212 Angular Momentum
A vector measure of rotational motion that is used to describe how a particle’s momentum is aimed about a chosen origin, where an origin is the reference point about which the turning is measured.
The angular momentum of a particle. The angular momentum of one particle about an origin is the cross product of its position and its linear momentum. This principle is used to assign a rotational measure to a moving particle.
The angular momentum of a particle is
\[ \mathbf{L} = \mathbf{r}\times\mathbf{p} \]
where
- \(\mathbf{L}\) is the angular momentum.
- \(\mathbf{r}\) is the position relative to the origin.
- \(\mathbf{p}\) is the linear momentum.
Origin dependence. Angular momentum depends on the choice of origin through the position vector. This principle is used to state which point the turning is measured about.
The direction of \(\mathbf{L}\). \(\mathbf{L}\) is perpendicular to the plane of \(\mathbf{r}\) and \(\mathbf{p}\) when those vectors are not parallel. This principle is used to read the sense of rotation from the right-hand rule.
The torque-angular-momentum theorem. The time rate of change of angular momentum about an origin equals the torque about the same origin. Torque is the turning analog of force. This principle is used to compute how a force changes rotational motion.
The torque-angular-momentum relation is
\[ \boldsymbol{\tau} = \dfrac{d\mathbf{L}}{dt} \]
where
- \(\boldsymbol{\tau}\) is the torque about the origin.
- \(\mathbf{L}\) is the angular momentum about the same origin.
- \(t\) is time.
The angular momentum of a system. The total angular momentum of a system is the sum of the angular momenta of its particles. This principle is used to treat many-body rotation as one vector.
The angular momentum of a system is
\[ \mathbf{L} = \sum_{i}\mathbf{r}_{i}\times\mathbf{p}_{i} \]
where
- \(\mathbf{L}\) is the total angular momentum.
- \(\mathbf{r}_{i}\) is the position of particle \(i\).
- \(\mathbf{p}_{i}\) is the momentum of particle \(i\).
Conservation of angular momentum. If the net external torque about an origin vanishes, the total angular momentum about that origin stays constant. This principle is used to analyze isolated rotation and motion under a central force.
Conservation of angular momentum is
\[ \boldsymbol{\tau}_{\mathrm{ext}} = 0 \implies \mathbf{L} = \mathrm{constant} \]
where
- \(\boldsymbol{\tau}_{\mathrm{ext}}\) is the net external torque.
- \(\mathbf{L}\) is the total angular momentum.
Note: Also denoted \(\mathbf{L}\).
212.1 References
- Feynman, R. P., Leighton, R. B., & Sands, M. The Feynman Lectures on Physics, Vol. I. Ch. 20 — \(\mathbf{L} = \mathbf{r} \times \mathbf{p}\); \(\boldsymbol{\tau} = \dfrac{d\mathbf{L}}{dt}\); conservation when external torque is zero.
- Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. Ch. 12 — angular momentum of a particle and of a system.