Physics in Motion

Angular Momentum

The angular momentum with respect to the origin of a particle with position \( \vec{r} \) and momentum \( \vec{p} = m \vec{v} \) is:

$$ \vec{L} = \vec{r} \times \vec{p} = \vec{r} \times m \vec{v} $$

If the angle between \( \vec{r} \) and \( \vec{p} \) is \( \theta \), then the magnitude of \( \vec{L} \) is:

$$ L = r p \sin \theta = m v r \sin \theta $$

The time rate of change of the angular momentum is:

$$ \frac{d \vec{L}}{d t} = \vec{r} \times \vec{F_{ext}} $$

$$ \frac{d \vec{L}}{d t} = \vec{\tau_{ext}} $$

If no external torque acts on a system, then the angular momentum of the system remains constant. This is the law of conservation of angular momentum.

An individual component of the angular momentum can be expressed in terms of the moment of inertia:

$$ L_z = I \omega $$