{"title":"Posters","description":"\u003cp\u003ePixel-art physics posters, one plate per experiment, law or constant. Each plate carries its key equation, a facts panel and sourced physics notes.\u003c\/p\u003e\n\u003cp\u003eEach physics poster is a numbered plate on one experiment, law or constant, from Kundt's tube and Ørsted's needle to Mercury at 4.2 K and the Mauna Loa CO₂ record. The facts panel gives the key numbers and their sources, so the wall art is accurate as well as good-looking. Physics posters make a good gift for classrooms, labs, offices and student rooms.\u003c\/p\u003e","products":[{"product_id":"katers-reversible-pendulum-plate-001-physics-poster","title":"Kater's Reversible Pendulum – Plate 001 Physics Poster","description":"\u003cp\u003ePlate 1 speaks for itself: \u003cem\u003e“Flip it. Same beat. That's g.”\u003c\/em\u003e\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eA compound pendulum with two knife-edges swings with the same period about either edge only when the edge separation equals the length of the equivalent simple pendulum. Kater used this to measure g without needing the centre of mass or the moment of inertia.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eg = 4π² (h₁ + h₂) \/ T²   (when T₁ = T₂ = T)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eg: local gravitational acceleration, m s⁻²\u003c\/li\u003e\n\u003cli\u003eh₁, h₂: distances from the centre of mass to each knife-edge, m\u003c\/li\u003e\n\u003cli\u003eT: common period about either edge, s\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eWhen both edges beat alike, the gap is the length. London, 1817–18: standard g = 9.80665 m s⁻² gives T ≈ 2.006 s for a 1 m equivalent length.\u003c\/p\u003e\u003ch3\u003eSources\u003c\/h3\u003e\u003cul\u003e\u003cli\u003eH. Kater, Phil. Trans. R. Soc. 108, 33 (1818).\u003c\/li\u003e\u003c\/ul\u003e\u003ch3\u003eOn the poster\u003c\/h3\u003e\u003cp\u003ePlate 001, “Kater's Reversible Pendulum”: the pixel-art plate with its slogan “Flip it. Same beat. That's g.”, the key equation and the facts panel shown above. Pixelated Physics. Physics. 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Answers sideways.”\u003c\/em\u003e\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eA symmetric top spinning with one point fixed feels gravity as a torque that turns its axis sideways (precession) with a small nodding (nutation). In Euler angles the Lagrangian contains neither φ nor ψ, so the matching momenta are conserved.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eL = (I₁\/2)(θ̇² + φ̇² sin²θ) + (I₃\/2)(ψ̇ + φ̇ cos θ)² − Mgℓ cos θ\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eL: Lagrangian, J\u003c\/li\u003e\n\u003cli\u003eI₁, I₃: moments of inertia about a transverse axis through the fixed point and about the symmetry axis, kg m²\u003c\/li\u003e\n\u003cli\u003eθ, φ, ψ: Euler angles: tilt, precession, spin, rad\u003c\/li\u003e\n\u003cli\u003eM: mass, kg\u003c\/li\u003e\n\u003cli\u003eg: gravitational acceleration, m s⁻²\u003c\/li\u003e\n\u003cli\u003eℓ: distance from fixed point to centre of mass, m\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eφ and ψ do not appear, so p_φ and p_ψ (kg m² s⁻¹) are conserved.\u003c\/p\u003e\u003ch3\u003eSources\u003c\/h3\u003e\u003cul\u003e\u003cli\u003eH. Goldstein, C. Poole, J. Safko, Classical Mechanics, 3rd ed., §5.7.\u003c\/li\u003e\u003c\/ul\u003e\u003ch3\u003eOn the poster\u003c\/h3\u003e\u003cp\u003ePlate 002, “The Heavy Top”: the pixel-art plate with its slogan “Pulled down. Answers sideways.”, the key equation and the facts panel shown above. Pixelated Physics. Physics. 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Atwood used it to test Newton's laws of uniformly accelerated motion.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ea = (m₁ − m₂) g \/ (m₁ + m₂)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003ea: acceleration of the masses, m s⁻²\u003c\/li\u003e\n\u003cli\u003em₁, m₂: the two masses, kg\u003c\/li\u003e\n\u003cli\u003eg: gravitational acceleration, m s⁻²\u003c\/li\u003e\n\u003cli\u003eT: string tension, N\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eTension T = 2 m₁ m₂ g \/ (m₁ + m₂). Cambridge, 1784.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The formula assumes a massless, frictionless pulley and string.\u003c\/p\u003e\u003ch3\u003eSources\u003c\/h3\u003e\u003cul\u003e\u003cli\u003eG. Atwood, A Treatise on the Rectilinear Motion and Rotation of Bodies (Cambridge, 1784).\u003c\/li\u003e\u003c\/ul\u003e\u003ch3\u003eOn the poster\u003c\/h3\u003e\u003cp\u003ePlate 003, “Atwood's Machine”: the pixel-art plate with its slogan “Gravity, slowed down enough to read.”, the key equation and the facts panel shown above. Pixelated Physics. Physics. 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For a central potential the Lagrangian does not depend on the angle φ, so the angular momentum m r²φ̇ is conserved; Kepler's equal areas in equal times is that promise kept.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ed\/dt (m r² φ̇) = ∂L\/∂φ = 0\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003em: mass, kg\u003c\/li\u003e\n\u003cli\u003er: radial distance, m\u003c\/li\u003e\n\u003cli\u003eφ: polar angle, rad\u003c\/li\u003e\n\u003cli\u003eφ̇: angular velocity, rad s⁻¹\u003c\/li\u003e\n\u003cli\u003eL: Lagrangian (1\/2)m(ṙ² + r²φ̇²) − V(r), J\u003c\/li\u003e\n\u003cli\u003eV(r): central potential energy, J\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eL lacks φ. Equal areas in equal times.\u003c\/p\u003e\u003ch3\u003eSources\u003c\/h3\u003e\u003cul\u003e\u003cli\u003eE. Noether, Nachr. Ges. Wiss. Göttingen, 235 (1918).\u003c\/li\u003e\u003c\/ul\u003e\u003ch3\u003eOn the poster\u003c\/h3\u003e\u003cp\u003ePlate 004, “Noether's Ledger”: the pixel-art plate with its slogan “Every symmetry keeps a promise.”, the key equation and the facts panel shown above. Pixelated Physics. Physics. 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The spring weakly couples the two motions, so when their frequencies are tuned equal the energy swaps slowly between bobbing and twisting, like two coupled pendulums beating.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ef_T = (1\/2π)√(k\/m) ≈ f_R = (1\/2π)√(κ\/I)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003ef_T, f_R: translational and rotational frequencies, Hz\u003c\/li\u003e\n\u003cli\u003ek: spring constant, N m⁻¹\u003c\/li\u003e\n\u003cli\u003em: bob mass, kg\u003c\/li\u003e\n\u003cli\u003eκ: torsion constant, N m rad⁻¹\u003c\/li\u003e\n\u003cli\u003eI: bob moment of inertia about the spring axis, kg m²\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eTuned so f_T ≈ f_R, the energy swaps from bobbing to twisting and back. Wilberforce, 1894.\u003c\/p\u003e\u003ch3\u003eSources\u003c\/h3\u003e\u003cul\u003e\n\u003cli\u003eL. R. Wilberforce, Phil. Mag. 38, 386 (1894).\u003c\/li\u003e\n\u003cli\u003eR. E. Berg, T. S. Marshall, Am. J. 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Faster. Then chaos.”\u003c\/em\u003e\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eReynolds injected a thin dye streak into water flowing through a glass tube. At low speed the streak stayed straight (laminar); past a critical value of one dimensionless number it broke up into eddies (turbulent).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eRe = ρ u D \/ μ\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eRe: Reynolds number, dimensionless\u003c\/li\u003e\n\u003cli\u003eρ: fluid density, kg m⁻³\u003c\/li\u003e\n\u003cli\u003eu: mean flow speed, m s⁻¹\u003c\/li\u003e\n\u003cli\u003eD: pipe diameter, m\u003c\/li\u003e\n\u003cli\u003eμ: dynamic viscosity, Pa s\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eLower critical value ≈ 2000; engineering onset ≈ 2300. 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