Classical Physics Vol. 1 Pixel Sticker Sheet

10 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Classical Physics Vol. 1. Every character below has its own sticker page with the full physics panel.

Adiabatix

At the 1911 Solvay conference Lorentz asked what happens to a pendulum if you shorten its string very slowly. Einstein answered: the energy changes, but the ratio of energy to frequency stays the same. Paul Ehrenfest built this into a principle (1916): for slow changes, the action J stays constant. That is why quantum numbers stay put when conditions change gently.

J = ∮p dq
J: action over one cycle (J·s); p: momentum; q: position coordinate; ∮: integral over one full oscillation

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)

Infrara

In 1800 William Herschel placed thermometers in the colours of a prism spectrum and found the temperature rose from violet to red. Then he moved one past the red end, where no light could be seen, and it rose further still (“Experiments on the Refrangibility of the Invisible Rays of the Sun”, Phil. Trans. R. Soc. 90, 284). The Sun emits invisible rays beyond the red: infrared.

λ_IR > λ_red ≈ 0.7 µm
λ_IR: infrared wavelengths, about 0.7 µm to 1 mm; λ_red: long-wavelength end of the visible spectrum

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

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)

Chainfount

Pile a long metal bead chain in a jar, pull one end over the rim and let it drop to the floor. The chain then pours itself out, and instead of sliding over the rim it leaps up into a self-supporting arch above the jar. It is also known as the Mould effect, after the video that made it famous. John Biggins and Mark Warner (Proc. R. Soc. A 470, 20130689, 2014) predicted and measured that the arch height is proportional to the drop height.

h₂ ∝ h₁
h₁: height of the jar above the floor, the drop height (m); h₂: height of the fountain above the jar rim (m)

Coriolix

In a rotating frame, moving objects appear to be pushed sideways by the Coriolis term, −2Ω × v. Something moving in a straight line as seen from outside traces a curve as seen from the rotating platform.

a′ = −2Ω×v′ − Ω×(Ω×r′)
a', v', r': acceleration, velocity and position seen on the turning disc (m/s², m/s, m); Ω: angular velocity of the disc (anticlockwise) (rad/s); v: speed of the ball, thrown from the centre (m/s); t: time since the throw (s); R: radius of the disc (m); Δφ: angle by which the ball misses the flag at the rim (rad)

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)

Janssena

In a tall column of water, pressure at the bottom keeps rising with depth. In a silo full of grain it does not. The grains push sideways on the walls, and friction with the walls carries part of the weight. Below a few silo diameters, the pressure at the bottom levels off. H. A. Janssen measured this with grain in model silos (“Versuche über Getreidedruck in Silozellen”, Zeitschrift des Vereines Deutscher Ingenieure 39, 1045, 1895).

p(z) = p_∞(1 − e^−z/λ), λ = R/(2μK)
p(z): vertical pressure at depth z (Pa); p_∞: saturation pressure, ρgλ (Pa); z: depth below the grain surface (m); λ: saturation length (m); R: silo radius (m); μ: wall friction coefficient; K: ratio of horizontal to vertical stress

Libratrix

Two bodies orbiting each other have five points where a small third body can keep pace with them. For the Sun and Earth, L2 lies a little beyond Earth, and the triangular points L4 and L5 are stable only when one mass greatly outweighs the other.

r_L2 ≈ R(M₂/(3M₁))^1/3
r_(L2): distance from the smaller body to L2 (second line: Sun–Earth L2, beyond Earth) (m); R: distance between the two bodies (m); M₁, M₂: masses of the larger and the smaller body (kg); M₁/M₂: mass ratio; above about 24.96, L4 and L5 are stable (Routh)