The mass moment of inertia (or simply moment of inertia), plays the role of mass in rotational dynamics, as we saw in Eq. (B.7) above.
The mass moment of inertia of a rigid body, relative to a given axis of rotation, is given by a weighted sum over its mass, with each mass-point weighted by the square of its distance from the rotation axis. Compare this with the center of mass (§B.4.1) in which each mass-point is weighted by its vector location in space (and divided by the total mass).
Equation (B.8) above gives the moment of inertia for a single point-mass rotating a distance from the axis to be . Therefore, for a rigid collection of point-masses , ,B.14 the moment of inertia about a given axis of rotation is obtained by adding the component moments of inertia:
For a continuous mass distribution, the moment of inertia is given by integrating the contribution of each differential mass element:
where denotes the mass density (kg/m ) at the point , and denotes a differential volume element located at .