Introduction to Angular Kinematics

What’s more fun than moving in a straight line? Moving on a plane. What’s more fun than that? Moving in space. What’s better than that?

They weren’t sure, so the Official Academy of Physicists, which does not actually exist, concocted moving in a circle, or angular motion. We touched on that in the article on uniform circular motion, where I gave the expression most fundamental to the study of angular motion: $\displaystyle \mathbf{s}=\mathbf{r}\times\boldsymbol{\theta}$

Every equation we use in studying angular motion either has a direct analogue in linear motion or is a variant on that equation. For example, you’ll surely remember the equation for torque: $ \boldsymbol{\tau}=\mathbf{r}\times\mathbf{F}$. Torque ($ \boldsymbol{\tau}$) is the angular analogue of (linear) force $ \mathbf{F}$, and all fundamental equations for angular analogues take the form $\displaystyle [\textbf{angular quantity}] = \mathbf{r}\times[\textbf{linear quantity}]$

Therefore, here’s a quick rundown of important angular quantities for a distance $ \mathbf{r}$ from a fixed axis of rotation:

  • Angular displacement (arc length): $ \mathbf{s}=\mathbf{r}\times\mathbf{\theta}$
  • Angular velocity: $ \boldsymbol{\omega}=\mathbf{r}\times\mathbf{v}$
  • Angular acceleration: $ \boldsymbol{\alpha}=\mathbf{r}\times\mathbf{a}$
  • Angular momentum: $ \mathbf{L}=\mathbf{r}\times\mathbf{p}$
  • Angular force, called torque: $ \boldsymbol{\tau}=\mathbf{r}\times\mathbf{F}$

Two quick notes — first, about units: we measure angles in radians $ [\mathrm{rad}]$, as you know, but radians really have no dimension because $ \theta=\frac{s}{r}$; since $ s$ and $ r$ both have units of $ [\mathrm{m}]$, the units cancel leaving none. Some quick dimensional analysis will show you, then, that the units for $ \boldsymbol{\omega}$ are either $ \mathrm{rad}\cdot\mathrm{s}^{-1}$ or just $ \mathrm{s}^{-1}$, and so on and so forth.

The second note regards the nature of angular velocity, acceleration, momentum, et cetera: since they are created by the cross product, it’s worth mentioning that these are not in fact true vectors but rather pseudovectors; that is, if we apply an improper rotation to them, they flip sign. This shouldn’t affect your calculations for the most part, but I think it merits mention.

Now that we’ve established the angular analogues of most quantities involved in rotation, we have one more to discuss—and a very important one it is—that of rotational inertia (also called moment of inertia). But it’s a somewhat involved topic, so we’ll save that one for next time.

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