Paradoxes & Puzzles Pixel Sticker Sheet

12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Paradoxes & Puzzles. Every character below has its own sticker page with the full physics panel.

Braessa

Two identical springs hang in series, joined by a short string, with the weight at the bottom. Two longer “safety” strings hang slack, one from the support to the lower spring and one from the upper spring to the weight. When the short string is cut, the load is carried by two parallel paths, each a spring plus a long string, so each spring carries half the weight. If the strings are long enough the weight ends up higher than before.

series: k_eff = k/2 → parallel: k_eff = 2k
k: stiffness of each spring (N/m); k_eff: effective stiffness of the support

Dalembertia

In 1752 Jean le Rond d'Alembert showed that for steady, incompressible, inviscid and irrotational (potential) flow, a body moving at constant speed feels zero drag. The streamlines and the pressure are symmetric front to back, so the pressure pushes on the rear as hard as on the front. Yet every real body in air or water feels drag.

φ = U (r + a³/2r²) cos θ ⇒ F_drag = 0
φ: velocity potential of steady flow past a sphere (m²/s); U: speed of the uniform flow far away (m/s); a: sphere radius (m); r, θ: distance from the centre and angle from the flow direction; F_drag: net force along the flow (N)

Teacupa

In a stirred cup, the swirl is held in balance by a pressure that rises toward the wall (the surface is higher there). Near the bottom, friction slows the swirl, but the pressure difference set by the surface is still there. So near the floor the inward pressure push wins, and water flows toward the centre, rises up the middle and returns outward near the top. This secondary flow carries the sunken leaves to the centre, where they settle.

∂p/∂r = ρ v_θ²/r
p: pressure (Pa); r: distance from the axis (m); ρ: density of the water (kg/m³); v_θ: swirl (azimuthal) speed (m/s)

Faintsol

As the Sun fuses hydrogen into helium, its core gets denser and hotter, so it slowly brightens. Standard solar models give about 70% of today's luminosity some 4.5 billion years ago. With today's atmosphere and albedo, that would leave the early Earth's average temperature below freezing for roughly its first two billion years. But the geological record shows liquid water on Earth more than 3.8 billion years ago. Carl Sagan and George Mullen stated this paradox in 1972.

L(t) ≈ [1 + (2/5)(1 − t/t_⊙)]⁻¹ L_⊙
L(t): solar luminosity at age t; t: age of the Sun at that time; t_⊙: present age of the Sun, about 4.57 billion years; L_⊙: present solar luminosity, 3.828 × 10²⁶ W (IAU nominal value)

Minkowskara

How much momentum does light carry inside a medium of refractive index n? Hermann Minkowski (1908) gave a momentum that grows with n; Max Abraham (1909–1910) gave one that shrinks as 1/n. Both follow from consistent electrodynamics, and experiments have seemed to support each in different setups.

p_Minkowski = nħω/c, p_Abraham = ħω/(nc)
p: momentum of one photon in the medium; n: refractive index of the medium; ħ: reduced Planck constant (exact in the 2019 SI); ω: angular frequency of the light; c: speed of light in vacuum (exact)

Homestaka

Raymond Davis Jr.'s experiment in the Homestake gold mine, South Dakota, 1478 m underground, used a tank of about 615 tonnes of perchloroethylene (tetrachloroethylene, C₂Cl₄). A solar electron neutrino can turn a chlorine-37 nucleus into argon-37 (threshold 0.814 MeV), and the few argon atoms were extracted and counted. From 1970 to 1994 it measured about a third of the rate predicted by John Bahcall's standard solar model.

measured 2.56 ± 0.16 ± 0.16 SNU vs predicted 7.6^+1.3_−1.1 SNU
SNU: solar neutrino unit: 10⁻³⁶ captures per target atom per second; 2.56: final Homestake result (Cleveland et al., ApJ 496, 505, 1998), statistical and systematic errors; 7.6: chlorine prediction of the BP2000 standard solar model (Bahcall, Pinsonneault and Basu, 2001)

Bucketta

In the Scholium of the Principia (1687), Isaac Newton described a bucket of water hung on a twisted rope. When the bucket starts spinning, the water at first stays flat. As the water picks up the rotation, its surface becomes concave, and it stays curved even after the bucket is stopped, while the water keeps turning. Newton argued that the curve shows rotation relative to absolute space, not relative to the bucket.

z = ω²r²/(2g)
z: height of the surface above its lowest point, on the axis (m); ω: angular speed of the water (rad/s); r: distance from the axis (m); g: gravitational acceleration (m/s²)

Capacitwin

Charge one capacitor C to voltage V and join it to an identical empty one. Charge is conserved, so each ends at V/2 with half the charge. But the stored energy is now Q²/4C instead of Q²/2C: half the energy is gone. This holds however small the wire resistance is.

ΔE = ½ · C₁C₂/(C₁ + C₂) · (V₁ − V₂)²
ΔE: energy lost when the capacitors are joined (J); C₁, C₂: capacitances (F); V₁, V₂: starting voltages (V)

Boltzbrain

In the 1890s Ludwig Boltzmann suggested that our ordered world might be a rare fluctuation in a universe otherwise in thermal equilibrium. The difficulty is that a small fluctuation is far more likely than a big one. A single brain, complete with false memories, needs a much smaller drop in entropy than a whole ordered universe, so in such a picture most observers would be lone “Boltzmann brains”. Andreas Albrecht and Lorenzo Sorbo (Phys. Rev. D 70, 063528, 2004) discussed it as the so-called “Boltzmann's brain” paradox.

P ∝ exp(ΔS/k_B)
P: relative probability of a fluctuation; ΔS: entropy change of the fluctuation, negative for a fluctuation into order (J/K); k_B: Boltzmann constant, 1.380 649 × 10⁻²³ J/K exactly in the 2019 SI

Delaychoice

In a Mach–Zehnder interferometer, a single photon meets a first beam splitter and then two paths. If a second beam splitter recombines the paths, the photon shows interference: which detector clicks depends on the phase difference. If the second splitter is removed, each detector clicks half the time, as if the photon took one path. John Wheeler proposed in 1978 making this choice after the photon has passed the first splitter. Jacques et al. (Science 315, 966, 2007) did it with single photons and a random choice that was space-like separated from the photon's entry.

with splitter: P₁ = cos²(φ/2), P₂ = sin²(φ/2); without: P₁ = P₂ = ½
P₁, P₂: probabilities that detector 1 or 2 clicks; φ: phase difference between the two arms (rad)

Magnetoconda

Move a magnet past a loop of wire and a current flows. Hold the magnet still and move the wire instead, at the same relative speed, and exactly the same current flows. Yet the textbook stories differ. If the magnet moves, a changing magnetic field makes an electric field that drives the charges. If the wire moves, there is no electric field in the magnet's frame, and the magnetic force on the moving charges does the job. A rod of length L moving at speed v across a field B gets the same emf either way.

ε = vBL
ε: induced emf (volt, V); v: relative speed of rod and magnet, perpendicular to the field (m/s); B: magnetic flux density (tesla, T); L: length of the rod across the field (m)

Ratchetta

The ratchet and pawl, analysed by Smoluchowski (1912) and made famous in the Feynman Lectures (Vol. I, ch. 46), is a vane in a gas joined by an axle to a ratchet wheel with a spring-loaded pawl. Molecules hitting the vane seem to push the wheel forward, while the pawl blocks it going back. But the pawl is also at a temperature and also jiggles. When both are at the same temperature, the pawl lifts and lets the wheel slip back as often as the vane pushes it forward.

rate forward ∝ e^{−ε/k_BT₁}, rate back ∝ e^{−ε/k_BT₂} ⇒ T₁ = T₂ gives ⟨W⟩ = 0
ε: energy needed to lift the pawl over one tooth (J); k_B: Boltzmann constant, exact in the 2019 SI; T₁: temperature of the gas around the vane (K); T₂: temperature of the ratchet and pawl (K); ⟨W⟩: average work delivered per unit time