moment of inertia
For a planar rigid body, rotating with angular speed ω about an axis perpendicular to the plane, the moment of inertia is given by
where the first expression is for a discrete collection of particles of mass mi at distance ri from the axis, and the second expression is for a continuous distribution of matter within a region R of density ρ(r), where r is the position vector of a point from the axis. The angular momentum of the body equals Iω, and its (rotational) kinetic energy equals ½Iω2.
Note that the same rigid body will have different moments of inertia for different axes. For example, the moment of inertia of a uniform disc will be greater for an axis through a point of the circumference than through the centre, as the mass is distributed farther from the axis. Moments of inertia about other axes may be calculated by using the parallel axis theorem and the perpendicular axis theorem. For a three-dimensional rigid body, moments of inertia Ixx, Iyy, and Izz can be defined for the x‐axis, the y‐axis, and the z‐axis. Here ri and r denote the distance and displacement from the axis. See inertia matrix, product of inertia.
For a list of various moments of inertia, see appendix 3.