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On 02/02/2010 07:45 AM, Edmiston, Mike wrote:
John Denker suggested saying "cycles per second" rather than "per
second" if frequency is what we are talking about. I would presume
we would also say "radians per second" rather than "per second" if
angular frequency is what we are talking about.
Agreed. (The only reason I emphasized the Hz example
is that it is where this thread started, and it is
the problem child on the NIST web page. It is not
a particularly special case.)
I whole-heartedly agree with this. We should explicitly say what we
mean. But this creates the usual problem with dimensionless units...
when students are making sure "the units come right" the units in
fact won't "come out right" because there is nothing to cancel the
dimensionless unit that we explicitly wrote into the paperwork.
Students have difficulty dealing with this, and I typically don't
have a good solution other than telling them just to learn to live
with it.
Sometimes it helps to remind them that there's more
to physics than unit analysis. See general remarks
below.
If we're going to explicitly put dimensionless units into our work
(which I indeed prefer), then how do we get students to realize they
can take them back out at the appropriate time.
This will never be easy, because distinguishing
"appropriate" from "inappropriate" requires judgment.
It requires understanding the physics. For example,
the formula
F = r / frequency^2
only makes sense for frequency in radians/second,
whereas the formula
wavelength = c / frequency
only makes sense for frequency in cycles/second.
We could make the formulas more explicit e.g. by
writing
F = r / (angular frequency)^2
wavelength = c / (circular frequency)
but in the interests of concision it is conventional
to just write formulas involving "ω" or "f" and to
require people to know what those symbols mean and
how they are used.
===========================================
More generally, this has to do with the question of
"what are units, anyway" ... and why do we multiply
units together the way we do? That is quite a deep
and serious metaphysical question.
It helps to keep in mind that equations are just
models that we have invented to represent various
things that go on in the real world. Sometimes
the models work well, and sometimes not so well.
In the real world, counting apples is not the same
as counting oranges, because apples are different
from oranges. The units are different.
On the other hand, to an excellent approximation,
counting apples is _isomorphic_ to counting oranges.
This isomorphism greatly simplifies the process
of counting ... at the cost of making it more
abstract.
If you want to count apples and oranges and kumquats
et cetera, the results are vectors in some abstract
high-dimensional space. The _units_ are basis vectors
in this space.
It's complicated and sophisticated, but it's not magic.
==========
It's always amusing when students have been successfully
using this-or-that technique for years, at the rote
level or the black-box level ... and then get to the
point where they want to and/or need to understand it,
and begin to appreciate the "moving parts" inside the
black box.
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