List of moments of inertia

The moment of inertia, denoted by I, measures the extent to which an object resists rotational acceleration about a particular axis; it is the rotational analogue to mass (which determines an object's resistance to linear acceleration). The moments of inertia of a mass have units of dimension ML² ([mass] × [length]²). It should not be confused with the second moment of area, which has units of dimension L⁴ ([length]⁴) and is used in beam calculations. The mass moment of inertia is often also known as the rotational inertia or sometimes as the angular mass.

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. In calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and the perpendicular axis theorems.

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.

Moments of inertia

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

List of 3D inertia tensors

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:

$\mathbf {n} \cdot \mathbf {I} \cdot \mathbf {n} \equiv n_{i}I_{ij}n_{j}\,,$

where the dots indicate tensor contraction and the Einstein summation convention is used. In the above table, n would be the unit Cartesian basis e_x, e_y, e_z to obtain I_x, I_y, I_z respectively.