13.4: Inertia Tensor

The square bracket term in $(13.3.9)$ is called the moment of inertia tensor, $\mathbf{I}$, which is usually referred to as the inertia tensor

$$I_{ij} \equiv \sum^{N}_{\alpha} m_{\alpha} \left[ \delta_{ij} \left( \sum^3_k x^2_{\alpha , k} \right) − x_{\alpha , i} x_{\alpha , j} \right] \label{13.12}$$

In most cases it is more useful to express the components of the inertia tensor in an integral form over the mass distribution rather than a summation for $N$ discrete bodies. That is,

$$I_{ij} = \int\rho (\mathbf{r}^{\prime} ) \left( \delta_{ij} \left( \sum^3_k x^2_{k} \right) − x_{i} x_{ j} \right) dV$$

The inertia tensor is easier to understand when written in cartesian coordinates $\mathbf{r}^{\prime}_{\alpha} = (x_{\alpha}, y_{\alpha}, z_{\alpha})$ rather than in the form $\mathbf{r}^{\prime}_{\alpha} = (x_{\alpha ,1}, x_{\alpha ,2}, x_{\alpha ,3})$. Then, the diagonal moments of inertia of the inertia tensor are

$$\begin{align} I_{x x} & \equiv \sum_{\alpha}^{N} m_{\alpha}\left[x_{\alpha}^{2}+y_{\alpha}^{2}+z_{\alpha}^{2}-x_{\alpha}^{2}\right]=\sum_{\alpha}^{N} m_{\alpha}\left[y_{\alpha}^{2}+z_{\alpha}^{2}\right] \\[4pt] \notag I_{y y} & \equiv \sum_{\alpha}^{N} m_{\alpha}\left[x_{\alpha}^{2}+y_{\alpha}^{2}+z_{\alpha}^{2}-y_{\alpha}^{2}\right]=\sum_{\alpha}^{N} m_{\alpha}\left[x_{\alpha}^{2}+z_{\alpha}^{2}\right] \\[4pt] I_{z z} & \equiv \sum_{\alpha}^{N} m_{\alpha}\left[x_{\alpha}^{2}+y_{\alpha}^{2}+z_{\alpha}^{2}-z_{\alpha}^{2}\right]=\sum_{\alpha}^{N} m_{\alpha}\left[x_{\alpha}^{2}+y_{\alpha}^{2}\right] \notag \end{align}$$

while the off-diagonal products of inertia are

$$\begin{align} I_{yx} & = I_{xy} \equiv - \sum^N_{\alpha} m_{\alpha} [x_{\alpha} y_{\alpha}] \\[4pt] \notag I_{zx} & = I_{xz} \equiv - \sum^N_{\alpha} m_{\alpha} [x_{\alpha} z_{\alpha}] \\[4pt] \notag I_{zy} & = I_{yz} \equiv - \sum^N_{\alpha} m_{\alpha} [y_{\alpha} z_{\alpha}] \end{align}$$

Note that the products of inertia are symmetric in that

$$I_{ij} = I_{ji}$$

The above notation for the inertia tensor allows the angular momentum $\ref{13.12}$ to be written as

$$L_i = \sum^3_j I_{ij} \omega_j$$

Expanded in cartesian coordinates

$$\begin{align} L_x & = I_{xx} \omega_x + I_{xy} \omega_y + I_{xz} \omega_z \\[4pt] \notag L_y & = I_{yx} \omega_x + I_{yy} \omega_y + I_{yz} \omega_z \\[4pt] \notag L_z & = I_{zx} \omega_x + I_{zy} \omega_y + I_{zz} \omega_z \end{align}$$

Note that every fixed point in a body has a specific inertia tensor. The components of the inertia tensor at a specified point depend on the orientation of the coordinate frame whose origin is located at the specified fixed point. For example, the inertia tensor for a cube is very different when the fixed point is at the center of mass compared with when the fixed point is at a corner of the cube.