The angle should be dimensionless

[jht320]

I encountered an error when trying to use the physical formula "linear velocity = radius * angular velocity". radians_per_second_t * meter_t cannot be assigned to meters_per_second_t.

[chiphogg]

The root cause is that the equation $v = \omega r$ is using a kind of "natural units" approach. It's analogous to picking units where $c=1$ and $\hbar=1$ in high energy physics, so that the factors of $c$ and $\hbar$ "drop out" and become invisible: that's why they can measure mass, energy, and momentum in $\text{MeV}$.

A telltale sign that you're using natural units is that the equation won't balance unless you use specific units for at least some of the quantities. In this case, you must use radians for $v = \omega r$ to be valid; it won't work if you use degrees.

Unfortunately, while it's very easy to go from fully general equations to "natural-unit" ones, going in the reverse direction is generally hard: it's an information-losing step. This is especially true when the entire scientific community has been using only the "natural-unit" equations for something on the order of centuries.

There were at least two prominent papers in 2021 that offered detailed solutions for how to fix the SI, and produce fully general equations that recognize dimensioned angles: Quincey (freely publicly available), and Leonard. Quincey's paper includes this helpful paragraph, that shows a practical way to think conceptually about any equations you encounter using this common "dimensionless angles" form:

The proposal that the radian convention could still be adopted within equations after angles are changed in status, when this is done explicitly, means that all the familiar equations could still be used. However, where they are different from the complete version, equations should be accompanied by a statement such as ‘where $\Theta_C$, which has a value of $1\, \text{rad}$, has been set equal to 1 (so that the angle unit must be the radian, and the units on either side of the equation do not match)’. This would be the only visible change. Of course, if only complete equations are used, the statement would not be needed.

(Emphasis in original.)

The two papers do disagree about which form some of the equations should take --- again, we're trying to undo an information-losing step, so this isn't totally surprising. But $v = \omega r$ isn't one where they differ: it's completely straightforward and unambiguous how to fix it. The fully general form is this:

$v = \omega r / \Theta_C$

Here, $\Theta_C$ is some mathematical constant whose value equals one radian. Now you can use any units you like, and the equation will automatically be correct.

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That's the conceptual background. Now what do we actually do about it in a C++ units library context? Well: we need to divide by that constant, $\Theta_C = 1\,\text{rad}$.

In the nholthaus library, I believe this would simply be radians_t{1}. Yes, this is a runtime mathematical operation, which is aesthetically displeasing. However, I strongly expect the compiler to optimize it out, especially if you make it an explicit constant, something like the following:

constexpr auto RAD = radians_t{1};

// ...

meter_t r;
radians_per_second_t omega;
meters_per_second_t v = r * omega / RAD;

Other units libraries have types that are pure compile-time constants, where multiplying and dividing simply affects the units in the type of the variable, without having any runtime effect at all. mp-units was the first place I saw this idea implemented. My own library, Au, has two of these kinds of entities: constants, and unit symbols. Either one would do.

Here's a godbolt example of using this kind of approach, via the rad "unit symbol", to make the equations balance. In this case, I measured angular velocity in "revolutions per minute", to drive home and emphasize the point that the fully general equation lets you use any units.

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I hope this gives you some helpful practical and theoretical tips!

[etsach]

The root cause is that the equation v = ω r is using a kind of "natural units" approach. It's analogous to picking units where c = 1 and ℏ = 1 in high energy physics, so that the factors of c and ℏ "drop out" and become invisible: that's why they can measure mass, energy, and momentum in MeV .

Radians are not a physical unit! Being defined by arc/radius they can be seen as a pure dimensionless unit without being a natural unit.

[chiphogg]

The root cause is that the equation v = ω r is using a kind of "natural units" approach. It's analogous to picking units where c = 1 and ℏ = 1 in high energy physics, so that the factors of c and ℏ "drop out" and become invisible: that's why they can measure mass, energy, and momentum in MeV .

Radians are not a physical unit! Being defined by arc/radius they can be seen as a pure dimensionless unit without being a natural unit.

This idea is discussed in the first and second "misconceptions" in Quincey, Mohr, and Phillips (2019), which I recommend reading. Appealing to $\theta = s/r$ is not a compelling argument that angles "must" be considered dimensionless. The $\theta$ in this equation doesn't necessarily refer to the angle itself, but it definitely refers to the value of that angle as measured in a particular unit.

And what makes the radian a "natural" unit in this case is that we have selected one preferred unit for a particular quantity, and then created equations that are only valid when we use that specific unit. It's got nothing to do with whether that unit requires physical measurements for its definition.