{"id":2606,"date":"2023-02-15T16:56:55","date_gmt":"2023-02-15T11:26:55","guid":{"rendered":"https:\/\/www.tutoroot.com\/blog\/?p=2606"},"modified":"2026-03-10T12:51:13","modified_gmt":"2026-03-10T07:21:13","slug":"parallel-perpendicular-axis-theorem-derivation-applications","status":"publish","type":"post","link":"https:\/\/www.tutoroot.com\/blog\/parallel-perpendicular-axis-theorem-derivation-applications\/","title":{"rendered":"State and Prove Parallel Axis and Perpendicular Axis Theorems"},"content":{"rendered":"<p data-path-to-node=\"4\">Understanding the mechanics of rotational motion is crucial in physics, and calculating the moment of inertia is at the heart of it. Let\u2019s dive deep into the <b data-path-to-node=\"4\" data-index-in-node=\"158\">Parallel and Perpendicular Axes Theorems<\/b>, exploring how to use them, their formulas, and real-world applications.<\/p>\n<h2 data-path-to-node=\"5\"><strong>What is the Parallel Axis Theorem?<\/strong><\/h2>\n<p data-path-to-node=\"6\">The <a href=\"https:\/\/www.tutoroot.com\/physics-online-tuition\"><b data-path-to-node=\"6\" data-index-in-node=\"4\">Parallel Axis Theorem<\/b><\/a> is used to calculate the moment of inertia of a rigid body around any given axis, provided that this axis is parallel to an axis passing through the object&#8217;s center of mass (where the moment of inertia is already known).<\/p>\n<p data-path-to-node=\"7\">By definition, the theorem states: The moment of inertia of a body about any axis is equal to the sum of its moment of inertia about a parallel axis passing through its center of mass and the product of its mass and the square of the perpendicular distance between the two axes.<\/p>\n<h3 data-path-to-node=\"8\"><strong>Parallel Axis Theorem Formula<\/strong><\/h3>\n<p data-path-to-node=\"9\">The mathematical expression for the parallel axis theorem is:<\/p>\n<div data-path-to-node=\"10\">\n<div class=\"math-block\" data-math=\"I = I_{c} + M h^{2}\">$$I = I_{c} + M h^{2}$$<\/div>\n<\/div>\n<p data-path-to-node=\"11\"><b data-path-to-node=\"11\" data-index-in-node=\"0\">Where:<\/b><\/p>\n<ul data-path-to-node=\"12\">\n<li>\n<p data-path-to-node=\"12,0,0\"><span class=\"math-inline\" data-math=\"I\" data-index-in-node=\"0\">$I$<\/span> = Moment of inertia of the body about the new parallel axis<\/p>\n<\/li>\n<li>\n<p data-path-to-node=\"12,1,0\"><span class=\"math-inline\" data-math=\"I_{c}\" data-index-in-node=\"0\">$I_{c}$<\/span> = Moment of inertia around the center of mass<\/p>\n<\/li>\n<li>\n<p data-path-to-node=\"12,2,0\"><span class=\"math-inline\" data-math=\"M\" data-index-in-node=\"0\">$M$<\/span> = Total mass of the body<\/p>\n<\/li>\n<li>\n<p data-path-to-node=\"12,3,0\"><span class=\"math-inline\" data-math=\"h\" data-index-in-node=\"0\">$h$<\/span> = Perpendicular distance between the two parallel axes<\/p>\n<\/li>\n<\/ul>\n<h3 data-path-to-node=\"13\"><strong>How to Derive the Parallel Axis Theorem<\/strong><\/h3>\n<p data-path-to-node=\"14\">Let <span class=\"math-inline\" data-math=\"I_{c}\" data-index-in-node=\"4\">$I_{c}$<\/span> be the moment of inertia of an axis passing through the center of mass <span class=\"math-inline\" data-math=\"AB\" data-index-in-node=\"81\">$AB$<\/span>, and <span class=\"math-inline\" data-math=\"I\" data-index-in-node=\"89\">$I$<\/span> be the moment of inertia around a new parallel axis <span class=\"math-inline\" data-math=\"A'B'\" data-index-in-node=\"143\">$A&#8217;B&#8217;$<\/span> located at a distance <span class=\"math-inline\" data-math=\"h\" data-index-in-node=\"170\">$h$<\/span>.<\/p>\n<p data-path-to-node=\"15\">Imagine a single particle with mass <span class=\"math-inline\" data-math=\"m\" data-index-in-node=\"36\">$m$<\/span> located at a distance <span class=\"math-inline\" data-math=\"r\" data-index-in-node=\"60\">$r$<\/span> from the body&#8217;s center of gravity. The distance of this particle from the new axis <span class=\"math-inline\" data-math=\"A'B'\" data-index-in-node=\"145\">$A&#8217;B&#8217;$<\/span> is therefore <span class=\"math-inline\" data-math=\"(r + h)\" data-index-in-node=\"163\">$(r + h)$<\/span>.<\/p>\n<p data-path-to-node=\"16\">Using the basic definition of moment of inertia:<\/p>\n<div data-path-to-node=\"17\">\n<div class=\"math-block\" data-math=\"I = \\Sigma m (r+h)^{2}\">$$I = \\Sigma m (r+h)^{2}$$<\/div>\n<\/div>\n<p data-path-to-node=\"18\">Expanding the binomial:<\/p>\n<div data-path-to-node=\"19\">\n<div class=\"math-block\" data-math=\"I = \\Sigma m( r^{2} + h^{2} + 2rh )\">$$I = \\Sigma m( r^{2} + h^{2} + 2rh )$$<\/div>\n<\/div>\n<p data-path-to-node=\"20\">Distributing the summation:<\/p>\n<div data-path-to-node=\"21\">\n<div class=\"math-block\" data-math=\"I = \\Sigma m r^{2} + \\Sigma m h^{2} + \\Sigma 2mrh\">$$I = \\Sigma m r^{2} + \\Sigma m h^{2} + \\Sigma 2mrh$$<\/div>\n<\/div>\n<p data-path-to-node=\"22\">Because the axis <span class=\"math-inline\" data-math=\"AB\" data-index-in-node=\"17\">$AB$<\/span> passes through the center of mass, the sum of the moments of mass about the center is zero (<span class=\"math-inline\" data-math=\"\\Sigma mr = 0\" data-index-in-node=\"112\">$\\Sigma mr = 0$<\/span>). Also, we know that <span class=\"math-inline\" data-math=\"\\Sigma m = M\" data-index-in-node=\"147\">$\\Sigma m = M$<\/span> (total mass) and <span class=\"math-inline\" data-math=\"\\Sigma m r^{2} = I_{c}\" data-index-in-node=\"177\">$\\Sigma m r^{2} = I_{c}$<\/span>. Substituting these into the equation:<\/p>\n<div data-path-to-node=\"23\">\n<div class=\"math-block\" data-math=\"I = I_{c} + M h^{2} + 0\">$$I = I_{c} + M h^{2} + 0$$<\/div>\n<\/div>\n<p data-path-to-node=\"24\">Therefore, the derivation concludes as:<\/p>\n<div data-path-to-node=\"25\">\n<div class=\"math-block\" data-math=\"I = I_{c} + M h^{2}\">$$I = I_{c} + M h^{2}$$<\/div>\n<\/div>\n<hr data-path-to-node=\"26\" \/>\n<h2 data-path-to-node=\"27\"><strong>What is the Perpendicular Axis Theorem?<\/strong><\/h2>\n<p data-path-to-node=\"28\">While the parallel theorem applies to 3D objects, the <b data-path-to-node=\"28\" data-index-in-node=\"54\">Perpendicular Axis Theorem<\/b> only applies to planar, flat bodies (lamina) that have very little or no thickness.<\/p>\n<p data-path-to-node=\"29\">It states that the moment of inertia of a planar body about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two mutually perpendicular axes lying in the plane of the body, provided all three axes intersect at a single point.<\/p>\n<h3 data-path-to-node=\"30\"><strong>Perpendicular Axis Theorem Formula<\/strong><\/h3>\n<p data-path-to-node=\"31\">If a flat body is situated in the <span class=\"math-inline\" data-math=\"xy\" data-index-in-node=\"34\">$xy$<\/span>-plane, and you know its moment of inertia about the <span class=\"math-inline\" data-math=\"x\" data-index-in-node=\"89\">$x$<\/span>-axis and <span class=\"math-inline\" data-math=\"y\" data-index-in-node=\"100\">$y$<\/span>-axis, you can calculate the moment of inertia about the perpendicular <span class=\"math-inline\" data-math=\"z\" data-index-in-node=\"172\">$z$<\/span>-axis using this expression:<\/p>\n<div data-path-to-node=\"32\">\n<div class=\"math-block\" data-math=\"I_{z} = I_{x} + I_{y}\">$$I_{z} = I_{x} + I_{y}$$<\/div>\n<\/div>\n<p data-path-to-node=\"33\"><b data-path-to-node=\"33\" data-index-in-node=\"0\">Where:<\/b><\/p>\n<ul data-path-to-node=\"34\">\n<li>\n<p data-path-to-node=\"34,0,0\"><span class=\"math-inline\" data-math=\"I_{x}\" data-index-in-node=\"0\">$I_{x}$<\/span> = Moment of inertia around the X-axis<\/p>\n<\/li>\n<li>\n<p data-path-to-node=\"34,1,0\"><span class=\"math-inline\" data-math=\"I_{y}\" data-index-in-node=\"0\">$I_{y}$<\/span> = Moment of inertia around the Y-axis<\/p>\n<\/li>\n<li>\n<p data-path-to-node=\"34,2,0\"><span class=\"math-inline\" data-math=\"I_{z}\" data-index-in-node=\"0\">$I_{z}$<\/span> = Moment of inertia around the Z-axis (perpendicular to the plane)<\/p>\n<\/li>\n<\/ul>\n<h3 data-path-to-node=\"35\"><strong>Example of the Perpendicular Axis Theorem<\/strong><\/h3>\n<p data-path-to-node=\"36\">Let&#8217;s calculate the moment of inertia of a uniform circular disc (like a perfectly round, flat piece of paper) about its diameter.<\/p>\n<p data-path-to-node=\"37\">Let <span class=\"math-inline\" data-math=\"M\" data-index-in-node=\"4\">$M$<\/span> represent the total mass and <span class=\"math-inline\" data-math=\"R\" data-index-in-node=\"35\">$R$<\/span> represent the radius. We already know the moment of inertia of a disc about an axis passing through its center and perpendicular to its plane (<span class=\"math-inline\" data-math=\"I_{z}\" data-index-in-node=\"180\">$I_{z}$<\/span>) is <span class=\"math-inline\" data-math=\"\\frac{M R^{2}}{2}\" data-index-in-node=\"190\">$\\frac{M R^{2}}{2}$<\/span>.<\/p>\n<p data-path-to-node=\"38\">According to the perpendicular axes theorem:<\/p>\n<div data-path-to-node=\"39\">\n<div class=\"math-block\" data-math=\"I_{z} = I_{x} + I_{y}\">$$I_{z} = I_{x} + I_{y}$$<\/div>\n<\/div>\n<p data-path-to-node=\"40\">Because the circular disc is completely uniform and symmetric, the moment of inertia along any diameter is the same. Therefore, <span class=\"math-inline\" data-math=\"I_{x} = I_{y}\" data-index-in-node=\"128\">$I_{x} = I_{y}$<\/span>.<\/p>\n<div data-path-to-node=\"41\">\n<div class=\"math-block\" data-math=\"I_{z} = 2 I_{x}\">$$I_{z} = 2 I_{x}$$<\/div>\n<\/div>\n<p data-path-to-node=\"42\">Substitute the known value of <span class=\"math-inline\" data-math=\"I_{z}\" data-index-in-node=\"30\">$I_{z}$<\/span>:<\/p>\n<div data-path-to-node=\"43\">\n<div class=\"math-block\" data-math=\"\\frac{M R^{2}}{2} = 2 I_{x}\">$$\\frac{M R^{2}}{2} = 2 I_{x}$$<\/div>\n<\/div>\n<p data-path-to-node=\"44\">Divide by 2 to solve for <span class=\"math-inline\" data-math=\"I_{x}\" data-index-in-node=\"25\">$I_{x}$<\/span>:<\/p>\n<div data-path-to-node=\"45\">\n<div class=\"math-block\" data-math=\"I_{x} = \\frac{M R^{2}}{4}\">$$I_{x} = \\frac{M R^{2}}{4}$$<\/div>\n<\/div>\n<p data-path-to-node=\"46\">Finally, the moment of inertia of a uniform disc about any of its diameters is equal to <span class=\"math-inline\" data-math=\"\\frac{M R^{2}}{4}\" data-index-in-node=\"88\">$\\frac{M R^{2}}{4}$<\/span>.<\/p>\n<hr data-path-to-node=\"47\" \/>\n<h2 data-path-to-node=\"48\"><strong>What are the Applications of Parallel and Perpendicular Axis Theorems?<\/strong><\/h2>\n<p data-path-to-node=\"49\">By combining these two theorems, physicists and engineers can solve complex rotational dynamics problems. Key applications include:<\/p>\n<ul data-path-to-node=\"50\">\n<li>\n<p data-path-to-node=\"50,0,0\"><b data-path-to-node=\"50,0,0\" data-index-in-node=\"0\">Calculating complex shapes:<\/b> Determining the moment of inertia of rigid bodies around <i data-path-to-node=\"50,0,0\" data-index-in-node=\"85\">any<\/i> arbitrary axis, not just the center of mass.<\/p>\n<\/li>\n<li>\n<p data-path-to-node=\"50,1,0\"><b data-path-to-node=\"50,1,0\" data-index-in-node=\"0\">3D Object Analysis:<\/b> While the perpendicular theorem is for 2D planes, combining it with the parallel theorem helps calculate the moment of inertia for 3D spinning objects like cylinders and spheres.<\/p>\n<\/li>\n<li>\n<p data-path-to-node=\"50,2,0\"><b data-path-to-node=\"50,2,0\" data-index-in-node=\"0\">Engineering and Mechanics:<\/b> Crucial for designing structural beams, flywheels, and rotating machinery where mass distribution impacts stability and kinetic energy.<\/p>\n<\/li>\n<\/ul>\n<hr data-path-to-node=\"51\" \/>\n<h2 data-path-to-node=\"52\"><strong>Final Notes<\/strong><\/h2>\n<p data-path-to-node=\"53\">Are you struggling to keep up with complex mechanics and rotational dynamics in your Physics class? Mastering these derivations takes time, but expert guidance makes it easier.<\/p>\n<p data-path-to-node=\"54\"><b data-path-to-node=\"54\" data-index-in-node=\"0\">Tutoroot<\/b> offers highly interactive, 1-on-1 online physics classes designed to help you thoroughly understand and apply these concepts for your board and competitive exams.<\/p>\n<p data-path-to-node=\"55\"><b data-path-to-node=\"55\" data-index-in-node=\"0\"><a class=\"ng-star-inserted\" href=\"https:\/\/www.tutoroot.com\/\" target=\"_blank\" rel=\"noopener\" data-hveid=\"0\" data-ved=\"0CAAQ_4QMahgKEwjvuuamzpSTAxUAAAAAHQAAAAAQ8gM\">Book a Free Demo with Tutoroot Today!<\/a><\/b><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/www.tutoroot.com\/physics-online-tuition?ref=parallel-and-perpendicular-axis-theorems\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" class=\"alignnone wp-image-6445 size-large\" src=\"https:\/\/www.tutoroot.com\/blog\/wp-content\/uploads\/2023\/02\/Parallel-Axis-and-Perpendicular-Axis-Theorems-CTA-1024x576.png\" alt=\"Parallel-Axis-and-Perpendicular-Axis-Theorems\" width=\"1024\" height=\"576\" srcset=\"https:\/\/www.tutoroot.com\/blog\/wp-content\/uploads\/2023\/02\/Parallel-Axis-and-Perpendicular-Axis-Theorems-CTA-1024x576.png 1024w, https:\/\/www.tutoroot.com\/blog\/wp-content\/uploads\/2023\/02\/Parallel-Axis-and-Perpendicular-Axis-Theorems-CTA-300x169.png 300w, https:\/\/www.tutoroot.com\/blog\/wp-content\/uploads\/2023\/02\/Parallel-Axis-and-Perpendicular-Axis-Theorems-CTA-768x432.png 768w, https:\/\/www.tutoroot.com\/blog\/wp-content\/uploads\/2023\/02\/Parallel-Axis-and-Perpendicular-Axis-Theorems-CTA-1536x864.png 1536w, https:\/\/www.tutoroot.com\/blog\/wp-content\/uploads\/2023\/02\/Parallel-Axis-and-Perpendicular-Axis-Theorems-CTA-2048x1152.png 2048w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/a><\/p>\n<hr data-path-to-node=\"56\" \/>\n<h2 data-path-to-node=\"57\"><strong>Frequently Asked Questions (FAQs)<\/strong><\/h2>\n<p data-path-to-node=\"58\"><b data-path-to-node=\"58\" data-index-in-node=\"0\">State the parallel axis theorem.<\/b><\/p>\n<p data-path-to-node=\"58\">The parallel axis theorem states that the moment of inertia of a body about any axis is equal to the sum of its moment of inertia about a parallel axis passing through its center of mass, plus the product of the body&#8217;s mass and the square of the perpendicular distance between the two axes.<\/p>\n<p data-path-to-node=\"59\"><b data-path-to-node=\"59\" data-index-in-node=\"0\">What is the main difference between the parallel and perpendicular axis theorems?<\/b><\/p>\n<p data-path-to-node=\"59\">The primary difference is their application. The parallel axis theorem can be applied to any 3D rigid body to find the moment of inertia about a parallel axis. The perpendicular axis theorem is strictly limited to 2D planar bodies (flat objects) to find the moment of inertia about a third perpendicular axis.<\/p>\n<p data-path-to-node=\"60\"><b data-path-to-node=\"60\" data-index-in-node=\"0\">What is <span class=\"math-inline\" data-math=\"I_c\" data-index-in-node=\"8\">$I_c$<\/span> in the parallel axis theorem?<\/b><\/p>\n<p data-path-to-node=\"60\">In the formula <span class=\"math-inline\" data-math=\"I = I_{c} + M h^{2}\" data-index-in-node=\"57\">$I = I_{c} + M h^{2}$<\/span>, <span class=\"math-inline\" data-math=\"I_{c}\" data-index-in-node=\"78\">$I_{c}$<\/span> represents the moment of inertia of the object about an axis that passes directly through its center of mass.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Understanding the mechanics of rotational motion is crucial in physics, and calculating the moment of inertia is at the heart of it. Let\u2019s dive deep into the Parallel and Perpendicular &hellip; <a href=\"https:\/\/www.tutoroot.com\/blog\/parallel-perpendicular-axis-theorem-derivation-applications\/\" class=\"more-link\">Read More<\/a><\/p>\n","protected":false},"author":7,"featured_media":6444,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[12],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v19.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Parallel, Perpendicular Axis Theorem - Derivation, Applications<\/title>\n<meta name=\"description\" content=\"Master the Parallel and Perpendicular Axis Theorems! 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