Relation of Angular to Linear Momentum

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Recall (§B.3) that the momentum of a mass traveling with velocity in a straight line is given by

$\displaystyle p = m v,$

while the angular momentum of a point-mass

rotating along a circle of radius

at $\omega$ rad/s is given by

$\displaystyle L \eqsp I\omega,$

where

. The tangential speed of the mass along the circle of radius

is given by

$\displaystyle v \eqsp R\omega.$

Expressing the angular momentum

in terms of

gives

$\displaystyle L \isdefs I\omega \eqsp I\frac{v}{R} \isdefs mR^2\frac{v}{R} \eqsp Rmv \eqsp Rp. \protect$

(B.18)

Thus, the angular momentum

times the linear momentum

Linear momentum can be viewed as a renormalized special case of angular momentum in which the radius of rotation goes to infinity.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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