What is Angular momentum? — Physics

Definition 15.3 University Physics — Year 1 · Chapter 15 — Angular Momentum

The angular momentum about $O$ of a particle of mass $m$ at $M$ with velocity $\vect v$ is

\vect L_O = \vect{OM}\wedge m\vect v \qquad (\mathrm{kg}\,\mathrm{m}^{2}/\mathrm{s}),

and about an axis $\Delta$ through $O$ , $L_\Delta = \vect L_O\cdot\vect e_\Delta$ . For a motion in a plane perpendicular to $\Delta$ , in polar coordinates about the axis, $L_\Delta = mr^2\dot\theta$ ; for a circle of radius $R$ , $L_\Delta = mR^2\omega = mRv$ .

Examples

Example 15.6 (The pendulum by moments)

Simple pendulum, axis $\Delta$ through the pivot perpendicular to the plane of swing: $L_\Delta = m\ell^2\dot\theta$ ; the tension meets the axis (no moment), the weight’s moment is $-mg\ell\sin\theta$ (lever arm $\ell\sin\theta$ , restoring). The theorem gives $m\ell^2\ddot\theta = -mg\ell\sin\theta$ , the equation of Example 12.12 without ever writing the tension — the chief advantage of moments: forces through the axis disappear.

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