Mass Moment of Inertia

The mass moment of inertia $I$ (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 $m$ rotating a distance $R$ from the axis to be $I=mR^2$ . Therefore, for a rigid collection of point-masses $m_i$ , $i=1,\ldots,N$ ,^B.14 the moment of inertia about a given axis of rotation is obtained by adding the component moments of inertia:

$$I = m_1 R_1^2 + m_2 R_2^2 + \cdots + m_N R_N^2, \protect$$ (B.9)

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

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

$$I \eqsp \int_M R^2 dm,$$ (B.10)

where $R$ is the distance from the axis of rotation to the mass element $dm$ . In terms of the density $\rho(\underline{x})$ of a continuous mass distribution, we can write

$$I \,\mathrel{\mathop=}\,\int_V R^2(\underline{x})\,\rho(\underline{x})\,dV,$$

where $\rho(\underline{x})$ denotes the mass density (kg/m$\null^3$ ) at the point $\underline {x}$ , and $dV=dx\,dy\,dz$ denotes a differential volume element located at $\underline{x}\in\mathbb{R}^3$ .