AcademyRigid-Body Rotation
Academy
Constant Angular Acceleration
Level 1 - Physics topic page in Rigid-Body Rotation.
Principle
Constant angular acceleration makes angular velocity change linearly with time.
Notation
\(\theta\)
angular position
\(\mathrm{rad}\)
\(\theta_0\)
initial angular position
\(\mathrm{rad}\)
\(\omega_0\)
initial angular velocity
\(\mathrm{rad\,s^{-1}}\)
\(\omega\)
final angular velocity
\(\mathrm{rad\,s^{-1}}\)
\(\alpha\)
constant angular acceleration
\(\mathrm{rad\,s^{-2}}\)
Method
Derivation 1: Integrate constant angular acceleration
The constant-\(\alpha\) equations come from the definitions of angular velocity and angular acceleration.
Start from constant acceleration
\[\alpha=\frac{d\omega}{dt}\]
Integrate once
\[\omega=\omega_0+\alpha t\]
Use angular velocity
\[\omega=\frac{d\theta}{dt}\]
Integrate again
\[\theta=\theta_0+\omega_0t+\frac12\alpha t^2\]
With constant \(\alpha\), the graph of \(\omega\) against time is a straight line. The slope is \(\alpha\), and the area under the line is angular displacement.
Derivation 2: Eliminate time when needed
If time is not needed, combine the velocity equation with the average angular velocity for constant acceleration.
Mean angular velocity
\[\Delta\theta=\frac12(\omega_0+\omega)t\]
Eliminate time
\[t=\frac{\omega-\omega_0}{\alpha}\]
No-time equation
\[\omega^2=\omega_0^2+2\alpha(\theta-\theta_0)\]
Rules
These are the compact results from the method above.
Angular velocity
\[\omega=\omega_0+\alpha t\]
Angular position
\[\theta=\theta_0+\omega_0t+\frac12\alpha t^2\]
No time
\[\omega^2=\omega_0^2+2\alpha(\theta-\theta_0)\]
Mean velocity
\[\theta-\theta_0=\frac12(\omega_0+\omega)t\]
Examples
Question
A wheel starts from rest with
\[\alpha=3.0\,\mathrm{rad\,s^{-2}}\]
for \[4.0\,\mathrm{s}\]
Find \(\omega\) and \[\Delta\theta\]
Answer
\[\omega=0+3.0(4.0)=12\,\mathrm{rad\,s^{-1}}\]
\[\Delta\theta=\frac12(3.0)(4.0)^2=24\,\mathrm{rad}\]
Checks
- Use angular displacement, not arc length.
- Convert revolutions to radians before substitution.
- A negative \(\alpha\) can either slow or speed rotation depending on \(\omega\).
- The constant-\(\alpha\) equations require \(\alpha\) to be constant.