Re: [Phys-L] unit for "pure ratio"

[John Denker <jsd@av8n.com>]

On 07/24/2016 06:11 AM, Marty Weiss wrote:

> using the coefficient of friction in the calculation yields a force
> which is in Newtons.  Ff = “mu" (x) Fnormal   But calling the “mu”
> with the unit “perun" will give the final answer perun-Newtons which
> doesn’t make any sense.  If you leave the perun unit out then why
> have it at all?

The problem with that argument is that it proves we don't
need radians.  It's bad luck to prove things that aren't true.
Specifically:

Suppose we have a particle on the rim of a wheel of radius
2 meters rotating with an angular rate 3 radians per second.
We multiply those together to find that the tangential speed
is 6 meters per second.  The radians drop out of the product.
If you can leave the radian unit out, then why have it at all?

Answer:  Is is absolutely "needed"?  Maybe not.  You are free
to write the angular speed as 3 s^-1 if you wish, instead of 
3 radians per second.  However, for communicating with real
humans, and especially with students, the unit has some value.

It helps by:
 a) giving us something to talk about in the case where it
  drops out, so we can delineate radians from cycles and
  degrees and other things that don't drop out, and
 b) providing a hook to which metric prefixes can be attached,
  e.g. milliradian.

Peruns are useful for the same reasons that radians are useful.

=======

Here's another argument leading to the same conclusion:  Special
relativity tells us that the speed of light must be dimensionless.
We can set it equal to 1 for all the same reasons that we can set
one radian equal to 1.

Experts do in fact routinely set it equal to 1.

However, IMHO just because we /can/ set it equal to 1 does not
mean we are /obliged/ to do so ... especially when talking to
non-experts, including students in the introductory course.