AcademyRotational Dynamics
Academy
Angular Momentum
Level 1 - Physics topic page in Rotational Dynamics.
Principle
Angular momentum measures rotational motion about a chosen point or axis.
Notation
\(\vec L\)
angular momentum
\(\mathrm{kg\,m^{2}\,s^{-1}}\)
\(\vec r\)
position from chosen origin
\(\mathrm{m}\)
\(\vec p\)
linear momentum
\(\mathrm{kg\,m\,s^{-1}}\)
\(I\)
moment of inertia about a fixed axis
\(\mathrm{kg\,m^{2}}\)
\(\omega\)
angular velocity
\(\mathrm{rad\,s^{-1}}\)
\(\vec\tau_{\mathrm{ext}}\)
net external torque
\(\mathrm{N\,m}\)
Method
Derivation 1: Define angular momentum for a particle
The origin matters. The same particle can have different angular momentum about different points because \(\vec r\) changes.
Linear momentum
\[\vec p=m\vec v\]
Angular momentum
\[\vec L=\vec r\times\vec p\]
Magnitude
\[L=mrv\sin\phi\]
Derivation 2: Reduce to fixed-axis rigid-body rotation
For a rigid body rotating about a fixed principal axis, all mass elements share the same angular speed.
Element contribution
\[dL=r_\perp^2\omega\,dm\]
Integrate over body
\[L=\left(\int r_\perp^2\,dm\right)\omega\]
Fixed-axis form
\[L=I\omega\]
Derivation 3: Connect torque to angular momentum change
Torque is the rate of change of angular momentum. It can change magnitude, direction, or both.
Torque law
\[\vec\tau_{\mathrm{ext}}=\frac{d\vec L}{dt}\]
Zero external torque
\[\vec\tau_{\mathrm{ext}}=\vec0\Rightarrow\vec L\ \text{is constant}\]
Rules
These are the compact results from the method above.
Particle angular momentum
\[\vec L=\vec r\times\vec p\]
Particle magnitude
\[L=mrv\sin\phi\]
Rigid fixed axis
\[L=I\omega\]
Torque law
\[\vec\tau_{\mathrm{ext}}=\frac{d\vec L}{dt}\]
Examples
Question
A
\[0.50\,\mathrm{kg}\]
particle moves at \[6.0\,\mathrm{m\,s^{-1}}\]
perpendicular to a radius \[0.80\,\mathrm{m}\]
Find \(L\).Answer
Use
\[L=mrv=0.50(0.80)(6.0)=2.4\,\mathrm{kg\,m^2\,s^{-1}}\]
Checks
- Angular momentum depends on the chosen origin.
- Direction follows the right-hand rule.
- \(L=I\\omega\) is not valid for every moving body.
- Torque changes angular momentum, not necessarily speed.