### MassMoment 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 massB.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: (B.9)

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 density of a continuous mass distribution, we can write where denotes the mass density (kg/m ) at the point , and denotes a differential volume element located at .

#### Circular Disk Rotating in Its Own Plane

For example, the moment of inertia for a uniform circular disk of total mass and radius , rotating in its own plane about a rotation axis piercing its center, is given by #### Circular Disk Rotating About Its Diameter

The moment of inertia for the same circular disk rotating about an axis in the plane of the disk, passing through its center, is given by Thus, the uniform disk's moment of inertia in its own plane is twice that about its diameter.

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