{"id":142,"date":"2021-07-30T17:20:09","date_gmt":"2021-07-30T17:20:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=142"},"modified":"2022-11-01T04:40:00","modified_gmt":"2022-11-01T04:40:00","slug":"summary-of-calculating-centers-of-mass-and-moments-of-inertia","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-calculating-centers-of-mass-and-moments-of-inertia\/","title":{"raw":"Summary of Calculating Centers of Mass and Moments of Inertia","rendered":"Summary of Calculating Centers of Mass and Moments of Inertia"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>Finding the mass, center of mass, moments, and moments of inertia in double integrals:\r\n<ul>\r\n \t<li>For a lamina\u00a0[latex]R[\/latex]\u00a0with a density function\u00a0[latex]\\rho(x,y)[\/latex] at any point\u00a0[latex](x,y)[\/latex]\u00a0in the plane, the mass is [latex]m=\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA[\/latex]<\/li>\r\n \t<li>The moments about the\u00a0[latex]x[\/latex]-axis and [latex]y[\/latex]-axis are [latex]M_{x}=\\underset{R}{\\displaystyle\\iint} y\\rho(x,y) dA[\/latex] and\u00a0[latex]M_{y}=\\underset{R}{\\displaystyle\\iint} x\\rho(x,y) dA[\/latex]<\/li>\r\n \t<li>The center of mass is given by\u00a0[latex]\\overline{x}=\\frac{M_{y}}{m}[\/latex],\u00a0[latex]\\overline{y}=\\frac{M_{x}}{m}[\/latex]<\/li>\r\n \t<li>The center of mass becomes the centroid of the plane when the density is constant.<\/li>\r\n \t<li>The moments of inertia about the [latex]x[\/latex]-axis,\u00a0[latex]y[\/latex]-axis, and the origin are\u00a0[latex]I_{x}=\\displaystyle\\iint_{R} y^{2}\\rho(x,y) dA[\/latex],\u00a0[latex]I_{y}=\\underset{R}{\\displaystyle\\iint} x^{2}\\rho(x,y) dA[\/latex], and\u00a0[latex]I_{0}=I_{x}+I_{y}=\\underset{R}{\\displaystyle\\iint} (x^{2}+y^{2})\\rho(x,y) dA[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Finding the mass, center of mass, moments, and moments of inertia in triple integrals:\r\n<ul>\r\n \t<li>For a solid object\u00a0[latex]Q[\/latex]\u00a0with a density function\u00a0[latex]\\rho(x,y,z)[\/latex] at any point\u00a0[latex](x,y,z)[\/latex]\u00a0in space, the mass is [latex]m=\\displaystyle\\iiint_{Q} \\rho(x,y,z) dV[\/latex]<\/li>\r\n \t<li>The moments about the\u00a0[latex]xy[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the [latex]yz[\/latex]-plane are\u00a0[latex]M_{xy}=\\displaystyle\\iiint_{Q} z\\rho(x,y,z) dV[\/latex],\u00a0[latex]M_{xz}=\\underset{Q}{\\displaystyle\\iiint} y\\rho(x,y,z) dV[\/latex],\u00a0[latex]M_{yz}=\\underset{Q}{\\displaystyle\\iiint} x\\rho(x,y,z) dV[\/latex]<\/li>\r\n \t<li>The center of mass is given by\u00a0[latex]\\overline{x}=\\frac{M_{yz}}{m}[\/latex],\u00a0[latex]\\overline{x}=\\frac{M_{xz}}{m}[\/latex],\u00a0[latex]\\overline{x}=\\frac{M_{xy}}{m}[\/latex]<\/li>\r\n \t<li>The center of mass becomes the centroid of the solid when the density is constant.<\/li>\r\n \t<li>The moments of inertia about the\u00a0[latex]yz[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the [latex]xy[\/latex]-plane are\u00a0[latex]I_{x}=\\underset{Q}{\\displaystyle\\iiint} (y^{2}+z^{2})\\rho(x,y,z) dV[\/latex],\u00a0[latex]I_{y}=\\underset{Q}{\\displaystyle\\iiint} (x^{2}+z^{2})\\rho(x,y,z) dV[\/latex],\u00a0[latex]I_{z}=\\underset{Q}{\\displaystyle\\iiint} (x^{2}+y^{2})\\rho(x,y,z) dV[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul>\r\n \t<li><strong>Mass of a lamina\r\n<\/strong>[latex]m=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}m_{ij}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}\\rho(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}=\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA[\/latex]<\/li>\r\n \t<li><strong>Moment about the [latex]x[\/latex]-axis\r\n<\/strong>[latex]M_{x}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(y_{ij}^{\\ast})m_{ij}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(y_{ij}^{\\ast})\\rho(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}=\\underset{R}{\\displaystyle\\iint} y\\rho(x,y) dA[\/latex]<\/li>\r\n \t<li><strong>Moment about the [latex]y[\/latex]-axis\r\n<\/strong>[latex]M_{y}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(x_{ij}^{\\ast})m_{ij}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(x_{ij}^{\\ast})\\rho(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}=\\underset{R}{\\displaystyle\\iint} x\\rho(x,y) dA[\/latex]<\/li>\r\n \t<li><strong>Center of mass of a lamina\r\n<\/strong>[latex]\\overline{x}=\\frac{M_{y}}{m}=\\dfrac{\\underset{R}{\\displaystyle\\iint} x\\rho(x,y) dA}{\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA}[\/latex] and [latex]\\overline{y}=\\frac{M_{x}}{m}=\\dfrac{\\underset{R}{\\displaystyle\\iint} y\\rho(x,y) dA}{\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>radius of gyration<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">the distance from an object\u2019s center of mass to its axis of rotation<\/span><\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>Finding the mass, center of mass, moments, and moments of inertia in double integrals:\n<ul>\n<li>For a lamina\u00a0[latex]R[\/latex]\u00a0with a density function\u00a0[latex]\\rho(x,y)[\/latex] at any point\u00a0[latex](x,y)[\/latex]\u00a0in the plane, the mass is [latex]m=\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA[\/latex]<\/li>\n<li>The moments about the\u00a0[latex]x[\/latex]-axis and [latex]y[\/latex]-axis are [latex]M_{x}=\\underset{R}{\\displaystyle\\iint} y\\rho(x,y) dA[\/latex] and\u00a0[latex]M_{y}=\\underset{R}{\\displaystyle\\iint} x\\rho(x,y) dA[\/latex]<\/li>\n<li>The center of mass is given by\u00a0[latex]\\overline{x}=\\frac{M_{y}}{m}[\/latex],\u00a0[latex]\\overline{y}=\\frac{M_{x}}{m}[\/latex]<\/li>\n<li>The center of mass becomes the centroid of the plane when the density is constant.<\/li>\n<li>The moments of inertia about the [latex]x[\/latex]-axis,\u00a0[latex]y[\/latex]-axis, and the origin are\u00a0[latex]I_{x}=\\displaystyle\\iint_{R} y^{2}\\rho(x,y) dA[\/latex],\u00a0[latex]I_{y}=\\underset{R}{\\displaystyle\\iint} x^{2}\\rho(x,y) dA[\/latex], and\u00a0[latex]I_{0}=I_{x}+I_{y}=\\underset{R}{\\displaystyle\\iint} (x^{2}+y^{2})\\rho(x,y) dA[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Finding the mass, center of mass, moments, and moments of inertia in triple integrals:\n<ul>\n<li>For a solid object\u00a0[latex]Q[\/latex]\u00a0with a density function\u00a0[latex]\\rho(x,y,z)[\/latex] at any point\u00a0[latex](x,y,z)[\/latex]\u00a0in space, the mass is [latex]m=\\displaystyle\\iiint_{Q} \\rho(x,y,z) dV[\/latex]<\/li>\n<li>The moments about the\u00a0[latex]xy[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the [latex]yz[\/latex]-plane are\u00a0[latex]M_{xy}=\\displaystyle\\iiint_{Q} z\\rho(x,y,z) dV[\/latex],\u00a0[latex]M_{xz}=\\underset{Q}{\\displaystyle\\iiint} y\\rho(x,y,z) dV[\/latex],\u00a0[latex]M_{yz}=\\underset{Q}{\\displaystyle\\iiint} x\\rho(x,y,z) dV[\/latex]<\/li>\n<li>The center of mass is given by\u00a0[latex]\\overline{x}=\\frac{M_{yz}}{m}[\/latex],\u00a0[latex]\\overline{x}=\\frac{M_{xz}}{m}[\/latex],\u00a0[latex]\\overline{x}=\\frac{M_{xy}}{m}[\/latex]<\/li>\n<li>The center of mass becomes the centroid of the solid when the density is constant.<\/li>\n<li>The moments of inertia about the\u00a0[latex]yz[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the [latex]xy[\/latex]-plane are\u00a0[latex]I_{x}=\\underset{Q}{\\displaystyle\\iiint} (y^{2}+z^{2})\\rho(x,y,z) dV[\/latex],\u00a0[latex]I_{y}=\\underset{Q}{\\displaystyle\\iiint} (x^{2}+z^{2})\\rho(x,y,z) dV[\/latex],\u00a0[latex]I_{z}=\\underset{Q}{\\displaystyle\\iiint} (x^{2}+y^{2})\\rho(x,y,z) dV[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul>\n<li><strong>Mass of a lamina<br \/>\n<\/strong>[latex]m=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}m_{ij}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}\\rho(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}=\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA[\/latex]<\/li>\n<li><strong>Moment about the [latex]x[\/latex]-axis<br \/>\n<\/strong>[latex]M_{x}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(y_{ij}^{\\ast})m_{ij}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(y_{ij}^{\\ast})\\rho(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}=\\underset{R}{\\displaystyle\\iint} y\\rho(x,y) dA[\/latex]<\/li>\n<li><strong>Moment about the [latex]y[\/latex]-axis<br \/>\n<\/strong>[latex]M_{y}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(x_{ij}^{\\ast})m_{ij}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(x_{ij}^{\\ast})\\rho(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}=\\underset{R}{\\displaystyle\\iint} x\\rho(x,y) dA[\/latex]<\/li>\n<li><strong>Center of mass of a lamina<br \/>\n<\/strong>[latex]\\overline{x}=\\frac{M_{y}}{m}=\\dfrac{\\underset{R}{\\displaystyle\\iint} x\\rho(x,y) dA}{\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA}[\/latex] and [latex]\\overline{y}=\\frac{M_{x}}{m}=\\dfrac{\\underset{R}{\\displaystyle\\iint} y\\rho(x,y) dA}{\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>radius of gyration<\/dt>\n<dd><span style=\"font-size: 1em;\">the distance from an object\u2019s center of mass to its axis of rotation<\/span><\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-142\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":27,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at 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