Re: Mass

[John Denker <jsd@AV8N.COM>]

Edmiston, Mike wrote:
> I appreciated the various ways of viewing mass that John D. itemized.

:-)

>  I
> was less thrilled with his comment that he is not comfortable defining
> mass in terms of amount of material.  I'm not saying to substitute
amount of material in place of all the other things in John's list.  I'm
just saying it should be added to the list.

How about a compromise?  I think we need two lists.
Here's where I'm coming from:

    mass --(gives us)--> a notion of "amount of stuff"
  volume --(gives us)--> another notion of "amount of stuff"

You cannot reliably turn the arrow around, i.e. you cannot
go from "amount of stuff" to a reliable notion of mass.

Let me rush to point out that one of the items I previously
put on the list has the same property:

    mass --(obeys)--> a local conservation law
  charge --(obeys)--> another local conservation law

So why don't we move both of these ideas (amount of stuff
and local conservation) to a second list, namely properties
that are associated with mass but not exclusively associated
with mass.

This is in contrast to the first list, namely items that
apply to mass alone, such as m = p/v.

===

I think we can benefit from an idea from high-school geometry.
They explicitly designate points, lines, and planes as
undefined quantities.  These terms acquire meaning solely from
the way they are used in the axioms of geometry.  IMHO trying
to define "mass" in a few words (e.g. "amount of stuff") is
just as unhelpful as trying to define "point" in a few words.

A related metaphysical/pedagogical parable:  Suppose you show
me a square red piece of paper with an "X" marked on it.  Are
you trying to teach me the concept of red, or the concept of
square, or the concept of "X", or the concept of paper???  I
can't tell from one example.  But if you show me a diverse
collection of red objects, I might be able to infer that
redness is the only thing they have in common.

The point of the parable is that given enough examples of
how the term "mass" is used, the student can presumably infer
a reliable notion of what mass is ... even if none of the
examples is by itself definitive.

So as M.E. said, the right approach is to add to the list(s),
rather than trying to replace the list with a single
allegedly-definitive example.

> My old edition of Webster's Collegiate Dictionary says all three of
> inertia, quantity of stuff, and gravity.  It's a rather nice statement
> so I will quote it here...
>
> "the property of a body that is a measure of its inertia, that is
> commonly taken as a measure of the amount of material it contains and
> causes it to have weight in a gravitational field, and that along with
> length and time constitutes one of the fundamental quantities on which
> all physical measurements are based"

I happily agree with the dictionary.  The way I read it,
connecting mass to "amount of material" is illustrative but
not by itself definitive.  The words "commonly taken" are
far removed from the usual definitive language.  The way I
read it, this is like saying that a hammer is commonly
used for driving nails ... which is true, but it is not
the only application for hammers, nor the only way of
driving nails.  It is an illustrative application, useful
as _part_ of a definition, but not definitive by itself.

> (1) It's the way students learned it in chemistry.  And at most places
> they still teach chemistry before physics.  So students are going to
> come into physics class thinking of mass as how you measure the quantity
> of something.

 a) The students might have learned to measure stuff by volume
  in HS chemistry.
 b) The students might have learned to measure stuff by mass
  in HS chemistry.

To repeat:  "amount of stuff" is a second-list concept.  There
is a reliable arrow _from_ mass _to_ "amount of stuff" but not
vice versa.

>  We already label packaged food items with the mass.

... as I sit here eating my pint of blueberries and drinking
my 8 fl.oz. carton of milk.

> (2) If we are ever successful at getting US households to have balances
> in their kitchens, that will really cement the idea that mass is a
> measure of how much stuff you have.

a) We cannot put our teaching on hold, waiting for kitchen scales
 to become common.
b) Even if they were common, they would not solve the problem.
 There remains a valid arrow _from_ volume _to_ "amount of stuff",
 especially for homogenous substances.  You cannot make this go
 away.  It will always be a distractor, a source of pedagogical
 difficulties.

> (3) I believe it is the way Newton viewed it, and that is why he was
> talking about volume and density.  He knew mass had something to do with
> size (volume) but he also knew the need to include density.  Had he
> known about protons and neutrons and what atoms are, I suspect he would
> have been able to better describe what he probably had in mind.

I find that to be a tortured extrapolation of what he
actually said.  I prefer to think he was tacitly
restricting the discussion to homogenous materials.
Density is important because it is an intrinsic quantity
and therefore characteristic of the type of material.
For a homogenous material (and not otherwise) both mass
and volume are well-behaved extensive quantities.  In
non-homogeneous situations (e.g. adding milk to cereal)
the density is ill-behaved and the volume is not additive.
The mass is robustly additive.

Defining a more-robust idea (mass) in terms of a less-
robust quantity (volume) is a bad idea.

The fact that Newton did so doesn't make it a good idea.

> (5) Depending on how the NIST experiments turn out, we may end up
> defining the standard kilogram as a certain number of atoms (probably of
> silicon).

I don't have a problem with that.  Counting Si atoms achieves
many of the same goals as counting C-clamps; it's just a little
harder for students to accomplish on their own.

Also note that a counted Si crystal, while it would make
a dandy standard of mass, could also be used as a standard
of volume, and perhaps a standard of hardness, voltage,
and various other observables ... so just having such a
thing does not immediately give students a precise
understanding of what mass is.  It is a very valuable part
of the story, but not the whole story.