Editing Angular acceleration (section)

=== Particle in three dimensions ===

In three dimensions, the orbital angular acceleration is the rate at which three-dimensional orbital angular velocity vector changes with time. The instantaneous angular velocity vector <math>\boldsymbol\omega</math> at any point in time is given by

: <math>\boldsymbol\omega =\frac{\mathbf r \times \mathbf v}{r^2} ,</math>

where <math>\mathbf r</math> is the particle's position vector, <math>r</math> its distance from the origin, and <math>\mathbf v</math> its velocity vector.<ref name="ref2">{{cite book |last1=Singh |first1=Sunil K. |title=Angular Velocity |url=https://cnx.org/contents/MymQBhVV@175.14:51fg7QFb@14/Angular-velocity |publisher=Rice University |ref=2}}</ref>

Therefore, the orbital angular acceleration is the vector <math>\boldsymbol\alpha</math> defined by

: <math>\boldsymbol\alpha = \frac{d}{dt} \left(\frac{\mathbf r \times \mathbf v}{r^2}\right).</math>

Expanding this derivative using the product rule for cross-products and the ordinary quotient rule, one gets:

: <math>\begin{align}
\boldsymbol\alpha &= \frac{1}{r^2} \left(\mathbf r\times \frac{d\mathbf v}{dt} + \frac{d\mathbf r}{dt} \times \mathbf v\right) - \frac{2}{r^3}\frac{dr}{dt} \left(\mathbf r\times\mathbf v\right)\\
\\ 
&= \frac{1}{r^2}\left(\mathbf r\times \mathbf a + \mathbf v\times \mathbf v\right) - \frac{2}{r^3}\frac{dr}{dt} \left(\mathbf r\times\mathbf v\right)\\
\\
&= \frac{\mathbf r\times \mathbf a}{r^2} - \frac{2}{r^3}\frac{dr}{dt}\left(\mathbf r\times\mathbf v\right).
\end{align}</math>

Since <math>\mathbf r\times\mathbf v</math> is just <math>r^2\boldsymbol{\omega}</math>, the second term may be rewritten as <math>-\frac{2}{r}\frac{dr}{dt} \boldsymbol{\omega}</math>. In the case where the distance <math>r</math> of the particle from the origin does not change with time (which includes circular motion as a subcase), the second term vanishes and the above formula simplifies to

: <math> \boldsymbol\alpha = \frac{\mathbf r\times \mathbf a}{r^2}.</math>

From the above equation, one can recover the cross-radial acceleration in this special case as:

: <math>\mathbf{a}_{\perp} = \boldsymbol{\alpha} \times\mathbf{r}.</math>

Unlike in two dimensions, the angular acceleration in three dimensions need not be associated with a change in the angular ''speed <math>\omega = |\boldsymbol{\omega}|</math>'': If the particle's position vector "twists" in space, changing its instantaneous plane of angular displacement, the change in the ''direction'' of the angular velocity <math>\boldsymbol{\omega}</math> will still produce a nonzero angular acceleration. This cannot not happen if the position vector is restricted to a fixed plane, in which case <math>\boldsymbol{\omega}</math> has a fixed direction perpendicular to the plane.

The angular acceleration vector is more properly called a [[pseudovector]]: It has three components which transform under rotations in the same way as the Cartesian coordinates of a point do, but which do not transform like Cartesian coordinates under reflections.