Angular Momentum

Angular momentum ($H$) is a measure of a body’s quantity of rotational motion. It applies equally to a particle orbiting a point as to a spinning rigid body.

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Angular Momentum of a Particle

For a particle with linear momentum $\text{m}\mathbf{v}$, angular momentum is defined as the “moment” of the particles linear motion about $O$:
${\vec{H}}_O=\mathbf{r}\times \text{m}\mathbf{v}$

In 2D, this reduces to a scalar where $d$ is the perpendicular distance from $O$:
$H_O=mv\times d$

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Angular Momentum of a Rigid Body

Integrating the particle definition over every point in a rigid body gives us equation with angular velocity $\omega$ and with mass moment of inertia $I$:
$H=I\omega$

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Principle of Angular Impulse and Momentum

Applying a moment about a point over a period of time changes a body’s angular momentum about that same point:
$(H_O{)}_1+\sum {\int }_{t_1}^{t_2}{\text{M}}_{O}dt=(H_O{)}_2$

For a constant net moment, this simplifies to:
$(H_O{)}_1+\sum {\text{M}}_O\Delta t=(H_O{)}_2$

Angular Impulse

$\sum {\int }_{t_1}^{t_2}{\text{M}}_{O}dt$ is known as the angular impulse. It represents the net torque accumulated over time.

If there are no meaningful angular impulses about point $O$, then angular momentum is conserved.

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