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The gravity, twist and period relations below are the standard torsion-balance equations.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eF = G M m \/ r²,   κθ = F L,   T = 2π √(I \/ κ)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eF: gravitational force (N)\u003c\/li\u003e\n\u003cli\u003eG: gravitational constant (m³ kg⁻¹ s⁻²)\u003c\/li\u003e\n\u003cli\u003eM, m: large and small masses (kg)\u003c\/li\u003e\n\u003cli\u003er: centre-to-centre distance (m)\u003c\/li\u003e\n\u003cli\u003eκ: torsion constant (N·m\/rad)\u003c\/li\u003e\n\u003cli\u003eθ: twist angle (rad)\u003c\/li\u003e\n\u003cli\u003eL: lever arm (m)\u003c\/li\u003e\n\u003cli\u003eT: swing period (s)\u003c\/li\u003e\n\u003cli\u003eI: moment of inertia (kg·m²)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eHenry Cavendish's paper was read to the Royal Society on 21 June 1798, using a torsion balance devised by John Michell. He printed Earth's density as 5.48 times water; his own data give 5.448, and today's value is about 5.51.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e He reported Earth's density, not G. Today's G (CODATA 2022) is 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻².\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Cavenda, the pixel-art character, with the title. Back: the three torsion-balance equations, Earth's density then and now, today's G and the 1798 paper. Printed on a unisex tee.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"S \/ White","offer_id":50936682250343,"sku":"37795753849947355436","price":29.99,"currency_code":"USD","in_stock":true},{"title":"S \/ Black","offer_id":50936682283111,"sku":"13090683425263620724","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ White","offer_id":50936682315879,"sku":"26289388106781649872","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ Black","offer_id":50936682348647,"sku":"12767194343473219221","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ White","offer_id":50936682381415,"sku":"32342451147733937124","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ Black","offer_id":50936682414183,"sku":"13307789072658842258","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ White","offer_id":50936682446951,"sku":"13082477554228574138","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ Black","offer_id":50936682479719,"sku":"16692024027180265100","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ White","offer_id":50936682512487,"sku":"17866232577370615908","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ Black","offer_id":50936682545255,"sku":"16745689790209441435","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ White","offer_id":50936682578023,"sku":"17605386382258146845","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ Black","offer_id":50936682610791,"sku":"72284303915766614462","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ White","offer_id":50936682643559,"sku":"32482620619026896891","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ Black","offer_id":50936682676327,"sku":"10582135256456441440","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ White","offer_id":50936682709095,"sku":"26086854514066238291","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ Black","offer_id":50936682741863,"sku":"19374001270203333237","price":29.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/cavendish-1798_front_01_white.png?v=1793595905"},{"product_id":"mossbra-the-mossbauer-effect-physics-t-shirt","title":"Mossbra – The Mössbauer Effect Physics T-Shirt","description":"\u003cp\u003eMossbra, armoured in crystal, looses a gamma arrow and does not rock back at all. The whole lattice takes the kick for her.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eA free nucleus that emits a gamma ray recoils and steals a little energy, so the photon misses the absorber's resonance. For iron-57's 14.4 keV line the recoil is about 2.0 × 10⁻³ eV, while the natural linewidth is only about 4.6 × 10⁻⁹ eV: the miss is roughly 400 000 linewidths.\u003c\/p\u003e\u003cp\u003eRudolf Mössbauer found in 1958, working with iridium-191, that a nucleus bound in a crystal can emit and absorb without recoil. The whole lattice takes the momentum, and with some probability no phonon is created. That gives a line of fractional width about 3 × 10⁻¹³, which made Mössbauer spectroscopy possible.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eE_R = E_γ² \/ (2Mc²)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eE_R: recoil energy of a free nucleus (eV)\u003c\/li\u003e\n\u003cli\u003eE_γ: gamma-ray energy (eV)\u003c\/li\u003e\n\u003cli\u003eM: mass of the nucleus (kg)\u003c\/li\u003e\n\u003cli\u003ec: speed of light (m\/s)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eMössbauer, Z. Phys. 151, 124 (1958). He shared the 1961 Nobel Prize in Physics for this discovery.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e Only a fraction of the gammas are recoil-free, not all of them. Even then, the crystal does recoil, just by a negligible amount, because it is so heavy.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Mossbra, the pixel-art character, with “NO RECOIL”. Back: the recoil formula, the iron-57 energy scales, the recoil-free crystal note and Mössbauer's 1958 paper. Printed on a unisex tee.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"S \/ White","offer_id":50960505340007,"sku":"22080069632733529928","price":29.99,"currency_code":"USD","in_stock":true},{"title":"S \/ Black","offer_id":50960505372775,"sku":"18682114461960211925","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ White","offer_id":50960505405543,"sku":"17816147222786280082","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ Black","offer_id":50960505438311,"sku":"22507863889991845611","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ White","offer_id":50960505471079,"sku":"19432604892048967509","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ Black","offer_id":50960505503847,"sku":"13295231039803914838","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ White","offer_id":50960505536615,"sku":"31457905340039369825","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ Black","offer_id":50960505569383,"sku":"16464241510723590195","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ White","offer_id":50960505602151,"sku":"21528728369061799935","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ Black","offer_id":50960505634919,"sku":"69182731222062708842","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ White","offer_id":50960505667687,"sku":"28935481259523774191","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ Black","offer_id":50960505700455,"sku":"17152973693257043763","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ White","offer_id":50960505733223,"sku":"15272189569026779188","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ Black","offer_id":50960505765991,"sku":"25153918117858663841","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ White","offer_id":50960505798759,"sku":"19273550696683064389","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ Black","offer_id":50960505831527,"sku":"15912267263844435409","price":29.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/mossbauer-effect_front_01_white.png?v=1793622513"},{"product_id":"glueba-yang-mills-and-the-mass-gap-physics-t-shirt","title":"Glueba – Yang–Mills and the Mass Gap Physics T-Shirt","description":"\u003cp\u003eGlueba is a knot of looped gluon flux: made of massless threads, and yet, it seems, heavy.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eYang and Mills wrote down gauge theories with non-commuting symmetry groups in 1954. Their quanta are massless at the classical level and interact with each other. In quantum theory, pure SU(3) Yang–Mills (QCD without quarks) is believed to confine and to have a mass gap: the lightest excitation has strictly positive mass.\u003c\/p\u003e\u003cp\u003eNobody has proved it. Lattice computer simulations support the gap: Morningstar and Peardon (1999) put the lightest pure-gauge glueball near 1.7 GeV.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eProve: quantum Yang–Mills exists on R⁴ and has a mass gap Δ \u0026gt; 0\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eR⁴: four-dimensional spacetime\u003c\/li\u003e\n\u003cli\u003eΔ: mass of the lightest excitation above the vacuum\u003c\/li\u003e\n\u003cli\u003eGauge group G: any compact simple group (Clay statement)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eYang \u0026amp; Mills, Phys. Rev. 96, 191 (1954). The Clay Mathematics Institute listed the problem among its Millennium Prize Problems in 2000, in a statement by Jaffe and Witten.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e This problem is UNSOLVED. The 1.7 GeV figure is a lattice estimate for a theory with no quarks, not a proof and not the gap itself. Glueballs in the real world, with quarks present, have not been conclusively identified.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Glueba, the pixel-art character, with “MIND THE GAP”. Back: the Clay problem statement, the lattice glueball hint, the UNSOLVED label and Yang \u0026amp; Mills 1954. 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