiffness adds a fourth-derivative term that has no place in the equation at all: with an inharmonicity coefficient of 2×10⁻⁴ the eighth harmonic of a piano string lands about 11 cents sharp of eight times the fundamental, which is why the top octave of every piano is deliberately stretched.

Tested in The equation that lets a shape travel · the wave motion reading path

“The set of frequencies a membrane produces is a fingerprint of its shape.”

The ratios are decided by the boundary alone, so the converse looks inevitable, and it is false. Kac asked in 1966 whether the shape could be recovered; in 1992 Gordon, Webb and Wolpert exhibited two eight-sided polygons with identical spectra — every one of infinitely many frequencies matching. The spectrum does fix the area and the perimeter, and it does not fix the shape.

Tested in The drum that has no harmonics · the standing waves reading path

“The energy available in a collision is the energy of the beam.”

What is available is the invariant mass of the whole system, √s. A 7 TeV proton striking a stationary proton gives √(2mc²E + 2m²c⁴) = 114.6 GeV; two 7 TeV protons meeting head-on give 14 TeV. The same accelerator, the same beam, a factor of 122 in what can be made — which is the entire reason high-energy machines collide.

Tested in The invariant that survives a boost · the Mass-energy reading path

“A rigid rod is a sound idealisation that relativity corrects only at high speed.”

Rigidity needs an infinite sound speed, so it fails at every speed. Steel carries a push at 5,100 m/s, which is 196 µs per metre; a 1 ms hammer blow is only five crossing times of a one-metre bar, and a 3 km drill string does not feel a torque change for 0.59 s. Relativity's contribution is a ceiling 59,000 times above the speed steel actually manages.

Tested in The pole that fits and does not fit · the length contraction reading path

“A system with a lower free energy available to it will go there.”

The criterion is about equilibrium and supplies no rate. Graphite is lower than diamond by 2.90 kJ/mol at 298 K, and a diamond lasts indefinitely because the rearrangement has a barrier of several electronvolts — at 300 K a 0.35 eV barrier alone already costs a factor of 1.3 × 10⁻⁶, and several eV is a factor no timescale reaches.

Tested in What a system actually minimises · the entropy reading path

“One latent heat is enough to draw a substance's whole vapour-pressure curve.”

It is enough over a modest interval and visibly not enough beyond it. Water's curve integrated at the constant 43.32 kJ/mol that its triple point and boiling point imply reaches 37.4 MPa at the critical temperature against a measured 22.1 — 70 per cent high, because the latent heat falls from 45.05 kJ/mol at the triple point to zero at the critical point.

Tested in A boiling point is a pressure, not a temperature · the phase change reading path

“Young's equation predicts the contact angle from the three surface tensions.”

Two of the three are not independently measurable, because a solid cannot rearrange to relieve its own surface stress, so the equation is almost always run backwards from a measured angle. The tensions drawn in the figure come from a stated closure — the geometric-mean rule, which returns γsv = 68.48 mN/m for the 20° solid — and clean soda-lime glass is in the hundreds of mJ/m². Below about a micrometre the equation also needs a line-tension term: at τ ≈ 10⁻¹¹ N and a 1 µm base radius that shifts cos θ by 0.14, which is 9° at 60°.

Tested in The angle a liquid makes with what it sits on · the capillarity reading path

“The innermost stable circular orbit sits at three Schwarzschild radii.”

That is the non-rotating answer alone. Around a maximally rotating hole a co-rotating orbit's innermost radius falls to GM/c², one sixth of 6GM/c², and the efficiency rises from 5.719% to 42.3%; a counter-rotating one goes out to 9GM/c² and the efficiency falls to 3.77%. The quoted range of efficiencies exists because 3 rs is one point in it.

Tested in The orbit that cannot be made smaller · the orbit stability reading path

“Experiments have broken Bell's inequality.”

A run is a statistical statement about a bound, not a demolition of it. At 5,000 simulated coincidences the estimate is 2.8528 ± 0.0396, which is 22 standard errors above 2; at 100,000 it is 2.8202 ± 0.0090, or 91. Real apparatus at a visibility of 0.97 reaches 2.744 rather than 2.828, and the first loophole-free run reported 2.42 ± 0.20 from 245 trials.

Tested in The correlation no instructions can produce · the entanglement reading path

“A body spinning about its greatest moment of inertia is stable, and that is the end of the matter.”

It is stable against small departures, which is what the linear analysis asks. It is not stable against *dissipation*, which the analysis has no term for: an energy-losing body at fixed angular momentum slides to the state of lowest energy at that momentum, which is rotation about the greatest moment. That is why the intermediate-axis flip is a rigid-body phenomenon and why Explorer 1, which flexed, ended up tumbling about an axis nobody chose.

Tested in The axis that will not hold · the rotation reading path

“The normal force acts at the centre of the base.”

It acts there only when nothing else is applied. The reaction of the floor is a pressure distributed over the whole contact patch, and its resultant sits wherever moment balance requires — a distance Py/W from the centre, walking toward the leading edge in proportion to the push. Tipping is not a separate phenomenon that begins at some angle; it is that point arriving at the edge of the base and having nowhere further to go. Every figure here plots its position, and the figure for a floating hull plots the same quantity for a body with no edge to reach.

Tested in Slide or topple · the Free-body reading path

“The induced charge on the plane is −q, spread out somewhere near the foot of the perpendicular.”

It is −q, and where it sits is computable and wider than that suggests. The density is −qd/2π(r² + d²) to the three-halves power, so half the induced charge lies inside a radius of √3 d — 34.6 mm for a charge 20 mm up — and a tenth of it is still outside 3d. The running total in the figure is that density integrated over the surface and reaches 99.999% of −q only at the edge of the arithmetic.

Tested in The charge that has to be somewhere else · the conductors reading path

“A small current loop is a magnetic dipole.”

Only at a distance, and the distance is larger than 'small' suggests. On the axis the exact field is μ₀Ia²/2(a² + z²) to the three-halves power and the ideal dipole's is μ₀m/2πz³; their ratio is (1 + (a/z)²) to the minus three-halves, so the approach is algebraic rather than exponential. A loop is still 28% below the dipole value at twice its own radius, 5.7% below at five times, and reaches one per cent only at 12.2 radii.

Tested in The loop that behaves like a needle · the magnetism reading path

“A pile of glass plates polarises the transmitted beam.”

Slowly, and the number of plates is the price. Each surface removes about 15% of the s-polarised light at Brewster's angle and none of the p, so after m plates — two surfaces each — the transmitted degree of polarisation is (1−R)^(2m) differenced against 1. Eight plates reach 85.7%; one plate reaches 26%. Reflection polarises perfectly and throws away most of the light; transmission keeps the light and polarises it by attrition.

Tested in The angle at which reflection picks a side · the polarisation reading path

“The gradient above a hot road must be enormous, since the effect is so obvious.”

The index deficit at the surface is 3 × 10⁻⁵ — thirty parts in a million — over a layer five centimetres deep. What makes it visible is not its size but its steepness: 6 × 10⁻⁴ per metre, which is 3,800 times the Earth's curvature, so a ray bends back up over a horizontal distance of a couple of hundred metres. The critical angle for the whole effect is 0.444°, a fiftieth of the width of the Moon.

Tested in The ray that bends without a surface · the fermat reading path

“Fringes fade because the two beams get out of step.”

They stay exactly in step for a monochromatic source at any delay whatever. What fades is the *agreement between the components*: each wavelength in the source produces its own fringe pattern with its own spacing, and at a large path difference those patterns are displaced relative to each other by more than a fringe and add to a uniform glow. The sodium doublet makes this visible by having only two components — its visibility does not decay at all but oscillates, returning to one every 0.58 mm.

Tested in How far a wave can remember · the coherence reading path

“The instability sits at twice the natural frequency.”

It sits at 2ω₀/n for every integer n, and the wedges lean as they are climbed. The figure locates each one by measuring rather than by assuming: the second opens at a depth of 40.0% and the third at 62.5%, against 8.0% for the first, so the higher ones are unreachable by anything a rider can do. The third has also drifted from 0.667 to about 0.63 by the time the modulation depth reaches 0.9 — an assumption that it stays under its own small-amplitude apex found nothing at all when it was made.

Tested in The swing that is pumped, not pushed · the resonance reading path

“At high enough frequency the compressions would become isothermal and Newton's speed would be right.”

There is nowhere in a gas it is right. The ratio of heat's reach in a period to the wavelength goes as √(Df)/c, so it reaches one at f = c²/D — 5.8 GHz for air at atmospheric pressure. But the sound wavelength falls to the mean free path at 4.9 GHz, and past that there is no continuum left to carry a wave. The two frequencies land in the same decade at every pressure drawn, and they must: the diffusivity is about a third of the mean free path times the molecular speed and the sound speed is about the molecular speed, so both are the collision rate in disguise.

Tested in The correction that took a century · the wave motion reading path

“The spreading is σ₀√(1 + (t/τ)²).”

That expression keeps only the curvature of ω(k) and drops everything beyond, so it is exact for a narrow spectrum and progressively wrong for a wide one. Measured against envelopes traced from the full superposition, it agrees to 0.3% at 10% bandwidth, 3.3% at 18% and 8.5% at 28% — an ordering that is itself the size of the cubic term, and the reason the closed form is drawn as a dashed check rather than as the curve.

Tested in The packet that will not keep its shape · the wave packets reading path

“Beaming makes an approaching source look brighter.”

By the fourth power of the Doppler factor, which is a larger statement than 'brighter'. One power comes from each photon's energy, one from the rate they arrive at, and two from the solid angle they are squeezed into. Front against back, that is ((1+β)/(1−β))², which is 9 at 0.5c, 361 at 0.9c and 3.96 × 10⁴ at 0.99c — so a source pointed away is not dimmed but effectively deleted, and any catalogue of such sources is a catalogue of the ones pointed at the observer.

Tested in The sky that crowds into a cone · the doppler reading path

“A higher orbit means a bigger correction.”

It means a bigger *potential* term and a smaller velocity term, and the two vary differently with radius — the potential term saturates because there is only so much potential to climb out of, while the velocity term falls as 1/r. Setting the sum to zero gives r = 1.5 R⊕, an altitude of 3,186 km, and that radius contains neither G nor the Earth's mass nor the speed of light. Below it a satellite's clock loses time; above it, gains.

Tested in The clock that is wrong in two directions · the time dilation reading path

“The heating and cooling coefficients are two separate performance figures.”

They differ by exactly one, at every temperature, and the figure measures the gap off the drawn curves at 1.8 × 10⁻¹⁵ across the whole range. Whatever is taken from the cold side is delivered to the warm side *plus* the work that moved it, so COP_heat = COP_cool + 1 is energy conservation and not a coincidence. A single machine quoted as a refrigerator and as a heat pump differs in specification by one.

Tested in The engine that pays back more than it takes · the heat engines reading path

“A parabolic mirror focuses better than a spherical one.”

It focuses perfectly, for rays parallel to its axis, and the figures check that on the drawn curve rather than quoting it: a vertical ray reflected at a quarter, a half, three quarters and the whole of the dish radius crosses the axis at the same height to better than one part in ten thousand. What a parabola does not do is focus rays arriving off-axis, which come to a comet-shaped blur — so a liquid mirror has a perfect image on its axis and a rapidly worsening one away from it, and can only ever point straight up.

Tested in The surface a spin decides · the hydrostatics reading path

“The equivalence principle holds locally.”

'Locally' is not a qualifier but a number, and it has two arguments. The box that stays undetectably flat has a size 2δ/[(GM/r³)t²] fixed by the instrument's resolution δ and by the duration t — so the principle is local in time on the same footing as in space, and with a worse exponent. Patience is more expensive than room.

Tested in The term free fall cannot remove · the equivalence principle reading path

“The three regimes — under-damped, critically damped and over-damped — are three different behaviours.”

They are one solution. Every trace here comes from a single fourth-order integration of ẍ + 2ζω₀ẋ + ω₀²x = 0 with no branch in the code at all, and what changes across ζ = 1 is the discriminant of the characteristic equation rather than anything in the physics. Nothing in the motion is discontinuous there: the number of zero crossings in a finite time falls to zero continuously, and the trace at ζ = 0.999 is indistinguishable from the trace at ζ = 1.001.

Tested in The three ways of coming to rest · the pendulum reading path

“The energy goes into the magnetic field.”

The field is the same at the end as at the beginning, so it holds the same energy. Every joule of the released gravitational energy appears as resistive heating in the wall, and the accounting closes exactly: with the magnet at terminal speed the rate of work done by gravity equals the rate of dissipation in the tube, since the kinetic energy is not changing. For the drop drawn here that is 118 millijoules, deposited in a few centimetres of copper travelling with the magnet.

Tested in The magnet that falls slowly · the induction reading path

“The residual colour error of a doublet is a defect that better manufacturing would remove.”

It is a property of the available glasses rather than of the workshop. Cancelling the second derivative as well needs a third condition and therefore a third glass with an anomalous partial dispersion, which is what an apochromat is and why it costs what it does. In the meantime the residual can be made irrelevant without being reduced, by stopping the lens down against the floor diffraction puts under any aperture: the depth of focus grows as the square of the focal ratio while the residual grows only in proportion to focal length, and the two cross at an aperture of about 12 mm for the lens drawn here.

Tested in Two glasses that cancel a derivative · the dispersion reading path

“Light scattered at ninety degrees is completely polarised.”

True only in the Rayleigh limit, where a dipole cannot radiate along its own axis. The full computation shows the polarisation at ninety degrees falling to a half by a size parameter of about 1.5 and oscillating about zero thereafter, so a droplet's ninety-degree scattering is essentially unpolarised and can even be polarised the other way. A polarising filter turned against the blue sky darkens it markedly; turned against a cloud in the same part of the sky it does almost nothing, and that difference is a measurement of particle size — a direction of shaking read as a length.

Tested in When the particle is the size of the wave · the scattering reading path

“Adding more layers to a quarter-wave stack makes its reflection band wider.”

It makes it deeper and its edges sharper, not wider. The half-height width of a titanium-dioxide and magnesium-fluoride stack measured on its own computed reflectance runs 0.517, 0.475 and 0.394 of the design wavelength at two, four and eight pairs, narrowing toward the 0.327 the index contrast alone predicts for an infinite stack. What sets the width is the ratio of the two indices; what the number of pairs sets is how close to unity the peak gets.

Tested in The layer that makes a reflection vanish · the impedance reading path

“Anomalous dispersion is a rare curiosity of exotic media.”

It happens wherever a medium absorbs, which is everywhere. The index rises with frequency between absorption lines and falls across each one, and the falls are what Kramers and Kronig showed must accompany any absorption at all. Ordinary water is anomalously dispersive in the infrared, and glass is in the ultraviolet. What is rare is not the anomaly but a place where it can be exploited without the accompanying absorption destroying the pulse — which is why the experiment is done with gain lines rather than absorption lines.

Tested in The speed that carries no signal · the wave packets reading path

“Dividing the count of arrangements by N! is a mathematical fix for making the entropy extensive.”

It is a physical claim with a measurable consequence, and it was made forty years before there was any justification for it. Without the division, entropy per particle grows as ln N: two identical flasks joined would have more than twice the entropy of one, and opening a tap between them would produce entropy from nothing. With it, the entropy per particle is flat to three parts in a hundred over a factor of forty in sample size, the residual being the Stirling correction ln(2πN)/2N. The vapour pressure of a solid depends on the absolute entropy of its gas, and the measured values agree with the divided count and not the undivided one.

Tested in Mixing what is already mixed · the entropy reading path

“The equal-area rule is a graphical trick for locating the tie line.”

It is the statement that the two phases have the same Gibbs energy, written as an area. Along an isotherm dg = v dp, so the difference in Gibbs energy between the two ends of the tie line is the integral of v dp round the loop, and setting it to zero is exactly the equal-area condition. The construction is a thermodynamic equilibrium condition drawn geometrically, and the same argument would locate the tie line for any equation of state whatever.

Tested in The part of the curve no fluid follows · the phase change reading path

“The uncertainty relation gives a lower bound on the ground-state energy, so the true value must be somewhat higher.”

For the harmonic oscillator the bound is attained exactly. Minimising p²/2m + ½kx² subject to ΔxΔp ≥ ħ/2 gives ħω/2, which is the exact ground-state energy — because the true ground state is a Gaussian and a Gaussian is the one shape that saturates the uncertainty relation. For other wells the argument gives an estimate rather than the answer: the same reasoning applied to hydrogen gives −13.6 eV as well, and applied to a linear potential it is out by a few per cent.

Tested in The motion that cannot be stopped · the uncertainty reading path

“Iron is the most tightly bound nucleus, which is why stellar fusion stops there.”

Nickel-62 is the most tightly bound per nucleon at 8.795 MeV, marginally above iron-56 at 8.790, and the semi-empirical formula evaluated at its best proton number puts the peak at mass number 58. That stellar burning ends near iron rather than nickel is a statement about which reactions are available at the temperatures reached, not about the curve. The peak's position and the endpoint of stellar nucleosynthesis are two different facts that agree approximately and are often stated as one.

Tested in The mass that is missing · the Mass-energy reading path

“Nothing can travel faster than light.”

Plenty of things can, and none of them carries information. The intersection point of two closing scissor blades, the spot of a searchlight sweeping a distant wall, the phase velocity of light in a waveguide below cutoff and the peak of a pulse in an amplifying medium all exceed c and none is a signal. The correct statement restricts what can be *sent*, not what can *move*: no influence propagates outside the light cone, and everything else is a pattern whose parts were arranged in advance.

Tested in Which came first, and who decides · the simultaneity reading path

“Archimedes' principle gives the buoyant force, so the problem is solved once the displaced volume is known.”

The principle is exactly right and says nothing about stability, because it is a statement at one depth rather than a statement about a derivative. Everything in this essay follows from differentiating it: the force is ρgV(z) − W, the weight does not depend on z, and the whole question is the sign of dV/dz. A principle that gives a force correctly still has to be differentiated before it says whether an equilibrium holds.

Tested in The depth past which it must sink · the buoyancy reading path

“Marangoni flows are a laboratory curiosity.”

They set the shape of every drying drop, they are why a soap film drains in the pattern it does, and they are the reason welding pools change shape when the sulphur content of the steel changes by a few dozen parts per million — because sulphur reverses the sign of the temperature coefficient, turning an outward surface flow into an inward one and doubling the penetration of the weld. Every process with a free surface and a temperature or composition difference has one.

Tested in The surface that pulls toward the stronger side · the surface tension reading path

“A distant observer sees the falling body freeze at the horizon, so it never actually crosses.”

Both statements are true in their own coordinates and neither is the whole account. In the falling clock's own time the crossing happens at a definite finite moment and the centre is reached shortly after; in the distant observer's coordinate time it takes for ever, while the light received redshifts exponentially with an e-folding time of about 2 × 10⁻⁵ seconds for a ten-solar-mass hole — so the image fades to nothing in well under a millisecond rather than lingering. What 'never crosses' describes is a coordinate, and the last photon is emitted at a finite proper time.

Tested in The horizon that nothing marks · the horizons reading path

“An orbit closes when its apsidal angle is a rational fraction of a full turn.”

That closes one orbit, not all of them. The apsidal angle generally depends on the eccentricity as well as on the force law, so a rational value at one amplitude becomes irrational at another: measured on integrated orbits, the angle drifts by 1.7 to 3.7 degrees across the range of eccentricities drawn for the exponents that do not close. Bertrand's theorem is about the laws whose angle does not move, and only two do.

Tested in The orbit that does not come back to itself · the orbit stability reading path

“A skater spins faster with her arms in because angular momentum is conserved.”

The conservation law fixes the rate and says nothing about the energy, which rises by the same factor — 4.33 in the figures here, from 24 joules to 104. A statement that accounts for the rate and leaves a fourfold increase in energy unexplained has answered half the question. Integrating the force needed to hold each arm on its circle, over the distance it is pulled in, gives exactly the increase, to seven decimal places in the drawn figure.

Tested in The quantity that survives a change of shape · the rotation reading path

“The precession rate is Mgl divided by Iω.”

That is one root of a quadratic that has two. Setting the tilt's second derivative to zero gives I₁cos θ φ̇² − I₃ω₃φ̇ + Mgl = 0, whose roots are a slow precession — of which Mgl/I₃ω₃ is the leading term — and a fast one that rises in proportion to the spin. The figures draw both, and the discriminant gives a third statement the usual formula cannot: below 1245 rpm at 30° for the top drawn here, neither root is real and no steady precession exists at any rate.

Tested in The push that comes out sideways · the rotation reading path

“A released top precesses steadily.”

Released from rest it does not: it falls first, and the fall is what generates the sideways motion. The path traced by the axis comes to a cusp each time it returns to the starting tilt, because at that instant the precession rate is momentarily zero. Steady precession is the one initial condition in which the axis is already moving at exactly the right rate, and it has to be arranged by hand.

Tested in The push that comes out sideways · the rotation reading path

“Diffraction is what happens when light goes through a small aperture.”

The same aperture diffracts differently at different distances, and the parameter is the Fresnel number — the number of half-period zones the aperture holds. The figures draw one slit 0.7 mm wide at four screen distances: at N = 12 the pattern fills the geometrical opening and is covered in ripples, and at N = 0.25 it is a single broad lobe with the aperture no longer recognisable. Nothing about the aperture changed.

Tested in The spiral that says how much light arrives · the huygens reading path

“Adding an absorber removes the resonance.”

It replaces one resonance with two, at 0.854 and 1.171 times the original frequency for a ten per cent absorber, and in an undamped system both are infinite. The device converts a machine that is dangerous at one frequency into a machine that is dangerous at two and safe between them, which is an improvement only if the drive stays where it was.

Tested in The mass that makes another stand still · the resonance reading path

“The extraordinary index is a property of the crystal.”

It is a property of the crystal and the direction. n(θ) runs from the ordinary value along the optic axis — where the two waves are identical and the crystal behaves exactly like glass — to its extreme value at right angles, following an ellipse. Quoting one number for it is quoting the endpoint of a curve.

Tested in The crystal that answers twice · the polarisation reading path

“The evanescent field reaches about a wavelength past the boundary.”

It reaches a wavelength in the middle of the range and diverges at the critical angle, where the exponent √(n₁²sin²θ − n₂²) goes to zero. For 550 nm light from glass into air the 1/e depth is 1017 nm one fifth of a degree past critical, 248 nm at 45°, and 83 nm at 75°. There is no single answer to how far, and the range over which it changes by a factor of ten is under four degrees.

Tested in The reflection that happens where the glass is not · the total internal reflection reading path

“An emf is produced by a changing magnetic flux.”

Two different mechanisms produce it. Charges dragged sideways through a field feel qv × B, which acts only on charge that is moving; charges at rest in a changing field feel a genuine electric field, which acts on charge that is not moving. The figures compute the two contributions separately for a loop that is both moving and in a changing field, and neither of them is the emf — their sum is, and it matches a numerical derivative of the flux to a fraction of a nanovolt.

Tested in The rule that is two laws wearing one coat · the induction reading path

“Viscosity is friction between layers of fluid.”

In a gas there is no contact between layers to have friction at. What crosses the surface between two layers is molecules, and what they carry is momentum: a molecule from the fast layer arriving in the slow one speeds it up. The prediction that follows is the density independence, which a friction picture cannot produce and does not survive.

Tested in The viscosity that does not care how much gas there is · the kinetic theory reading path

“The electrons in a metal carry a heat capacity of 3/2 k each, by equipartition.”

They carry about a hundredth of it — 1.2 per cent for copper at room temperature — and the shortfall is not a small correction but a different functional form: the electronic heat capacity is proportional to temperature rather than constant. Equipartition assumes every degree of freedom can take up energy, and an electron deep in the sea cannot, because every state it could be promoted to is occupied. This discrepancy stood over the classical theory of metals from 1900 to 1927.

Tested in The pressure that is not a temperature · the exclusion reading path

“A sphere moving at close to light speed is flattened into an ellipsoid.”

Its outline photographs as a circle of its proper radius at every speed, because a rotated sphere is the same sphere. The figure traces the markings on it and shows them turned while the outline never moves. In the simultaneous-positions sense the sphere really is an oblate ellipsoid, and no camera at any exposure can record that shape.

Tested in The contraction no photograph shows · the length contraction reading path

“The spin–orbit splitting is the energy of the electron's magnetic moment in the magnetic field it sees in its own frame.”

That calculation gives twice the measured splitting. The frame it is done in is being carried round the nucleus, and a frame carried round a closed path in velocity space comes back rotated — by 2π(γ − 1) per orbit, which is 34.5 arcseconds for hydrogen's ground state. Subtracting that precession halves the answer and makes it right, and it is not a new interaction but a statement about how the calculation's own frame behaves.

Tested in The turn that two pushes leave behind · the Velocity-addition reading path

“The ring forms because the drop is thinnest at the edge and so dries there first.”

The thinness is not what decides it. The evaporation rate per unit area is set by how vapour diffuses away from a lens-shaped object, which is the same boundary-value problem as the electrostatic field around a charged lens, and it *diverges* at the contact line for any contact angle below ninety degrees. Above ninety it is smallest at the edge and the flow reverses, and a drop above ninety degrees is still thinnest at its rim.

Tested in The ring the drop leaves behind · the surface tension reading path

“The maximum height of a siphon is 10.3 metres.”

It is (p_ambient − p_vapour)/ρg, which is 10.11 m for water at 20 °C, 5.65 m at 80 °C, 12.32 m for ethanol and 0.76 m for mercury. Every one of those is a property of the liquid and the ambient pressure, and none is a property of the tube. Quoting a single number for it hides both dependences.

Tested in The height a siphon cannot pass · the hydrostatics reading path

“The Shapiro delay is a test of general relativity like the deflection is.”

It is a test of a different part of it. The deflection is set by the sum of the metric's time and space curvature terms; the delay, to leading order, is set by the time term alone. A theory could get the deflection right and the delay wrong. The parametrised comparison of theories was built for exactly this reason, and the delay is currently the sharpest constraint on it — measured by the Cassini spacecraft to about two parts in a hundred thousand.

Tested in The delay that is not a bend · the light deflection reading path

“Radiation pressure blows small grains away and gravity holds large ones.”

Only between two sizes. The ratio β = 3LQ/16πGMcρa rises as grains are made smaller only while the efficiency Q stays near one, and Q falls as the fourth power of the size below the wavelength of the light. So β turns over — for silicate grains in sunlight it peaks at 1.87 near 115 nm — and the blow-out band runs from 48 nm to 574 nm, with grains on both sides of it staying.

Tested in The size the light cannot blow away · the radiation pressure reading path

“A grain that is not blown away stays in its orbit.”

It spirals in. In the grain's own frame the starlight arrives from slightly ahead of the star — aberration by v/c — so it delivers momentum against the motion, and the grain re-emits that energy isotropically in its own frame and does not get the momentum back. The resulting drag takes a 10 μm grain from 1 AU into the Sun in 13,900 years, which is short compared with anything astronomical.

Tested in The size the light cannot blow away · the radiation pressure reading path

“Stick-slip happens because static friction is larger than kinetic friction.”

That gap is real and it is not the mechanism. A model with two coefficients and nothing else contains no length and no stiffness, so it predicts a sawtooth at every stiffness a spring can have — and the figures here show a real contact sliding silently the moment its holder passes a computed threshold, with every friction parameter unchanged. What is missing from the two-coefficient account is how far the surface has to move to lose its grip.

Tested in The chatter a stiffer holder removes · the friction reading path

“Energy is conserved, so the books balance.”

They balance in each frame separately, and the entries differ between frames. A rifle firing a bullet gives the bullet almost all the kinetic energy in the frame of the ground and gives the recoiling shooter most of it in a frame moving with the bullet. Both accounts are correct and neither is a correction of the other; what is common to them is the energy the propellant released, which is the only figure the chemist could have supplied.

Tested in The energy that depends on the observer · the energy reading path

“Below cutoff the wave is reflected.”

It is reflected in the sense that almost no energy gets through, and the mechanism is not reflection at a surface. The field enters the guide and decays exponentially along it, over a length that diverges as the cutoff is approached from below. That is the same solution — a real exponential where a travelling wave would have an imaginary one — that appears past a critical angle and inside a quantum barrier, and it is the reason a short enough length of undersized guide passes a signal it has no travelling mode for.

Tested in The pipe that will not carry a low note · the guided waves reading path

“Fermat's principle explains why light reflects at equal angles.”

It picks out the equal-angle point and says nothing about why light should care. The reason lives one level down, in the sum over paths: neighbouring routes that differ in length by much more than a wavelength arrive with scrambled phases and cancel, and only near a stationary point do a band of them agree well enough to survive. Which is why the criterion is stationarity — a first derivative vanishing is exactly the condition for a band of paths to share a phase, and the sign of the second derivative does not enter it.

Tested in The path that takes the longest time · the fermat reading path

“Quantum tunnelling lets particles pass through barriers.”

It lets a fraction pass, and the fraction is what the subject is about. For a 5 MeV alpha in a heavy nucleus the escape probability per attempt is about 10⁻³⁰, and the nucleus makes 10²¹ attempts a second — which gives a lifetime of years rather than the immediate escape the word suggests. The interesting content of tunnelling is always the exponent, and the exponent is an integral over the shape of the barrier rather than a property of its height.

Tested in A wall that a factor of two makes impassable · the tunnelling reading path

“Polarisation is the material's response to the field.”

It is the material's response to the field *inside*, which the polarisation itself has changed. That circularity is the whole of the subject: the bound charge appearing on the surface makes a field opposing the polarisation, which reduces the polarisation, which reduces the opposing field. Solving it is a single line for an ellipsoid and is not solvable in closed form for anything else, because only an ellipsoid polarises uniformly.

Tested in The field the matter takes away · the dielectrics reading path

“Hysteresis happens because the domains get stuck.”

Pinning is one mechanism among several and is not needed for the effect. The figures compute the loop of a single particle with no domain walls in it at all, purely from following a local minimum of its energy as the field is swept — and it is square, with a remanence and a coercivity. What produces hysteresis is a state that persists after it has stopped being the lowest one, and any system with more than one minimum has that available.

Tested in The magnet that has to fight its own field · the magnetisation reading path

“The harmonic approximation is good enough for a solid at ordinary temperatures.”

It is good enough for the frequencies and useless for everything else. It gives the vibrational spectrum, the heat capacity and the speed of sound to a few per cent, and it gives an expansion coefficient of exactly zero and a thermal conductivity of exactly infinity — both of them qualitatively wrong rather than numerically off. Any property that requires modes to be coupled requires the term the approximation drops.

Tested in Why heating a perfect spring changes nothing · the harmonic approximation reading path

“Critical opalescence happens because the fluid becomes cloudy.”

The fluid remains a single homogeneous substance with no droplets in it. What grows is the *correlation length* — the distance over which a density fluctuation extends — and when it reaches the wavelength of light divided by 2π the fluid scatters strongly at every angle. The figures put that crossing at twenty millikelvin from carbon dioxide's critical temperature, which is why the effect appears to switch on rather than to develop.

Tested in The point at which the two become one · the phase change reading path

“A steadily accelerating rocket eventually outruns any signal sent after it.”

It does, and calling that outrunning misdescribes it. A body under constant proper acceleration has a worldline whose asymptote is a light ray, so signals emitted from its starting point after a certain moment never catch it — not because the rocket is faster than they are, but because it started early enough. The figures draw that asymptote and the signals on either side of it, and the boundary is a horizon produced by acceleration alone, with no mass anywhere near it.

Tested in The push that does not point where the body goes · the relativistic dynamics reading path

“Electric and magnetic fields are two aspects of one field, so the distinction is arbitrary.”

The split is observer-dependent and the classification is not. Two numbers built from the fields — E² − c²B² and E·B — take the same value for every observer, and between them they decide which of three kinds a field is: one that can be made purely magnetic, one that can be made purely electric, and one that can be made neither. A light wave is permanently in the third class, and no boost in any direction moves it out.

Tested in The field nobody can transform away · the field transformation reading path

“Whether something floats is decided by comparing its density to the fluid's.”

That comparison decides whether a fully wetted body rises, and it does not decide whether an object floats. A steel needle denser than water rests on it, held by surface tension, and a paraffin-coated block on a wet floor stays down. Density settles the question only when the surface has no say — which is most engineering cases and almost none of the interesting ones.

Tested in The block the water does not lift · the buoyancy reading path

“Viscosity measures how runny a fluid is.”

For a Newtonian fluid it does. For every other kind the number an instrument reports depends on how fast it was stirred, and for a viscoelastic one it depends on how long the measurement lasted as well. The figures compute the apparent viscosity of four fluids across a decade of shear rate and it varies by more than a factor of ten for three of them — so a single quoted viscosity is a quoted measurement condition as much as a material property.

Tested in The liquid that remembers · the rheology reading path

“A gravitational-wave signal measures the masses of the two bodies.”

The inspiral measures one combination of them and not the two. The rate at which the frequency sweeps upward depends only on the chirp mass, a particular weighted product, so a pair of 10 and 40 solar masses and a pair of 20 and 20 produce nearly identical sweeps. Separating the two masses requires the later part of the signal, where post-Newtonian corrections enter and the degeneracy is broken — which is why published mass estimates have much larger uncertainties than the chirp mass does.

Tested in The orbit that has to shrink · the gravitational waves reading path

“Gravitational contraction could power the Sun.”

For nine million years, which is the number Kelvin computed and it is 480 times too small. The figures divide the energy given up in reaching a given radius by the rate of radiation and place the Sun's Kelvin–Helmholtz time at 9.4 million years against an Earth 4.5 billion years old. Nothing in that arithmetic was wrong; a term was missing, and no amount of care with the terms present could have found it.

Tested in The ball of gas that heats up as it cools · the Self-gravity reading path

“The load in a packing is distributed among the grains.”

It is distributed, and the distribution is the interesting part. A model in which every grain weighs the same and passes its load to two neighbours in a ratio drawn at random produces chains carrying several grains' worth past their neighbours and quiet regions between them, with the heaviest grain in the bottom row of a twelve-row packing at nearly twice the mean and the lightest at a seventh of it. Nothing in the rule is heterogeneous; the heterogeneity is what a random split does when it is iterated.

Tested in The silo that does not weigh what it holds · the granular matter reading path

“Reverse osmosis pushes water through a filter fine enough to stop salt.”

A filter has a pore size and a pressure drop that scales with flow; this has a threshold. Below 27 bar for seawater nothing crosses at all, in either direction — the flux is not small, it is absent — and the threshold is thermodynamic rather than geometric, being the same osmotic pressure the solution would generate on its own. The figures locate that intercept on the plotted flux line, and no improvement in the membrane will move it.

Tested in The pressure that comes from counting · the osmosis reading path

“A pendulum measures g because its period is 2π√(L/g).”

That formula is for a point mass on a weightless string, which no apparatus is. A real bar has its mass spread out, and its period is 2π√((k² + h²)/gh) with k the radius of gyration about the centre of mass and h the distance from that centre to the pivot. Both of those are properties of the mass distribution, both are hard to measure to five figures, and the figures show that the naive formula and the true one differ by tens of per cent for any real bar.

Tested in The length nobody has to measure · the pendulum reading path

“The gap comes from the difference between the two materials.”

The materials set the width and the periodicity sets the position, and only the second is structural. The figures fold a uniform medium's dispersion into an imagined unit cell and the branches cross at the zone centre and boundary; switching on a twelve per cent index contrast opens gaps at those crossings and moves nothing else perceptibly. A weak modulation does nothing whatever except where two travelling waves already shared a frequency.

Tested in The gap a repeat opens · the periodic media reading path

“A waveguide has a cutoff frequency below which nothing propagates.”

A hollow pipe does, because its walls impose a boundary condition that a long wavelength cannot satisfy. An index-guided channel does not: the transverse problem is a potential well, a symmetric one-dimensional well always has at least one bound state however shallow it is, and the fundamental mode of a fibre is guided at every frequency. The figures show the lowest mode's transverse profile spreading into the cladding as the contrast falls, and never unbinding.

Tested in The channel with no walls · the guided waves reading path

“Concentrating light means collecting a large area onto a small one.”

It means trading area against angle, and the product is what is conserved. The figures take a beam of half-angle 12° through an aperture three times smaller and the exit half-angle is 38.6°, not 12° — the area and the sine of the angle multiply to the same number at both ends. A concentrator that shrank the area without opening the angle would be increasing the radiance, which is the thing no passive system can do.

Tested in The brightness no lens can increase · the etendue reading path

“An isolator is a polariser that only lets light through one way.”

A polariser is perfectly symmetric — it passes the same component in both directions — and no combination of polarisers, waveplates, lenses and mirrors can pass light one way and stop it the other, because every one of them looks identical run backwards. The device needs a component that does not, and the only one available in ordinary optics is a magnetic field. The figures show the transmission forwards and backwards against the rotator's angle, and the backward leak is exactly zero at 45 degrees.

Tested in The rotation a return trip doubles · the polarisation reading path

“A 360° rotation returns a spin-½ to its original state, up to an unobservable sign.”

The sign is unobservable on the whole state and perfectly observable on half of one. Split a neutron beam, rotate the spin in one arm only, and recombine: the intensity is ½(1 + cos θ/2), at its minimum after one full turn and back to full brightness after two. The period is 720 degrees, the experiment was done, and the measured period is 720 degrees.

Tested in The angular momentum that is not a rotation · the spin reading path

“The interaction-free measurement works one time in four, and that is the limit.”

That is the limit of asking the whole question at once. Ask it in N gentle pieces, rotating the state by π/2N each time, and the chance of success is cos^2N(π/2N), which the figures plot: 53 per cent at four interrogations, 96 per cent at sixty-four, with the shortfall falling as π²/4N. There is no ceiling in the physics; the quarter was the price of impatience.

Tested in The measurement that never touched it · the measurement reading path

“An accelerating charge radiates because it is accelerating.”

True and silent on the mechanism. What radiates is the *join*: outside a sphere of radius ct the field still points at where the charge would have been had nothing changed, inside it points at where the charge is, and the two must be connected because a field line cannot stop in empty space. The connection is transverse and falls as 1/r rather than 1/r², and the figures measure its ratio to the radial field as a·r·sinθ/c² — from which Larmor's formula follows with no wave equation used anywhere.

Tested in The field that points where the charge is now · the retardation reading path

“The field is a convenient way of bookkeeping the forces between distant charges.”

A bookkeeping device does not have a conserved quantity of its own that can be handed to matter on demand. Here the field holds angular momentum in a completely static configuration and gives it up when the configuration changes, with the transfer traceable instant by instant. The same argument runs for energy and for linear momentum, and in each case the alternative — action at a distance with a delay — fails to conserve the quantity during the delay.

Tested in The angular momentum that is in nothing at all · the field energy reading path

“Entropy is a measure of disorder.”

It is a count of the microscopic arrangements consistent with what is known, which is why it depends on what is known. Szilard's engine is the sharpest demonstration: the gas is identical before and after the measurement, and its entropy falls by k·ln2 because an observer has learnt which half the molecule is in. Nothing about the molecule's arrangement changed. The figures compute the work that reduction makes available, and it is entirely real.

Tested in The bit that has to be paid for · the entropy reading path

“The second law limits how good an engine can be.”

It limits the work per unit of heat and says nothing whatever about the rate. Every quantity in Carnot's argument is a state function and no time appears in it, so the bound holds equally for an engine that completes a cycle in a millisecond and one that takes a century. What limits a real plant is a different optimisation — the most power rather than the most work — and it produces 1 − √(Tc/Th), a bound the second law does not mention.

Tested in The engine that has to finish · the heat engines reading path

“A rigid rod moved from rest to speed v ends up contracted by γ.”

It does, and only if it is moved in a particular way. Holding the proper length fixed — Born's definition of rigidity — requires the trailing end to accelerate harder than the leading end, by the ratio of their distances from the horizon the acceleration creates, and to go on doing so for ever. Give both ends the same acceleration instead and the proper length grows as γ. The two programmes differ by γ², and a rod made of matter follows the first only because its own elasticity supplies the difference.

Tested in The string that breaks between two rockets · the length contraction reading path

“Lensing magnification can be arbitrarily large.”

In the point-mass model it diverges at exact alignment, and the divergence is the model's rather than the world's. Every real source has a finite angular size, so the magnification is averaged over its face and comes out large and finite; a real lens has structure, which breaks the perfect symmetry the divergence needs. The figures show the total brightness running away as the alignment closes, and the divergence is the honest signal that the idealisation has been carried past its range.

Tested in The lens with no focal length · the light deflection reading path

“A measured Curie law confirms that the moments are independent, and the constant gives their size.”

It gives J(J+1) and no more. The figures separate the four Brillouin curves by their initial slope (J+1)/3J alone: a spin-half moment is three times as responsive per unit saturation as the classical dipole of the same size, and every curve saturates at one. Distinguishing them needs the bend, which needs a field over the temperature that a room-temperature magnet never reaches.

Tested in The magnetism classical physics forbids · the magnetisation reading path

“Absorption and dispersion are two properties of a material, related by the fact that both come from the same resonances.”

They are one property. The Kramers–Kronig reconstruction here computes ε′ at nine frequencies from the absorption alone — no resonant frequency, no damping, no oscillator strength — and lands on the directly computed curve to a part in ten million. Knowing where a substance absorbs fixes how it refracts everywhere, including where it is perfectly transparent, and the reason is causality rather than a shared origin.

Tested in The constant that depends on how fast it is asked · the dielectrics reading path

“The fine-structure constant measures the strength of the electromagnetic interaction.”

It does, and the statement leaves out what the number is doing in this calculation. Here α is the speed of the innermost electron in units of c, so (Zα)² is how relativistic the atom is — and the figure of (Zα)² against nuclear charge shows the same quantity passing a half for hydrogen-like uranium, where the splitting stops being a correction and the atom it corrects stops being a starting point.

Tested in The line that is really two · the atomic spectra reading path

“Equipartition fails at low temperature because quantum mechanics freezes modes out.”

That is one of its two failures and the better known one. The other has no quantum mechanics in it at all: equipartition also fails whenever a term in the energy is not quadratic, which is a purely classical fact and is why the figure of kT/n against n is drawn with no ħ anywhere on it. A classical gas of ultrarelativistic particles obeys equipartition exactly and gets 3kT per particle.

Tested in The share that is not half a kT · the equipartition reading path

“Kirchhoff's law says a good absorber is a good emitter.”

It says a body's emissivity equals its absorptivity at each wavelength, each direction and each polarisation separately, and the totals only follow when the spectra overlap. A selective solar absorber has absorptance 0.95 at half a micrometre and emittance 0.05 at ten, breaks no law, and reaches 761 K in sunlight where a grey surface reaches 364 K. The stronger, spectral, form is the one that does the work.

Tested in The glow that says nothing about the surface · the blackbody reading path

“The quantum of circulation is h over the mass of a helium atom, so it is a property of helium.”

It is a property of whatever carries the phase. In helium-4 that is a single atom and the quantum is h/m; in helium-3 and in every superconductor it is a pair, and the quantum is h/2m. The factor of two is a measurement of how many particles are bound into the object whose phase is winding, and it is how pairing was confirmed in both cases.

Tested in The whirlpool that comes in one size · the superfluidity reading path

“General relativity is needed to predict the gravitational redshift.”

The redshift follows from the equivalence principle and energy conservation alone, and this collection derives it that way at the rung below. What needs more is the converse: the argument here runs the implication backwards, from a measured redshift to the impossibility of a flat spacetime, and it establishes that some curved geometry is required without saying which one. Choosing among the curved geometries is what a field equation is for.

Tested in The parallelogram that will not close · the gravitational redshift reading path

“Magnetic reconnection converts magnetic energy into heat and motion.”

It does, and the sentence omits the only difficult part. The conversion needs the frozen-in condition to fail, and it fails only in a layer of thickness L over the square root of the Lundquist number — three metres inside a flare ten thousand kilometres across. What decides the rate is not how much energy is available but how fast plasma can be pushed through a sheet of negligible volume, and the answer computed here is five orders of magnitude too slow.

Tested in The knot the field cannot untie · the flux freezing reading path

“The self-image happens because the light from adjacent slits arrives in step at that distance.”

That is the two-slit account and it gives the wrong distance. The revival is a property of the whole spectrum at once: propagating a distance z multiplies the m-th order by exp(−iπλzm²/d²), and what makes the pattern reassemble is that every one of those phases is a multiple of 2π together, for every m, at z = 2d²/λ. Two orders would revive at four times as often as they do.

Tested in The grating that photographs itself · the diffraction reading path

“Staging helps because a smaller rocket is easier to build.”

It helps because a spent stage is dead mass carried by everything that follows, and dropping it changes the mass ratio of what is left — which is the only quantity the logarithm contains. The figure computes payload fraction against stage count for a fixed velocity budget and finds one stage impossible at any size whatever, two giving 2.13 per cent and three giving 2.76.

Tested in The push that needs nothing to push against · the momentum reading path

“The defect state sits inside the gap because the defect layer has a different resonant frequency.”

It sits inside the gap because the surrounding stack is the only thing preventing it from radiating away, and its own frequency has to be one the stack forbids. The figures show the field decaying away from the defect at 0.693 nepers per cell, which is the attenuation the perfect crystal's own band structure gives at that frequency — the same evanescent wave, measured two ways, and the reason a crystal with no gap can trap nothing.

Tested in The mode that lives in the mistake · the periodic media reading path

“A steady electric field accelerates a band electron, which is why a metal conducts.”

A steady field makes the electron *oscillate*. The integrated trajectory at 10 MV/m turns round after 0.69 ps and returns to its starting point 1.379 ps after release, over an amplitude of 180 nm, because the force acts on k and the band is periodic in k. Conduction exists because that orbit is interrupted: a collision every 10 femtoseconds arrives 138 times before one Bloch period is complete, and a metal has a resistance instead of an oscillation.

Tested in The mass a curve decides · the bands reading path

“Radioactive decay is exponential, which is what it means for a nucleus to have no memory.”

Exponential decay requires an energy distribution running over the whole real line, and no bound system has one. Transforming a resonance's own spectrum gives a survival probability that is flat below Γτ_z² — 1.57 × 10⁻² lifetimes for the case drawn — and an inverse square beyond 23 lifetimes, with the exponential in between good to a part in ten thousand.

Tested in The exponential that is only true in the middle · the decay reading path

“A solenoid confines its field, which is why it is used where a magnet must not disturb its surroundings.”

A solenoid with no external field would have no magnetic moment, and a coil with no moment cannot be a magnet. Beyond forty solenoid lengths the summed field agrees with a point dipole of moment NIπR² to 0.08 per cent and falls as the inverse cube with a fitted exponent of −3.001. The confinement is a matter of degree, and the degree is what the aspect ratio buys.

Tested in The field outside the solenoid, which is not zero · the ampere law reading path

“Air is blue because each molecule scatters short wavelengths more strongly, and there are a lot of molecules.”

N molecules with random phases scatter N times one molecule's worth; N molecules in phase scatter N². Both happen at once, in the same gas, to the same light: the sideways sum is incoherent and gives Rayleigh's answer, and the forward sum is coherent and gives the refractive index. Air's n − 1 = 2.78 × 10⁻⁴ fixes a polarisability that returns a 550 nm optical depth of 0.0877 against a published 0.0973, with no scattering measurement used anywhere.

Tested in Why a litre of water is not blue for the reason the sky is · the scattering reading path

“Adiabatic demagnetisation gets colder every time it is repeated, so enough repetitions get arbitrarily close.”

In the model, yes: at a field ratio of four, five stages take 1 K to 0.98 mK. In a real salt the spins make a field of a few millitesla for each other, and below μb/k they are ordered by it whatever the coil is doing — 3.4 mK for 5 mT — so the entropy is already spent and there is nothing left to pump. The refrigerant runs out before the arithmetic does, which is why a multi-stage machine changes salt rather than repeating with one.

Tested in The staircase that never reaches the floor · the third law reading path

“An accelerating charge radiates energy at the Larmor rate, and that is the whole of the classical theory of radiation.”

Larmor's formula is a statement about the field far away and contains no force on the charge at all. Requiring the charge to pay for what leaves gives F = mτȧ with τ = μ₀q²/6πmc = 6.27 × 10⁻²⁴ s, and that equation has solutions Larmor's does not: with no applied force at all, an acceleration growing by thirteen decades in 2 × 10⁻²² seconds.

Tested in The force a charge exerts on itself · the radiating charge reading path

“Reflection happens at the surface.”

It happens over a thickness c/ωp, which is 12.5 nm for aluminium and 5.3 m for the F2 layer, and diverges as the cutoff is approached — 17 m at 0.95 of the plasma frequency for the ionosphere. The field enters as an evanescent tail, stores energy in that layer and returns every joule of it, and the layer is a property of the medium rather than of the wavelength.

Tested in The frequency below which nothing gets in · the plasma oscillation reading path

“A distant sound is a quiet sound — the difference is the level.”

Spreading takes every frequency alike and absorption does not. Over a kilometre of air at 20 °C the loss to absorption is 3.7 dB at 1 kHz and 117 dB at 8 kHz; over ten kilometres, 37 dB and 1,170. A flat broadband source ten kilometres away has nothing left above about 500 Hz, so what arrives is not the same sound made smaller.

Tested in The distance that takes the treble out · the attenuation reading path

“The cone angle depends on the shape of the aircraft and the strength of the shock.”

The envelope of the wavefronts has sin θ = 1/M and nothing else in it — no pressure, no density, no size, no shape. Measuring the tangent from the apex to each of six constructed wavefronts at one speed gives the same angle to a part in 10¹², because the ratio of a circle's radius to its distance from the apex is c/v for every one of them. A finite-strength shock is slightly steeper than the Mach angle, and it tends to it as the disturbance weakens.

Tested in The cone the source leaves behind · the doppler reading path

“A coefficient of friction cannot exceed one, because that would mean friction stronger than the force pressing the surfaces together.”

Nothing forbids it. A clean metal pair in vacuum seizes, with an effective coefficient of several; a rubber tyre on dry asphalt runs at 1.0 to 1.5; a gecko's foot holds against gravity with no normal load at all. Each of those is a case where the friction is not shear of load-bearing junctions, so the load has stopped being the quantity it is divided by.

Tested in The grip that is not a coefficient · the friction reading path

“Any system with a nonlinearity in it will be chaotic.”

The same code, the same integrator and the same nudge at 3° instead of 120° return an exponent of 2.1 × 10⁻⁴ per second — four orders of magnitude smaller, and consistent with zero. A double pendulum near its rest position is two coupled normal modes and is as predictable as anything in mechanics. Nonlinearity is necessary and it is a long way from sufficient.

Tested in The error that doubles on a schedule · the chaos reading path

“Liquid water cannot be at a negative pressure, because it would boil.”

It would, if a bubble could form, and forming one costs energy: a nucleus of radius r has a surface energy going as r² and a volume term going as r³, so there is a barrier, and below a critical radius any bubble collapses. At −1 MPa the critical radius is 146 nm and the barrier is enormous; homogeneous nucleation needs around −140 MPa. What sets the real limit is a pre-existing gas pocket, which is why the pores of the conduit walls are the relevant length.

Tested in The column that is pulled, not pushed · the capillarity reading path

“Buoyancy is decided by density, so a parcel denser than its surroundings sinks.”

It is decided by the density a parcel would have if it were moved, which is not the density it has where it is. The whole classification is a comparison of two slopes with no density in it anywhere — the environment's lapse rate against the adiabatic one — and the same cold, dense air is stable under one profile and unstable under another.

Tested in The layer a parcel cannot leave · the stratification reading path

“A fibre gyroscope works because light takes longer to go round the coil the way it is turning.”

It does, and the useful part is what the expression does not contain. The refractive index of the fibre cancels out, so a glass gyroscope and a vacuum one of the same enclosed area read the same; the shape does not appear, so a coil can be wound to any outline; and the axis need not be inside the loop's centre or even inside the loop's plane. What is measured is the enclosed area times the rotation rate, and nothing else about the instrument.

Tested in The ring where the two beams disagree · the simultaneity reading path

“Uniform acceleration means the speed increases at a constant rate.”

It means the acceleration measured on board is constant, which is what an accelerometer reads and what a passenger feels as weight. The speed measured in the launch frame does not rise linearly — it approaches c — while the quantity that does rise linearly with proper time is the rapidity, whose hyperbolic sine and cosine are the two coordinates of the worldline.

Tested in The wall of silence behind a rocket that never stops · the accelerated frames reading path

“Chirikov's overlap criterion gives the point at which a system becomes globally chaotic.”

It gives π²/4 = 2.467 for the standard map and the measured value is 0.971635, so it is too large by a factor of 2.54. The scaling it predicts is right — resonance widths really do grow as the square root of the coupling — and the coefficient is wrong because the criterion counts the two largest resonances and ignores the infinitely many smaller ones that have already eaten most of the gap.

Tested in The last curve to go · the chaos reading path

“The energy released by the falling chain ends up as kinetic energy of the pile.”

None of it does. The chain arrives at rest, so the pile has no kinetic energy at all, and the ledger shows the released potential energy and the dissipated energy agreeing to four decimal places at 0.5 λgL². Every joule is destroyed at the impacts. A chain lowered gently releases the same energy and destroys none of it.

Tested in The pile that lands heavier than it weighs · the momentum reading path

“A resonance curve's width tells you the damping.”

Only if the width is homogeneous. A collection of resonators with slightly different frequencies has a broad response and a long ringdown at the same time, so the two measurements disagree — and the disagreement is the measurement, because its size is what says how much of the width is spread rather than loss.

Tested in The width that is a lifetime · the resonance reading path

“Reflections happen at surfaces.”

Reflections happen where the impedance changes, and how much reflects depends on how fast it changes compared with a wavelength. A boundary a hundred wavelengths thick reflects nothing at all and is still a boundary; a boundary a thousandth of a wavelength thick reflects the full Fresnel amount. What a surface supplies is abruptness, and abruptness is relative to the wave.

Tested in The taper that matches every note · the impedance reading path

“Diffusion is the same process in any number of dimensions, with a different constant.”

The mean square displacement grows as t in every dimension, so on that measure they are identical. The recurrence property is not a matter of a constant at all: it changes from probability one to probability 0.3405 between two dimensions and three, and every rate that depends on two diffusing objects meeting inherits the discontinuity.

Tested in The walk that comes home · the diffusion reading path

“The third law is a statement about entropy.”

Its working content is a statement about heat capacity. Entropy is the integral of C/T from absolute zero upward, and that integral converges only because C goes to zero — for a solid as T³, which the Debye model computes and which is what makes an absolute entropy definable at all. A heat capacity that stayed at 3R would leave every absolute entropy infinite.

Tested in The entropy that is still there at zero · the third law reading path

“The plug in the middle of the pipe shrinks away as the driving is increased.”

It shrinks and never disappears. The stress in a pipe is proportional to the distance from the axis and is therefore exactly zero on the axis, so there is always a core below the yield stress. The computed plug is 67 per cent of the diameter at 1.5 times the threshold and 20 per cent at five times, and the fraction is exactly the reciprocal of the overdrive.

Tested in The paste that holds up its own hill · the rheology reading path

“Driving a fluid harder makes a bigger wave.”

Driving it faster than its own buoyancy frequency makes no wave whatever. There is no angle whose cosine exceeds one, so above N the response is trapped at the source and decays away from it rather than radiating. The generator refuses such a request rather than drawing a beam that does not exist.

Tested in The wave that picks an angle · the stratification reading path