Re: [Phys-L] form of Newtons 2nd law

[John Denker <jsd@av8n.com>]

On 07/27/2012 12:40 AM, Ian Sefton wrote:
> If we use the convention of what is sometimes called physical algebra
> and let each symbol represent  both the number and the unit (as in  F
> = 5 N) then F= ma works just fine in any system of units.

1) Welcome to the list.

2) I had not previously heard the term "physical algebra".  That is a
useful term.  For one thing, googling it turns up some interesting
stuff:
   http://www.google.com/search?q=%22physical+algebra%22+units

3) I would argue that teaching "physical algebra" is not a big deal
because it is not different from any other kind of algebra.  That is
to say, the methods that allow you to multiply 3xz by 4y to obtain 
12xyz work the same, no matter whether x, y, and z are "variables"
or units.

Essentially, "physical algebra" is just algebra.

4) Perhaps more importantly, I learned something from the first word
quoted above, namely "If".  That word introduces a hypothesis ... and
one of the rules of science is to consider all the plausible hypotheses,
so the question arises, what if we do *not* use physical algebra?  What
is the alternative?

I'm so addicted to units that it took me a moment to realize that yes,
there is an alternative.  The alternative to algebra is grade-school
arithmetic.  Suppose you are in 4th grade.  You know basic arithmetic,
but not algebra.  You could represent force, mass, and acceleration by 
/pure numbers/ devoid of units.  In this scenario, to calculate the
force you might well need a fudge factor to make the answer come out 
right, so it makes a certain amount of sense to write F = k m a, where 
k depends on the choice of units.

At the start of this thread, on 07/26/2012 02:11 PM, Larry Smith wrote:
> > My engineering colleague says I should teach N2 as F = kma where k
> > can make it work in non-SI-unit systems.  How do you respond to such
> > requests?

I answered this before, but I can give a more thoughtful answer
now:

Note the contrast:
 -- For persons who can handle only arithmetic, not algebra, then
  F = k m a makes a certain amount of sense.
 -- However, any person who has advanced to the level of algebra 
  or beyond would be verrrry well advised to write F = m a (without 
  the k) and to consider the units as "baked in" to the definition 
  of F, m, a, and every other physical quantity.

Despite the Subject: line of this thread, the real issue has got 
nothing to do with Newton's laws!  To see what I mean, suppose you 
measure length and width in feet, and multiply them together to get 
area in acres.  Then you have to write
  area = k2 * length * width

where the fudge factor k2 depends on the choice of units.

That is to say, it makes no sense to pick on Newton's law, unless
you are going to inflict the same overcomplication on every other 
equation known to man.

There exist calculator programs that will automagically keep track
of the units for you.
  http://www.google.com/search?q=1000+feet+*+1000+feet+in+acres

On the other hand, at the present day, most hand-calculators and
most spreadsheets don't do this.  Therefore you have to follow a 
two-step procedure:  First do the algebra by hand, including the 
factor-label factors ... then crunch the numbers in the calculator.

If you get this wrong, it might cost you a third of a billion dollars.
  http://en.wikipedia.org/wiki/Mars_Climate_Orbiter

=====================

The algebraic properties of units seems like a basic thing, in the 
sense that no deep physics is involved ... but sometimes the basics 
are incredibly important.  I would say this is a basic thing, but 
not a minor thing.

This provides a good answer to the question of what high-school algebra,
chemistry, and physics are good for.  If you know algebra, you can
learn the factor-label method easily, and then you calculate force, 
area, and everything else properly, no matter what units are involved.
Conversely, anybody who doesn't know those things is seriously 
handicapped when dealing with simple real-world calculations.

As a related point:  This is an easy thing to teach, if the students
have any clue about algebra.  Otherwise, it's an opportunity to
"remind" them of the axioms of algebra.  This gets back to the main
point of this section:  Some things that are easy to teach are
incredibly important.  Do not undervalue the low-hanging fruit.

Pedagogical remark:  This also illustrates why I think the value 
of "discovery learning" is often grossly exaggerated.  If you have
students who are functioning at the arithmetical pre-algebraic level,
there is virtually no chance that they are going to spontaneously
"discover" that there is a better way of doing things.  Sometimes
you have to take the direct approach.  Introducing factor-label by
direct instruction doesn't need to take longer than 15 seconds, but
those 15 seconds are important:

    "The axioms of algebra say we can always multiply by unity,
     so let's multiply by (1 foot) / (12 inches) and see what
     happens.  This trick is called the factor-label method. It
     is as easy as π and it's incredibly useful."

I cannot imagine making a big fuss when introducing the idea.  The
idea is going to get used 1000 times during the course of the year.
Inculcation works better than fussing.

This actually came up in conversation yesterday.  I was working with
a person who remembered "nothing" from high-school physics.  I asked
her about unit conversions, and the factor-label method.  She said
she remembered that ... but it didn't count because it was "easy".
I say yeah, it's easy, but it's still important.  Do not undervalue 
the low-hanging fruit.

This cuts to the core of what I think education is about.  I am
horrified when I hear teachers talk about trying to increase "time
on task" or when they tell students we are going to solve "hard
problems".  I say:
 -- The goal is to solve _important_ problems as easily as possible.
 -- We are going to solve problems that _would have been hard_ if
  you hadn't learned the right techniques.

There is a management aka politics angle to this, as well.  Joshua 
Bell makes it look easy to play the violin.  Derek Jeter makes it 
look easy to play second base.  Sometimes good teachers don't get 
sufficient credit for teaching important stuff ... and making it 
look easy.  When people don't get sufficient credit, it becomes 
hard to get needed support, resources, et cetera.

There's a lot more that could be said about this, including classic
tried-and-true ways of dealing with such problems, which we can 
discuss if anybody is interested.