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In physics and applied mathematics, the mass moment of inertia, usually denoted I, measures the extent to which an object resists rotational acceleration about an axis, and is the rotational analogue to mass. Mass moments of inertia have units of dimension [mass] × [length]^{2}. It should not be confused with the second moment of area, which is used in bending calculations.

For simple objects with geometric symmetry, one can often determine the moment of inertia in an exact closed-form expression. Typically this occurs when the mass density is constant, but in some cases the density can vary throughout the object as well. In general, it may not be straightforward to symbolically express the moment of inertia of shapes with more complicated mass distributions and lacking symmetry. When calculating moments of inertia, it is helpful to exploit the properties of the moment of inertia, namely it is an additive quantity and the parallel axis theorem, and perpendicular axis theorem.

This article considers mainly symmetric mass distributions, with constant density throughout the object, and the axis of rotation is taken to be through the center of mass, unless otherwise specified.

Following are scalar moments of inertia. In general, the moment of inertia is a tensor, see below.

Description | Figure | Moment(s) of inertia |
---|---|---|

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 μ 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 | ^{[1]} | |

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 | ^{[1]} | |

Thin circular hoop of radius r and mass m. This is a special case of a torus for | ||

Thin, solid disk of radius r and mass m. This is a special case of the solid cylinder, with | ||

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 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. | ^{[1]} | |

Solid cylinder of radius r, height h and mass m. This is a special case of the thick-walled cylindrical tube, with | ^{[1]} | |

Thick-walled cylindrical tube with open ends, of inner radius r_{1}, outer radius r_{2}, length h and mass m. With a density of | ^{[1]}^{[2]}or when defining the normalized thickness t = _{n}t/r and letting r = r_{2},then | |

Tetrahedron of side s and mass m | ||

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 | ^{[1]} | |

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 | ^{[1]} | |

Sphere (shell) of radius r_{2}, with centered spherical cavity of radius r_{1} and mass m. When the cavity radius When | ^{[1]} | |

Right circular cone with radius r, height h and mass m | ^{[3]}^{[3]} | |

Torus of tube radius a, cross-sectional radius b and mass m. About the vertical axis: | About a diameter: ^{[4]} | |

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 | ^{[1]} | |

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 , . | ||

Triangle with vertices at the origin and at P and Q, with mass m, rotating about an axis perpendicular to the plane and passing through the origin. | ||

Plane polygon with vertices P_{1}, P_{2}, P_{3}, ..., P_{N} and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin. | ||

Plane regular polygon with n-vertices and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origina stands for side length. | ^{[5]} | |

Infinite disk with mass normally distributed on two axes around the axis of rotation with mass-density as a function of x and y: |

This list of moment of inertia tensors is given for principal axes of each object.

To obtain the scalar moments of inertia *I* above, the tensor moment of inertia **I** is projected along some axis defined by a unit vector **n** according to the formula:

where the dots indicate tensor contraction and we have used the Einstein summation convention. In the above table, **n** would be the unit Cartesian basis **e**_{x}, **e**_{y}, **e**_{z} to obtain *I _{x}*,

Description | Figure | Moment of inertia tensor |
---|---|---|

Solid sphere of radius r and mass m | ||

Hollow sphere of radius r and mass m | ||

Solid ellipsoid of semi-axes a, b, c and mass m | ||

Right circular cone with radius r, height h and mass m, about the apex | ||

Solid cuboid of width w, height h, depth d, and mass m | ||

Slender rod along y-axis of length l and mass m about end | ||

Slender rod along y-axis of length l and mass m about center | ||

Solid cylinder of radius r, height h and mass m | ||

Thick-walled cylindrical tube with open ends, of inner radius r_{1}, outer radius r_{2}, length h and mass m |

- Parallel axis theorem
- Perpendicular axis theorem
- List of area moments of inertia
- The inertia tensor of a triangle in three-dimensional space

- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}Raymond A. Serway (1986).*Physics for Scientists and Engineers, second ed*. Saunders College Publishing. p. 202. ISBN 0-03-004534-7. **^**Classical Mechanics - Moment of inertia of a uniform hollow cylinder. LivePhysics.com. Retrieved on 2008-01-31.- ^
^{a}^{b}Ferdinand P. Beer and E. Russell Johnston, Jr (1984).*Vector Mechanics for Engineers, fourth ed*. McGraw-Hill. p. 911. ISBN 0-07-004389-2. - ^
^{a}^{b}Eric W. Weisstein. "Moment of Inertia — Ring". Wolfram Research. Retrieved 2010-03-25. **^**Karel Rektorys (1994).*Survey of Applicable Mathematics, second ed., Vol. II*. Kluwer Academic Publisher. p. 942. ISBN 0-7923-0681-3.