DescriptionFigureMoment(s) of inertiaComment

Point massmat a distancerfrom the axis of rotation.A point mass does not have a moment of inertia around its own axis, but by using theparallel axis theorema moment of inertia around a distant axis of rotation is achieved.

Two point masses,Mandm, withreduced massand separated by a distance,x.

Rodof lengthLand massm(Axis of rotation at the end of the rod)[1]This expression assumes that the rod is an infinitely thin (but rigid) wire. This is also a special case of the thin rectangular plate with axis of rotation at the end of the plate, withh=Landw=0.

Rodof lengthLand massm[1]This expression assumes that the rod is an infinitely thin (but rigid) wire. This is a special case of the thin rectangular plate with axis of rotation at the center of the plate, withw=Landh=0.

Thin circularhoopof radiusrand massm

This is a special case of atorusforb= 0. (See below.), as well as of a thick-walled cylindrical tube with open ends, withr1=r2andh= 0.

Thin, soliddiskof radiusrand massm

This is a special case of the solid cylinder, withh= 0. Thatis a consequence of thePerpendicular axis theorem.

Thincylindricalshell with open ends, of radiusrand massm[1]This expression assumes the shell thickness is negligible. It is a special case of the thick-walled cylindrical tube forr1=r2.Also, a point mass (m) at the end of a rod of lengthrhas this same moment of inertia and the valueris called theradius of gyration.

Solid cylinder of radiusr, heighthand massm[1]

This is a special case of the thick-walled cylindrical tube, withr1= 0. (Note: X-Y axis should be swapped for a standard right handed frame)

Thick-walled cylindrical tube with open ends, of inner radiusr1, outer radiusr2, lengthhand massm[1][2]

or when defining the normalized thicknesstn=t/rand lettingr=r2,thenWith a density ofand the same geometry

Tetrahedronof sidesand massm

Octahedron(hollow) of sidesand massm

Octahedron(solid) of sidesand massm

Sphere(hollow) of radiusrand massm[1]A hollow sphere can be taken to be made up of two stacks of infinitesimally thin, circular hoops, where the radius differs from0tor(or a single stack, , where the radius differs from-rtor).

Ball(solid) of radiusrand massm[1]A sphere can be taken to be made up of two stacks of infinitesimally thin, solid discs, where the radius differs from 0 tor(or a single stack, where the radius differs from-rtor).Also, it can be taken to be made up of infinitesimally thin, hollow spheres, where the radius differs from 0 tor.

Sphere(shell) of radiusr2, with centered spherical cavity of radiusr1and massm[1]When the cavity radiusr1= 0, the object is a solid ball (above).Whenr1=r2,, and the object is a hollow sphere.

Rightcircularconewith radiusr, heighthand massm[3][3]

Torusof tube radiusa, cross-sectional radiusband massm.About a diameter:[4]About the vertical axis:[4]

Ellipsoid(solid) of semiaxesa,b, andcwith axis of rotationaand massm

Thin rectangular plate of heighthand of widthwand massm(Axis of rotation at the end of the plate)

Thin rectangular plate of heighthand of widthwand massm[1]

Solidcuboidof heighth, widthw, and depthd, and massm

For a similarly orientedcubewith sides of length,.

Solidcuboidof heightD, widthW, and lengthL, and massmwith the longest diagonal as the axis.For a cube with sides,.

Planepolygonwith vertices,,, ...,andmassuniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin.This expression assumes that the polygon isstar-shaped. The vectors,,, ...,areposition vectorsof the vertices.

Infinitediskwith massnormally distributedon two axes around the axis of rotation(i.e.Where:is the mass-density as a function of x and y).