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

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

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Aug 1, 2021
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Shape with Volume and Center of Mass Location Shown	Mass Moments of Inertia
Slender Rod	$I_{xx} = I_{zz} = \dfrac{1}{12} ml^2$ $I_{yy} = 0$ $I_{xx'} = I_{zz'} = \dfrac{1}{3} ml^2$
Flat Rectangular Plate	$I_{xx} = \dfrac{1}{12} mh^2$ $I_{yy} = \dfrac{1}{12} m (h^2 + b^2)$ $I_{zz} = \dfrac{1}{12} mb^2$
Flat Circular Plate	$I_{xx} = I_{zz} = \dfrac{1}{4} mr^2$ $I_{yy} = \dfrac{1}{2} mr^2$
Thin Circular Ring	$I_{xx} = I_{zz} = \dfrac{1}{2} mr^2$ $I_{yy} = mr^2$
Rectangular Prism $Volume = dwh$	$\begin{align} I_{xx} &= \frac{1}{12} m(h^2 + d^2) \\[4pt] I_{yy} &= \frac{1}{12} m(d^2 + w^2) \\[4pt] I_{zz} &= \frac{1}{12} m(h^2 + w^2) \end{align}$
Cylinder $Volume = \pi r^2 h$	$I_{xx} = I_{zz} = \dfrac{1}{12} m(3r^2 + h^2)$ $I_{yy} = \dfrac{1}{2} mr^2$
Thin Cylindrical Shell	$I_{xx} = I_{zz} = \dfrac{1}{6} m(3r^2 + h^2)$ $I_{yy} = mr^2$
Half Cylinder $Volume = \dfrac{1}{2} \pi r^2h$	$I_{xx} = I_{zz} = \left( \dfrac{1}{4} - \dfrac{16}{9 \pi^2} \right) mr^2 + \dfrac{1}{12} mh^2$ $I_{yy} = \left( \dfrac{1}{2} - \dfrac{16}{9 \pi^2} \right) mr^2$ $I_{xx'} = I_{zz'} = \dfrac{1}{12} m(3r^2 + h^2)$ $I_{yy'} = \dfrac{1}{2} mr^2$
Sphere $Volume = \dfrac{4}{3} \pi r^3$	$I_{xx} = I_{yy} = I_{zz} = \dfrac{2}{5} mr^2$
Spherical Shell	$I_{xx} = I_{yy} = I_{zz} = \dfrac{2}{3} m r^2$
Hemisphere $Volume = \dfrac{2}{3} \pi r^3$	$I_{xx} = I_{zz} = \dfrac{83}{320} mr^2$ $I_{yy} = \dfrac{2}{5} mr^2$ $I_{xx'} = I_{zz'} = \dfrac{2}{5} mr^2$
Hemispherical Shell	$I_{xx} = I_{zz} = \dfrac{5}{12} mr^2$ $I_{yy} = \dfrac{2}{3} mr^2$ $I_{xx'} = I_{zz'} = \dfrac{2}{3} mr^2$
Right Circular Cone $Volume = \dfrac{1}{3} \pi r^2 h$	$I_{xx} = I_{zz} = \dfrac{3}{80} m(4r^2 + h^2)$ $I_{yy} = \dfrac{3}{10} mr^2$ $I_{xx'} = I_{zz'} = \dfrac{1}{20} m(3r^2 + 2h^2)$

Shape with Volume and Center of Mass Location Shown

Mass Moments of Inertia

Slender Rod

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

$I_{yy} = 0$

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

Flat Rectangular Plate

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

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

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

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

$Volume = dwh$

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

Cylinder

$Volume = \pi r^2 h$

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

$I_{yy} = \dfrac{1}{2} mr^2$

Thin Cylindrical Shell

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

$I_{yy} = mr^2$

Half Cylinder

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

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

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

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

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

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

$Volume = \dfrac{2}{3} \pi r^3$

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

$I_{yy} = \dfrac{2}{5} mr^2$

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

Hemispherical Shell

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

$I_{yy} = \dfrac{2}{3} mr^2$

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

Right Circular Cone

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

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

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

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

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