Moments of Inertia
| Description | Figure | Moment(s) of inertia | Comment |
|---|---|---|---|
| Point mass m at a distance r from the axis of rotation. | A point mass does not have a moment of inertia around its own axis, but by using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved. | ||
| Two point masses, M and m, with reduced mass and separated by a distance, x. | — | ||
| Rod of length L and mass m (Axis of rotation at the end of the rod) |
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, with h = L and w = 0. | ||
| Rod of length L and mass m | 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, with w = L and h = 0. | ||
| Thin circular hoop of radius r and mass m | This is a special case of a torus for b = 0. (See below.), as well as of a thick-walled cylindrical tube with open ends, with r1 = r2 and h = 0. | ||
| Thin, solid disk of radius r and mass m | This is a special case of the solid cylinder, with h = 0. That is a consequence of the Perpendicular axis theorem. | ||
| Thin cylindrical shell with open ends, of radius r and mass m | This expression assumes the shell thickness is negligible. It is a special case of the thick-walled cylindrical tube for r1 = r2.
Also, a point mass (m) at the end of a rod of length r has this same moment of inertia and the value r is called the radius of gyration. |
||
| Solid cylinder of radius r, height h and mass m | |
This is a special case of the thick-walled cylindrical tube, with r1 = 0. (Note: X-Y axis should be swapped for a standard right handed frame) | |
| Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m | or when defining the normalized thickness tn = t/r and letting r = r2, then |
With a density of ρ and the same geometry | |
| Tetrahedron of side s and mass m |
f=ma |
— | |
| Octahedron (hollow) of side s and mass m | — | ||
| Octahedron (solid) of side s and mass m | — | ||
| Sphere (hollow) of radius r and mass m | A hollow sphere can be taken to be made up of two stacks of infinitesimally thin, circular hoops, where the radius differs from 0 to r (or a single stack, where the radius differs from -r to r). | ||
| Ball (solid) of radius r and mass m | A sphere can be taken to be made up of two stacks of infinitesimally thin, solid discs, where the radius differs from 0 to r (or a single stack, where the radius differs from -r to r).
Also, it can be taken to be made up of infinitesimally thin, hollow spheres, where the radius differs from 0 to r. |
||
| Sphere (shell) of radius r2, with centered spherical cavity of radius r1 and mass m | When the cavity radius r1 = 0, the object is a solid ball (above).
When r1 = r2, and the object is a hollow sphere. |
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| Right circular cone with radius r, height h and mass m | |
— | |
| Torus of tube radius a, cross-sectional radius b and mass m. | About a diameter: About the vertical axis: |
— | |
| Ellipsoid (solid) of semiaxes a, b, and c with axis of rotation a and mass m | — | ||
| Thin rectangular plate of height h and of width w and mass m (Axis of rotation at the end of the plate) |
— | ||
| Thin rectangular plate of height h and of width w and mass m | — | ||
| Solid cuboid of height h, width w, and depth d, and mass m | For a similarly oriented cube with sides of length, . | ||
| Solid cuboid of height D, width W, and length L, and mass m with the longest diagonal as the axis. | For a cube with sides, . | ||
| Plane polygon with vertices, ..., and
mass uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin. |
This expression assumes that the polygon is star-shaped. The vectors, ..., are position vectors of the vertices. | ||
| Infinite disk with mass normally distributed on two axes around the axis of rotation
(i.e. Where : is the mass-density as a function of x and y). |
— |
Read more about this topic: List Of Moments Of Inertia
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