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# Rotation & translation

• Stuff can move – like a falling stone – ‘some metres’ and with ‘some metres-per-second’ etc.
• Stuff can also rotate – like a spinning merry-go-round. But it doesn’t rotate ‘“some metres”’ or with ‘“some metres-per-second”’. Rather, it rotate ‘some degrees’ and with ‘some degrees-per-second’ – or radians and radians-per-second.

Our usual motion is about length, but rotational motion is clearly about angle. They seem to be two different types of motion. We can call the usual type linear motion or translation.1 And the rotating one angular motion or simply rotation.[2]

These are two types of motion we can easily spot.2 And the properties?

• Translation has position $s$, as rotation has, shall we call it angular position $\theta$.
• Translation has speed $v$, as rotation has angular speed $\omega$.3
• Translation has acceleration $a$, as rotation has angular acceleration $\alpha$.4

These $\theta$, $\omega$ and $\alpha$ are simply ‘how far’, ‘how fast’ and ‘how much fasteraround. And just like ‘acceleration is change in speed’, which is ‘change in position’,

$$a=v’\qquad\text{and}\qquad v=s’$$

surely ‘angular acceleration is change in angular speed’, which is ‘change in angular position’.

$$\alpha=\omega’ \qquad\text{and}\qquad \omega=\theta’$$

Spin a top on a tray, then lift and move the tray around. The top keeps spinning! Unaltered. As if rotation and translation are separate, independent types of motions.[2,6] 5

Possibly we can treat the two types of motion entirely separately. We can use our motion equations that work for linear motion (translation), even if the object is also rotating at the same time.[2]

References:

1. Online Etymology Dictionary’ (dictionary), Douglas Harper, www.etymonline.com
2. Sears and Zemansky’s Univesity Physics with Modern Physics’ (book), Hugh D. Young & Roger A. Freedman, Pearson Education, 13th ed., 2012
3. Maths - Reflection’ (web page), Martin John Baker, Euclidean Space, 2007, www.euclideanspace.com/maths/geometry/affine/reflection/index.htm (accessed Mar. 12th, 2020)
4. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics’ (book), David Hestenes, Springer, 1987, www.amazon.co.uk/Clifford-Algebra-Geometric-Calculus-Mathematics/dp/9027725616, ISBN 978-9027725615
5. Geometric Transformations’ (web page), Michigan Technological University (MTU), pages.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/geo-tran.html (accessed Mar. 12th, 2020)
6. Planning Algorithms: Combining translation and rotation’ (web page), Steven M. LaValle , Cambridge University Press, 2012, planning.cs.uiuc.edu/node99.html (accessed Mar. 12th, 2020)