1: Centers of Mass
- Page ID
- 6931
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 1.1: Introduction and Some Definitions
- This page offers a comprehensive overview of calculating the center of mass for different geometric bodies, explaining key terms and their importance. It details calculations for plane areas and curves in both Cartesian and polar coordinates, as well as in three-dimensional contexts.
- 1.2: Plane Triangular Lamina
- This page defines a triangle's median and presents two important theorems: Theorem I states that the three medians intersect at a point two-thirds from a vertex to the midpoint of the opposite side, while Theorem II identifies this point as the centroid of a uniform triangular lamina. It also includes a vector proof for these theorems and leads into calculus topics.
- 1.3: Plane Areas
- This page covers the calculation of centroids for areas defined by curves and polar coordinates. It explains how to determine the centroid coordinates \((\overline{x}, \overline{y})\) using integrals of area and first moments, with examples including semicircular laminae. Additionally, it outlines the process for finding the centroid in polar coordinates, specifically for a triangular sector, and provides relevant formulas and an example with a semicircle for better understanding.
- 1.4: Plane Curves
- This page covers calculating mass and centers of mass for plane curves using Cartesian and polar coordinates. It explains deriving elemental length \( \delta s \) and expressing mass and first moments about axes, alongside formulas for total mass and centers of mass (\(\overline{x}\) and \(\overline{y}\)) with uniform linear density. It includes examples, such as finding the center of mass for a semicircular wire in both coordinate systems, illustrating the conversion between them.
- 1.5: Summary of the Formulas for Plane Laminas and Curves
- This page covers the calculation of centroids for uniform plane laminae and curves, detailing the use of integrals based on area and length. It provides specific formulas for determining centroid coordinates (\(\overline{x}, \overline{y}\)) through area integration for laminae and length integration for curves. Additionally, it addresses the distinctions between Cartesian and polar coordinate systems in the calculations.
- 1.6: The Theorems of Pappus
- This page covers theorems of Pappus, linking the volume of solids of revolution and the area of surfaces of revolution to the centroid of a plane figure. When a plane area rotates around a non-intersecting axis, the volume is the area multiplied by the centroid's distance traveled. For curves, the rotated area equals the curve's length times the centroid's distance. Examples include semicircles, triangles, and circles to demonstrate deriving volumes and surface areas of different solids.
- 1.7: Uniform Solid Tetrahedron, Pyramid and Cone
- This page defines the median of a tetrahedron and presents two key theorems about their concurrency and the center of mass of a uniform solid tetrahedron, which is located at the intersection of the medians. It also discusses similar properties in right pyramids and circular cones, illustrating relationships using calculus to derive expressions for volume and mass in these three-dimensional shapes.
- 1.8: Hollow Cone
- This page explains that the surface of a hollow cone can be conceptualized as made up of numerous slender isosceles triangles, leading to the conclusion that the center of mass of the cone, not including the base, is positioned two-thirds of the way from the vertex to the midpoint of the base.
- 1.9: Hemispheres
- This page explains the calculation of the center of mass for a uniform solid hemisphere and a hollow hemispherical shell, deriving positions of \(\overline{x} = \frac{3a}{8}\) for the solid and \(\overline{x} = \frac{a}{2}\) for the hollow shell. It highlights the difference in their mass distributions, with the hollow shell's center of mass being positioned further from the base compared to that of the solid hemisphere.
- 1.S: Centers of Mass (Summary)
- This page details the locations of centroids for various geometric shapes, identifying specific placements such as two-thirds from a vertex for triangular laminas and three-quarters for solids like tetrahedrons and cones. It also provides formulas for centroids of hollow cones, semicircular laminas, and semicircular wires, outlining the distances depending on geometric parameters.
Thumbnail: This toy uses the principles of center of mass to keep balance on a finger. (CC BY-SA 3.0; APN MJM).