{"product_id":"braessa-spring-paradox","title":"Braessa – Braess’s Paradox with Springs \/ Cut It, It Rises Physics Sticker","description":"\u003cp\u003eBraessa, a trickster rogue of the Paradox Parlour, cuts the middle string, and the hanging weight goes up.\u003c\/p\u003e\u003ch3\u003eThe Physics\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\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003ek: stiffness of each spring (N\/m)\u003c\/li\u003e\n\u003cli\u003ek\u003csub\u003eeff\u003c\/sub\u003e: effective stiffness of the support\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eThis is the mechanical analogue of Braess's paradox from traffic networks (Dietrich Braess, 1968), where adding a road can slow everyone down. The spring and string version is by Joel E. Cohen and Paul Horowitz (Nature 352, 699, 1991), who also gave an electrical version.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e No energy rule is broken. The rise depends on the string lengths. The weight rises because the support becomes stiffer, and the energy comes from rearranging the springs' stored energy.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003eBraessa holds the blade that cuts the string, with a glowing arrow pointing up. Printed text: \u003cstrong\u003eBRAESSA\u003c\/strong\u003e with the line \u003cstrong\u003eCUT IT, IT RISES\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":50963658244199,"sku":"25857580215884059501","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50963658276967,"sku":"50359105463427221890","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50963658309735,"sku":"23958554426126811336","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50963658342503,"sku":"20950758754173600962","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/15399159310832802490_1200.jpg?v=1793665287","url":"https:\/\/pixelatedphysics.com\/fr\/products\/braessa-spring-paradox","provider":"Pixelated Physics","version":"1.0","type":"link"}