AcademyRigid-Body Rotation
Academy
Angular Acceleration
Level 1 - Physics topic page in Rigid-Body Rotation.
Principle
Angular acceleration is the signed rate of change of angular velocity.
Notation
\(\alpha\)
angular acceleration
\(\mathrm{rad\,s^{-2}}\)
\(\omega\)
angular velocity
\(\mathrm{rad\,s^{-1}}\)
\(\theta\)
angular position
\(\mathrm{rad}\)
\(a_t\)
tangential acceleration
\(\mathrm{m\,s^{-2}}\)
\(r\)
distance from rotation axis
\(\mathrm{m}\)
Method
Derivation 1: Differentiate angular velocity
Angular acceleration is a slope on an angular-velocity graph. It tells how \(\\omega\) changes, not whether rotation is clockwise or counterclockwise by itself.
Average angular acceleration
\[\alpha_{\mathrm{av}}=\frac{\Delta\omega}{\Delta t}\]
Instantaneous angular acceleration
\[\alpha=\lim_{\Delta t\to0}\frac{\Delta\omega}{\Delta t}=\frac{d\omega}{dt}\]
Second derivative
\[\alpha=\frac{d^2\theta}{dt^2}\]
The graph shows the geometric reading: the slope of \(\omega(t)\) is \(\alpha\). A straight line means constant angular acceleration.
Derivation 2: Interpret sign and tangential acceleration
Angular speed increases when \(\omega\) and \(\alpha\) have the same sign. A point farther from the axis has the same \(\alpha\) but larger tangential acceleration.
Speeding up
\[\omega\alpha>0\]
Slowing down
\[\omega\alpha<0\]
Tangential acceleration
\[a_t=r\alpha\]
Rules
These are the compact results from the method above.
Average angular acceleration
\[\alpha_{\mathrm{av}}=\frac{\Delta\omega}{\Delta t}\]
Instantaneous angular acceleration
\[\alpha=\frac{d\omega}{dt}\]
Second derivative
\[\alpha=\frac{d^2\theta}{dt^2}\]
Tangential acceleration
\[a_t=r\alpha\]
Examples
Question
A rotor changes from
\[4\,\mathrm{rad\,s^{-1}}\]
to \[16\,\mathrm{rad\,s^{-1}}\]
in \[3\,\mathrm{s}\]
Find \[\alpha_{\mathrm{av}}\]
Answer
\[\alpha_{\mathrm{av}}=\frac{16-4}{3}=4.0\,\mathrm{rad\,s^{-2}}\]
Checks
- Angular acceleration is not automatically in the same direction as angular velocity.
- Positive \(\alpha\) means increasing \(\omega\), not necessarily increasing angular speed.
- Tangential acceleration grows with radius.
- Centripetal acceleration is separate from angular acceleration.