{"id":216925,"date":"2025-05-20T16:45:52","date_gmt":"2025-05-20T16:45:52","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=216925"},"modified":"2025-05-20T16:45:54","modified_gmt":"2025-05-20T16:45:54","slug":"calculate-the-moment-of-inertia-of-a-rectangular-prism-that-is-150-mm-high-100-mm-wide-and-250-mm-long-with-respect-to-its-longest-centroidal-axis","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/20\/calculate-the-moment-of-inertia-of-a-rectangular-prism-that-is-150-mm-high-100-mm-wide-and-250-mm-long-with-respect-to-its-longest-centroidal-axis\/","title":{"rendered":"Calculate the moment of inertia of a rectangular prism that is 150 mm high, 100 mm wide, and 250 mm long, with respect to its longest centroidal axis"},"content":{"rendered":"\n<p>Calculate the moment of inertia of a rectangular prism that is 150 mm high, 100 mm wide, and 250 mm long, with respect to its longest centroidal axis. Use a density of 640 kg\/m3.<\/p>\n\n\n\n<p>Find the moment of inertia of an aluminum sphere 200 mm in diameter with respect to a centroidal axis. Aluminum has a density of 2560 kg\/m3.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Let&#8217;s solve each problem step by step.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">1. Moment of inertia of the rectangular prism<\/h3>\n\n\n\n<p><strong>Given:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Height h=150\u00a0mm=0.150\u00a0mh = 150 \\text{ mm} = 0.150 \\text{ m}<\/li>\n\n\n\n<li>Width w=100\u00a0mm=0.100\u00a0mw = 100 \\text{ mm} = 0.100 \\text{ m}<\/li>\n\n\n\n<li>Length l=250\u00a0mm=0.250\u00a0ml = 250 \\text{ mm} = 0.250 \\text{ m} \u2014 this is the longest dimension<\/li>\n\n\n\n<li>Density \u03c1=640\u00a0kg\/m3\\rho = 640 \\text{ kg\/m}^3<\/li>\n<\/ul>\n\n\n\n<p><strong>Goal:<\/strong> Find the moment of inertia II of the prism about its <strong>longest centroidal axis<\/strong>, which is along the length ll.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">Step 1: Find the mass mm of the prism.<\/h4>\n\n\n\n<p>Volume V=l\u00d7w\u00d7h=0.250\u00d70.100\u00d70.150=0.00375&nbsp;m3V = l \\times w \\times h = 0.250 \\times 0.100 \\times 0.150 = 0.00375 \\text{ m}^3<\/p>\n\n\n\n<p>Mass m=\u03c1\u00d7V=640\u00d70.00375=2.4&nbsp;kgm = \\rho \\times V = 640 \\times 0.00375 = 2.4 \\text{ kg}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">Step 2: Moment of inertia for rectangular prism about its longest centroidal axis<\/h4>\n\n\n\n<p>The longest axis is along the length ll, so the axis is along the length direction.<\/p>\n\n\n\n<p>For a rectangular prism, the moments of inertia about its centroidal axes are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>About the x-axis (length axis):<\/li>\n<\/ul>\n\n\n\n<p>Ix=112m(h2+w2)I_x = \\frac{1}{12} m (h^2 + w^2)<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>About the y-axis (width axis):<\/li>\n<\/ul>\n\n\n\n<p>Iy=112m(l2+h2)I_y = \\frac{1}{12} m (l^2 + h^2)<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>About the z-axis (height axis):<\/li>\n<\/ul>\n\n\n\n<p>Iz=112m(l2+w2)I_z = \\frac{1}{12} m (l^2 + w^2)<\/p>\n\n\n\n<p>Since the longest axis is length ll, the moment of inertia about the length axis is: I=Ix=112\u00d72.4\u00d7(0.1502+0.1002)I = I_x = \\frac{1}{12} \\times 2.4 \\times (0.150^2 + 0.100^2)<\/p>\n\n\n\n<p>Calculate the terms inside parentheses: 0.1502=0.0225,0.1002=0.0100,\u21d20.0225+0.0100=0.03250.150^2 = 0.0225, \\quad 0.100^2 = 0.0100, \\quad \\Rightarrow 0.0225 + 0.0100 = 0.0325<\/p>\n\n\n\n<p>Calculate moment of inertia: Ix=112\u00d72.4\u00d70.0325=0.2\u00d70.0325=0.0065&nbsp;kg\\cdotpm2I_x = \\frac{1}{12} \\times 2.4 \\times 0.0325 = 0.2 \\times 0.0325 = 0.0065 \\text{ kg\u00b7m}^2<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">2. Moment of inertia of an aluminum sphere<\/h3>\n\n\n\n<p><strong>Given:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Diameter d=200\u00a0mm=0.200\u00a0md = 200 \\text{ mm} = 0.200 \\text{ m}<\/li>\n\n\n\n<li>Radius r=0.100\u00a0mr = 0.100 \\text{ m}<\/li>\n\n\n\n<li>Density \u03c1=2560\u00a0kg\/m3\\rho = 2560 \\text{ kg\/m}^3<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">Step 1: Calculate the volume VV of the sphere<\/h4>\n\n\n\n<p>V=43\u03c0r3=43\u03c0(0.1)3=43\u03c0\u00d70.001=0.00419&nbsp;m3V = \\frac{4}{3} \\pi r^3 = \\frac{4}{3} \\pi (0.1)^3 = \\frac{4}{3} \\pi \\times 0.001 = 0.00419 \\text{ m}^3<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">Step 2: Calculate the mass mm<\/h4>\n\n\n\n<p>m=\u03c1\u00d7V=2560\u00d70.00419=10.726&nbsp;kgm = \\rho \\times V = 2560 \\times 0.00419 = 10.726 \\text{ kg}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">Step 3: Moment of inertia for solid sphere about centroidal axis<\/h4>\n\n\n\n<p>For a solid sphere rotating about any diameter: I=25mr2=25\u00d710.726\u00d7(0.1)2=25\u00d710.726\u00d70.01=0.0429&nbsp;kg\\cdotpm2I = \\frac{2}{5} m r^2 = \\frac{2}{5} \\times 10.726 \\times (0.1)^2 = \\frac{2}{5} \\times 10.726 \\times 0.01 = 0.0429 \\text{ kg\u00b7m}^2<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Summary:<\/strong><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Rectangular prism moment of inertia about longest centroidal axis (length):<\/li>\n<\/ul>\n\n\n\n<p>0.0065&nbsp;kg\\cdotpm2\\boxed{0.0065 \\text{ kg\u00b7m}^2}<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Aluminum sphere moment of inertia about centroidal axis:<\/li>\n<\/ul>\n\n\n\n<p>0.0429&nbsp;kg\\cdotpm2\\boxed{0.0429 \\text{ kg\u00b7m}^2}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Explanation:<\/h2>\n\n\n\n<p>The moment of inertia is a physical quantity that measures how difficult it is to change the rotational motion of an object about a given axis. It depends on the mass distribution relative to the axis of rotation. For a rectangular prism, the mass is spread out unevenly depending on the axis: rotating about the longest axis means the mass is closer to the axis, leading to a smaller moment of inertia. The formula Ix=112m(h2+w2)I_x = \\frac{1}{12} m (h^2 + w^2) shows that only the perpendicular dimensions to the axis contribute, not the length itself.<\/p>\n\n\n\n<p>In contrast, the solid sphere has a symmetric mass distribution in all directions. Its moment of inertia about any diameter is given by I=25mr2I = \\frac{2}{5} m r^2, which reflects how mass is uniformly distributed at varying distances from the center. The higher density of aluminum (2560 kg\/m\u00b3) compared to the prism&#8217;s material (640 kg\/m\u00b3) results in a significantly higher mass for the sphere, despite its smaller radius relative to the prism&#8217;s longest dimension, resulting in a larger moment of inertia.<\/p>\n\n\n\n<p>These calculations are essential in mechanical engineering and physics, especially when analyzing rotational dynamics, structural stability, or designing components subject to rotational motion. Knowing the moment of inertia helps predict how an object will behave when subjected to torques, aiding in design optimization for safety and performance.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the moment of inertia of a rectangular prism that is 150 mm high, 100 mm wide, and 250 mm long, with respect to its longest centroidal axis. Use a density of 640 kg\/m3. Find the moment of inertia of an aluminum sphere 200 mm in diameter with respect to a centroidal axis. Aluminum has [&hellip;]<\/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-216925","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/216925","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=216925"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/216925\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=216925"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=216925"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=216925"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}