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[Phys-l] fundamentals versus corollaries (was: work done by Static Friction)



On 03/15/2007 10:53 AM, LaMontagne, Bob wrote:
... blanket statement in many texts (Serway is
the one I'm currently using) that static friction can do no work because
there is no displacement involved.

The responses so far confirm my conclusion that the blanket statement in
the texts is simply wrong - that they were only considering a narrow set
of examples when formulating the statement.


It is amusing to note that this erroneous statement has a sibling, namely
the blanket assertion that
"a force of constraint does no work."

As I have said before, I generally don't like cataloging mistakes one by
one; it's like hunting bacteria with a sniper rifle. So a better agenda
would be: What can we learn that will kill off these two mistakes /and/
some much larger /category/ of mistakes?

I see several points; the big payoff is in item (4) below.

1) One of the goals of science is to avoid mistakes, and certain categories
of mistake are so common as to deserve special attention. We are reminded
that:
"Scientific rules generally have a limited domain of applicability.
To state the rule without stating its limits of validity is improper."
http://www.av8n.com/physics/scientific-methods.htm#sec-provisos


In this case we have rules that are valid in one reference frame but not
in another.
-- We have a rule that is valid /provided/ the static force is stationary,
not just static.
-- The other rule is valid /provided/ the force of constraint is stationary.



2) Another angle stems from the following observation:
"Life can be so nonlinear."

Consider the following sequence:
a) Linear: delta(position) is zeroth order in velocity, and therefore
commutes with spacelike rotations and with boosts.
b) Linear: delta(momentum) is first order in velocity, and therefore
commutes with spacelike rotations and with boosts.
c) Nonlinear: delta(KE) is second order in velocity, and therefore
commutes with spacelike rotations /but not with boosts/.

Students get the idea that they can calculate stuff in one frame and
then transfer the result to another frame. Students are always forming
habits. Habits are /implicit/ rules. I doubt many students would say
/explicitly/ that results from one frame are always valid in other
frames, but they may develop the habit of transferring results.


3) Item (1) and item (2) are not the same idea. I think they intersect
according to the following Venn diagram:

______________________________|__________________|_____
| |
assume point of application | X |
of force is stationary | |
______________________________|__________________|_____
| |
| |
| assume results |
| can be xferred |
| from frame to |
| frame |
| |
| |


The questions that prompted this thread lie at point X, where either of
the two fallacies sufficies to give you the wrong answer.



4) Here is some more advice that may help extirpate these two bugs and
a whole lot more:
Keep track of what is fundamental and what is not.

In this case, the rule that
work = force dot dx [1]
is fundamental. In contrast, the "rules" that
static friction work = 0 [2]
or
force of constraint work = 0 [3]
are far from fundamental. They are less-than-general corollaries of
equation [1].


So, rather than learning [2] or [3] /with the necessary provisos/ it
strikes me as simpler to learn something like this:
static friction force ==> think what force dot dx tells you [4]
force of constraint ==> think what force dot dx tells you [5]


In cases where [2] is valid, it gives you the answer in one step,
whereas [4] gives you the same answer in two steps. Similarly
in cases where [3] is valid, it gives you the answer in one step,
whereas [5] gives you the same answer in two steps.

The punch line is that in cases where [2] and [3] do not apply,
[4] and [5] still give the right answer!

YMMV, but as for me, I would happily pay the price of the extra
step -- a super-easy step -- in order to buy more generality and
more reliability.

Remark: There is quite a fundamental difference between [2,3] and
[4,5]. It involves a fundamentally different way of doing business.
Loosely speaking, [2,3] involves memorizing a rule, whereas [4,5]
involves rederiving the rule from first principles every time it
is needed. Don't get me wrong; I'm not saying that memorization
is always bad; there are some things that should be memorized and
some things that /must/ be memorized. I'm just saying memorization
shouldn't be carried too far. There are some cases where rederiving
the result is just plain easier -- and more reliable -- than trying
to memorize the corollary /with all the required provisos/.

There's a lot more that could be said about this, but I reckon you
get the idea, so I'll stop here.