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On Aristotle, Newton and a 500 Pound Gorilla (Friction)

The physics/astronomy of Plato/Aristotle/Ptolmey/(and later the Church) is often
held up as famously wrong in many introductory physics/astronomy courses.  However, in
some ways this may be a bad rap [see the Footnote below].  Newton's [and the giants
(Copernicus, Galileo, and Kepler) on whose shoulders he said he stood] triumph was in
some sense the ability to temporarily put the 500 pound gorilla of friction in the
closet and imagine motion in the absence of friction. (Once done the systematic influence
of the gorilla could be incorporated.)
Consider Newton's Law of Inertia which says, in part, that a body in uniform
[constant velocity = speed + direction] motion will naturally remain in this state unless acted
upon by an outside influence.  Aristotle, on the other hand, said that the natural state of an
object is at rest, that all objects seek this state, and that a constant outside action is required to
keep an object in motion.
In the words of W. Wolf "Lets go to the video tape" and consider what we observe in
real life.  If I want to move a chair across the room I have to push on it continuously or else it
quickly comes to rest.  Indeed to keep anything moving horizontally, near the surface of the
earth, one has to keep pushing/pulling it or it will come to rest.  Our every day world is
dominated by friction (of various kinds) that robs the energy of motion from an object
(turning it to heat that is often difficult to account for) until it inexorably grinds to a halt.
Thus Aristotle indeed summarized the world, as he knew it and as most of us know it.
The leap of intellect that Newton et. al. realized was that: 1) the reason such objects
came to rest could be attributed to an external frictional forces that always opposed the
motion; and 2) in the absence of  such messy (every day) forces, the object could proceed in
uniform motion never coming to rest.
Really the only place that the ancients could observe objects where friction was
negligible was in the regular motions of the planets across the sky.  Although the planets
showed variations in their motion (speeding up, slowing down or even reversing direction for
brief periods) these motions were exquisitely reproducible and regular and showed no signs of
slowing down.  Aristotelian logic dictated that some constant outside action must be at work
to keep these objects moving.  The church, which embraced the Aristotelian/Ptolemaic
universe, could perhaps see the hand of god or angels in this constant outside push that kept
the planets moving.
Newton et. al. realized that the planets didn't slow down because there was no
friction acting on them (there is no air and hence no air friction in space).  [Newton's precise
ideas on this are not known to this author.]  Thus the planets were free to continue their
motion undiminished.  At the same time Newton et. al. realized that the planets moved,
approximately, in circular orbits about the sun.  A circle is not straight-line motion, so
Newton knew there must be an external action (force) that bends the planets motion into a
circle, and this force must pull toward the sun (the circular orbits center).  The quantification
of this force was provided Newton's Universal Law of Gravity.  This law of gravity allows
the understanding (or rather the mathematical prediction) of the dances of the planets in all
their exquisite detail.  It is important to realize however, while the law of gravity provides us
with the mathematical ability to explain and predict the motions of celestial bodies, it gives us
no hint as to why is there gravity, and why it obeys this specific mathematics.
This last point was emphasized beautifully by Robert Feynman (a true geniuses, and
wonderful character in the history of modern physics).  He said, in effect, that as far as
fundamental understanding goes, we had not come that far.  He noted that the ancients
believed that angels pushed the planets along their paths in the sky and that now we only
know that the angels actually push toward the sun.
In introductory physics courses there is a tendency to relegate the 500 lb. friction-
gorilla, for the most part, to the closet and to discuss in detail the easier case of frictionless
motion.  The closet door is often opened only to consider the easiest case of a constant sliding
surface frictional force.  As emphasized by the experience-based philosophy of Aristotle, true
frictionless motion is rather uncommon in every day life.  This is done for the very good
reason that the inclusion of air friction in the equations of motion for an object moving
through the air (for example) makes them mathematically very difficult to solve.  Indeed in a
full treatment (including the variation of air density with height) the equations becomes
impossible to solve exactly analytically (that is to say with a paper and pencil using
algebra/calculus).  As a mater of fact it was to solve the problem of a bullet traveling through
air that the first electronic computer was constructed the at the U. of Penn. in the 2nd World
War era (near the same time one was constructed in England to crack the German Enigma
code-machine messages).
If another part of this discussion we will consider some simple cases of air-frictional
(drag) forces on objects.  Some typical approximations for the drag force will be solved
analytically.  Also a simple numerical method of solving the equations of motion will be
illustrated.  The numerical method could be generalized to cases that can not be solved with
calculus.

Footnote.  The Platonic philosophy of science elevated pure abstract reasoning to a holy
pedestal.  At the same time it viewed experimental observation with extreme prejudice.  This
stance is unequivocally deserving of the label as a very bad idea. Consider the classic Plutonic
example of the observation of the shadow of a hand cast against a cave wall by a flickering
campfire providing a poor guide to understanding the shape of the hand.  Plato used this to
argue that observation was horribly fallible, could not be trusted and that pure abstract reason,
regarding the essence of the hand, was the only true way to proceed.  Clearly the choice here
is of a horrendously bad experiment being chosen to discredit the entire method of inquire.
For a counter example consider the use of a distant, well defined light source (like the sun)
and the observation of the shadow cast on a flat surface close to the hand.  This shadow would
provide a rather good image of the morphology of the hand (given the constraint that one is
precluded from looking directly at the hand).
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