Why dimension units of radius is not $\rm m/rad$ or $\rm cm/rad$?

Question

Radius is not a just simple size or length between the two points. The radius shows the connection of linear and angular values. Something must indicate the information about a perpendicularity of the radius to the angular values. This something may be radian at dimension units of radius. So if$$\left[ r \right] = \frac{\text{m}}{\text{rad}}$$then for example $$\left[ {\omega = \frac{v}{r}} \right] = \frac{\text{m/s}}{\text{m/rad}} = \frac{\text{rad}}{\text{s}} \ne \frac{1}{\text{s}} = \left[ f= \frac{v}{C} \right]$$ $$\left[ {\tau = F \cdot r} \right] = \frac{\text{N} \cdot {\text{m}}}{\text{rad}} = \frac{\text{J}}{\text{rad}} \ne \text{J} = \left[ W = \int {Fdx} \right]$$

The reference to a radian as a primary issue may be logically incorrect, because often dimension units of the radius is used in the proof of a radian dimensionless . As a result, the issue of dimension units of the radius becomes primary before a considering of a radian.

Why dimension units of a radius is not $\rm m/rad $ or $\rm cm/rad $? Where $\rm m $ or $\rm cm $ is from a ruler (or a line gauge or a scale) and $\rm /rad $ is from a goniometer (or a 90° protractor or something similar), which is necessary additionally to a ruler for the definition of a perpendicularity of the radius to the angular values.

Answer 1

You can achieve something like what you are trying to do here if you consider "metres in the x direction" and "metres in the y direction" to be distinct units with vectorial properties.

If we write $\text{m}_x$ for "metres in the x direction", $\text{m}_y$ for "metres in the y direction", then

$$\left[ {\omega = \frac{v}{r}} \right] = \frac{\text{m}_x/\text{s}}{\text{m}_y} = \frac{\text{m}_x \text{m}_y^{-1}}{\text{s}}= \frac{\text{rad}_{xy}}{\text{s}} $$

Here $\text{rad}_{xy}$ means "radians measured in the $xy$ plane". It's a signed quantity, positive as you rotate (either clockwise or anticlockwise) from $x$ to $y$.

We can do this a little more formally if we consider spatial units to be standardised quantities that can have vector, bivector, etc. geometrical properties. They are not all scalars. When we talk about "length" now we are talking about the ratio of the vector between the two points being measured and a standardised unit vector pointing in the same direction. We can legitimately divide vectors pointing in the same direction by one another and get a scalar quantity, because a scalar times a vector gives another vector that is some multiple of it. But when we divide a vector by another vector pointing in a different direction, this is no longer true. Instead we get a bivector quantity that represents a plane area in the same way vectors represent linear lengths. Bivectors are the generators of rotations. And in the case of circular motion, comparing a velocity vector in the tangential direction to the radius in the radial direction gives us a bivector quantity with an orientation 'pointing' in the plane of rotation.

Then making the claim that the units of radius are $\text{m/rad}$ is actually just saying $\text{m}_x/\text{m}_x\text{m}_y^{-1}=\text{m}_y$, we are converting metres in the $x$ direction to metres in the $y$ direction. The magnitudes of the standard metre vectors in every direction are the same, so the radian unit has a scalar magnitude of $1$, but is not actually the scalar $1$, but instead a bivector quantity with an orientation in space.

Using oriented units usually just complicates things, and it is normal to use only the scalar magnitudes of units. But it can sometimes be useful or insightful. If you do dimensional analysis keeping length units in different directions separate, it can sometimes resolve ambiguities that the scalar version of dimensional analysis fails to distinguish.

Geometric Algebra is a system that is capable of doing this correctly, keeping everything straight, even when mixing directions that are not orthogonal to one another, although in practice practioners of Geometric Algebra usually do it the same way everybody else does.

Answer 2

I'd start by saying that the way "frequency vs angular frequency" and the unit "radian" is taught in high school/secondary school is really awful... they make it sound like they're going to use radians as a unit all over the place but then they forget it in a lot of equations, leaving students confused.

Can you measure it with a ruler? Yes. Then it has units of length. As people have said in comments, if you use a convention where $1\text{ rad}=1$, then the problem is solved altogether.

If you desperately want to use an inconvenient system where the "unit" radian is carried with you all over the place, then you can say "the radius is $r=1\text{ m}$, and the arc length of an arc with angle $\theta=\pi\text{ rad}$ is $r\theta/\text{rad}$." If you even more desperately want to use a system where radius is measured in $\text{m}/\text{rad}$, that makes the arc length equation nicer ($r\theta$), but it makes other equations like the area of a slice of circle pretty ugly (normally $r^2\theta/2$ becomes $r^2\theta(\text{rad})/2$).

Both systems that explicitly use "rad" become really annoying when you think of simple harmonic motion. You'd have to say the angular frequency of a pendulum is $\omega=3\text{ rad}/s$. Then the angular position as a function of time is $\theta=\theta_{\text{max}}\cos(\omega t/\text{rad})$, assuming cosine has its usual mathematical definition and can be written as: $$ \cos(x)=1-x^2/2+O(x^4) $$ I personally use a system where $\text{rad}=1$, and $1\text{ whole cycle}=2\pi$. I then assert that, for example, $\hbar=h=1.054\times10^{-34}\text{ Js}=6.626\times10^{-34}\text{ Js}/\text{cycle}$ and $\omega=f$ (angular frequency and frequency are the same thing). I accept this isn't a commonly used system but I think it makes things a lot simpler. Suffice to say you can choose which system to use with regards to how you deal with angles and frequencies, but you shouldn't make an inconvenient choice that introduces a whole bunch of cumbersome notation.