Moment of inertia tensor

Consider a rigid body rotating with fixed angular velocity $\omega$ about an axis that passes through the origin. See Figure 8.1. Let ${\bf r}_i$ be the position vector of the $i$th mass element, whose mass is $m_i$. We expect this position vector to precess about the axis of rotation (which is parallel to $\omega$) with angular velocity $\omega$. It, therefore, follows from Section A.7 that

$$\frac{d{\bf r}_i}{dt} = \mbox{\boldmath$\omega$} \times {\bf r}_i.$$ (8.4)

Thus, Equation (8.4) specifies the velocity, ${\bf v}_i = d{\bf r}_i/dt$, of each mass element as the body rotates with fixed angular velocity $\omega$ about an axis passing through the origin.

Figure 8.1: A rigid rotating body.

The total angular momentum of the body (about the origin) is written

$${\bf L} = \sum_{i=1,N} m_i\,{\bf r}_i\times\frac{d{\bf r}_i}{dt} = \sum_{i=1,N}m_i\,{\bf r}_i\times( \mbox{\boldmath$\omega$} \times {\bf r}_i) = \sum_{i=1,N}m_i\left[r_i^{\,2}\,\mbox{\boldmath$\omega$}- ({\bf r}_i\cdot\mbox{\boldmath$\omega$})\,{\bf r}_i\right],$$ (8.5)

where use has been made of Equation (8.4), and some standard vector identities. (See Section A.4.) The preceding formula can be written as a matrix equation of the form

$$\left(\begin{array}{c}L_x\\ L_y\\ L_z\end{array}\right)= \left(\b... ...}\right)\left(\begin{array}{c}\omega_x\\ \omega_y\\ \omega_z\end{array}\right),$$ (8.6)

where

$${\cal I}_{xx} = \sum_{i=1,N}(y_i^{\,2}+z_i^{\,2}) \,m_i= \int(y^{\,2}+ z^{\,2})\,dm,$$ (8.7)

$${\cal I}_{yy} = \sum_{i=1,N}(x_i^{\,2}+z_i^{\,2}) \,m_i= \int(x^{\,2}+ z^{\,2})\,dm,$$ (8.8)

$${\cal I}_{zz} = \sum_{i=1,N}(x_i^{\,2}+y_i^{\,2}) \,m_i= \int(x^{\,2}+ y^{\,2})\,dm,$$ (8.9)

$${\cal I}_{xy}={\cal I}_{yx} = - \sum_{i=1,N}x_i\,y_i \,m_i=- \int x\,y\,dm,$$ (8.10)

$${\cal I}_{yz}={\cal I}_{zy} = - \sum_{i=1,N}y_i\,z_i \,m_i= -\int y\,z\,dm,$$ (8.11)

$${\cal I}_{xz}={\cal I}_{zx} = - \sum_{i=1,N}x_i\,z_i \,m_i= -\int x\,z\,dm.$$ (8.12)

Here, ${\cal I}_{xx}$ is called the moment of inertia about the $x$-axis, ${\cal I}_{yy}$ the moment of inertia about the $y$-axis, ${\cal I}_{xy}$ the $xy$ product of inertia, ${\cal I}_{yz}$ the $yz$ product of inertia, and so on. The matrix of the ${\cal I}_{ij}$ values is known as the moment of inertia tensor. Each component of the moment of inertia tensor can be written as either a sum over separate mass elements, or as an integral over infinitesimal mass elements. In the integrals, $dm = \rho\,d^{\,3}{\bf r}$, where $\rho$ is the mass density, and $d^{\,3}{\bf r}$ a volume element. Equation (8.6) can be written more succinctly as

$${\bf L} = \textbf{\em I}\, \mbox{\boldmath$\omega$} .$$ (8.13)

Here, it is understood that ${\bf L}$ and $\omega$ are both column vectors, and $\textbf{\em I}$ is the matrix of the ${\cal I}_{ij}$ values. Note that $\textbf{\em I}$ is a real symmetric matrix; that is, ${\cal I}_{ij}^{\,\ast} = {\cal I}_{ij}$ and ${\cal I}_{ji} = {\cal I}_{ij}$.

In general, the angular momentum vector, ${\bf L}$, obtained from Equation (8.13), points in a different direction to the angular velocity vector, $\omega$. In other words, ${\bf L}$ is generally not parallel to $\omega$.

Finally, although the preceding results were obtained assuming a fixed angular velocity they remain valid, at each instant in time, if the angular velocity varies.