{"id":191010,"date":"2025-02-14T05:08:03","date_gmt":"2025-02-14T05:08:03","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=191010"},"modified":"2025-02-14T05:08:06","modified_gmt":"2025-02-14T05:08:06","slug":"find-the-expression-for-mass-moment-of-inertia-of-a-sphere","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/14\/find-the-expression-for-mass-moment-of-inertia-of-a-sphere\/","title":{"rendered":"Find the expression for mass moment of inertia of a sphere"},"content":{"rendered":"\n
    \n
  1. Find the expression for mass moment of inertia of a sphere.<\/li>\n\n\n\n
  2. Find the mass moment of inertia of a hollow cylinder about its axis passing through the centroid.<\/li>\n\n\n\n
  3. Differentiate area moment of inertia from the mass moment of inertia.<\/li>\n\n\n\n
  4. From a circular plate of diameter 100 mm, a circular part is cut out whose diameter is 50 mm as shown in Figure 11.25. Find the centroid of the remaining part.<\/li>\n<\/ol>\n\n\n\n
    \"\"<\/figure>\n\n\n\n

    The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

    1. Mass Moment of Inertia of a Sphere<\/strong><\/h3>\n\n\n\n

    The mass moment of inertia (I)<\/strong> of a solid sphere of mass ( M ) and radius ( R ) about an axis passing through its center is:<\/p>\n\n\n\n

    [
    I = \\frac{2}{5} M R^2
    ]<\/p>\n\n\n\n

    Derivation:<\/strong><\/h4>\n\n\n\n
      \n
    • The differential mass element ( dm ) is taken as a thin shell at radius ( r ).<\/li>\n\n\n\n
    • The moment of inertia of each shell is integrated over the entire sphere volume.<\/li>\n\n\n\n
    • The result of this integral gives ( I = \\frac{2}{5} M R^2 ).<\/li>\n<\/ul>\n\n\n\n

      For a hollow sphere<\/strong>, where mass is distributed only on the outer surface:<\/p>\n\n\n\n

      [
      I = \\frac{2}{3} M R^2
      ]<\/p>\n\n\n\n

      \n\n\n\n

      2. Mass Moment of Inertia of a Hollow Cylinder about its Centroidal Axis<\/strong><\/h3>\n\n\n\n

      For a hollow cylinder<\/strong> (thin-walled) with mass ( M ), inner radius ( R_1 ), and outer radius ( R_2 ), the moment of inertia about the central axis is:<\/p>\n\n\n\n

      [
      I = \\frac{1}{2} M (R_1^2 + R_2^2)
      ]<\/p>\n\n\n\n

      If it is a thin-walled hollow cylinder<\/strong>, then ( R_1 \\approx R_2 = R ), and:<\/p>\n\n\n\n

      [
      I \\approx M R^2
      ]<\/p>\n\n\n\n

      \n\n\n\n

      3. Difference between Area Moment of Inertia and Mass Moment of Inertia<\/strong><\/h3>\n\n\n\n
      Property<\/th>Mass Moment of Inertia<\/strong><\/th>Area Moment of Inertia<\/strong><\/th><\/tr><\/thead>
      Definition<\/td>Resistance to rotational motion (angular acceleration).<\/td>Resistance to bending or deflection.<\/td><\/tr>
      Depends on<\/td>Mass distribution about the axis.<\/td>Area distribution about the axis.<\/td><\/tr>
      Units<\/td>( \\text{kg} \\cdot \\text{m}^2 )<\/td>( \\text{mm}^4 ) or ( \\text{m}^4 )<\/td><\/tr>
      Used in<\/td>Dynamics, angular momentum calculations.<\/td>Beam bending, structural analysis.<\/td><\/tr>
      Example Formula<\/td>( I = \\int r^2 dm )<\/td>( I = \\int y^2 dA )<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
      \n\n\n\n

      4. Centroid of the Remaining Circular Plate<\/strong><\/h3>\n\n\n\n

      We analyze the problem where a circular plate of diameter 100 mm<\/strong> has a 50 mm diameter circular cut-out<\/strong>. The centroid of the remaining shape can be determined using the method of composite areas<\/strong>.<\/p>\n\n\n\n

      Step 1: Define the System<\/strong><\/h4>\n\n\n\n
        \n
      • Original Plate<\/strong>: A circle of radius ( R = 50 ) mm<\/strong> centered at ( (0,0) ).<\/li>\n\n\n\n
      • Cut-out Circle<\/strong>: A circle of radius ( r = 25 ) mm<\/strong>, removed at a given location.<\/li>\n<\/ul>\n\n\n\n

        Step 2: Apply the Centroid Formula<\/strong><\/h4>\n\n\n\n

        The centroid of the remaining area is:<\/p>\n\n\n\n

        [
        \\bar{x} = \\frac{A_1 x_1 – A_2 x_2}{A_1 – A_2}
        ]<\/p>\n\n\n\n

        [
        \\bar{y} = \\frac{A_1 y_1 – A_2 y_2}{A_1 – A_2}
        ]<\/p>\n\n\n\n

        Where:<\/p>\n\n\n\n

          \n
        • ( A_1 = \\pi (50^2) ) (original plate),<\/li>\n\n\n\n
        • ( A_2 = \\pi (25^2) ) (cut-out),<\/li>\n\n\n\n
        • ( x_1 = 0, y_1 = 0 ) (original center),<\/li>\n\n\n\n
        • ( x_2 = 25, y_2 = 0 ) (assuming the cut-out is at ( (25,0) )).<\/li>\n<\/ul>\n\n\n\n

          Step 3: Compute the Values<\/strong><\/h4>\n\n\n\n

          [
          A_1 = 2500\\pi, \\quad A_2 = 625\\pi
          ]<\/p>\n\n\n\n

          [
          \\bar{x} = \\frac{(2500\\pi \\cdot 0) – (625\\pi \\cdot 25)}{2500\\pi – 625\\pi}
          ]<\/p>\n\n\n\n

          [
          \\bar{x} = \\frac{-15625\\pi}{1875\\pi} = -8.33 \\text{ mm}
          ]<\/p>\n\n\n\n

          [
          \\bar{y} = 0
          ]<\/p>\n\n\n\n

          Final Answer:<\/strong><\/h4>\n\n\n\n

          The centroid of the remaining shape shifts 8.33 mm left<\/strong> from the original center.<\/p>\n","protected":false},"excerpt":{"rendered":"

          The Correct Answer and Explanation is : 1. Mass Moment of Inertia of a Sphere The mass moment of inertia (I) of a solid sphere of mass ( M ) and radius ( R ) about an axis passing through its center is: [I = \\frac{2}{5} M R^2] Derivation: For a hollow sphere, where mass […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-191010","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191010","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=191010"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191010\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=191010"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=191010"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=191010"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}