Comparison of Simple Harmonic Motion and Angular Simple Harmonic Motion

Linear SHM and angular SHM are structurally identical, differing only in which physical quantities play the roles of displacement, inertia, and restoring constant. In linear SHM the displacement is the linear vector $\vec r$, the inertia factor is the mass $m$ of the oscillating particle, the restoring force is $\vec F=-k\vec r$ with $k$ the force (spring) constant, and $\omega=\sqrt{k/m}$. In angular SHM the displacement is the angular vector $\vec\theta$ (the angle of twist), the inertia factor is the moment of inertia $I$ of the oscillating body, the restoring torque is $\vec\tau=-\kappa\vec\theta$ with $\kappa$ the torsion constant, and $\omega=\sqrt{\kappa/I}$. Correspondingly, acceleration $\vec a=-\omega^2\vec r$ in the linear case matches angular acceleration $\vec\alpha=-\omega^2\vec\theta$ in the angular case, and Newton's second law $\vec F=m\vec a$ matches its rotational analogue $\vec\tau=I\vec\alpha$. Table 10.2 lays these fi …