Rotation is the Cinderella of the dimensional world; always in the background, doing the heavy housework, but never getting any of the credit.
This chapter is more about limited displacement than it is about RPM’s. We’ll talk about them later. Rotation (and direction) is one of our most useful and necessary measurement tools, and for some reason we generally treat it with disdain, calling it ‘unit-less’ and sometimes losing its dimension in the calculations.
This line of thinking probably, again, dates back to the Greeks, who invented trigonometry and viewed the angles as ratios of the measurement of the sides of a triangle, which are truly dimensionless. Rectilinear thinking does not necessarily want or need to include true curves.
But the fact remains that, by necessity, we need to include rotation in our measurements and calculations of so many things. Think about it. Rotation figures into the design of plumbing valves, door hinges, astrophysics, engineering design, even the old, regular analog clock depends on a rotational location to provide for the keeping of time. And yet, in physicists and engineering circles (where we must account for such things) the units for the measurement of the angular displacement (degrees, radians) are considered essentially dimensionless, and those units do not need to be tracked when doing dimensional analysis. How does that make sense?
Engineering has a whole set of forces, called ‘moments’ that are forces experienced by an object due to the rotational force applied to it (times the moment arm, of course) that is completely separate from the linearly applied forces. They have their own standard symbols, MX, MY and MZ, yet, the fundamental aspect of the calculation, the unit for rotation, is left out of the equations. Engineers use that naming convention to make sure not to mix the rotational forces with the linear ones accidently. But they have to consider that aspect of the loads when designing any object.
Why did Descartes, when he was codifying geometry, not choose to add a notation for the direction of the locating vector, quantified in degrees, radians or whatever, instead of choosing to create negative numbers in order to specify the direction for a location? By choosing to invent negative numbers instead of creating a naming convention that referenced direction, he did a great disservice to future generations, including ours.
Rotation is a real measurable quantity, and when utilized in theories, calculations or designs, it should always be noted and accounted for. Our failure to do so has helped to obscure the true nature of what a dimension really is, and as we shall see in the following chapter, refusing to utilize it as a preferred measurement can lead to fundamental, systematic errors.