{"id":704,"date":"2026-10-04T10:55:17","date_gmt":"2026-10-04T09:55:17","guid":{"rendered":"https:\/\/wpcalc.com\/pl\/fizyka\/kalkulator-energii-kinetycznej-obrotowej\/"},"modified":"2026-10-04T10:55:17","modified_gmt":"2026-10-04T09:55:17","slug":"kalkulator-energii-kinetycznej-obrotowej","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pl\/fizyka\/kalkulator-energii-kinetycznej-obrotowej\/","title":{"rendered":"Kalkulator energii kinetycznej obrotowej"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Oblicz energi\u0119 kinetyczn\u0105 obracaj\u0105cego si\u0119 obiektu lub ko\u0142a zamachowego<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"704\" data-favorite-title=\"Energia kinetyczna obrotowa\" data-favorite-url=\"https:\/\/wpcalc.com\/pl\/fizyka\/kalkulator-energii-kinetycznej-obrotowej\/\" data-favorite-excerpt=\"Oblicz energi\u0119 kinetyczn\u0105 obrotow\u0105 wiruj\u0105cego obiektu lub ko\u0142a zamachowego na podstawie momentu bezw\u0142adno\u015bci i pr\u0119dko\u015bci k\u0105towej albo z masy i promienia pe\u0142nego dysku. Opcjonalnie dodaj energi\u0119 kinetyczn\u0105 post\u0119pow\u0105 dla obiekt\u00f3w tocz\u0105cych si\u0119.\" aria-label=\"Dodaj do ulubionych\" data-favorite-icon=\"icon icon-physic\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/pl\/fizyka\/kalkulator-energii-kinetycznej-obrotowej\/\" method=\"POST\" class=\"calculator\" id=\"calculator-704\" data-post=\"704\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Pola danych<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Tryb wprowadzania danych<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"tryb_wprowadzania_danych\"><option value=\"disk\" selected>Masa i promie\u0144 (pe\u0142ny dysk \/ ko\u0142o zamachowe)<\/option><option value=\"iomega\">Moment bezw\u0142adno\u015bci i pr\u0119dko\u015b\u0107 k\u0105towa<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field is-single has-term\" id=\"input-2\" hidden><label for=\"field-2\">Moment bezw\u0142adno\u015bci (I)<\/label><div class=\"term\">I<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"moment_bezwadnoci_i\" id=\"field-2\" aria-describedby=\"calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 2.5\" min=\"0\" value=\"2.5\"\/><div class=\"grid grid-cols-1\"><select id=\"field-3\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"kgm2\" selected>kg\u00b7m\u00b2<\/option><option value=\"lbft2\">lb\u00b7ft\u00b2<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-2\">Op\u00f3r cia\u0142a wobec obrotu. Dla typowych bry\u0142: pe\u0142ny dysk \u00bdmr\u00b2, pier\u015bcie\u0144 mr\u00b2, pe\u0142na kula \u2156mr\u00b2, pr\u0119t wzgl\u0119dem \u015brodka \u215f\u2081\u2082ml\u00b2.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-3\" hidden><label for=\"field-4\">Pr\u0119dko\u015b\u0107 k\u0105towa (\u03c9)<\/label><div class=\"term\">\u03c9<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"prdko_ktowa_\" id=\"field-4\" aria-describedby=\"calc-info-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 3000\" min=\"0\" value=\"3000\"\/><div class=\"grid grid-cols-1\"><select id=\"field-5\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"rpm\" selected>rpm<\/option><option value=\"rads\">rad\/s<\/option><option value=\"degs\">deg\/s<\/option><option value=\"hz\">rev\/s (Hz)<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-4\">Jak szybko obraca si\u0119 obiekt. Znak jest ignorowany \u2014 kierunek nie zmienia energii.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-4\"><label for=\"field-6\">Masa (m)<\/label><div class=\"term\">m<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"masa_m\" id=\"field-6\" aria-describedby=\"calc-info-6\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 20\" min=\"0\" value=\"20\"\/><div class=\"grid grid-cols-1\"><select id=\"field-7\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"kg\" selected>kg<\/option><option value=\"lb\">lb<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-6\">Ca\u0142kowita masa dysku lub ko\u0142a zamachowego.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-5\"><label for=\"field-8\">Promie\u0144 (r)<\/label><div class=\"term\">r<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"promie_r\" id=\"field-8\" aria-describedby=\"calc-info-8\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 0.5\" min=\"0\" value=\"0.5\"\/><div class=\"grid grid-cols-1\"><select id=\"field-9\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"m\" selected>m<\/option><option value=\"cm\">cm<\/option><option value=\"ft\">ft<\/option><option value=\"in\">in<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-8\">Promie\u0144 zewn\u0119trzny dysku (po\u0142owa \u015brednicy).<\/small><\/div><div class=\"is-single field\" id=\"input-6\" hidden><span class=\"font-semibold\">Ruch post\u0119powy \u2014 opcjonalnie (ruch prostoliniowy)<\/span><\/div><div class=\"field is-checkbox is-single\" id=\"input-7\"><input type=\"checkbox\" id=\"field-11\" value=\"1\" checked=\"checked\"\/><label for=\"field-11\">Dodaj energi\u0119 kinetyczn\u0105 ruchu post\u0119powego<\/label><\/div><div class=\"field is-single has-term\" id=\"input-8\"><label for=\"field-12\">Masa w ruchu post\u0119powym (m)<\/label><div class=\"term\">m<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"masa_w_ruchu_postpowym_m\" id=\"field-12\" aria-describedby=\"calc-info-12\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 20\" min=\"0\" value=\"20\"\/><div class=\"grid grid-cols-1\"><select id=\"field-13\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"kg\" selected>kg<\/option><option value=\"lb\">lb<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-12\">Masa poruszaj\u0105ca si\u0119 prostoliniowo \u2014 zwykle to samo cia\u0142o, kt\u00f3re si\u0119 obraca.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-9\"><label for=\"field-14\">Pr\u0119dko\u015b\u0107 liniowa (v)<\/label><div class=\"term\">v<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"prdko_liniowa_v\" id=\"field-14\" aria-describedby=\"calc-info-14\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 10\" min=\"0\" value=\"10\"\/><div class=\"grid grid-cols-1\"><select id=\"field-15\" name=\"jednostka\" aria-label=\"Jednostka\"><option value=\"ms\" selected>m\/s<\/option><option value=\"kmh\">km\/h<\/option><option value=\"mph\">mph<\/option><option value=\"fts\">ft\/s<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-14\">Pr\u0119dko\u015b\u0107 \u015brodka masy wzd\u0142u\u017c toru ruchu.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Przelicz ponownie\" id=\"calculate-button\" data-post=\"704\">Oblicz<\/button><button type=\"reset\">Wyczy\u015b\u0107<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Obliczaj automatycznie<\/label><small>Po w\u0142\u0105czeniu wynik aktualizuje si\u0119 automatycznie po zmianie dowolnej warto\u015bci.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Wyniki energii kinetycznej obrotowej<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Energia kinetyczna obrotowa<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-2\"><span class=\"field-title\"><span>Wzory, moment bezw\u0142adno\u015bci i dane o obrocie<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Kopiuj link do tego obliczenia<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Zostaw komentarz<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"704\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Jak oceniasz ten wynik?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Lubi\u0119 to <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Pomocne <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Niejasne <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Rozczarowanie <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Niedok\u0142adne <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Wz\u00f3r na energi\u0119 kinetyczn\u0105 obrotow\u0105<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Wz\u00f3r<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\"><\/div>\r\n\t<\/figure>\r\n\t[formula]\\[ E_{rot} = \\tfrac{1}{2} I \\omega^2 \\\\ I = \\tfrac{1}{2} m r^2 \\quad \\text{(pe\u0142ny dysk)} \\\\[ E_{total} = E_{rot} + \\tfrac{1}{2} m v^2 \\\n][\/formula]<\/p>\n<p><strong>Gdzie:<\/strong><\/p>\n<ul>\n<li>\\(E_{rot}\\) \u2014 energia kinetyczna obrotowa (J)<\/li>\n<li>\\(I\\) \u2014 moment bezw\u0142adno\u015bci cia\u0142a wzgl\u0119dem osi obrotu (kg\u00b7m\u00b2)<\/li>\n<li>\\(\\omega\\) \u2014 pr\u0119dko\u015b\u0107 k\u0105towa (rad\/s)<\/li>\n<li>\\(m\\) \u2014 masa dysku (kg)<\/li>\n<li>\\(r\\) \u2014 promie\u0144 dysku (m)<\/li>\n<li>\\(v\\) \u2014 pr\u0119dko\u015b\u0107 liniowa \u015brodka masy (m\/s), tylko je\u015bli obiekt porusza si\u0119 tak\u017ce post\u0119powo<\/li>\n<\/ul>\n<p>Wz\u00f3r m\u00f3wi, \u017ce energia zgromadzona w wiruj\u0105cym ciele ro\u015bnie wraz z momentem bezw\u0142adno\u015bci i z <em>kwadratem<\/em> pr\u0119dko\u015bci k\u0105towej: podwojenie pr\u0119dko\u015bci obrotowej czterokrotnie zwi\u0119ksza zgromadzon\u0105 energi\u0119. Dla cia\u0142a, kt\u00f3re porusza si\u0119 tak\u017ce ruchem prostoliniowym, np. tocz\u0105cego si\u0119 ko\u0142a, cz\u0119\u015b\u0107 obrotowa i post\u0119powa po prostu si\u0119 sumuj\u0105.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Energia kinetyczna obrotowa \u2013 przyk\u0142ad obliczenia<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Przyk\u0142ad:<\/strong> pe\u0142ne ko\u0142o zamachowe o masie 20 kg i promieniu 0,5 m obraca si\u0119 z pr\u0119dko\u015bci\u0105 3000 obr.\/min; ca\u0142e ko\u0142o porusza si\u0119 te\u017c do przodu z pr\u0119dko\u015bci\u0105 10 m\/s.<\/p>\n<p>\\[ I = \\tfrac{1}{2} m r^2 = \\tfrac{1}{2} \\times 20 \\times 0{,}5^2 = 2{,}5\\ \\text{kg\u00b7m}^2 \\\\ \\omega = 3000 \\times \\frac{2\\pi}{60} = 314{,}16\\ \\text{rad\/s} \\\\ E_{rot} = \\tfrac{1}{2} \\times 2{,}5 \\times 314{,}16^2 = 123\\,370\\ \\text{J} \\\\ E_{trans} = \\tfrac{1}{2} \\times 20 \\times 10^2 = 1000\\ \\text{J} \\\\ E_{total} = 123\\,370 + 1000 = 124\\,370\\ \\text{J} \\<br \/>\n]<\/p>\n\r\n<\/section>\n<p><strong>Wynik:<\/strong> ko\u0142o zamachowe magazynuje oko\u0142o 123,37 kJ energii obrotowej; po uwzgl\u0119dnieniu ruchu post\u0119powego ca\u0142kowita energia kinetyczna wynosi oko\u0142o 124,37 kJ (\u2248 34,5 Wh) \u2014 wystarczy to do zasilania \u017car\u00f3wki 60 W przez oko\u0142o 34 minuty.<\/p>\n<p>[\/wp_code]<\/p>\n<h2>Jak korzysta\u0107 z kalkulatora<\/h2>\n<ol>\n<li>Wybierz <strong>Tryb wprowadzania danych<\/strong>: albo \u201eMoment bezw\u0142adno\u015bci i pr\u0119dko\u015b\u0107 k\u0105towa\u201d, je\u015bli znasz ju\u017c \\(I\\), albo \u201eMasa i promie\u0144 (pe\u0142ny dysk \/ ko\u0142o zamachowe)\u201d, a kalkulator wyznaczy za Ciebie \\(I = \\tfrac{1}{2}mr^2\\).<\/li>\n<li>Je\u015bli wybra\u0142e\u015b tryb dysku, wpisz <strong>Mas\u0119 (m)<\/strong> i <strong>Promie\u0144 (r)<\/strong> dysku wraz z jednostkami.<\/li>\n<li>Wpisz <strong>Pr\u0119dko\u015b\u0107 k\u0105tow\u0105 (\u03c9)<\/strong> w obr.\/min, rad\/s, stopniach\/s lub obrotach\/s. Znak jest ignorowany, poniewa\u017c kierunek obrotu nie wp\u0142ywa na energi\u0119.<\/li>\n<li>Je\u015bli obiekt porusza si\u0119 tak\u017ce ruchem prostoliniowym (np. tocz\u0105ce si\u0119 ko\u0142o), zaznacz <strong>Dodaj energi\u0119 kinetyczn\u0105 post\u0119pow\u0105<\/strong> i wype\u0142nij pola <strong>Masa w ruchu post\u0119powym (m)<\/strong> oraz <strong>Pr\u0119dko\u015b\u0107 liniowa (v)<\/strong>.<\/li>\n<li>Odczytaj wyniki: energi\u0119 kinetyczn\u0105 obrotow\u0105 w d\u017culach, kilod\u017culach i watogodzinach, wraz z rozpis\u0105 pokazuj\u0105c\u0105 moment bezw\u0142adno\u015bci i wszystkie kroki po\u015brednie.<\/li>\n<\/ol>\n<h2>Czym jest energia kinetyczna obrotowa?<\/h2>\n<p>Energia kinetyczna obrotowa to energia, kt\u00f3r\u0105 posiada cia\u0142o dlatego, \u017ce si\u0119 obraca, a nie porusza po linii prostej. Podobnie jak jad\u0105cy samoch\u00f3d niesie energi\u0119 kinetyczn\u0105 \\(\\tfrac{1}{2}mv^2\\), tak wiruj\u0105ce ko\u0142o zamachowe, wirnik turbiny, wentylator sufitowy czy tocz\u0105ce si\u0119 ko\u0142o magazynuje energi\u0119 opisan\u0105 wzorem \\(\\tfrac{1}{2}I\\omega^2\\). Oba wzory s\u0105 dok\u0142adnymi analogiami: moment bezw\u0142adno\u015bci \\(I\\) odgrywa rol\u0119 masy, a pr\u0119dko\u015b\u0107 k\u0105towa \\(\\omega\\) \u2014 rol\u0119 pr\u0119dko\u015bci.<\/p>\n<p>Moment bezw\u0142adno\u015bci opisuje, jak masa cia\u0142a jest rozmieszczona wok\u00f3\u0142 osi obrotu. Masa oddalona od osi wnosi znacznie wi\u0119cej ni\u017c masa blisko niej, dlatego ko\u0142o zamachowe z obci\u0105\u017conym wie\u0144cem magazynuje oko\u0142o dwa razy wi\u0119cej energii ni\u017c jednorodny pe\u0142ny dysk o tej samej masie i promieniu. Z tego samego powodu \u0142y\u017cwiarze figurowi przyci\u0105gaj\u0105 ramiona, aby wirowa\u0107 szybciej: przy braku zewn\u0119trznego momentu si\u0142 moment p\u0119du \\(I\\omega\\) pozostaje sta\u0142y, wi\u0119c zmniejszenie \\(I\\) zwi\u0119ksza \\(\\omega\\) \u2014 a poniewa\u017c energia ro\u015bnie z \\(\\omega^2\\), przyci\u0105gni\u0119cie ramion wymaga wykonania pracy.<\/p>\n<p>Poniewa\u017c energia ro\u015bnie z <em>kwadratem<\/em> pr\u0119dko\u015bci obrotowej, szybkie ko\u0142a zamachowe mog\u0105 magazynowa\u0107 du\u017ce ilo\u015bci energii w niewielkiej obj\u0119to\u015bci \u2014 dlatego stosuje si\u0119 je do magazynowania energii w sieciach, w hamowaniu rekuperacyjnym oraz w zasilaczach bezprzerwowych. Podwojenie pr\u0119dko\u015bci obrotowej ko\u0142a zamachowego czterokrotnie zwi\u0119ksza magazynowan\u0105 energi\u0119 \u2014 ale te\u017c czterokrotnie zwi\u0119ksza napr\u0119\u017cenia w materiale, co jest praktycznym ograniczeniem pr\u0119dko\u015bci ko\u0142a zamachowego.<\/p>\n<p>Je\u015bli cia\u0142o jednocze\u015bnie obraca si\u0119 i porusza post\u0119powo, ca\u0142kowita energia kinetyczna jest zwyk\u0142\u0105 sum\u0105 obu cz\u0119\u015bci. Tocz\u0105ce si\u0119 ko\u0142o niesie na przyk\u0142ad \\(\\tfrac{1}{2}mv^2\\) z ruchu post\u0119powego plus \\(\\tfrac{1}{2}I\\omega^2\\) z obrotu. Ka\u017cd\u0105 cz\u0119\u015b\u0107 mo\u017cesz przeanalizowa\u0107 osobno w naszym kalkulatorze <a href=\"https:\/\/wpcalc.com\/pl\/fizyka\/energia-kinetyczna\/\">Energia kinetyczna<\/a>, a przemiany energii mi\u0119dzy postaciami prze\u015bledzi\u0107 w kalkulatorze <a href=\"https:\/\/wpcalc.com\/pl\/fizyka\/kalkulator-zachowania-energii\/\">Zasada zachowania energii<\/a>.<\/p>\n<h2>Za\u0142o\u017cenia, ograniczenia i praktyczne wskaz\u00f3wki<\/h2>\n<ul>\n<li><strong>Tylko pe\u0142ny dysk w trybie dysku.<\/strong> Tryb \u201emasa i promie\u0144\u201d zak\u0142ada jednorodny pe\u0142ny dysk z \\(I = \\tfrac{1}{2}mr^2\\). Dla innych kszta\u0142t\u00f3w \u2014 pier\u015bcienia lub ko\u0142a zamachowego z cienkim wie\u0144cem (\\(I = mr^2\\)), kuli pe\u0142nej (\\(I = \\tfrac{2}{5}mr^2\\)) lub pr\u0119ta wzgl\u0119dem \u015brodka (\\(I = \\tfrac{1}{12}ml^2\\)) \u2014 oblicz \\(I\\) samodzielnie i u\u017cyj trybu \u201eMoment bezw\u0142adno\u015bci i pr\u0119dko\u015b\u0107 k\u0105towa\u201d.<\/li>\n<li><strong>Cia\u0142o sztywne, sta\u0142a pr\u0119dko\u015b\u0107 obrotowa.<\/strong> Wz\u00f3r dotyczy cia\u0142, kt\u00f3re si\u0119 nie odkszta\u0142caj\u0105 i obracaj\u0105 ze sta\u0142\u0105 pr\u0119dko\u015bci\u0105; pomija tarcie, op\u00f3r powietrza i straty na drganiach wewn\u0119trznych.<\/li>\n<li><strong>We wzorze \u03c9 musi by\u0107 w rad\/s.<\/strong> Kalkulator automatycznie przelicza obr.\/min, stopnie\/s i obroty\/s; przy r\u0119cznych obliczeniach pami\u0119taj, \u017ce 1 obr.\/min \u2248 0,10472 rad\/s.<\/li>\n<li><strong>Jednostki maj\u0105 znaczenie.<\/strong> Mieszanie uk\u0142ad\u00f3w jednostek (np. lb\u00b7ft\u00b2 z obr.\/min) to cz\u0119ste \u017ar\u00f3d\u0142o b\u0142\u0119d\u00f3w \u2014 kalkulator przelicza wszystko wewn\u0119trznie na uk\u0142ad SI, wi\u0119c jego wyniki s\u0105 zawsze w d\u017culach.<\/li>\n<li><strong>Uwaga bezpiecze\u0144stwa.<\/strong> Energia zgromadzona w szybkim, ci\u0119\u017ckim kole zamachowym uwalnia si\u0119 niemal natychmiast w przypadku jego zniszczenia. Ko\u0142o zamachowe o masie 20 kg i pr\u0119dko\u015bci 3000 obr.\/min magazynuje ponad 123 kJ \u2014 por\u00f3wnywalnie z ma\u0142ym samochodem jad\u0105cym z pr\u0119dko\u015bci\u0105 50 km\/h \u2014 dlatego rzeczywiste konstrukcje k\u00f3\u0142 zamachowych musz\u0105 uwzgl\u0119dnia\u0107 zabezpieczenie przed rozerwaniem.<\/li>\n<\/ul>\n<p>Powi\u0105zane narz\u0119dzia: aby sprawdzi\u0107, jak szybko silnik musi dostarczy\u0107 t\u0119 energi\u0119, u\u017cyj kalkulatora <a href=\"https:\/\/wpcalc.com\/pl\/fizyka\/kalkulator-mocy\/\">Moc<\/a>; dla energii zgromadzonej w spr\u0119\u017cynie zamiast w ruchu obrotowym zobacz kalkulator <a href=\"https:\/\/wpcalc.com\/pl\/fizyka\/kalkulator-energii-sprezystosci\/\">Energia potencjalna spr\u0119\u017cysto\u015bci<\/a> lub og\u00f3lny kalkulator <a href=\"https:\/\/wpcalc.com\/pl\/fizyka\/energia-potencjalna\/\">Energia potencjalna<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Oblicz energi\u0119 kinetyczn\u0105 obrotow\u0105 wiruj\u0105cego obiektu lub ko\u0142a zamachowego na podstawie momentu bezw\u0142adno\u015bci i pr\u0119dko\u015bci k\u0105towej albo z masy i promienia pe\u0142nego dysku. Opcjonalnie dodaj energi\u0119 kinetyczn\u0105 post\u0119pow\u0105 dla obiekt\u00f3w tocz\u0105cych si\u0119.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[10],"tags":[54],"class_list":["post-704","post","type-post","status-publish","format-standard","hentry","category-fizyka","tag-mechanika"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/704","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/comments?post=704"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/704\/revisions"}],"predecessor-version":[{"id":1644,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/704\/revisions\/1644"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/media?parent=704"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/categories?post=704"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/tags?post=704"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}