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.
Following are scalar moments of inertia.
In general, the moment of inertia is a tensor: see below.
π
ρ
is a normalized thickness ratio;
[citation needed] which coincides with the geometric center of the cylinder.
If the xy plane is at the base of the cylinder, i.e. offset by
then by the parallel axis theorem the following formula applies:
π ρ h
, and the object is a hollow sphere.
About an axis passing through the tip:
[4] About an axis passing through the base:
About an axis passing through the center of mass:
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 the Einstein summation convention is used.
In the above table, n would be the unit Cartesian basis ex, ey, ez to obtain Ix, Iy, Iz respectively.
{\displaystyle I={\begin{bmatrix}{\frac {2}{3}}mr^{2}&0&0\\0&{\frac {2}{3}}mr^{2}&0\\0&0&{\frac {2}{3}}mr^{2}\end{bmatrix}}}
{\displaystyle I={\begin{bmatrix}{\frac {1}{3}}ml^{2}&0&0\\0&0&0\\0&0&{\frac {1}{3}}ml^{2}\end{bmatrix}}}
{\displaystyle I={\begin{bmatrix}{\frac {1}{12}}ml^{2}&0&0\\0&0&0\\0&0&{\frac {1}{12}}ml^{2}\end{bmatrix}}}
{\displaystyle I={\begin{bmatrix}{\frac {1}{12}}m(3r^{2}+h^{2})&0&0\\0&{\frac {1}{12}}m(3r^{2}+h^{2})&0\\0&0&{\frac {1}{2}}mr^{2}\end{bmatrix}}}
{\displaystyle I={\begin{bmatrix}{\frac {1}{12}}m(3(r_{2}^{2}+r_{1}^{2})+h^{2})&0&0\\0&{\frac {1}{12}}m(3(r_{2}^{2}+r_{1}^{2})+h^{2})&0\\0&0&{\frac {1}{2}}m(r_{2}^{2}+r_{1}^{2})\end{bmatrix}}}
