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Center of Mass and Mass Moments of Inertia for Homogeneous 3D Bodies

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    Shape with Volume and Center of Mass Location Shown Mass Moments of Inertia

    Slender Rod

    A three-dimensional Cartesian coordinate system with the z'-axis pointing out of the screen, the x'-axis lying horizontally in the plane of the screen, and the y-axis lying vertically in the plane of the screen. A rod of length l lies along the positive y-axis, with one end at the origin of this system. The rod's center of mass G lies l/2 units above the origin. Point G forms the origin of another Cartesian system, with the z-axis pointing out of the screen, the x-axis lying horizontally in the plane of the screen, and the y-axis shared with the existing y-axis.

    \(I_{xx} = I_{zz} = \dfrac{1}{12} ml^2\)

    \(I_{yy} = 0\)

    \(I_{xx'} = I_{zz'} = \dfrac{1}{3} ml^2\)

    Flat Rectangular Plate

    A three-dimensional Cartesian coordinate system with the z-axis pointing out of the screen, the x-axis lying horizontally in the plane of the screen, and the y-axis lying vertically in the plane of the screen. A flat rectangular plate lies in the xz-plane, with its center of mass G at the origin of this system. The plate has a length of b parallel to the x-axis, and and a width of h, parallel to the z-axis.

    \(I_{xx} = \dfrac{1}{12} mh^2\)

    \(I_{yy} = \dfrac{1}{12} m (h^2 + b^2) \)

    \(I_{zz} = \dfrac{1}{12} mb^2\)

    Flat Circular Plate

    A three-dimensional Cartesian coordinate system with the z-axis pointing out of the screen, the x-axis lying horizontally in the plane of the screen, and the y-axis lying vertically in the plane of the screen. A flat circular plate of radius r lies in the xz-plane, with its center of mass G located at the origin of the system.

    \(I_{xx} = I_{zz} = \dfrac{1}{4} mr^2\)

    \(I_{yy} = \dfrac{1}{2} mr^2\)

    Thin Circular Ring

    A three-dimensional Cartesian coordinate plane with the z-axis pointing out of the screen, the x-axis lying horizontally in the plane of the screen, and the y-axis lying vertically in the plane of the screen. A thin circular ring of radius r lies in the xz-plane, with its center of mass G lying at the origin of this system.

    \(I_{xx} = I_{zz} = \dfrac{1}{2} mr^2\)

    \(I_{yy} = mr^2\)

    Rectangular Prism

    A three-dimensional Cartesian coordinate plane with the z-axis pointing out of the screen, the x-axis lying horizontally in the plane of the screen, and the y-axis lying vertically in the plane of the screen. A rectangular prism is centered on this system, with the center of mass G lying at the origin. The prism has a width of w units lying parallel to the x-axis, a height of h units lying parallel to the y-axis, and a depth of d units lying parallel to the z-axis.

    \(Volume = dwh\)

    \begin{align*} I_{xx} &= \frac{1}{12} m(h^2 + d^2) \\[4pt] I_{yy} &= \frac{1}{12} m(d^2 + w^2) \\[4pt] I_{zz} &= \frac{1}{12} m(h^2 + w^2) \end{align*}

    Cylinder

    A three-dimensional Cartesian coordinate plane with the z-axis pointing out of the screen, the x-axis lying horizontally in the plane of the screen, and the y-axis lying vertically in the plane of the screen. A cylinder lies centered in this system, with its center of mass G at the origin. The cylinder's base has a radius of r and lies parallel to the xz-plane, and the cylinder has a height h that is measured parallel to the y-axis.

    \(Volume = \pi r^2 h\)

    \( I_{xx} = I_{zz} = \dfrac{1}{12} m(3r^2 + h^2) \)

    \(I_{yy} = \dfrac{1}{2} mr^2\)

    Thin Cylindrical Shell

    A three-dimensional Cartesian coordinate plane with the z-axis pointing out of the screen, the x-axis lying horizontally in the plane of the screen, and the y-axis lying vertically in the plane of the screen. A hollow cylindrical shell lies centered in this system, with its center of mass G at the origin. The cylindrical shell's base has a radius of r and lies parallel to the xz-plane, and the shell has a height h that is measured parallel to the y-axis.

    \(I_{xx} = I_{zz} = \dfrac{1}{6} m(3r^2 + h^2)\)

    \(I_{yy} = mr^2\)

    Half Cylinder

    A three-dimensional Cartesian coordinate plane with the z-axis pointing out of the screen, the x'-axis lying horizontally in the plane of the screen, and the y'-axis lying vertically in the plane of the screen. A half-cylinder lies with its rectangular face in the x'y'-plane and the centroid of that rectangular face at the origin of this system. The half-cylinder has a radius of r and a height of h, measured parallel to the y'-axis. The center of mass G of the half-cylinder lies a distance of 4r/(3 pi) from the origin of this system, in the negative z-direction. Point G forms the origin of another three-dimensional Cartesian system, with the x-axis lying horizontally inside the plane of the screen, the y-axis lying vertically inside the plane of the screen, and the z-axis shared with the original z-axis.

    \(Volume = \dfrac{1}{2} \pi r^2h\)

    \(I_{xx} = I_{zz} = \left( \dfrac{1}{4} - \dfrac{16}{9 \pi^2} \right) mr^2 + \dfrac{1}{12} mh^2\)

    \(I_{yy} = \left( \dfrac{1}{2} - \dfrac{16}{9 \pi^2} \right) mr^2\)

    \(I_{xx'} = I_{zz'} = \dfrac{1}{12} m(3r^2 + h^2)\)

    \(I_{yy'} = \dfrac{1}{2} mr^2\)

    Sphere

    A three-dimensional Cartesian coordinate plane with the z-axis pointing out of the screen, the x-axis lying horizontally in the plane of the screen, and the y-axis lying vertically in the plane of the screen. A sphere of radius r lies with its center of mass G at the origin of this system.

    \(Volume = \dfrac{4}{3} \pi r^3\)

    		 \(I_{xx} = I_{yy} = I_{zz} = \dfrac{2}{5} mr^2\)

    Spherical Shell

    A three-dimensional Cartesian coordinate plane with the z-axis pointing out of the screen, the x-axis lying horizontally in the plane of the screen, and the y-axis lying vertically in the plane of the screen. A thin spherical shell of radius r with a hollow interior lies in this system, with its center of mass G located at the origin.

    		 \(I_{xx} = I_{yy} = I_{zz} = \dfrac{2}{3} m r^2\)

    Hemisphere

    A three-dimensional Cartesian coordinate plane with the z'-axis pointing out of the screen, the x'-axis lying horizontally in the plane of the screen, and the y-axis lying vertically in the plane of the screen. A solid hemisphere of radius r lies in this system, with its flat base located in the x'z'-plane and centered at the origin. The hemisphere extends upwards along the positive y-axis, and its center of mass G is located 3r/8 units above the origin. Point G forms the origin of another coordinate system, with the z-axis pointing out of the screen, the x-axis lying horizontally in the plane of the screen, and the y-axis shared with the y-axis of the existing system.

    \(Volume = \dfrac{2}{3} \pi r^3\)

    \(I_{xx} = I_{zz} = \dfrac{83}{320} mr^2\)

    \(I_{yy} = \dfrac{2}{5} mr^2\)

    \(I_{xx'} = I_{zz'} = \dfrac{2}{5} mr^2\)

    Hemispherical Shell

    A three-dimensional Cartesian coordinate plane with the z'-axis pointing out of the screen, the x'-axis lying horizontally in the plane of the screen, and the y-axis lying vertically in the plane of the screen. A hollow, bowl-like hemispherical shell of radius r lies in this system, with its flat base located in the x'z'-plane and centered at the origin. The hemisphere extends upwards along the positive y-axis, and its center of mass G is located r/2 units above the origin. Point G forms the origin of another coordinate system, with the z-axis pointing out of the screen, the x-axis lying horizontally in the plane of the screen, and the y-axis shared with the y-axis of the existing system.

    \(I_{xx} = I_{zz} = \dfrac{5}{12} mr^2\)

    \(I_{yy} = \dfrac{2}{3} mr^2\)

    \(I_{xx'} = I_{zz'} = \dfrac{2}{3} mr^2\)

    Right Circular Cone

    A three-dimensional Cartesian coordinate plane with the z'-axis pointing out of the screen, the x'-axis lying horizontally in the plane of the screen, and the y-axis lying vertically in the plane of the screen. A right circular cone lies in this system, with its base of radius r located in the x'z'-plane and centered at the origin and its height h extending along the positive y-axis. The cone's center of mass G is located on the y-axis, h/4 units above the origin. Point G forms the origin of another coordinate system, with the z-axis pointing out of the screen, the x-axis lying horizontally in the plane of the screen, and the y-axis shared with the y-axis of the existing system.

    \(Volume = \dfrac{1}{3} \pi r^2 h\)

    \(I_{xx} = I_{zz} = \dfrac{3}{80} m(4r^2 + h^2) \)

    \(I_{yy} = \dfrac{3}{10} mr^2\)

    \(I_{xx'} = I_{zz'} = \dfrac{1}{20} m(3r^2 + 2h^2) \)
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