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On 8/8/24 1:42 PM, Matthew Heaney wrote in part:
If a student asks why a ball drops to the ground when you release it,what
do you tell them?
As usual, it helps to sharpen the question before answering it.
First, as a fundamental point of metaphysics: The word "why"
is not particularly scientific. It is at best ambiguous. As
Galileo pointed out on Day One of modern science,
— The laws of physics have to say what happens.
— Sometimes they say how it happens.
— The fundamental laws virtually never say "why" it happens.
Usually questions about "why" are best answered in terms of:
*) What will happen?
*) How do we know that will happen?
*) How is that related to other things we know?
As another fundamental point of reasoning: Don't ask "why"
something happens unless you know /whether/ it will happen.
If you release a ball /from rest/ it will drop straight down.
If you release a ball from a trebuchet, it will go up and
sideways, possibly quite far.
In the general case without friction:
1) Integrate the force to get the momentum, or equivalently
integrate the acceleration to get the velocity.
2) Integrate the velocity to get the position.
3) Use the position to calculate the force.
4) Go back to step 1 and turn the crank, iteratively.
It's not hard to use a spreadsheet to implement a symplectic
integrator to solve the equations of motion.
A planet in an elliptical orbit will go round and round for
a billion years. Half the time it is going uphill and half
the time it is going downhill. So again, assuming that a
ball always drops "down" is a bad assumption.
You can build small mechanical oscillators with very little
friction, i.e. very high Q, as you find for example in a
wristwatch. Again the system spends half its time going up
the potential hill and half going down.
If you want the ball to go down and *stay* down, then you
need some sort of friction, some sort of shock absorber,
i.e. something to create entropy.
This introduces a damping term in the equations of motion.
There are lots of ways of doing it badly. Doing it properly
is a tour de force. There are back-to-back papers in the
1928 Physical Review by the Bell Labs guys: Harry Nyquist
(theory) and his buddy John Johnson (experiment).
======
Physicists since Galileo have usually started by considering
the zero-friction case: A body at rest remains at rest, and
a body in motion remains in motion. Later they add in a little
bit of damping.
Chemists, in contrast, usually assume — implicitly — that
everything is grossly overdamped. This is more-or-less the
physics of Aristotle: A body at rest remains at rest, and a
body in motion comes to rest. As a consequence, they treat
spontaneity and irreversibility as essentially the same
concept.
So beware: The terminology and concepts of chemistry do not
reliably extrapolate to anything other than chemistry.
What's worse, the assumption that everything is overdamped
is not 100% reliable even for chemical reactions. Consider
for example the world's simplest chemical reaction:
H + H —> H₂ [1]
You'd think that would be the world's most energetic and
vigorous reaction. In fact, though, in the gas phase, it
does not proceed as written, because there is no relevant
damping mechanism. The force is strongly attractive, but
the atoms whiz past each other like a comet whizzing past
the sun, and then they separate again.
Reaction [1] is a second-order reaction, but what actually
happens is a third-order reaction:
H + H + H —> H₂ + H [2]
where the two atoms collide with the third. They unload
their binding energy into the third atom, which flies off
at high speed.
Bottom line: When you drop a ball into a potential well,
there are two analyses that need to be done. Start by
figuring out the undamped case, then figure out what
happens when you add a little bit of damping.
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