Everyday Physics Pixel Sticker Sheet

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

Cheeriosa

Floating cereal clumps together and sticks to the bowl's edge. It is not magnetic or electric: each floating ring bends the liquid surface into a meniscus. A neighbouring ring sitting on that tilted surface is pushed along it, by buoyancy and surface tension, towards the other, so rings drift together. Dominic Vella and L. Mahadevan analysed this in “The ‘Cheerios effect’” (Am. J. Phys. 73, 817, 2005). The distance over which a meniscus reaches is set by the capillary length.

ℓ_c = √(γ / ρg)
ℓ_c: capillary length (m); γ: surface tension (N/m); ρ: liquid density (kg/m³); g: gravitational acceleration (m/s²)

Winetear

Swirl a glass of strong wine and a thin film climbs the wall above the liquid, gathers into a ridge and falls back as drops: tears, or legs. Alcohol evaporates from the film faster than water, so the film becomes more watery, and water has a higher surface tension than alcohol. The difference in surface tension pulls liquid up the glass from the bulk below, where there is more alcohol. James Thomson, Lord Kelvin's brother, described it in 1855 (“On certain curious motions observable at the surfaces of wine and other alcoholic liquors”, Phil. Mag. 10, 330). A flow driven like this is now called a Marangoni flow.

τ = ∇γ
τ: shear stress on the liquid surface (Pa); ∇γ: gradient of surface tension along the surface (N/m²)

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)

Scatterina

Small particles and molecules scatter short wavelengths much more strongly, in proportion to one over wavelength to the fourth power. Blue light is scattered about 5.9 times more than red, which is why the sky looks blue.

I_s ∝ 1/λ⁴
I_s: power scattered by a molecule much smaller than the wavelength (W); λ: wavelength of the light (m)

Halotra

The real halo is made by millions of tiny hexagonal ice crystals in thin, high cirrus cloud, not by water droplets. Light entering one side face of a crystal and leaving by another passes through what is in effect a 60° prism. Every such prism has a minimum angle of deviation, and light piles up near it, so a bright ring appears at that angle from the Sun.

D_min = 2 arcsin(n sin 30°) − 60°
D_min: minimum deviation, the halo radius (degrees); n: refractive index of ice, about 1.31 for visible light; 30°: half the 60° prism angle between alternate faces

Terminara

A falling object speeds up until air drag, which grows with the square of speed, balances its weight. From then on it falls at a steady terminal velocity, approaching it along a tanh curve.

m dv/dt = mg − ½ρC_dAv²
m: mass of the skydiver (here 80 kg) (kg); v: falling speed (m/s); t: time since leaving the plane, falling from rest (s); g: gravitational acceleration (m/s²); ρ: air density (here 1.225 kg/m^3, sea-level standard) (kg/m³); C_d A: drag coefficient times frontal area (here 0.45 m^2, belly-down) (m²)

Magnara

A spinning ball moving through air is pushed sideways, toward the side where its surface moves with the airflow. That Magnus force, proportional to spin crossed with velocity, makes spinning balls curve.

F_M = α ω×v
F_M: Magnus force on the spinning ball (N); α: empirical coefficient (depends on size, spin regime, fluid) (kg); ω: spin angular-velocity vector (rad/s); v: centre-of-mass velocity through the air (m/s); m: mass of the ball (kg); b: linear drag coefficient (kg/s)

Saltara

A fast, shallow flow can suddenly jump to a slow, deep one. Momentum conservation across the jump gives the ratio of the two depths in terms of the incoming Froude number, which compares flow speed with the speed of shallow-water waves.

Fr = v/√(gh)
Fr: Froude number: flow speed over the speed of shallow-water waves; v: depth-averaged flow speed (m/s); g: gravitational acceleration (m/s²); h: depth of the flow (m); h₁, h₂: depths just before and just after a planar jump in a straight channel (m); Fr₁: Froude number of the incoming flow

Orbitick

GPS clocks feel weaker gravity than clocks on the ground, so they run fast, and their orbital speed makes them run slightly slow. The net effect would build up timing errors quickly, so the satellite clocks are deliberately set to tick slightly slower before launch.

Δf/f ≈ ΔΦ/c² − v²/(2c²)
Δ f/f: fractional rate of a GPS satellite clock relative to a clock on the geoid (positive: it runs fast); ΔΦ: gravitational potential at the satellite minus the effective potential on the geoid, which includes Earth's rotation term (J/kg); v: orbital speed of the satellite (m/s); c: speed of light in vacuum (m/s); f₀: factory-specified frequency of the clock's 10.23 MHz reference before launch (Hz)

Lenzor

A changing magnetic flux induces a voltage, and the induced current always flows so as to oppose the change. A magnet dropped through a copper pipe therefore drives eddy currents that brake it, so it drifts down at a steady terminal speed.

ℰ = −dΦ_B/dt
mathcalE: EMF induced around one ring of the pipe (V); Φ_B: magnetic flux through that ring (Wb); t: time (s); m: mass of the falling magnet (kg); g: gravitational acceleration (m/s²); F_(eddy): upward drag from the eddy currents; it grows with the magnet's speed (N)

Spinwell

Angular momentum is moment of inertia times angular speed, and without an outside torque it stays constant. Pull your arms in and your moment of inertia drops, so you spin faster.

L = Iω
L: angular momentum about the spin axis (kg m² / s); I: moment of inertia about that axis (kg m²); ω: angular speed (rad/s); τ: external torque about the axis (N m)

Vortessa

Flow past a cylinder sheds vortices from alternate sides, at a frequency set by the flow speed divided by the cylinder's diameter, times the Strouhal number. Von Kármán showed that the staggered vortex street is stable only for a particular spacing ratio.

f = St·U/D, St ≈ 0.2
f: shedding frequency: full left-right cycles per second (Hz); St: Strouhal number, about 0.2 to 0.21 for Re from about 300 to 200,000 (lower near onset); U: speed of the oncoming flow (m/s); D: diameter of the cylinder (m); Re: Reynolds number; ν: kinematic viscosity of the fluid (m²/s)