Center of Mass and Mass Moments of Inertia for Homogeneous Bodies


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

Slender Rod

Centroid of a Slender Rod 		\[I_{xx}=I_{zz}=\frac{1}{12}ml^{2}\] \[I_{yy}=0\]
\[I_{xx'}=I_{zz'}=\frac{1}{3}ml^{2}\]

Flat Rectangular Plate

Centroid of a Flat Rectangular Plate 		\[I_{xx}=\frac{1}{12}mh^{2}\] \[I_{yy}=\frac{1}{12}m(h^{2}+b^{2})\] \[I_{zz}=\frac{1}{12}mb^{2}\]

Flat Circular Plate

Centroid of a Flat Circular Plate 		\[I_{xx}=I_{zz}=\frac{1}{4}mr^{2}\] \[I_{yy}=\frac{1}{2}mr^{2}\]

Thin Circular Ring

Centroid of a Thin Ring 		\[I_{xx}=I_{zz}=\frac{1}{2}mr^{2}\] \[I_{yy}=mr^{2}\]

Rectangular Prism

Centroid of a Rectangular Prism \[Volume=dhw\] 		\[I_{xx}=\frac{1}{12}m(h^{2}+d^{2})\] \[I_{yy}=\frac{1}{12}m(d^{2}+w^{2})\] \[I_{zz}=\frac{1}{12}m(h^{2}+w^{2})\]

Cylinder

Centroid of a Cylinder \[Volume=\pi r^{2}h\] 		\[I_{xx}=I_{zz}=\frac{1}{12}m(3r^{2}+h^{2})\] \[I_{yy}=\frac{1}{2}mr^{2}\]

Thin Cylindrical Shell

Centroid of a Cylindrical Shell		 \[I_{xx}=I_{zz}=\frac{1}{6}m(3r^{2}+h^{2})\] \[I_{yy}=mr^{2}\]

Half Cylinder

Centroid of a Half Cylinder \[Volume=\frac{1}{2}\pi r^{2}h\] 		\[I_{xx}=I_{zz}=\left( \frac{1}{4}-\frac{16}{9\pi^{2}} \right)mr^{2}+\frac{1}{12}mh^{2}\] \[I_{yy}=\left(\frac{1}{2}-\frac{16}{9\pi^{2}}\right)mr^{2}\]
\[I_{xx'}=I_{zz'}=\frac{1}{12}m(3r^{2}+h^{2})\] \[I_{yy'}=\frac{1}{2}mr^{2}\]

Sphere

Centroid of a Sphere \[Volume=\frac{4}{3}\pi r^{3}\] 		\[I_{xx}=I_{yy}=I_{zz}=\frac{2}{5}mr^{2}\]

Spherical Shell

Centroid of a Spherical Shell 		\[I_{xx}=I_{yy}=I_{zz}=\frac{2}{3}mr^{2}\]

Hemisphere

Centroid of a Hemisphere \[Volume=\frac{2}{3}\pi r^{3}\] 		\[I_{xx}=I_{zz}=\frac{83}{320}mr^{2}\] \[I_{yy}=\frac{2}{5}mr^{2}\]
\[I_{xx'}=I_{zz'}=\frac{2}{5}mr^{2}\]

Hemispherical Shell

Centroid of a Hemispherical Shell 		\[I_{xx}=I_{zz}=\frac{5}{12}mr^{2}\] \[I_{yy}=\frac{2}{3}mr^{2}\]
\[I_{xx'}=I_{zz'}=\frac{2}{3}mr^{2}\]

Right Circular Cone

Centroid of a Cone \[Volume=\frac{1}{3}\pi r^{2}h\] 		\[I_{xx}=I_{zz}=\frac{3}{80}m(4r^{2}+h^{2})\] \[I_{yy}=\frac{3}{10}mr^{2}\]
\[I_{xx'}=I_{zz'}=\frac{1}{20}m(3r^{2}+2h^{2})\]