{"product_id":"ehrenwheel-the-disk-that-cant-stay-flat-physics-t-shirt","title":"Ehrenwheel – The Disk That Can't Stay Flat Physics T-Shirt","description":"\u003cp\u003eEhrenwheel spins her discus, and its rim ripples as it turns.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eSpin a disk. Seen from the lab, each rim element moves along its own length and is Lorentz-contracted, while each radial element moves sideways and is not. So a body cannot be spun up from rest and stay Born-rigid. On a disk already turning steadily, co-rotating rulers laid along the rim are contracted as seen from the lab, so more of them fit: the rim measures more than 2πR while the radius still measures R. Geometry on the turning disk is not Euclidean.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eC′ = 2πR \/ √(1 − ω²R²\/c²)  \u0026gt;  2πR   (co-rotating rulers)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eC′: rim circumference measured with co-rotating rulers (m)\u003c\/li\u003e\n\u003cli\u003eR: disk radius (m)\u003c\/li\u003e\n\u003cli\u003eω: angular speed of the disk (rad\/s)\u003c\/li\u003e\n\u003cli\u003ec: speed of light, 299 792 458 m\/s (exact)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003ePaul Ehrenfest posed the puzzle in Physikalische Zeitschrift 10, 918 (1909).\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e How to interpret the spinning disk's geometry was argued over for decades. The \"more rulers fit\" result is now standard, but finer points of interpretation are still discussed.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Ehrenwheel, the pixel-art character, with the title. Back: the rim and radial rulers, the circumference formula, the Born-rigidity result and Ehrenfest's 1909 paper. Printed on a unisex tee.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"S \/ White","offer_id":50960826990695,"sku":"22067833997013228035","price":29.99,"currency_code":"USD","in_stock":true},{"title":"S \/ Black","offer_id":50960827023463,"sku":"95856947282112740551","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ White","offer_id":50960827056231,"sku":"18097731012605962052","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ Black","offer_id":50960827088999,"sku":"30509276845831767228","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ White","offer_id":50960827121767,"sku":"32210444248381588960","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ Black","offer_id":50960827154535,"sku":"13284773459098170255","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ White","offer_id":50960827187303,"sku":"13739116625723589018","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ Black","offer_id":50960827220071,"sku":"36892317689299078395","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ White","offer_id":50960827252839,"sku":"15883264808045382896","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ Black","offer_id":50960827285607,"sku":"77815059285073500183","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ White","offer_id":50960827318375,"sku":"14672283979814255675","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ Black","offer_id":50960827351143,"sku":"23012849143775129215","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ White","offer_id":50960827383911,"sku":"20060918562650361350","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ Black","offer_id":50960827416679,"sku":"16353662961984651977","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ White","offer_id":50960827449447,"sku":"21920692290435574327","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ Black","offer_id":50960827482215,"sku":"31621360463962783234","price":29.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/ehrenfest-paradox_front_16_white.png?v=1793628596","url":"https:\/\/pixelatedphysics.com\/ja\/products\/ehrenwheel-the-disk-that-cant-stay-flat-physics-t-shirt","provider":"Pixelated Physics","version":"1.0","type":"link"}