List of moments of inertia

**Moment of inertia**, denoted by I, measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass (which determines an object's resistance to *linear* acceleration). Mass moments of inertia have units of dimension ML2([mass] × [length]2). It should not be confused with the second moment of area, which is used in beam calculations. The mass moment of inertia is often also known as the **rotational inertia**, and 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. When calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and perpendicular axis theorems.

This article mainly considers 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.

## Rotational momentum

Rotational momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of rotational momentum, and treat it as a scalar (more precisely, a pseudoscalar). Rotational momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum p is proportional to mass m and linear speed v,

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rotational momentum L is proportional to rotational mass I and angular speed ω measured in radians per second.

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Unlike mass, which depends only on amount of matter, rotational mass is also dependent on the position of the axis of rotation and the shape of the matter. Unlike linear velocity, which does not depend upon the choice of origin, orbital rotational velocity is always measured with respect to a fixed origin. Therefore, strictly speaking, L should be referred to as the rotational momentum *relative to that center*.

## 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:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \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 *Ix*, *Iy*, *Iz* respectively.

## See also

* Parallel axis theorem
* Perpendicular axis theorem
* List of second moments of area

## References

1. ↑ Width perpendicular to the axis of rotation (side of plate); height (parallel to axis) is irrelevant.