[Math] Angular speed to linear speed: Arbitrarily discarding Radians units

calculusdimensional analysisunit-of-measure

I asked this question long ago, but wanted to see if I can get some new perspectives this time.

Why is the radians implicitly cancelled? Somehow, the feet just trumps the numerator unit.
For all other cases, you need to introduce the dimensional analysis / unit conversion fraction, and cancel explicitly. Is it because radians and angles have no relevance to linear speed ($v$), so they are simply discarded?

In the 2nd equation, you are simply multiplying radius by how much angle it sweeps per minute. 2 times $360\pi$ somethings. No need to describe what there are $360\pi$'s of? $360\pi$ seems meaningless without the units to describe what you're counting (radians)

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Best Answer

There are two r's here (unfortunate choice of variables by the author)

$$\text{r} = r \omega $$

Better to choose $s$ for linear speed

$$s = r \omega $$

The radius is two feet per radian

$$r = \frac{2 \ feet}{radian} $$

So now we easily see that the 'dimensionless' unit radian(s) cancels properly.

To put it another way , every radian is 2 feet because the radius is 2 feet. Therefore , 2 feet per radian. The fact that a 'radian' can represent an angle OR a length ... is a minor detail.

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