Chapter 10.4, Problem 8P

For the mass distributions in Problems 5 to 7, find the inertia tensor about the origin, and find the principal moments of inertia and the principal axes.

For the point mass $m$ we considered in (4.2) to (4.4), the velocity is $v=\omega \times r$ so the kinetic energy is $T=\frac{1}{2}m{v}^2=\frac{1}{2}m(\omega \times r)\cdot (\omega \times r).$ Show that $T$ can be written in matrix notation as $T=\frac{1}{2}{\omega }^{T}I\omega$ where $I$ is the inertia matrix, $\omega$ is a column matrix, and ${\omega }^{\text{T}}$ is a row matrix with elements equal to the components of $\omega$.