{"id":65874,"date":"2026-09-30T13:56:45","date_gmt":"2026-09-30T10:56:45","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65874"},"modified":"2026-10-13T12:03:45","modified_gmt":"2026-10-13T09:03:45","slug":"rotational-kinetic-energy-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/physics\/rotational-kinetic-energy-calculator\/","title":{"rendered":"Rotational Kinetic Energy Calculator"},"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\">Find the Kinetic Energy of a Spinning Object or Flywheel<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"65874\" data-favorite-title=\"Rotational Kinetic Energy\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/physics\/rotational-kinetic-energy-calculator\/\" data-favorite-excerpt=\"Calculate the rotational kinetic energy of a spinning object or flywheel from its moment of inertia and angular velocity, or from the mass and radius of a solid disk. Optionally add translational kinetic energy for rolling objects.\" aria-label=\"Add to Favorites\" 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\/en\/physics\/rotational-kinetic-energy-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65874\" data-post=\"65874\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Input mode<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"input_mode\"><option value=\"disk\" selected>Mass and radius (solid disk \/ flywheel)<\/option><option value=\"iomega\">Moment of inertia and angular velocity<\/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 of inertia (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_of_inertia_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=\"unit\" aria-label=\"Unit\"><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\">Resistance of the body to spinning. For common shapes: solid disk \u00bdmr\u00b2, ring mr\u00b2, solid sphere \u2156mr\u00b2, rod about centre \u215f\u2081\u2082ml\u00b2.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-3\" hidden><label for=\"field-4\">Angular velocity (\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=\"angular_velocity_\" 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=\"unit\" aria-label=\"Unit\"><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\">How fast the object spins. The sign is ignored \u2014 direction does not change the energy.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-4\"><label for=\"field-6\">Mass (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=\"mass_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=\"unit\" aria-label=\"Unit\"><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\">Total mass of the disk or flywheel.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-5\"><label for=\"field-8\">Radius (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=\"radius_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=\"unit\" aria-label=\"Unit\"><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\">Outer radius of the disk (half the diameter).<\/small><\/div><div class=\"is-single field\" id=\"input-6\" hidden><span class=\"font-semibold\">Optional translation (linear motion)<\/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\">Add translational kinetic energy<\/label><\/div><div class=\"field is-single has-term\" id=\"input-8\"><label for=\"field-12\">Translating mass (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=\"translating_mass_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=\"unit\" aria-label=\"Unit\"><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\">Mass that moves linearly \u2014 usually the same body that spins.<\/small><\/div><div class=\"field is-single has-term\" id=\"input-9\"><label for=\"field-14\">Linear speed (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=\"linear_speed_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=\"unit\" aria-label=\"Unit\"><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\">Speed of the centre of mass along its path.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"65874\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/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> Rotational Kinetic Energy Results<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Rotational kinetic energy<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><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>Formulas, moment of inertia and spin data<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><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\">Copy link to this calculation<\/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>Leave a Comment<\/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=\"65874\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/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\">Like <\/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\">Helpful <\/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\">Confused <\/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\">Disappointed <\/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\">1<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/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>Rotational Kinetic Energy Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[\n\\begin{aligned}\nE_{rot} &= \\tfrac{1}{2} I \\omega^2 \\\\\nI &= \\tfrac{1}{2} m r^2 \\quad \\text{(solid disk)} \\\\\nE_{total} &= E_{rot} + \\tfrac{1}{2} m v^2\n\\end{aligned}\n\\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(E_{rot}\\) \u2014 rotational kinetic energy (J)<\/li>\n<li>\\(I\\) \u2014 moment of inertia of the body about the spin axis (kg\u00b7m\u00b2)<\/li>\n<li>\\(\\omega\\) \u2014 angular velocity (rad\/s)<\/li>\n<li>\\(m\\) \u2014 mass of the disk (kg)<\/li>\n<li>\\(r\\) \u2014 radius of the disk (m)<\/li>\n<li>\\(v\\) \u2014 linear speed of the centre of mass (m\/s), only if the object also translates<\/li>\n<\/ul>\n<p>The formula says that the energy stored in a spinning body grows with the moment of inertia and with the <em>square<\/em> of the angular velocity: doubling the spin speed stores four times more energy. For a body that also moves in a straight line, such as a rolling wheel, the rotational and translational parts simply add up.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.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>Rotational Kinetic Energy \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a 20 kg solid flywheel of radius 0.5 m spins at 3000 rpm; the whole wheel also moves forward at 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>Result:<\/strong> the flywheel stores about 123.37 kJ of rotational energy; including its forward motion, the total kinetic energy is about 124.37 kJ (\u2248 34.5 Wh) \u2014 enough to run a 60 W bulb for roughly 34 minutes.<\/p>\n<p>[\/wp_code]<\/p>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Choose the <strong>Input mode<\/strong>: either \"Moment of inertia and angular velocity\" if you already know \\(I\\), or \"Mass and radius (solid disk \/ flywheel)\" and the calculator will find \\(I = \\tfrac{1}{2}mr^2\\) for you.<\/li>\n<li>If you picked the disk mode, enter the <strong>Mass (m)<\/strong> and <strong>Radius (r)<\/strong> of the disk with their units.<\/li>\n<li>Enter the <strong>Angular velocity (\u03c9)<\/strong> in rpm, rad\/s, deg\/s or rev\/s. The sign is ignored, since the direction of rotation does not affect the energy.<\/li>\n<li>If the object also moves in a straight line (for example a rolling wheel), tick <strong>Add translational kinetic energy<\/strong> and fill in the <strong>Translating mass (m)<\/strong> and <strong>Linear speed (v)<\/strong>.<\/li>\n<li>Read the results: the rotational kinetic energy in joules, kilojoules and watt-hours, plus a breakdown showing the moment of inertia and every intermediate step.<\/li>\n<\/ol>\n<h2>What is rotational kinetic energy?<\/h2>\n<p>Rotational kinetic energy is the energy a body possesses because it is spinning rather than moving in a straight line. Just as a moving car carries kinetic energy \\(\\tfrac{1}{2}mv^2\\), a spinning flywheel, turbine rotor, ceiling fan or rolling wheel stores energy described by \\(\\tfrac{1}{2}I\\omega^2\\). The two formulas are exact analogues: the moment of inertia \\(I\\) plays the role of mass, and the angular velocity \\(\\omega\\) plays the role of speed.<\/p>\n<p>The moment of inertia measures how the body's mass is distributed around the spin axis. Mass far from the axis contributes much more than mass near it, which is why a rim-weighted flywheel stores about twice the energy of a uniform solid disk of the same mass and radius. For the same reason, Figure skaters pull their arms in to spin faster: with no external torque, angular momentum \\(I\\omega\\) stays constant, so reducing \\(I\\) increases \\(\\omega\\) \u2014 and because energy grows with \\(\\omega^2\\), they must do work to pull their arms in.<\/p>\n<p>Because energy scales with the <em>square<\/em> of the spin rate, high-speed flywheels can store large amounts of energy in a small package, which is why they are used for grid energy storage, regenerative braking and uninterruptible power supplies. Doubling the rpm of a flywheel quadruples the energy it stores \u2014 and also quadruples the stresses in the material, which is the practical limit on flywheel speed.<\/p>\n<p>If a body both spins and translates, the total kinetic energy is the plain sum of the two parts. A rolling wheel, for instance, carries \\(\\tfrac{1}{2}mv^2\\) from its forward motion plus \\(\\tfrac{1}{2}I\\omega^2\\) from its rotation. You can explore each part separately with our <a href=\"https:\/\/wpcalc.com\/en\/physics\/kinetic-energy\/\">Kinetic Energy calculator<\/a> and see how energy transforms between forms with the <a href=\"https:\/\/wpcalc.com\/en\/physics\/conservation-of-energy-calculator\/\">Conservation of Energy calculator<\/a>.<\/p>\n<h2>Assumptions, limitations and practical tips<\/h2>\n<ul>\n<li><strong>Solid disk only in disk mode.<\/strong> The mass-and-radius mode assumes a uniform solid disk with \\(I = \\tfrac{1}{2}mr^2\\). For other shapes \u2014 a ring or thin-rim flywheel (\\(I = mr^2\\)), a solid sphere (\\(I = \\tfrac{2}{5}mr^2\\)), or a rod about its centre (\\(I = \\tfrac{1}{12}ml^2\\)) \u2014 compute \\(I\\) yourself and use the \"Moment of inertia and angular velocity\" mode instead.<\/li>\n<li><strong>Rigid body, constant spin rate.<\/strong> The formula applies to bodies that do not deform and spin at a steady rate; it ignores friction, air resistance and internal vibration losses.<\/li>\n<li><strong>\u03c9 must be in rad\/s in the formula.<\/strong> The calculator converts rpm, deg\/s and rev\/s automatically; when doing it by hand, remember 1 rpm \u2248 0.10472 rad\/s.<\/li>\n<li><strong>Units matter.<\/strong> Mixing unit systems (e.g. lb\u00b7ft\u00b2 with rpm) is a common source of error \u2014 the calculator converts everything to SI internally, so its results are always in joules.<\/li>\n<li><strong>Safety note.<\/strong> The energy stored in a fast, heavy flywheel is released almost instantly if it fails. A 20 kg flywheel at 3000 rpm stores over 123 kJ \u2014 comparable to a small car at 50 km\/h \u2014 so real flywheel designs must account for burst containment.<\/li>\n<\/ul>\n<p>Related tools: to find how quickly a motor must supply this energy, use the <a href=\"https:\/\/wpcalc.com\/en\/physics\/power-calculator\/\">Power<\/a> calculator; for the energy stored in a spring instead of a spin, see the <a href=\"https:\/\/wpcalc.com\/en\/physics\/elastic-potential-energy-calculator\/\">Elastic Potential Energy<\/a> calculator or the general <a href=\"https:\/\/wpcalc.com\/en\/physics\/potential-energy\/\">Potential Energy<\/a> calculator.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the rotational kinetic energy of a spinning object or flywheel from its moment of inertia and angular velocity, or from the mass and radius of a solid disk. Optionally add translational kinetic energy for rolling objects.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[62],"tags":[86],"class_list":["post-65874","post","type-post","status-publish","format-standard","hentry","category-physics","tag-rotational-motion"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65874","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=65874"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65874\/revisions"}],"predecessor-version":[{"id":67100,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65874\/revisions\/67100"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65874"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65874"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65874"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}