{"id":559,"date":"2026-03-31T06:35:42","date_gmt":"2026-03-31T06:35:42","guid":{"rendered":"https:\/\/blueroads.in\/blog\/?p=559"},"modified":"2026-03-31T06:35:43","modified_gmt":"2026-03-31T06:35:43","slug":"moment-of-inertia-of-a-disk-formula-derivation","status":"publish","type":"post","link":"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/","title":{"rendered":"Moment of Inertia of a Disk: Formula &amp; Derivation"},"content":{"rendered":"\n<p>The&nbsp;<strong>moment of inertia of a disk<\/strong>&nbsp;is one of the most important concepts in&nbsp;<strong>rotational mechanics<\/strong>, and students in physics and engineering repeatedly encounter it in exams, problem sets, and real\u2011world design work. In this detailed 2026\u2011style article, we\u2019ll answer the core questions people ask:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>What is the\u00a0<strong>moment of inertia of a disk formula<\/strong>?<\/li>\n\n\n\n<li>How do you\u00a0<strong>derive the moment of inertia of a disk<\/strong>\u00a0about different axes?<\/li>\n\n\n\n<li>How does the formula change for a\u00a0<strong>disk with a central hole<\/strong>\u00a0(annular disc)?<\/li>\n\n\n\n<li>What are the\u00a0<strong>key formulas and numerical\u2011style results<\/strong>\u00a0you can plug into your homework or exams?<\/li>\n<\/ul>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_80 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#1_What_is_the_moment_of_inertia_of_a_disk\" >1. What is the moment of inertia of a disk?<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#11_Basic_idea_of_moment_of_inertia\" >1.1 Basic idea of moment of inertia<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#12_Moment_of_inertia_of_a_disk_Why_it_matters\" >1.2 Moment of inertia of a disk: Why it matters<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#Moment_of_Inertia_of_Common_Shapes_Disk_Ring_Rod\" >Moment of Inertia of Common Shapes (Disk, Ring, Rod)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#2_Moment_of_inertia_of_a_disk_formula\" >2. Moment of inertia of a disk formula<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#21_Standard_case_Disk_about_central_perpendicular_axis\" >2.1 Standard case: Disk about central perpendicular axis<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#22_About_a_diameter_in%E2%80%91plane\" >2.2 About a diameter (in\u2011plane)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#23_Annular_disc_disk_with_a_central_hole\" >2.3 Annular disc (disk with a central hole)<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#Moment_of_Inertia_of_Disk_Formulas_2026%E2%80%91Ready\" >Moment of Inertia of Disk Formulas (2026\u2011Ready)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#3_Moment_of_inertia_of_disk_derivation_Central_perpendicular_axis\" >3. Moment of inertia of disk derivation: Central perpendicular axis<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#31_Conceptual_roadmap\" >3.1 Conceptual roadmap<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#32_Step%E2%80%91by%E2%80%91step_derivation\" >3.2 Step\u2011by\u2011step derivation<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#Step_1_Define_surface_mass_density\" >Step 1: Define surface mass density<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#Step_2_Divide_the_disk_into_thin_concentric_rings\" >Step 2: Divide the disk into thin concentric rings<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#Step_3_Moment_of_inertia_of_a_thin_ring\" >Step 3: Moment of inertia of a thin ring<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#Step_4_Integrate_from_centre_to_edge\" >Step 4: Integrate from centre to edge<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#33_Why_concentric_rings\" >3.3 Why concentric rings?<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#Moment%E2%80%91of%E2%80%91Inertia_Derivation_%E2%80%9CCompass%E2%80%9D_%E2%80%94_Solid_Disk\" >Moment\u2011of\u2011Inertia Derivation \u201cCompass\u201d \u2014 Solid Disk<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#4_Non%E2%80%91uniform_and_annular_disks_in_2026%E2%80%91style_problems\" >4. Non\u2011uniform and annular disks in 2026\u2011style problems<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#41_Non%E2%80%91uniform_thin_disk\" >4.1 Non\u2011uniform thin disk<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#42_Annular_disc_disk_with_hole\" >4.2 Annular disc (disk with hole)<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"1-what-is-the-moment-of-inertia-of-a-disk\"><span class=\"ez-toc-section\" id=\"1_What_is_the_moment_of_inertia_of_a_disk\"><\/span>1. What is the moment of inertia of a disk?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Before jumping into formulas, it helps to understand what&nbsp;<strong>\u201cmoment of inertia of a disk\u201d<\/strong>&nbsp;actually means.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"11-basic-idea-of-moment-of-inertia\"><span class=\"ez-toc-section\" id=\"11_Basic_idea_of_moment_of_inertia\"><\/span>1.1 Basic idea of moment of inertia<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>The&nbsp;<strong>moment of inertia (I)<\/strong>&nbsp;is the&nbsp;<strong>rotational analogue of mass<\/strong>&nbsp;in linear motion.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For\u00a0<strong>linear motion<\/strong>, inertia is just\u00a0<strong>mass (m)<\/strong>.<\/li>\n\n\n\n<li>For\u00a0<strong>rotational motion<\/strong>, inertia is\u00a0<strong>I<\/strong>, and it depends not only on\u00a0<strong>mass<\/strong>\u00a0but also on\u00a0<strong>how the mass is distributed relative to the axis of rotation<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>So, two objects with the&nbsp;<strong>same mass<\/strong>&nbsp;can have&nbsp;<strong>different moments of inertia<\/strong>&nbsp;if one is&nbsp;<strong>spread out farther from the axis<\/strong>&nbsp;and the other is&nbsp;<strong>closer in<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"12-moment-of-inertia-of-a-disk-why-it-matters\"><span class=\"ez-toc-section\" id=\"12_Moment_of_inertia_of_a_disk_Why_it_matters\"><\/span>1.2 Moment of inertia of a disk: Why it matters<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>A&nbsp;<strong>uniform disk (or disc)<\/strong>&nbsp;is a common ideal shape in physics problems:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Think of a\u00a0<strong>pulley, gear, flywheel, CD, or circular platform<\/strong>.<\/li>\n\n\n\n<li>In many textbook and exam questions, the\u00a0<strong>\u201cdisk\u201d<\/strong>\u00a0is assumed to be\u00a0<strong>thin, flat, and of uniform density<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>For such a disk, we are typically interested in the&nbsp;<strong>moment of inertia about three main axes<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>About the\u00a0<strong>central axis perpendicular to the plane<\/strong>\u00a0(spinning like a CD).<\/li>\n\n\n\n<li>About an\u00a0<strong>axis along a diameter in the plane<\/strong>\u00a0(like a rolling wheel leaning sideways).<\/li>\n\n\n\n<li>About a\u00a0<strong>parallel axis through the rim<\/strong>\u00a0(using the\u00a0<strong>parallel\u2011axis theorem<\/strong>).<\/li>\n<\/ul>\n\n\n\n<p>Each of these has its own&nbsp;<strong>formula<\/strong>, and all of them start from the&nbsp;<strong>same underlying idea of calculus and integration<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"table-1-moment-of-inertia-of-common-shapes-disk-ri\"><span class=\"ez-toc-section\" id=\"Moment_of_Inertia_of_Common_Shapes_Disk_Ring_Rod\"><\/span>Moment of Inertia of Common Shapes (Disk, Ring, Rod)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\">Object<\/th><th class=\"has-text-align-left\" data-align=\"left\">Axis of rotation<\/th><th class=\"has-text-align-left\" data-align=\"left\">Moment of inertia (I)<\/th><\/tr><\/thead><tbody><tr><td>Solid disk (uniform)<\/td><td>Perpendicular to plane, through centre<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>M<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/semantics><\/math><em>I<\/em>=21\u200b<em>M<\/em><em>R<\/em>2<\/td><\/tr><tr><td>Solid disk (uniform)<\/td><td>Along a diameter (in\u2011plane)<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mi>M<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/semantics><\/math><em>I<\/em>=41\u200b<em>M<\/em><em>R<\/em>2<\/td><\/tr><tr><td>Thin ring \/ hoop<\/td><td>Perpendicular to plane, through centre<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mi>M<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/semantics><\/math><em>I<\/em>=<em>M<\/em><em>R<\/em>2<\/td><\/tr><tr><td>Thin rod<\/td><td>Through centre, perpendicular to length<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>12<\/mn><\/mfrac><mi>M<\/mi><msup><mi>L<\/mi><mn>2<\/mn><\/msup><\/mrow><\/semantics><\/math><em>I<\/em>=121\u200b<em>M<\/em><em>L<\/em>2<\/td><\/tr><tr><td>Thin rod<\/td><td>Through one end, perpendicular to length<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><mi>M<\/mi><msup><mi>L<\/mi><mn>2<\/mn><\/msup><\/mrow><\/semantics><\/math><em>I<\/em>=31\u200b<em>M<\/em><em>L<\/em>2<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>This table is a&nbsp;<strong>quick reference<\/strong>&nbsp;for the standard formulas you\u2019ll see in 2026\u2011style physics and engineering exams. For this article, we\u2019ll focus on the&nbsp;<strong>disk rows<\/strong>&nbsp;(top two), especially the&nbsp;<strong>perpendicular\u2011through\u2011centre<\/strong>&nbsp;case, which is the&nbsp;<strong>most common \u201cmoment of inertia of a disk formula\u201d<\/strong>&nbsp;everyone memorises.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"2-moment-of-inertia-of-a-disk-formula\"><span class=\"ez-toc-section\" id=\"2_Moment_of_inertia_of_a_disk_formula\"><\/span>2. Moment of inertia of a disk formula<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Let\u2019s define the&nbsp;<strong>standard formula<\/strong>&nbsp;that you\u2019ll write in your 2026 answer boxes.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"21-standard-case-disk-about-central-perpendicular\"><span class=\"ez-toc-section\" id=\"21_Standard_case_Disk_about_central_perpendicular_axis\"><\/span>2.1 Standard case: Disk about central perpendicular axis<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Consider a&nbsp;<strong>uniform thin solid disk<\/strong>&nbsp;of:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Mass<\/strong>\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>M<\/mi><\/mrow><\/semantics><\/math><em>M<\/em>,<\/li>\n\n\n\n<li><strong>Radius<\/strong>\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><\/mrow><\/semantics><\/math><em>R<\/em>,<\/li>\n\n\n\n<li>Rotating about an axis\u00a0<strong>through its centre and perpendicular to its plane<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>Then, the&nbsp;<strong>moment of inertia<\/strong>&nbsp;is:<\/p>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>I<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>M<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><\/semantics><\/math><em>I<\/em>=21\u200b<em>M<\/em><em>R<\/em>2\u200b<\/p>\n\n\n\n<p>This is the&nbsp;<strong>most famous \u201cmoment of inertia of a disk formula\u201d<\/strong>&nbsp;and appears in virtually every rotational\u2011mechanics syllabus.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"22-about-a-diameter-inplane\"><span class=\"ez-toc-section\" id=\"22_About_a_diameter_in%E2%80%91plane\"><\/span>2.2 About a diameter (in\u2011plane)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>For the same disk, if the axis lies&nbsp;<strong>in the plane of the disk and passes through its centre along a diameter<\/strong>, the moment of inertia is:<\/p>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mi>M<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/semantics><\/math><strong><em>I<\/em>=41\u200b<em>MR<\/em>2<\/strong><\/p>\n\n\n\n<p>This is less common in day\u2011to\u2011day use but is very important conceptually because it shows how&nbsp;<strong>axis\u2011orientation changes the moment of inertia<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"23-annular-disc-disk-with-a-central-hole\"><span class=\"ez-toc-section\" id=\"23_Annular_disc_disk_with_a_central_hole\"><\/span>2.3 Annular disc (disk with a central hole)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Now suppose the disk has a&nbsp;<strong>central hole<\/strong>: an&nbsp;<strong>annular disc<\/strong>&nbsp;with:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Outer radius<\/strong>\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>R<\/mi><mn>2<\/mn><\/msub><\/mrow><\/semantics><\/math><em>R<\/em>2\u200b,<\/li>\n\n\n\n<li><strong>Inner radius<\/strong>\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>R<\/mi><mn>1<\/mn><\/msub><\/mrow><\/semantics><\/math><em>R<\/em>1\u200b,<\/li>\n\n\n\n<li><strong>Total mass<\/strong>\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>M<\/mi><\/mrow><\/semantics><\/math><em>M<\/em>.<\/li>\n<\/ul>\n\n\n\n<p>Then, the moment of inertia about the&nbsp;<strong>central perpendicular axis<\/strong>&nbsp;is:<\/p>\n\n\n\n<p><strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>M<\/mi><mrow><mo fence=\"true\">(<\/mo><msubsup><mi>R<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><mo>+<\/mo><msubsup><mi>R<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><\/semantics><\/math><em>I<\/em>=21\u200b<em>M<\/em>(<em>R<\/em>22\u200b+<em>R<\/em>12\u200b)<\/strong><\/p>\n\n\n\n<p>or, if you start from two solid disks:<\/p>\n\n\n\n<p><strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><msub><mi>I<\/mi><mtext>big&nbsp;disk<\/mtext><\/msub><mo>\u2212<\/mo><msub><mi>I<\/mi><mtext>hole<\/mtext><\/msub><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msub><mi>M<\/mi><mtext>big<\/mtext><\/msub><msubsup><mi>R<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msub><mi>M<\/mi><mtext>hole<\/mtext><\/msub><msubsup><mi>R<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><\/mrow><\/semantics><\/math><em>I<\/em>=<em>I<\/em>big\u00a0disk\u200b\u2212<em>I<\/em>hole\u200b=21\u200b<em>M<\/em>big\u200b<em>R<\/em>22\u200b\u221221\u200b<em>M<\/em>hole\u200b<em>R<\/em>12\u200b<\/strong><\/p>\n\n\n\n<p>with&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>M<\/mi><mtext>hole<\/mtext><\/msub><\/mrow><\/semantics><\/math><em>M<\/em>hole\u200b&nbsp;proportional to the area of the hole.<a rel=\"noreferrer noopener\" target=\"_blank\" href=\"https:\/\/www.vedantu.com\/jee-main\/physics-moment-of-inertia-of-a-disc\"><\/a><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"table-2-moment-of-inertia-of-disk-formulas-2026rea\"><span class=\"ez-toc-section\" id=\"Moment_of_Inertia_of_Disk_Formulas_2026%E2%80%91Ready\"><\/span>Moment of Inertia of Disk Formulas (2026\u2011Ready)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\">Scenario<\/th><th class=\"has-text-align-left\" data-align=\"left\">Axis of rotation<\/th><th class=\"has-text-align-left\" data-align=\"left\">Moment of inertia formula<\/th><\/tr><\/thead><tbody><tr><td>Uniform solid disk<\/td><td>Perpendicular to plane, through centre<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>M<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/semantics><\/math><em>I<\/em>=21\u200b<em>M<\/em><em>R<\/em>2<\/td><\/tr><tr><td>Uniform solid disk<\/td><td>Along a diameter (in\u2011plane)<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mi>M<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/semantics><\/math><em>I<\/em>=41\u200b<em>M<\/em><em>R<\/em>2<\/td><\/tr><tr><td>Uniform solid disk<\/td><td>About a rim point (parallel\u2011axis)<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>M<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>M<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mfrac><mn>3<\/mn><mn>2<\/mn><\/mfrac><mi>M<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/semantics><\/math><em>I<\/em>=21\u200b<em>M<\/em><em>R<\/em>2+<em>M<\/em><em>R<\/em>2=23\u200b<em>M<\/em><em>R<\/em>2<\/td><\/tr><tr><td>Annular disc (hole of radius&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>R<\/mi><mn>1<\/mn><\/msub><\/mrow><\/semantics><\/math><em>R<\/em>1\u200b, outer&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>R<\/mi><mn>2<\/mn><\/msub><\/mrow><\/semantics><\/math><em>R<\/em>2\u200b)<\/td><td>Central perpendicular axis<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>M<\/mi><mo stretchy=\"false\">(<\/mo><msubsup><mi>R<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><mo>+<\/mo><msubsup><mi>R<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><mo stretchy=\"false\">)<\/mo><\/mrow><\/semantics><\/math><em>I<\/em>=21\u200b<em>M<\/em>(<em>R<\/em>22\u200b+<em>R<\/em>12\u200b)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>These are the&nbsp;<strong>main formulas<\/strong>&nbsp;you\u2019ll need in 2026\u2011style problems. The&nbsp;<strong>derivation<\/strong>&nbsp;of the first one (solid disk, central axis) is what you\u2019ll usually be asked to write in long\u2011answer or \u201cderive the expression for the moment of inertia of a disk\u201d questions.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-moment-of-inertia-of-disk-derivation-central-per\"><span class=\"ez-toc-section\" id=\"3_Moment_of_inertia_of_disk_derivation_Central_perpendicular_axis\"><\/span>3. Moment of inertia of disk derivation: Central perpendicular axis<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Now let\u2019s go through the&nbsp;<strong>step\u2011by\u2011step derivation<\/strong>&nbsp;of the formula<\/p>\n\n\n\n<p><strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>M<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/semantics><\/math><em>I<\/em>=21\u200b<em>MR<\/em>2<\/strong><\/p>\n\n\n\n<p>for a&nbsp;<strong>uniform thin solid disk<\/strong>&nbsp;rotating about an axis&nbsp;<strong>through its centre and perpendicular to its plane<\/strong>. This is the classic&nbsp;<strong>\u201cmoment of inertia of disk derivation\u201d<\/strong>&nbsp;you\u2019ll see in 2026\u2011style textbooks and exam instructions.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"31-conceptual-roadmap\"><span class=\"ez-toc-section\" id=\"31_Conceptual_roadmap\"><\/span>3.1 Conceptual roadmap<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>To compute the moment of inertia of a&nbsp;<strong>continuous body<\/strong>, we:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Divide the body into tiny mass elements<\/strong>\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>m<\/mi><\/mrow><\/semantics><\/math><em>d<\/em><em>m<\/em>.<\/li>\n\n\n\n<li>For each element, compute its contribution to the moment of inertia:\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>I<\/mi><mo>=<\/mo><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mtext>\u2009<\/mtext><mi>d<\/mi><mi>m<\/mi><\/mrow><\/semantics><\/math><em>d<\/em><em>I<\/em>=<em>r<\/em>2<em>d<\/em><em>m<\/em>, where\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><\/semantics><\/math><em>r<\/em>\u00a0is the\u00a0<strong>distance from the axis<\/strong>.<\/li>\n\n\n\n<li><strong>Integrate<\/strong>\u00a0over the whole body:\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mo>\u222b<\/mo><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mtext>\u2009<\/mtext><mi>d<\/mi><mi>m<\/mi><\/mrow><\/semantics><\/math><em>I<\/em>=\u222b<em>r<\/em>2<em>d<\/em><em>m<\/em>.<\/li>\n<\/ol>\n\n\n\n<p>For a&nbsp;<strong>disk<\/strong>, the trick is to slice it into&nbsp;<strong>thin concentric rings<\/strong>&nbsp;and integrate over radius rather than area.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"32-stepbystep-derivation\"><span class=\"ez-toc-section\" id=\"32_Step%E2%80%91by%E2%80%91step_derivation\"><\/span>3.2 Step\u2011by\u2011step derivation<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>We\u2019ll work in&nbsp;<strong>2026\u2011friendly notation<\/strong>&nbsp;you can copy directly into notes.<\/p>\n\n\n\n<p>Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Uniform thin disk:\n<ul class=\"wp-block-list\">\n<li>Total mass:\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>M<\/mi><\/mrow><\/semantics><\/math><em>M<\/em><\/li>\n\n\n\n<li>Radius:\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><\/mrow><\/semantics><\/math><em>R<\/em><\/li>\n\n\n\n<li>Axis: through centre, perpendicular to the plane.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"step-1-define-surface-mass-density\"><span class=\"ez-toc-section\" id=\"Step_1_Define_surface_mass_density\"><\/span>Step 1: Define surface mass density<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Because the disk is&nbsp;<strong>uniform and flat<\/strong>, mass is spread over area.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Surface mass density<\/strong>:<\/li>\n<\/ul>\n\n\n\n<p><strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>\u03c3<\/mi><mo>=<\/mo><mfrac><mtext>mass<\/mtext><mtext>area<\/mtext><\/mfrac><mo>=<\/mo><mfrac><mi>M<\/mi><mrow><mi>\u03c0<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><\/mrow><\/semantics><\/math><em>\u03c3<\/em>=areamass\u200b=<em>\u03c0R<\/em>2<em>M<\/em>\u200b<\/strong><\/p>\n\n\n\n<p>We\u2019ll later relate this to tiny area elements&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>A<\/mi><\/mrow><\/semantics><\/math><em>d<\/em><em>A<\/em>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"step-2-divide-the-disk-into-thin-concentric-rings\"><span class=\"ez-toc-section\" id=\"Step_2_Divide_the_disk_into_thin_concentric_rings\"><\/span>Step 2: Divide the disk into thin concentric rings<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Imagine slicing the disk into&nbsp;<strong>thin concentric rings<\/strong>&nbsp;of:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Radius\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><\/semantics><\/math><em>r<\/em>\u00a0(from 0 to\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><\/mrow><\/semantics><\/math><em>R<\/em>),<\/li>\n\n\n\n<li>Thickness\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>r<\/mi><\/mrow><\/semantics><\/math><em>d<\/em><em>r<\/em>,<\/li>\n<\/ul>\n\n\n\n<p>Each ring is so thin that every point on it is&nbsp;<strong>approximately the same distance&nbsp;r<em>r<\/em><\/strong>&nbsp;from the central axis.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Area of one ring<\/strong>:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>d<\/mi><mi>A<\/mi><mo>=<\/mo><mtext>circumference<\/mtext><mo>\u00d7<\/mo><mtext>thickness<\/mtext><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mtext>\u2009<\/mtext><mi>d<\/mi><mi>r<\/mi><\/mrow><\/semantics><\/math><em>d<\/em><em>A<\/em>=circumference\u00d7thickness=(2<em>\u03c0<\/em><em>r<\/em>)<em>d<\/em><em>r<\/em><\/li>\n\n\n\n<li><strong>Mass of that ring<\/strong>:<\/li>\n<\/ul>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>d<\/mi><mi>m<\/mi><mo>=<\/mo><mi>\u03c3<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>A<\/mi><mo>=<\/mo><mi>\u03c3<\/mi><mo>\u22c5<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mi>r<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>r<\/mi><\/mrow><\/semantics><\/math><em>d<\/em><em>m<\/em>=<em>\u03c3<\/em><em>d<\/em><em>A<\/em>=<em>\u03c3<\/em>\u22c52<em>\u03c0<\/em><em>r<\/em><em>d<\/em><em>r<\/em><\/p>\n\n\n\n<p>Substitute&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c3<\/mi><mo>=<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mi>M<\/mi><mrow><mi>\u03c0<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><\/mstyle><\/mrow><\/semantics><\/math><em>\u03c3<\/em>=<em>\u03c0<\/em><em>R<\/em>2<em>M<\/em>\u200b:<\/p>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>d<\/mi><mi>m<\/mi><mo>=<\/mo><mfrac><mi>M<\/mi><mrow><mi>\u03c0<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><mo>\u22c5<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mi>r<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>r<\/mi><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>M<\/mi><\/mrow><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mfrac><mtext>\u2009<\/mtext><mi>r<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>r<\/mi><\/mrow><\/semantics><\/math><em>d<\/em><em>m<\/em>=<em>\u03c0<\/em><em>R<\/em>2<em>M<\/em>\u200b\u22c52<em>\u03c0<\/em><em>r<\/em><em>d<\/em><em>r<\/em>=<em>R<\/em>22<em>M<\/em>\u200b<em>r<\/em><em>d<\/em><em>r<\/em><\/p>\n\n\n\n<p>This is the&nbsp;<strong>\u201cmass element\u201d<\/strong>&nbsp;for our integral.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"step-3-moment-of-inertia-of-a-thin-ring\"><span class=\"ez-toc-section\" id=\"Step_3_Moment_of_inertia_of_a_thin_ring\"><\/span>Step 3: Moment of inertia of a thin ring<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>For a&nbsp;<strong>thin ring of mass&nbsp;dm<em>d<\/em><em>m<\/em><\/strong>&nbsp;rotating about its symmetry axis, the&nbsp;<strong>whole ring<\/strong>&nbsp;is at distance&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><\/semantics><\/math><em>r<\/em>&nbsp;from the axis. So:<\/p>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>d<\/mi><mi>I<\/mi><mo>=<\/mo><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mtext>\u2009<\/mtext><mi>d<\/mi><mi>m<\/mi><\/mrow><\/semantics><\/math><em>d<\/em><em>I<\/em>=<em>r<\/em>2<em>d<\/em><em>m<\/em><\/p>\n\n\n\n<p>Substitute&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>m<\/mi><mo>=<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mrow><mn>2<\/mn><mi>M<\/mi><\/mrow><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mfrac><\/mstyle><mtext>\u2009<\/mtext><mi>r<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>r<\/mi><\/mrow><\/semantics><\/math><em>d<\/em><em>m<\/em>=<em>R<\/em>22<em>M<\/em>\u200b<em>r<\/em><em>d<\/em><em>r<\/em>:<\/p>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>d<\/mi><mi>I<\/mi><mo>=<\/mo><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mo>\u22c5<\/mo><mfrac><mrow><mn>2<\/mn><mi>M<\/mi><\/mrow><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mfrac><mtext>\u2009<\/mtext><mi>r<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>r<\/mi><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>M<\/mi><\/mrow><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mfrac><mtext>\u2009<\/mtext><msup><mi>r<\/mi><mn>3<\/mn><\/msup><mtext>\u2009<\/mtext><mi>d<\/mi><mi>r<\/mi><\/mrow><\/semantics><\/math><em>d<\/em><em>I<\/em>=<em>r<\/em>2\u22c5<em>R<\/em>22<em>M<\/em>\u200b<em>r<\/em><em>d<\/em><em>r<\/em>=<em>R<\/em>22<em>M<\/em>\u200b<em>r<\/em>3<em>d<\/em><em>r<\/em><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"step-4-integrate-from-centre-to-edge\"><span class=\"ez-toc-section\" id=\"Step_4_Integrate_from_centre_to_edge\"><\/span>Step 4: Integrate from centre to edge<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>The total moment of inertia is the sum (integral) of all these thin\u2011ring contributions:<\/p>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mo>\u222b<\/mo><mi>d<\/mi><mi>I<\/mi><mo>=<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>R<\/mi><\/msubsup><mfrac><mrow><mn>2<\/mn><mi>M<\/mi><\/mrow><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mfrac><mtext>\u2009<\/mtext><msup><mi>r<\/mi><mn>3<\/mn><\/msup><mtext>\u2009<\/mtext><mi>d<\/mi><mi>r<\/mi><\/mrow><\/semantics><\/math><em>I<\/em>=\u222b<em>d<\/em><em>I<\/em>=\u222b0<em>R<\/em>\u200b<em>R<\/em>22<em>M<\/em>\u200b<em>r<\/em>3<em>d<\/em><em>r<\/em><\/p>\n\n\n\n<p>Factor out constants:<\/p>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>M<\/mi><\/mrow><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mfrac><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>R<\/mi><\/msubsup><msup><mi>r<\/mi><mn>3<\/mn><\/msup><mtext>\u2009<\/mtext><mi>d<\/mi><mi>r<\/mi><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>M<\/mi><\/mrow><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo>\u22c5<\/mo><msubsup><mrow><mo fence=\"true\">[<\/mo><mfrac><msup><mi>r<\/mi><mn>4<\/mn><\/msup><mn>4<\/mn><\/mfrac><mo fence=\"true\">]<\/mo><\/mrow><mn>0<\/mn><mi>R<\/mi><\/msubsup><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>M<\/mi><\/mrow><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo>\u22c5<\/mo><mfrac><msup><mi>R<\/mi><mn>4<\/mn><\/msup><mn>4<\/mn><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>M<\/mi><\/mrow><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo>\u22c5<\/mo><mfrac><msup><mi>R<\/mi><mn>4<\/mn><\/msup><mn>4<\/mn><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>M<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><mn>4<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>M<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/semantics><\/math><em>I<\/em>=<em>R<\/em>22<em>M<\/em>\u200b\u222b0<em>R<\/em>\u200b<em>r<\/em>3<em>d<\/em><em>r<\/em>=<em>R<\/em>22<em>M<\/em>\u200b\u22c5[4<em>r<\/em>4\u200b]0<em>R<\/em>\u200b=<em>R<\/em>22<em>M<\/em>\u200b\u22c54<em>R<\/em>4\u200b=<em>R<\/em>22<em>M<\/em>\u200b\u22c54<em>R<\/em>4\u200b=42<em>M<\/em><em>R<\/em>2\u200b=21\u200b<em>M<\/em><em>R<\/em>2<\/p>\n\n\n\n<p><strong>Final result:<\/strong><\/p>\n\n\n\n<p><strong><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>I<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>M<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><\/semantics><\/math><em>I<\/em>=21\u200b<em>MR<\/em>2\u200b<\/strong><\/p>\n\n\n\n<p>This is the&nbsp;<strong>moment of inertia of a uniform solid disk about its central perpendicular axis<\/strong>. The derivation is&nbsp;<strong>identical in 2026\u2011style syllabi<\/strong>&nbsp;and is the same one taught in&nbsp;<strong>CBSE, JEE\u2011style, and university\u2011mechanics<\/strong>&nbsp;courses.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"33-why-concentric-rings\"><span class=\"ez-toc-section\" id=\"33_Why_concentric_rings\"><\/span>3.3 Why concentric rings?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>You might wonder: why slice into&nbsp;<strong>rings<\/strong>&nbsp;instead of squares or small boxes?<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Because the\u00a0<strong>axis is the centre<\/strong>, every ring is\u00a0<strong>rotationally symmetric<\/strong>\u00a0and every point on the ring is at the\u00a0<strong>same distance\u00a0r<em>r<\/em><\/strong>\u00a0from the axis.<\/li>\n\n\n\n<li>This makes the integral\u00a0<strong>tractable with a single variable\u00a0r<em>r<\/em><\/strong>\u00a0instead of a 2D areal integral.<\/li>\n<\/ul>\n\n\n\n<p>If you tried to do it as a&nbsp;<strong>double integral in Cartesian coordinates<\/strong>, it would be much messier and is usually mentioned only for conceptual completeness.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"table-3-momentofinertia-derivation-compass--solid\"><span class=\"ez-toc-section\" id=\"Moment%E2%80%91of%E2%80%91Inertia_Derivation_%E2%80%9CCompass%E2%80%9D_%E2%80%94_Solid_Disk\"><\/span>Moment\u2011of\u2011Inertia Derivation \u201cCompass\u201d \u2014 Solid Disk<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Use this table as a&nbsp;<strong>memory\u2011aid compass<\/strong>&nbsp;when you practice writing the derivation in 2026\u2011style exam answers.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th class=\"has-text-align-left\" data-align=\"left\">Step (in your written answer)<\/th><th class=\"has-text-align-left\" data-align=\"left\">What to write<\/th><th class=\"has-text-align-left\" data-align=\"left\">Key formula \/ idea<\/th><\/tr><\/thead><tbody><tr><td>1. State the idea<\/td><td>\u201cMoment of inertia of a continuous body is&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mo>\u222b<\/mo><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mtext>\u2009<\/mtext><mi>d<\/mi><mi>m<\/mi><\/mrow><\/semantics><\/math><em>I<\/em>=\u222b<em>r<\/em>2<em>d<\/em><em>m<\/em>.\u201d<\/td><td>Definition of rotational inertia<\/td><\/tr><tr><td>2. Slice the disk<\/td><td>\u201cDivide the disk into thin concentric rings of radius&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>r<\/mi><\/mrow><\/semantics><\/math><em>r<\/em>&nbsp;and thickness&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>r<\/mi><\/mrow><\/semantics><\/math><em>d<\/em><em>r<\/em>.\u201d<\/td><td>Geometry choice<\/td><\/tr><tr><td>3. Area of ring<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>A<\/mi><mo>=<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mi>r<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>r<\/mi><\/mrow><\/semantics><\/math><em>d<\/em><em>A<\/em>=2<em>\u03c0<\/em><em>r<\/em><em>d<\/em><em>r<\/em><\/td><td>Ring circumference \u00d7 thickness<\/td><\/tr><tr><td>4. Surface density<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c3<\/mi><mo>=<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mi>M<\/mi><mrow><mi>\u03c0<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><\/mstyle><\/mrow><\/semantics><\/math><em>\u03c3<\/em>=<em>\u03c0<\/em><em>R<\/em>2<em>M<\/em>\u200b<\/td><td>Uniform mass distribution<\/td><\/tr><tr><td>5. Mass of ring<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>m<\/mi><mo>=<\/mo><mi>\u03c3<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>A<\/mi><mo>=<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mrow><mn>2<\/mn><mi>M<\/mi><\/mrow><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mfrac><\/mstyle><mtext>\u2009<\/mtext><mi>r<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>r<\/mi><\/mrow><\/semantics><\/math><em>d<\/em><em>m<\/em>=<em>\u03c3<\/em><em>d<\/em><em>A<\/em>=<em>R<\/em>22<em>M<\/em>\u200b<em>r<\/em><em>d<\/em><em>r<\/em><\/td><td>Combine 3 &amp; 4<\/td><\/tr><tr><td>6. Ring\u2019s moment of inertia<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>I<\/mi><mo>=<\/mo><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mtext>\u2009<\/mtext><mi>d<\/mi><mi>m<\/mi><mo>=<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mrow><mn>2<\/mn><mi>M<\/mi><\/mrow><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mfrac><\/mstyle><mtext>\u2009<\/mtext><msup><mi>r<\/mi><mn>3<\/mn><\/msup><mtext>\u2009<\/mtext><mi>d<\/mi><mi>r<\/mi><\/mrow><\/semantics><\/math><em>d<\/em><em>I<\/em>=<em>r<\/em>2<em>d<\/em><em>m<\/em>=<em>R<\/em>22<em>M<\/em>\u200b<em>r<\/em>3<em>d<\/em><em>r<\/em><\/td><td>Apply&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><mi>I<\/mi><mo>=<\/mo><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mtext>\u2009<\/mtext><mi>d<\/mi><mi>m<\/mi><\/mrow><\/semantics><\/math><em>d<\/em><em>I<\/em>=<em>r<\/em>2<em>d<\/em><em>m<\/em><\/td><\/tr><tr><td>7. Integrate<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>R<\/mi><\/msubsup><mi>d<\/mi><mi>I<\/mi><mo>=<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>R<\/mi><\/msubsup><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mrow><mn>2<\/mn><mi>M<\/mi><\/mrow><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mfrac><\/mstyle><mtext>\u2009<\/mtext><msup><mi>r<\/mi><mn>3<\/mn><\/msup><mtext>\u2009<\/mtext><mi>d<\/mi><mi>r<\/mi><\/mrow><\/semantics><\/math><em>I<\/em>=\u222b0<em>R<\/em>\u200b<em>d<\/em><em>I<\/em>=\u222b0<em>R<\/em>\u200b<em>R<\/em>22<em>M<\/em>\u200b<em>r<\/em>3<em>d<\/em><em>r<\/em><\/td><td>Set up the integral<\/td><\/tr><tr><td>8. Evaluate<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mrow><mn>2<\/mn><mi>M<\/mi><\/mrow><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mfrac><\/mstyle><mo>\u22c5<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><msup><mi>R<\/mi><mn>4<\/mn><\/msup><mn>4<\/mn><\/mfrac><\/mstyle><mo>=<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mstyle><mi>M<\/mi><msup><mi>R<\/mi><mn>2<\/mn><\/msup><\/mrow><\/semantics><\/math><em>I<\/em>=<em>R<\/em>22<em>M<\/em>\u200b\u22c54<em>R<\/em>4\u200b=21\u200b<em>M<\/em><em>R<\/em>2<\/td><td>Final result<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>If you can walk through these 8 logical steps, you\u2019ve fully shown the&nbsp;<strong>\u201cmoment of inertia of a disk derivation\u201d<\/strong>&nbsp;in a form that\u2019s very suitable for 8\u201310\u2011mark exam questions.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-nonuniform-and-annular-disks-in-2026style-proble\"><span class=\"ez-toc-section\" id=\"4_Non%E2%80%91uniform_and_annular_disks_in_2026%E2%80%91style_problems\"><\/span>4. Non\u2011uniform and annular disks in 2026\u2011style problems<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>In some 2026\u2011style problems, the disk is not uniform, or has a&nbsp;<strong>hole in the centre<\/strong>, so let\u2019s sketch how the same idea generalises.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"41-nonuniform-thin-disk\"><span class=\"ez-toc-section\" id=\"41_Non%E2%80%91uniform_thin_disk\"><\/span>4.1 Non\u2011uniform thin disk<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Suppose the surface mass density is&nbsp;<strong>not constant<\/strong>&nbsp;but a function of radius:&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c3<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/semantics><\/math><em>\u03c3<\/em>(<em>r<\/em>). Then:<\/p>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><mo>\u222b<\/mo><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mtext>\u2009<\/mtext><mi>d<\/mi><mi>m<\/mi><mo>=<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>R<\/mi><\/msubsup><msup><mi>r<\/mi><mn>2<\/mn><\/msup><mo>\u22c5<\/mo><mrow><mo fence=\"true\">(<\/mo><mi>\u03c3<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mtext>\u2009<\/mtext><mn>2<\/mn><mi>\u03c0<\/mi><mi>r<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>r<\/mi><mo fence=\"true\">)<\/mo><\/mrow><mo>=<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>R<\/mi><\/msubsup><mi>\u03c3<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><mtext>\u2009<\/mtext><msup><mi>r<\/mi><mn>3<\/mn><\/msup><mtext>\u2009<\/mtext><mi>d<\/mi><mi>r<\/mi><\/mrow><\/semantics><\/math><em>I<\/em>=\u222b<em>r<\/em>2<em>d<\/em><em>m<\/em>=\u222b0<em>R<\/em>\u200b<em>r<\/em>2\u22c5(<em>\u03c3<\/em>(<em>r<\/em>)2<em>\u03c0<\/em><em>r<\/em><em>d<\/em><em>r<\/em>)=2<em>\u03c0<\/em>\u222b0<em>R<\/em>\u200b<em>\u03c3<\/em>(<em>r<\/em>)<em>r<\/em>3<em>d<\/em><em>r<\/em><\/p>\n\n\n\n<p>Once you know&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c3<\/mi><mo stretchy=\"false\">(<\/mo><mi>r<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/semantics><\/math><em>\u03c3<\/em>(<em>r<\/em>)&nbsp;(e.g., linear, quadratic, etc.), you plug it in and integrate numerically or analytically. This is a common&nbsp;<strong>advanced\u2011style exam problem<\/strong>&nbsp;where you are told, \u201cdensity varies with radius as&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03c3<\/mi><mo>=<\/mo><mi>k<\/mi><mi>r<\/mi><\/mrow><\/semantics><\/math><em>\u03c3<\/em>=<em>k<\/em><em>r<\/em>\u201d and asked to compute the moment of inertia.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"42-annular-disc-disk-with-hole\"><span class=\"ez-toc-section\" id=\"42_Annular_disc_disk_with_hole\"><\/span>4.2 Annular disc (disk with hole)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>For an annular disc with:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Inner radius\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>R<\/mi><mn>1<\/mn><\/msub><\/mrow><\/semantics><\/math><em>R<\/em>1\u200b,<\/li>\n\n\n\n<li>Outer radius\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>R<\/mi><mn>2<\/mn><\/msub><\/mrow><\/semantics><\/math><em>R<\/em>2\u200b,<\/li>\n\n\n\n<li>Mass\u00a0<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>M<\/mi><\/mrow><\/semantics><\/math><em>M<\/em>,<\/li>\n<\/ul>\n\n\n\n<p>you can think of it as a&nbsp;<strong>large solid disk of radius&nbsp;R2<em>R<\/em>2\u200b<\/strong>&nbsp;minus a&nbsp;<strong>small solid disk of radius&nbsp;R1<em>R<\/em>1\u200b<\/strong>.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Moment of inertia of big disk about centre<\/strong>:<\/li>\n<\/ul>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>I<\/mi><mtext>big<\/mtext><\/msub><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msub><mi>M<\/mi><mtext>big<\/mtext><\/msub><msubsup><mi>R<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><\/mrow><\/semantics><\/math><em>I<\/em>big\u200b=21\u200b<em>M<\/em>big\u200b<em>R<\/em>22\u200b<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Mass of the hole<\/strong>:<\/li>\n<\/ul>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>M<\/mi><mtext>hole<\/mtext><\/msub><mo>=<\/mo><mi>M<\/mi><mo>\u22c5<\/mo><mfrac><mrow><mi>\u03c0<\/mi><msubsup><mi>R<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><\/mrow><mrow><mi>\u03c0<\/mi><msubsup><mi>R<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><\/mrow><\/mfrac><mo>=<\/mo><mi>M<\/mi><mo>\u22c5<\/mo><mfrac><msubsup><mi>R<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><msubsup><mi>R<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><\/mfrac><\/mrow><\/semantics><\/math><em>M<\/em>hole\u200b=<em>M<\/em>\u22c5<em>\u03c0<\/em><em>R<\/em>22\u200b<em>\u03c0<\/em><em>R<\/em>12\u200b\u200b=<em>M<\/em>\u22c5<em>R<\/em>22\u200b<em>R<\/em>12\u200b\u200b<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Moment of inertia of the hole<\/strong>:<\/li>\n<\/ul>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msub><mi>I<\/mi><mtext>hole<\/mtext><\/msub><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msub><mi>M<\/mi><mtext>hole<\/mtext><\/msub><msubsup><mi>R<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><\/mrow><\/semantics><\/math><em>I<\/em>hole\u200b=21\u200b<em>M<\/em>hole\u200b<em>R<\/em>12\u200b<\/p>\n\n\n\n<p>So the total moment of inertia of the annular disc is:<\/p>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>I<\/mi><mo>=<\/mo><msub><mi>I<\/mi><mtext>big<\/mtext><\/msub><mo>\u2212<\/mo><msub><mi>I<\/mi><mtext>hole<\/mtext><\/msub><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msub><mi>M<\/mi><mtext>big<\/mtext><\/msub><msubsup><mi>R<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msub><mi>M<\/mi><mtext>hole<\/mtext><\/msub><msubsup><mi>R<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><\/mrow><\/semantics><\/math><em>I<\/em>=<em>I<\/em>big\u200b\u2212<em>I<\/em>hole\u200b=21\u200b<em>M<\/em>big\u200b<em>R<\/em>22\u200b\u221221\u200b<em>M<\/em>hole\u200b<em>R<\/em>12\u200b<\/p>\n\n\n\n<p>Substitute&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>M<\/mi><mtext>big<\/mtext><\/msub><mo>=<\/mo><mi>M<\/mi><\/mrow><\/semantics><\/math><em>M<\/em>big\u200b=<em>M<\/em>&nbsp;and&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>M<\/mi><mtext>hole<\/mtext><\/msub><mo>=<\/mo><mi>M<\/mi><mo>\u22c5<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><msubsup><mi>R<\/mi><mn>1<\/mn><mn>2<\/mn><\/msubsup><msubsup><mi>R<\/mi><mn>2<\/mn><mn>2<\/mn><\/msubsup><\/mfrac><\/mstyle><\/mrow><\/semantics><\/math><em>M<\/em>hole\u200b=<em>M<\/em>\u22c5<em>R<\/em>22\u200b<em>R<\/em>12\u200b\u200b:<\/p>\n\n\n\n<p>I = \\frac{1}{2}M R_2^2 &#8211; \\frac{1}{2} \\left(M<\/p>\n\n\n\n<p><strong>You may also like<\/strong>&nbsp;:&nbsp;<strong><a href=\"https:\/\/blueroads.in\/\">Blueroads<\/a><\/strong>,&nbsp;<strong><a href=\"https:\/\/blueroads.in\/\">Travel Agency in Agra<\/a><\/strong><\/p>\n\n\n\n<p><strong>Follow Us On<\/strong>&nbsp;:&nbsp;<strong><a href=\"https:\/\/www.facebook.com\/BlueRoads\/\">Facebook<\/a><\/strong>,&nbsp;<strong><a href=\"https:\/\/www.instagram.com\/BlueRoadsOfficial\">Instagram<\/a><\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The&nbsp;moment of inertia of a disk&nbsp;is one of the most important concepts in&nbsp;rotational mechanics, and students in physics and engineering repeatedly encounter it in exams, problem sets, and real\u2011world design work. In this detailed 2026\u2011style article, we\u2019ll answer the core questions people ask: 1. What is the moment of inertia of a disk? Before jumping [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":560,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ocean_post_layout":"","ocean_both_sidebars_style":"","ocean_both_sidebars_content_width":0,"ocean_both_sidebars_sidebars_width":0,"ocean_sidebar":"","ocean_second_sidebar":"","ocean_disable_margins":"enable","ocean_add_body_class":"","ocean_shortcode_before_top_bar":"","ocean_shortcode_after_top_bar":"","ocean_shortcode_before_header":"","ocean_shortcode_after_header":"","ocean_has_shortcode":"","ocean_shortcode_after_title":"","ocean_shortcode_before_footer_widgets":"","ocean_shortcode_after_footer_widgets":"","ocean_shortcode_before_footer_bottom":"","ocean_shortcode_after_footer_bottom":"","ocean_display_top_bar":"default","ocean_display_header":"default","ocean_header_style":"","ocean_center_header_left_menu":"","ocean_custom_header_template":"","ocean_custom_logo":0,"ocean_custom_retina_logo":0,"ocean_custom_logo_max_width":0,"ocean_custom_logo_tablet_max_width":0,"ocean_custom_logo_mobile_max_width":0,"ocean_custom_logo_max_height":0,"ocean_custom_logo_tablet_max_height":0,"ocean_custom_logo_mobile_max_height":0,"ocean_header_custom_menu":"","ocean_menu_typo_font_family":"","ocean_menu_typo_font_subset":"","ocean_menu_typo_font_size":0,"ocean_menu_typo_font_size_tablet":0,"ocean_menu_typo_font_size_mobile":0,"ocean_menu_typo_font_size_unit":"px","ocean_menu_typo_font_weight":"","ocean_menu_typo_font_weight_tablet":"","ocean_menu_typo_font_weight_mobile":"","ocean_menu_typo_transform":"","ocean_menu_typo_transform_tablet":"","ocean_menu_typo_transform_mobile":"","ocean_menu_typo_line_height":0,"ocean_menu_typo_line_height_tablet":0,"ocean_menu_typo_line_height_mobile":0,"ocean_menu_typo_line_height_unit":"","ocean_menu_typo_spacing":0,"ocean_menu_typo_spacing_tablet":0,"ocean_menu_typo_spacing_mobile":0,"ocean_menu_typo_spacing_unit":"","ocean_menu_link_color":"","ocean_menu_link_color_hover":"","ocean_menu_link_color_active":"","ocean_menu_link_background":"","ocean_menu_link_hover_background":"","ocean_menu_link_active_background":"","ocean_menu_social_links_bg":"","ocean_menu_social_hover_links_bg":"","ocean_menu_social_links_color":"","ocean_menu_social_hover_links_color":"","ocean_disable_title":"default","ocean_disable_heading":"default","ocean_post_title":"","ocean_post_subheading":"","ocean_post_title_style":"","ocean_post_title_background_color":"","ocean_post_title_background":0,"ocean_post_title_bg_image_position":"","ocean_post_title_bg_image_attachment":"","ocean_post_title_bg_image_repeat":"","ocean_post_title_bg_image_size":"","ocean_post_title_height":0,"ocean_post_title_bg_overlay":0.5,"ocean_post_title_bg_overlay_color":"","ocean_disable_breadcrumbs":"default","ocean_breadcrumbs_color":"","ocean_breadcrumbs_separator_color":"","ocean_breadcrumbs_links_color":"","ocean_breadcrumbs_links_hover_color":"","ocean_display_footer_widgets":"default","ocean_display_footer_bottom":"default","ocean_custom_footer_template":"","ocean_post_oembed":"","ocean_post_self_hosted_media":"","ocean_post_video_embed":"","ocean_link_format":"","ocean_link_format_target":"self","ocean_quote_format":"","ocean_quote_format_link":"post","ocean_gallery_link_images":"on","ocean_gallery_id":[],"footnotes":""},"categories":[710],"tags":[711,713,712],"class_list":["post-559","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-moment-of-inertia-of-a-disk","tag-moment-of-inertia-of-a-disk","tag-moment-of-inertia-of-a-disk-derivation","tag-moment-of-inertia-of-a-disk-formula","entry","has-media"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Moment of Inertia of a Disk : Formula &amp; Derivation<\/title>\n<meta name=\"description\" content=\"The moment of inertia of a disk is one of the most important concepts in rotational mechanics, and students in physics and engineering\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Moment of Inertia of a Disk : Formula &amp; Derivation\" \/>\n<meta property=\"og:description\" content=\"The moment of inertia of a disk is one of the most important concepts in rotational mechanics, and students in physics and engineering\" \/>\n<meta property=\"og:url\" content=\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/\" \/>\n<meta property=\"og:site_name\" content=\"Blog\" \/>\n<meta property=\"article:published_time\" content=\"2026-03-31T06:35:42+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-03-31T06:35:43+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/03\/Moment-of-Inertia-of-a-Disk.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1200\" \/>\n\t<meta property=\"og:image:height\" content=\"628\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"admin\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Moment of Inertia of a Disk : Formula &amp; Derivation\" \/>\n<meta name=\"twitter:description\" content=\"The moment of inertia of a disk is one of the most important concepts in rotational mechanics, and students in physics and engineering\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/03\/Moment-of-Inertia-of-a-Disk.jpg\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"admin\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"12 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/\"},\"author\":{\"name\":\"admin\",\"@id\":\"https:\/\/blueroads.in\/blog\/#\/schema\/person\/3ba1131f47c674c3aa152bd987ae091c\"},\"headline\":\"Moment of Inertia of a Disk: Formula &amp; Derivation\",\"datePublished\":\"2026-03-31T06:35:42+00:00\",\"dateModified\":\"2026-03-31T06:35:43+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/\"},\"wordCount\":2148,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/blueroads.in\/blog\/#organization\"},\"image\":{\"@id\":\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/03\/Moment-of-Inertia-of-a-Disk.jpg\",\"keywords\":[\"Moment of Inertia of a Disk\",\"Moment of Inertia of a Disk Derivation\",\"Moment of Inertia of a Disk Formula\"],\"articleSection\":[\"Moment of Inertia of a Disk\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/\",\"url\":\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/\",\"name\":\"Moment of Inertia of a Disk : Formula & Derivation\",\"isPartOf\":{\"@id\":\"https:\/\/blueroads.in\/blog\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/03\/Moment-of-Inertia-of-a-Disk.jpg\",\"datePublished\":\"2026-03-31T06:35:42+00:00\",\"dateModified\":\"2026-03-31T06:35:43+00:00\",\"description\":\"The moment of inertia of a disk is one of the most important concepts in rotational mechanics, and students in physics and engineering\",\"breadcrumb\":{\"@id\":\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#primaryimage\",\"url\":\"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/03\/Moment-of-Inertia-of-a-Disk.jpg\",\"contentUrl\":\"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/03\/Moment-of-Inertia-of-a-Disk.jpg\",\"width\":1200,\"height\":628,\"caption\":\"Moment of Inertia of a Disk\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/blueroads.in\/blog\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Moment of Inertia of a Disk: Formula &amp; Derivation\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/blueroads.in\/blog\/#website\",\"url\":\"https:\/\/blueroads.in\/blog\/\",\"name\":\"Blog\",\"description\":\"\",\"publisher\":{\"@id\":\"https:\/\/blueroads.in\/blog\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/blueroads.in\/blog\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/blueroads.in\/blog\/#organization\",\"name\":\"Blog\",\"url\":\"https:\/\/blueroads.in\/blog\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/blueroads.in\/blog\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/02\/logo.png\",\"contentUrl\":\"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/02\/logo.png\",\"width\":616,\"height\":106,\"caption\":\"Blog\"},\"image\":{\"@id\":\"https:\/\/blueroads.in\/blog\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/blueroads.in\/blog\/#\/schema\/person\/3ba1131f47c674c3aa152bd987ae091c\",\"name\":\"admin\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/blueroads.in\/blog\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/7ed8b1b6c448ce30da4842e55edbd40d51cf12a7d857873bd517a8936aa253dd?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/7ed8b1b6c448ce30da4842e55edbd40d51cf12a7d857873bd517a8936aa253dd?s=96&d=mm&r=g\",\"caption\":\"admin\"},\"sameAs\":[\"https:\/\/blueroads.in\/blog\"],\"url\":\"https:\/\/blueroads.in\/blog\/author\/pingmedia\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Moment of Inertia of a Disk : Formula & Derivation","description":"The moment of inertia of a disk is one of the most important concepts in rotational mechanics, and students in physics and engineering","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/","og_locale":"en_US","og_type":"article","og_title":"Moment of Inertia of a Disk : Formula & Derivation","og_description":"The moment of inertia of a disk is one of the most important concepts in rotational mechanics, and students in physics and engineering","og_url":"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/","og_site_name":"Blog","article_published_time":"2026-03-31T06:35:42+00:00","article_modified_time":"2026-03-31T06:35:43+00:00","og_image":[{"width":1200,"height":628,"url":"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/03\/Moment-of-Inertia-of-a-Disk.jpg","type":"image\/jpeg"}],"author":"admin","twitter_card":"summary_large_image","twitter_title":"Moment of Inertia of a Disk : Formula & Derivation","twitter_description":"The moment of inertia of a disk is one of the most important concepts in rotational mechanics, and students in physics and engineering","twitter_image":"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/03\/Moment-of-Inertia-of-a-Disk.jpg","twitter_misc":{"Written by":"admin","Est. reading time":"12 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#article","isPartOf":{"@id":"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/"},"author":{"name":"admin","@id":"https:\/\/blueroads.in\/blog\/#\/schema\/person\/3ba1131f47c674c3aa152bd987ae091c"},"headline":"Moment of Inertia of a Disk: Formula &amp; Derivation","datePublished":"2026-03-31T06:35:42+00:00","dateModified":"2026-03-31T06:35:43+00:00","mainEntityOfPage":{"@id":"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/"},"wordCount":2148,"commentCount":0,"publisher":{"@id":"https:\/\/blueroads.in\/blog\/#organization"},"image":{"@id":"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#primaryimage"},"thumbnailUrl":"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/03\/Moment-of-Inertia-of-a-Disk.jpg","keywords":["Moment of Inertia of a Disk","Moment of Inertia of a Disk Derivation","Moment of Inertia of a Disk Formula"],"articleSection":["Moment of Inertia of a Disk"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/","url":"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/","name":"Moment of Inertia of a Disk : Formula & Derivation","isPartOf":{"@id":"https:\/\/blueroads.in\/blog\/#website"},"primaryImageOfPage":{"@id":"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#primaryimage"},"image":{"@id":"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#primaryimage"},"thumbnailUrl":"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/03\/Moment-of-Inertia-of-a-Disk.jpg","datePublished":"2026-03-31T06:35:42+00:00","dateModified":"2026-03-31T06:35:43+00:00","description":"The moment of inertia of a disk is one of the most important concepts in rotational mechanics, and students in physics and engineering","breadcrumb":{"@id":"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#primaryimage","url":"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/03\/Moment-of-Inertia-of-a-Disk.jpg","contentUrl":"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/03\/Moment-of-Inertia-of-a-Disk.jpg","width":1200,"height":628,"caption":"Moment of Inertia of a Disk"},{"@type":"BreadcrumbList","@id":"https:\/\/blueroads.in\/blog\/moment-of-inertia-of-a-disk-formula-derivation\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/blueroads.in\/blog\/"},{"@type":"ListItem","position":2,"name":"Moment of Inertia of a Disk: Formula &amp; Derivation"}]},{"@type":"WebSite","@id":"https:\/\/blueroads.in\/blog\/#website","url":"https:\/\/blueroads.in\/blog\/","name":"Blog","description":"","publisher":{"@id":"https:\/\/blueroads.in\/blog\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/blueroads.in\/blog\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/blueroads.in\/blog\/#organization","name":"Blog","url":"https:\/\/blueroads.in\/blog\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/blueroads.in\/blog\/#\/schema\/logo\/image\/","url":"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/02\/logo.png","contentUrl":"https:\/\/blueroads.in\/blog\/wp-content\/uploads\/2026\/02\/logo.png","width":616,"height":106,"caption":"Blog"},"image":{"@id":"https:\/\/blueroads.in\/blog\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/blueroads.in\/blog\/#\/schema\/person\/3ba1131f47c674c3aa152bd987ae091c","name":"admin","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/blueroads.in\/blog\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/7ed8b1b6c448ce30da4842e55edbd40d51cf12a7d857873bd517a8936aa253dd?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/7ed8b1b6c448ce30da4842e55edbd40d51cf12a7d857873bd517a8936aa253dd?s=96&d=mm&r=g","caption":"admin"},"sameAs":["https:\/\/blueroads.in\/blog"],"url":"https:\/\/blueroads.in\/blog\/author\/pingmedia\/"}]}},"_links":{"self":[{"href":"https:\/\/blueroads.in\/blog\/wp-json\/wp\/v2\/posts\/559","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blueroads.in\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blueroads.in\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blueroads.in\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blueroads.in\/blog\/wp-json\/wp\/v2\/comments?post=559"}],"version-history":[{"count":1,"href":"https:\/\/blueroads.in\/blog\/wp-json\/wp\/v2\/posts\/559\/revisions"}],"predecessor-version":[{"id":561,"href":"https:\/\/blueroads.in\/blog\/wp-json\/wp\/v2\/posts\/559\/revisions\/561"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blueroads.in\/blog\/wp-json\/wp\/v2\/media\/560"}],"wp:attachment":[{"href":"https:\/\/blueroads.in\/blog\/wp-json\/wp\/v2\/media?parent=559"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blueroads.in\/blog\/wp-json\/wp\/v2\/categories?post=559"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blueroads.in\/blog\/wp-json\/wp\/v2\/tags?post=559"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}