Mass Moment of Inertia

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:

where is the distance from the axis of rotation to the th mass.

For a continuous mass distribution, the moment of inertia is given by integrating the contribution of each differential mass element:

(B.10) |

where is the distance from the axis of rotation to the mass element . In terms of the

where denotes the mass density (kg/m ) at the point , and denotes a differential volume element located at .

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