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 .