{"id":65166,"date":"2026-05-02T11:14:57","date_gmt":"2026-05-02T08:14:57","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65166"},"modified":"2026-10-01T13:47:50","modified_gmt":"2026-10-01T10:47:50","slug":"atwood-machine-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/physics\/atwood-machine-calculator\/","title":{"rendered":"Atwood Machine 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 Acceleration and Tension<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"65166\" data-favorite-title=\"Atwood Machine\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/physics\/atwood-machine-calculator\/\" data-favorite-excerpt=\"Calculate the acceleration and rope tension in an Atwood machine. Enter the two masses (kg or g) and gravitational acceleration to instantly solve this classic two-mass pulley system.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-cogwheel\">\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\/atwood-machine-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65166\" data-post=\"65166\"><fieldset class=\"fieldset-input is-2\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Mass 1 (m\u2081)<\/label><div class=\"field-group\"><input type=\"text\" name=\"mass_1_m\" id=\"field-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"5\"\/><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"unit\" aria-label=\"Unit\"><option value=\"kg\">kg<\/option><option value=\"g\">g<\/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><div class=\"field is-single\" id=\"input-2\"><label for=\"field-3\">Mass 2 (m\u2082)<\/label><div class=\"field-group\"><input type=\"text\" name=\"mass_2_m\" id=\"field-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"3\"\/><div class=\"grid grid-cols-1\"><select id=\"field-4\" name=\"unit\" aria-label=\"Unit\"><option value=\"kg\">kg<\/option><option value=\"g\">g<\/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><div class=\"field is-single\" id=\"input-3\"><label for=\"field-5\">Gravitational Acceleration (g)<\/label><input type=\"text\" name=\"gravitational_acceleration_g\" id=\"field-5\" aria-describedby=\"calc-unit-5\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"9.81\"\/><div class=\"unit\" id=\"calc-unit-5\">m\/s\u00b2<\/div><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"65166\">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> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Acceleration<\/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>Tension in Rope<\/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=\"65166\" 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\">0<\/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>Atwood Machine 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\\[ a = \\frac{(m_1 - m_2)\\,g}{m_1 + m_2} \\\\ T = \\frac{2\\,m_1\\,m_2\\,g}{m_1 + m_2} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(a\\) \u2014 acceleration of the system (m\/s\u00b2)<\/li>\n<li>\\(T\\) \u2014 tension in the rope (N)<\/li>\n<li>\\(m_1\\), \\(m_2\\) \u2014 the two hanging masses (kg)<\/li>\n<li>\\(g\\) \u2014 gravitational acceleration (m\/s\u00b2)<\/li>\n<\/ul>\n<p>The first equation comes from applying Newton's second law to the two-mass system as a whole: the net driving force is the weight difference \\((m_1 - m_2)g\\), and it accelerates the combined inertia \\(m_1 + m_2\\). The second equation gives the rope tension, which is the same everywhere in an ideal (massless, inextensible) rope. By convention, \\(a\\) is positive when \\(m_1\\) moves downward and \\(m_2\\) moves upward.<\/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>Atwood Machine \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> an Atwood machine carries a mass of \\(m_1 = 5\\ \\text{kg}\\) on one side and \\(m_2 = 3\\ \\text{kg}\\) on the other, with \\(g = 9.81\\ \\text{m\/s}^2\\).<\/p>\n<p>\\[ a = \\frac{(5 - 3) \\times 9.81}{5 + 3} = \\frac{19.62}{8} = 2.4525\\ \\text{m\/s}^2 \\]<\/p>\n<p>\\[ T = \\frac{2 \\times 5 \\times 3 \\times 9.81}{5 + 3} = \\frac{294.3}{8} = 36.7875\\ \\text{N} \\]<\/p>\n<p><strong>Result:<\/strong> the system accelerates at 2.4525 m\/s\u00b2 (the heavier 5 kg mass moves down), and the rope tension is 36.7875 N.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Atwood Machine Calculator<\/h2>\n<ol>\n<li>Enter the first mass in the <strong>Mass 1 (m\u2081)<\/strong> field and choose its unit \u2014 <code>kg<\/code> or <code>g<\/code> \u2014 from the dropdown. This is the mass expected to move downward.<\/li>\n<li>Enter the second mass in the <strong>Mass 2 (m\u2082)<\/strong> field and select its unit the same way.<\/li>\n<li>Set the <strong>Gravitational Acceleration (g)<\/strong> in m\/s\u00b2. Leave the default 9.81 for Earth, or use 1.62 for the Moon, 3.71 for Mars, and so on.<\/li>\n<li>Read the results: <strong>Acceleration<\/strong> is shown in m\/s\u00b2 and <strong>Tension in Rope<\/strong> in newtons. If both masses are equal, the calculator reports that the system is in equilibrium and shows the tension \\(T = m \\cdot g\\).<\/li>\n<\/ol>\n<h2>What Is an Atwood Machine?<\/h2>\n<p>An Atwood machine, invented by English mathematician George Atwood in 1784, is one of the simplest and most instructive setups in classical mechanics: two masses connected by a light rope that runs over a pulley. Because the masses are unequal, gravity produces a net force on the system and both masses accelerate \u2014 the heavier one downward, the lighter one upward \u2014 with the same magnitude of acceleration.<\/p>\n<p>The machine is a favorite teaching tool because it \"dilutes\" gravity. The system accelerates more slowly than a freely falling object (\\(a &lt; g\\) always), which made it possible to measure acceleration with the timing instruments of the 18th century. It also isolates the key ideas of Newton&#039;s second law: net force, total inertial mass, and the constraint that a rope transmits a single tension to both sides.<\/p>\n<p>Interpreting the results is straightforward. The acceleration is zero when the masses are equal, and it approaches \\(g\\) when one mass is much larger than the other (the small mass is then dragged up almost like a free-falling object). The tension always lies between the two weights: \\(m_2 g &lt; T  m_2\\). Note that the tension is <em>not<\/em> simply the average of the weights \u2014 it depends on both masses and on the acceleration they share.<\/p>\n<p>The Atwood machine is also the building block of more complex systems: half-pulleys in elevators and cable systems, multi-pulley arrangements, and laboratory experiments that measure \\(g\\) or verify Newton's second law. If you want to compare it with related setups, see the <a href=\"https:\/\/wpcalc.com\/en\/physics\/pulley-mechanical-advantage-calculator\/\">Pulley Advantage Calculator<\/a> for systems where pulleys trade speed for force, or the <a href=\"https:\/\/wpcalc.com\/en\/physics\/free-fall-calculator\/\">Free Fall Calculator<\/a> for the limiting case \\(m_2 \\to 0\\).<\/p>\n<h2>Assumptions, Limitations and Practical Tips<\/h2>\n<ul>\n<li><strong>Frictionless, massless pulley:<\/strong> a real pulley has mass and friction; part of the driving force goes into spinning it, so measured acceleration is slightly lower than predicted. With a pulley of moment of inertia \\(I\\) and radius \\(r\\), replace \\(m_1 + m_2\\) with \\(m_1 + m_2 + I\/r^2\\).<\/li>\n<li><strong>Massless, inextensible rope:<\/strong> the rope's own weight and stretch are neglected, so both masses share the same acceleration and the tension is uniform along the rope.<\/li>\n<li><strong>Equal masses:<\/strong> if \\(m_1 = m_2\\), the net force is zero, the system stays at rest (or moves at constant speed), and the tension equals the weight of either mass, \\(T = mg\\).<\/li>\n<li><strong>Sign convention:<\/strong> the calculator reports the magnitude of the acceleration; the heavier mass always moves downward. Enter the larger mass as \\(m_1\\) if you want the positive result to match the direction of motion.<\/li>\n<li><strong>Other planets:<\/strong> because \\(a\\) and \\(T\\) both scale linearly with \\(g\\), you can model the Moon or Mars simply by changing the gravitational acceleration field \u2014 or check values with our <a href=\"https:\/\/wpcalc.com\/en\/physics\/weight-on-other-planets-calculator\/\">Weight on Other Planets Calculator<\/a>.<\/li>\n<li><strong>Related physics:<\/strong> to explore the forces involved further, try the <a href=\"https:\/\/wpcalc.com\/en\/physics\/gravitational-force-calculator\/\">Gravitational Force Calculator<\/a> for Newton's law of gravitation, or the <a href=\"https:\/\/wpcalc.com\/en\/physics\/hookes-law-calculator\/\">Hooke's Law Calculator<\/a> if the connection is a spring instead of a rope.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the acceleration and rope tension in an Atwood machine. Enter the two masses (kg or g) and gravitational acceleration to instantly solve this classic two-mass pulley system.<\/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":[90],"class_list":["post-65166","post","type-post","status-publish","format-standard","hentry","category-physics","tag-simple-machines"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65166","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=65166"}],"version-history":[{"count":5,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65166\/revisions"}],"predecessor-version":[{"id":66353,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65166\/revisions\/66353"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65166"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65166"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65166"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}