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For small perturbations the surviving set has large measure.\u003c\/p\u003e\u003cp\u003eFor one model, the Chirikov standard map, Greene found numerically in 1979 that the last surviving rotational torus has the golden mean as its rotation number, and breaks at K ≈ 0.971635.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e|k · ω| ≥ γ \/ |k|^τ\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003ek: any nonzero integer vector\u003c\/li\u003e\n\u003cli\u003eω: frequency vector of the torus\u003c\/li\u003e\n\u003cli\u003eγ, τ: positive constants of the Diophantine condition\u003c\/li\u003e\n\u003cli\u003eGolden mean: (1 + √5)\/2 = 1.618…\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eKolmogorov 1954; Arnold, Russ. Math. Surveys 18(5), 9 (1963); Moser 1962 (area-preserving twist maps); Greene, J. Math. 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Summarising those measurements, Schnitzer, Visscher and Block (2000) put the motor's maximum load at about 5.5–7.5 pN.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eW = F × d = 6 pN × 8 nm = 48 pN·nm ≈ 12 k_BT\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eW: work per step (pN·nm)\u003c\/li\u003e\n\u003cli\u003eF: load force, near stall (pN)\u003c\/li\u003e\n\u003cli\u003ed: step size (nm)\u003c\/li\u003e\n\u003cli\u003ek_BT: thermal energy, 4.1 pN·nm at 298 K\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eSvoboda et al., Nature 365, 721 (1993); Visscher et al., Nature 400, 184 (1999); Schnitzer et al., Nat. Cell Biol. 2, 718 (2000); Yildiz et al., Science 303, 676 (2004).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The stall force is not always 6 pN; it depends on conditions. 48\/4.1 is 11.7, so “about 12 k_BT”.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Kinessa, the pixel-art character, with “8 NM PER STEP”. Back: the one-ATP step, the stall force, the work per step in k_BT and the 1993, 1999 and 2004 sources. 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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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eh\u003csub\u003e2\u003c\/sub\u003e ∝ h\u003csub\u003e1\u003c\/sub\u003e\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eh\u003csub\u003e1\u003c\/sub\u003e: height of the jar above the floor, the drop height (m)\u003c\/li\u003e\n\u003cli\u003eh\u003csub\u003e2\u003c\/sub\u003e: height of the fountain above the jar rim (m)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eBiggins and Warner argued that the pile of chain itself gives each link an upward push as it is picked up, an extra reaction force from the jar on top of the pull from the chain above. J. Pantaleone (Am. J. Phys. 85, 414, 2017) measured the fountain and supported a force from the container.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The mechanism is still debated. Later work argues that the upward “kicks” from the pile are not the main cause and that the chain's own inertia, its tension and the way it leaves the pile can drive the fountain (for example arXiv:1810.13008 and arXiv:1912.08682). The proportionality of the two heights is the measured result; the reason for it is not settled.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003eChainfount cradles a glowing jar as a white-gold bead chain arcs up out of it. Printed text: \u003cstrong\u003eCHAINFOUNT\u003c\/strong\u003e with the line \u003cstrong\u003eh1 ∝ h2\u003c\/strong\u003e. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"2\" × 2\" \/ White","offer_id":50963791675495,"sku":"10032179872111843333","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50963791708263,"sku":"22453231643325521438","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50963791741031,"sku":"12475949852883742093","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50963791773799,"sku":"26411628576487071996","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/13169392163644520325_1200.jpg?v=1793655208"},{"product_id":"janssena-silo-pressure","title":"Janssena – The Janssen Effect \/ Silo Pressure Saturates Physics Sticker","description":"\u003cp\u003eJanssena, a stone giant, guards a glowing bin of grain whose floor feels far less weight than the grain above it.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eIn 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).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ep(z) = p\u003csub\u003e∞\u003c\/sub\u003e(1 − e\u003csup\u003e−z\/λ\u003c\/sup\u003e), λ = R\/(2μK)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003ep(z): vertical pressure at depth z (Pa)\u003c\/li\u003e\n\u003cli\u003ep\u003csub\u003e∞\u003c\/sub\u003e: saturation pressure, ρgλ (Pa)\u003c\/li\u003e\n\u003cli\u003ez: depth below the grain surface (m)\u003c\/li\u003e\n\u003cli\u003eλ: saturation length (m)\u003c\/li\u003e\n\u003cli\u003eR: silo radius (m)\u003c\/li\u003e\n\u003cli\u003eμ: wall friction coefficient\u003c\/li\u003e\n\u003cli\u003eK: ratio of horizontal to vertical stress\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eThis is why an hourglass flows at a nearly steady rate whatever its fill height, and why silos are designed for wall loads rather than for a hydrostatic floor load.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The Janssen model assumes K is a constant and the friction is fully mobilised everywhere. Real silos, especially while being filled or emptied, can show sudden higher pressures.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003eJanssena, a stone giant, holds a glowing grain bin; the pressure curve itself is not drawn. Printed text: \u003cstrong\u003eJANSSENA\u003c\/strong\u003e with the line \u003cstrong\u003ep SATURATES\u003c\/strong\u003e. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"2\" × 2\" \/ White","offer_id":50964003258471,"sku":"13646733390686753444","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50964003291239,"sku":"18349300816672497031","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50964003324007,"sku":"88479958355741094851","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50964003356775,"sku":"31149168358270318792","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/13526934923732840832_1200.jpg?v=1793657918"},{"product_id":"meldea-melde-experiment","title":"Meldea – Melde’s Experiment \/ f = n√(T\/μ)\/(2L) Physics Sticker","description":"\u003cp\u003eMeldea, a young wave-tamer, holds a tuning fork that shakes a glowing string into waves.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eAround 1859 Franz Melde tied a string to the prong of a tuning fork and pulled it taut (Ann. Phys. 185, 193, 1860). At the right tensions the string broke into steady standing waves, with nodes that stood still and loops that blurred. Change the tension and the number of loops jumps.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ef = n√(T\/μ)\/(2L)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003ef: frequency of the string vibration (Hz)\u003c\/li\u003e\n\u003cli\u003en: number of loops\u003c\/li\u003e\n\u003cli\u003eT: tension (N)\u003c\/li\u003e\n\u003cli\u003eμ: mass per unit length (kg\/m)\u003c\/li\u003e\n\u003cli\u003eL: length of the string (m)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eWith the fork driving along the string, the string vibrates at half the fork's frequency; driven sideways, at the same frequency.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e Real strings stretch and have stiffness, so loops are not perfectly evenly spaced.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003eMeldea holds a tuning fork at one end of a glowing orange wavy string. Printed text: \u003cstrong\u003eMELDEA\u003c\/strong\u003e with the line \u003cstrong\u003ef = n√(T\/μ)\/(2L)\u003c\/strong\u003e. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"2\" × 2\" \/ White","offer_id":50964268744807,"sku":"20156624504413259799","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50964268777575,"sku":"30363279111021151857","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50964268810343,"sku":"24451687177238950452","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50964268843111,"sku":"31374344795563401959","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/6059535193102558249_1200.jpg?v=1793663764"},{"product_id":"adiabatix-adiabatic-invariant","title":"Adiabatix – The Adiabatic Invariant \/ J = ∮p dq Physics Sticker","description":"\u003cp\u003eAdiabatix, a white-bearded sage, slowly draws up the string of a glowing pendulum.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eAt 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eJ = ∮p dq\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eJ: action over one cycle (J·s)\u003c\/li\u003e\n\u003cli\u003ep: momentum\u003c\/li\u003e\n\u003cli\u003eq: position coordinate\u003c\/li\u003e\n\u003cli\u003e∮: integral over one full oscillation\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eFor a harmonic oscillator J = E\/f, which is why the old quantum rule set E = nhf.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e J is only approximately conserved; it holds better the slower the change compared with one period.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003eAdiabatix pinches a thin string from which a small glowing bob hangs. Printed text: \u003cstrong\u003eADIABATIX\u003c\/strong\u003e with the line \u003cstrong\u003eJ = ∮p dq\u003c\/strong\u003e. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"2\" × 2\" \/ White","offer_id":50964270219367,"sku":"18372733272431524979","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50964270252135,"sku":"25926292825269668903","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50964270284903,"sku":"17146217689410735086","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50964270317671,"sku":"27081779628282022638","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/10886878321522334611_1200.jpg?v=1793664030"},{"product_id":"classical-physics-1-sticker-sheet","title":"Classical Physics Vol. 1 Pixel Sticker Sheet – 10 Physics Stickers","description":"\u003cp\u003e10 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.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/adiabatix-adiabatic-invariant\"\u003eAdiabatix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAt 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eJ = ∮p dq\u003c\/strong\u003e\u003cbr\u003eJ: action over one cycle (J·s); p: momentum; q: position coordinate; ∮: integral over one full oscillation\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/teacupa-tea-leaf-paradox\"\u003eTeacupa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e∂p\/∂r = ρ v\u003csub\u003eθ\u003c\/sub\u003e²\/r\u003c\/strong\u003e\u003cbr\u003ep: pressure (Pa); r: distance from the axis (m); ρ: density of the water (kg\/m³); v\u003csub\u003eθ\u003c\/sub\u003e: swirl (azimuthal) speed (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/infrara-herschel-infrared\"\u003eInfrara\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eλ\u003csub\u003eIR\u003c\/sub\u003e \u0026gt; λ\u003csub\u003ered\u003c\/sub\u003e ≈ 0.7 µm\u003c\/strong\u003e\u003cbr\u003eλ\u003csub\u003eIR\u003c\/sub\u003e: infrared wavelengths, about 0.7 µm to 1 mm; λ\u003csub\u003ered\u003c\/sub\u003e: long-wavelength end of the visible spectrum\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/braessa-spring-paradox\"\u003eBraessa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTwo 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eseries: k\u003csub\u003eeff\u003c\/sub\u003e = k\/2   →   parallel: k\u003csub\u003eeff\u003c\/sub\u003e = 2k\u003c\/strong\u003e\u003cbr\u003ek: stiffness of each spring (N\/m); k\u003csub\u003eeff\u003c\/sub\u003e: effective stiffness of the support\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/bucketta-newtons-bucket\"\u003eBucketta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ez = ω²r²\/(2g)\u003c\/strong\u003e\u003cbr\u003ez: 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)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/chainfount-chain-fountain\"\u003eChainfount\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003ePile 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eh\u003csub\u003e2\u003c\/sub\u003e ∝ h\u003csub\u003e1\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eh\u003csub\u003e1\u003c\/sub\u003e: height of the jar above the floor, the drop height (m); h\u003csub\u003e2\u003c\/sub\u003e: height of the fountain above the jar rim (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/coriolix-coriolis-effect-pixel-sticker\"\u003eCoriolix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e\u003cb\u003ea\u003c\/b\u003e′ = −2\u003cb\u003eΩ\u003c\/b\u003e×\u003cb\u003ev\u003c\/b\u003e′ − \u003cb\u003eΩ\u003c\/b\u003e×(\u003cb\u003eΩ\u003c\/b\u003e×\u003cb\u003er\u003c\/b\u003e′)\u003c\/strong\u003e\u003cbr\u003ea', 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)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/dalembertia-zero-drag\"\u003eDalembertia\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eφ = U (r + a³\/2r²) cos θ   ⇒   F\u003csub\u003edrag\u003c\/sub\u003e = 0\u003c\/strong\u003e\u003cbr\u003eφ: 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\u003csub\u003edrag\u003c\/sub\u003e: net force along the flow (N)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/janssena-silo-pressure\"\u003eJanssena\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 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).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ep(z) = p\u003csub\u003e∞\u003c\/sub\u003e(1 − e\u003csup\u003e−z\/λ\u003c\/sup\u003e), λ = R\/(2μK)\u003c\/strong\u003e\u003cbr\u003ep(z): vertical pressure at depth z (Pa); p\u003csub\u003e∞\u003c\/sub\u003e: 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\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/libratrix-lagrange-points-pixel-sticker\"\u003eLibratrix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTwo 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003er\u003csub\u003eL2\u003c\/sub\u003e ≈ R(M\u003csub\u003e2\u003c\/sub\u003e\/(3M\u003csub\u003e1\u003c\/sub\u003e))\u003csup\u003e1\/3\u003c\/sup\u003e\u003c\/strong\u003e\u003cbr\u003er_(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)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50964877443175,"sku":"33317769571062772207","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/3764902297399688554_2048.jpg?v=1793674501"},{"product_id":"classical-physics-2-sticker-sheet","title":"Classical Physics Vol. 2 Pixel Sticker Sheet – 10 Physics Stickers","description":"\u003cp\u003e10 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Classical Physics Vol. 2. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/magnara-magnus-effect-pixel-sticker\"\u003eMagnara\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e\u003cb\u003eF\u003c\/b\u003e\u003csub\u003eM\u003c\/sub\u003e = α \u003cb\u003eω\u003c\/b\u003e×\u003cb\u003ev\u003c\/b\u003e\u003c\/strong\u003e\u003cbr\u003eF_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)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/meldea-melde-experiment\"\u003eMeldea\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAround 1859 Franz Melde tied a string to the prong of a tuning fork and pulled it taut (Ann. Phys. 185, 193, 1860). At the right tensions the string broke into steady standing waves, with nodes that stood still and loops that blurred. Change the tension and the number of loops jumps.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ef = n√(T\/μ)\/(2L)\u003c\/strong\u003e\u003cbr\u003ef: frequency of the string vibration (Hz); n: number of loops; T: tension (N); μ: mass per unit length (kg\/m); L: length of the string (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/pendulux-foucault-pendulum-pixel-sticker\"\u003ePendulux\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA pendulum keeps swinging in a fixed plane while the Earth turns beneath it. The swing plane appears to rotate at the Earth's rotation rate times the sine of the latitude: once a sidereal day at the poles, and not at all on the equator.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eω\u003csub\u003eprec\u003c\/sub\u003e = Ω\u003csub\u003e⊕\u003c\/sub\u003e sin φ\u003c\/strong\u003e\u003cbr\u003eω_(prec): rate at which the swing plane turns relative to the floor (clockwise seen from above in the Northern Hemisphere) (°\/h); Ω_(oplus): Earth's rotation rate relative to the stars: one turn per sidereal day of 23.93 h (°\/h); φ: latitude (°)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/spinwell-angular-momentum-pixel-sticker\"\u003eSpinwell\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAngular 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eL = Iω\u003c\/strong\u003e\u003cbr\u003eL: 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)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/terminara-terminal-velocity-pixel-sticker\"\u003eTerminara\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003em dv\/dt = mg − ½ρC\u003csub\u003ed\u003c\/sub\u003eAv²\u003c\/strong\u003e\u003cbr\u003em: 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)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/barkhausa-barkhausen-noise\"\u003eBarkhausa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eMagnetise a piece of iron slowly and smoothly and its magnetisation does not change smoothly. It grows in many tiny, sudden jumps as the walls between magnetic domains snag on defects and then snap free. Heinrich Barkhausen heard this in 1919 by winding a coil around the iron and amplifying the induced pulses into a telephone receiver: a crackle (“Zwei mit Hilfe der neuen Verstärker entdeckte Erscheinungen”, Physikalische Zeitschrift 20, 401, 1919). Each jump induces a small voltage pulse in the coil.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eV = −N dΦ\/dt\u003c\/strong\u003e\u003cbr\u003eV: voltage induced in the pick-up coil (V); N: number of turns of the coil; Φ: magnetic flux through the coil (Wb); t: time (s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/displacera-displacement-current\"\u003eDisplacera\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAmpère's law broke down at a charging capacitor: current flows in the wire but not across the gap. James Clerk Maxwell fixed it in 1861–65 by adding a term for the changing electric field in the gap, which makes a magnetic field just as a real current does (“A Dynamical Theory of the Electromagnetic Field”, Phil. Trans. 155, 459, 1865). With it, the equations predicted electromagnetic waves travelling at the speed of light.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eJ\u003csub\u003ed\u003c\/sub\u003e = ε\u003csub\u003e0\u003c\/sub\u003e ∂E\/∂t\u003c\/strong\u003e\u003cbr\u003eJ\u003csub\u003ed\u003c\/sub\u003e: displacement current density (A\/m²); ε\u003csub\u003e0\u003c\/sub\u003e: permittivity of free space (F\/m); ∂E\/∂t: rate of change of the electric field (V m⁻¹ s⁻¹)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/faradra-faraday-rotation\"\u003eFaradra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn September 1845 Michael Faraday sent polarised light through a block of heavy lead-borate glass placed between the poles of a strong electromagnet, with the field along the beam. When the magnet was switched on, the plane of polarisation turned. It was the first experimental link between light and magnetism. He reported it in the Nineteenth Series of his Experimental Researches in Electricity (Phil. Trans. R. Soc. 136, 1, 1846). The rotation angle grows with the field and with the path length.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eβ = VBd\u003c\/strong\u003e\u003cbr\u003eβ: rotation angle of the plane of polarisation (rad); V: Verdet constant of the material (rad T⁻¹ m⁻¹); B: magnetic flux density along the beam (tesla, T); d: path length through the material (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/galvana-mirror-galvanometer\"\u003eGalvana\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eWilliam Thomson, later Lord Kelvin, developed the mirror galvanometer around the 1858 transatlantic telegraph cable, whose signals were too faint for ordinary needles. A tiny mirror carrying small magnets hangs inside a coil; a current twists it slightly, and a beam of light reflected from it sweeps across a scale. By the law of reflection, turning a mirror by an angle θ turns the reflected beam by twice that angle.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ebeam deflection = 2θ\u003c\/strong\u003e\u003cbr\u003eθ: rotation angle of the mirror (rad); 2θ: rotation angle of the reflected beam (rad)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/inertix-tolman-stewart\"\u003eInertix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eSpin a coil of wire fast and stop it suddenly: the free charges inside keep going for a moment, like passengers in a braking bus, and a pulse of current flows. Richard Tolman and Thomas Stewart measured it in 1916 (Phys. Rev. 8, 97). From the charge that flowed they got the carriers' charge-to-mass ratio, close to that of the electron, showing that current in metals is carried by electrons.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eQ = (m\/e) v\u003csub\u003e0\u003c\/sub\u003e L \/ R\u003c\/strong\u003e\u003cbr\u003eQ: charge that flows after stopping (C); m\/e: mass-to-charge ratio of the carriers (kg\/C); v\u003csub\u003e0\u003c\/sub\u003e: rim speed of the wire before stopping (m\/s); L: length of the wire (m); R: resistance of the coil (Ω)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966844276839,"sku":"28301990384207583364","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/5152942607117677361_2048.jpg?v=1793676395"}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/collections\/classical-mechanics.png?v=1793678000","url":"https:\/\/pixelatedphysics.com\/ko\/collections\/classical-mechanics\/domain:forces.oembed","provider":"Pixelated Physics","version":"1.0","type":"link"}