Editing Angular acceleration (section)

=== Particle in two dimensions ===

In two dimensions, the orbital angular acceleration is the rate at which the two-dimensional orbital angular velocity of the particle about the origin changes. The instantaneous angular velocity ''ω'' at any point in time is given by

: <math qid=Q161635>\omega = \frac{v_{\perp}}{r},</math>

where <math>r</math> is the distance from the origin and <math>v_{\perp}</math> is the cross-radial component of the instantaneous velocity (i.e. the component perpendicular to the position vector), which by convention is positive for counter-clockwise motion and negative for clockwise motion.

Therefore, the instantaneous angular acceleration ''α'' of the particle is given by<ref name="ref2" />

: <math qid=Q186300>\alpha = \frac{d}{dt} \left(\frac{v_{\perp}}{r}\right).</math>

Expanding the right-hand-side using the product rule from differential calculus, this becomes

: <math>\alpha = \frac{1}{r} \frac{dv_\perp}{dt} - \frac{v_\perp}{r^2} \frac{dr}{dt}.</math>

In the special case where the particle undergoes circular motion about the origin, <math>\frac{dv_{\perp}}{dt}</math> becomes just the tangential acceleration <math>a_{\perp}</math>, and <math>\frac{dr}{dt}</math> vanishes (since the distance from the origin stays constant), so the above equation simplifies to

: <math>\alpha = \frac{a_{\perp}}{r}. </math>

In two dimensions, angular acceleration is a number with plus or minus sign indicating orientation, but not pointing in a direction. The sign is conventionally taken to be positive if the angular speed increases in the counter-clockwise direction or decreases in the clockwise direction, and the sign is taken negative if the angular speed increases in the clockwise direction or decreases in the counter-clockwise direction. Angular acceleration then may be termed a [[pseudoscalar]], a numerical quantity which changes sign under a [[parity (physics)|parity inversion]], such as inverting one axis or switching the two axes.