Re: [Phys-L] spring energy and other equation-hunting puzzles

[John Denker <jsd@av8n.com>]

On 02/09/2015 05:32 PM, Anthony Lapinski wrote:
> 3. You hang an object on a spring (with spring constant k). At the
> equilibrium position of the object, where it hangs at rest, the spring has
> stretched a distance x. You wish to find the object's mass, and ask two
> classmates for help. Equating forces, student A sets mg = kx to find that
> m = kx/g. Equating energies, student B sets mgx = 0.5kx2 to find that m =
> kx/2g -- half of student A's value. Whose method is correct and why?
> What's wrong with the other method?

That's brilliant.  I love it.

> I always get interesting answers for this one. 

Here's my answer:  Mr. B is wrong.  Mr. A is also wrong,
not because of the numerical answer, but because of the
method (or lack thereof) used to obtain it.

As I like to say, the problem with mindless equation-hunting
isn't the equation-hunting;  it is the mindlessness.  Both
A and B used mindless approaches.  Both failed to CHECK THE
WORK.

The prize goes to student C, who analyzed the problem both
ways.  This allowed him to CHECK THE WORK and notice the
discrepancy.  Thereupon he went back to see what other
forces need to be considered, and what other energies
need to be considered.

> Really shows who
> understands the physics of springs. Try it with your kids!

This is much bigger than just springs.  Similar plausible-but-
wrong analyses crop up all the time.

A lot of teachers go to great lengths to stomp out equation-
hunting, but IMHO that's a mistake.  Dimensional analysis 
is basically equation-hunting on a grand scale.  Note that 
answer A and answer B above were dimensionally correct,
and for purposes of a Fermi estimate either one would do
nicely.

Note that Day One of quantum mechanics was the day Max Planck
equation-hunted the black body formula.  He didn't derive it, 
and didn't pretend to.  The principles of QM were figured out 
from the equation, not vice versa.  Similarly, Kepler equation-
hunted his three laws.  He didn't derive them.  There was 
nothing he could have derived them from.

The point is, after hunting up an equation, you have to
CHECK THE WORK.  Check that the equation fits the facts.

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

As another example of the same thing:  I was perusing the
excellent web site
  Kyle Forinash
  "Waves: An Interactive Tutorial"
  http://homepages.ius.edu/kforinas/WJS/WavesJS.html

One one of the later pages it asks
  21.6. You are buying a stero system and a set of speakers. 
  The stereo has an output imedance of 10 ohms. What impedance
  speakers do you need to buy to get the loudest sound?
     http://homepages.ius.edu/kforinas/WJS/TransmissionJS.html

It is an interesting idea to take the idea of impedance
matching (as we understand it from wave mechanics) and apply
it to resistive networks.  From the students' point of view,
this is equation hunting, pure and simple.  The wave medium
has an impedance, and the Allen-Bradley resistor has an
impedance, so why not use the same equation?  This is what
we call "proof by pun".  It is not an acceptable substitute
for actual factual logic.  Now it turns out that if you 
look deeply enough, there are some good physics reasons why 
thinking of a resistor as a transmission line is actually 
/better/ than the usual Ohm's-law description ... but kids
in the introductory class could not possibly know that.

I mention this in order to make the point that 
 1) Nobody seems to object to equation-hunting 
  /when it gets the right answer/.
 2) The rest of the time they complain about it.

IMHO both (1) and (2) are wrong.  It's perfectly OK to go 
hunting for equations.  However, before you can promote
the equation to anything more than a hypothesis, you have
to check it.  Dimensional analysis is not guaranteed to
produce the right answer.  Proof by pun is not guaranteed
to produce the right answer.

The way geometry has been taught for most of the last 2000
years, it is traditional to derive virtually everything.
I think that's a mistake.  Physics sometimes tries to follow
the same pattern, but we also pull some stuff out of thin 
air.  Hypotheses non fingo.  My main point for today is 
that we ought to be upfront about which is which.  I might 
say to students:
 a) We derived this thing.
 b) We did not derive this other thing.  Here's some
  empirical evidence for it.  If you think it resembles 
  the first thing, that's a deceptive coincidence, so
  don't go there.
 c) We did not derive this third thing.  It resembles
  the first thing, which is actually not a coincidence,
  but explaining the connection would be beyond the
  scope of the course.  Come back after you've taken
  ten more physics courses and I'll tell you about
  this non-coincidence.  In the meantime, here is a
  pile of empirical evidence that shows we are on the
  right track......