AcademyRigid-Body Rotation
Academy
Angular Velocity
Level 1 - Physics topic page in Rigid-Body Rotation.
Principle
Angular velocity is the signed rate of change of angular position.
Notation
\(\theta\)
angular position
\(\mathrm{rad}\)
\(\Delta\theta\)
angular displacement
\(\mathrm{rad}\)
\(\omega\)
angular velocity
\(\mathrm{rad\,s^{-1}}\)
\(T\)
period for one revolution
\(\mathrm{s}\)
\(f\)
rotation frequency
\(\mathrm{Hz}\)
Method
Derivation 1: Define signed angular displacement
Choose a positive rotation direction first. Angular displacement is then a signed change in angle, not the total angle swept.
Angular change
\[\Delta\theta=\theta_f-\theta_i\]
Average rate
\[\omega_{\mathrm{av}}=\frac{\Delta\theta}{\Delta t}\]
Instantaneous rate
\[\omega=\lim_{\Delta t\to0}\frac{\Delta\theta}{\Delta t}=\frac{d\theta}{dt}\]
The sketch fixes what \(\\Delta\\theta\) means: both radius vectors are measured from the same reference direction, and the sign comes from the chosen positive sense.
Derivation 2: Connect period and frequency
Uniform rotation repeats the same angular displacement, \(2\\pi\), every period.
One revolution
\[1\,\mathrm{rev}=2\pi\,\mathrm{rad}\]
Period relation
\[\omega=\frac{2\pi}{T}\]
Frequency relation
\[f=\frac1T\Rightarrow\omega=2\pi f\]
Rules
These are the compact results from the method above.
Average angular velocity
\[\omega_{\mathrm{av}}=\frac{\Delta\theta}{\Delta t}\]
Instantaneous angular velocity
\[\omega=\frac{d\theta}{dt}\]
One revolution
\[1\,\mathrm{rev}=2\pi\,\mathrm{rad}\]
Period relation
\[\omega=\frac{2\pi}{T}=2\pi f\]
Examples
Question
A wheel turns through
\[18\,\mathrm{rad}\]
in \[3.0\,\mathrm{s}\]
Find average angular velocity.Answer
\[\omega_{\mathrm{av}}=\frac{18}{3.0}=6.0\,\mathrm{rad\,s^{-1}}\]
Checks
- Use radians in angular equations.
- Angular velocity can be negative.
- Average angular velocity uses net angular displacement.
- The angular-velocity vector direction follows the right-hand rule.