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Readers here have a chance to test their own scientific knowledge, wisdom,
integrity...
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Do lighter and heavier bodies — a la Galileo
— fall at the same rate?
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How sure are you?
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Would you consider science a failure, or yourself a
failure as a scientist, if lighter and heavier bodies did fall at
different rates?
Would you if Newton’s theory of gravity predicted that
they would fall at different rates?
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What should be done about people who question or try
to find serious fault with such
fundamental scientific beliefs?
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Would you rather look at
pictures and plots?!
Newton’s
Great... Oversight
Galileo’s Falling Bodies and
Lagrange’s Trojan Asteroids
With Their Tadpole and Horseshoe Orbits
by Michael Hugh Knowles
Sir Isaac Newton, of all people,
made a fascinating... oversight.
He failed to notice that
by his own theory of gravity Galileo was
scientifically wrong when he determined — by both physical and gedanken experiments — that lighter and heavier bodies fall at precisely the
same rate. It is more than fascinating that every scientist since Newton has so failed as well,
including Einstein (and even Eddington). It is a simple consequence of Newton’s
laws that the asymmetrical gravitational interactions cause them to fall at
different rates, unless... unless one of them is at a Lagrangian point (L4 or
L5) with respect to the other and the necessary third body (e.g. Jupiter and the
Sun) that they fall toward/together with. This falling rate difference is the underlying dynamic of the Trojan
asteroids discovered-predicted theoretically by
Lagrange, of
which hundreds/thousands have since been found by astronomers in the 20th Century.
(See
Figure 5.)
The two symmetric “tadpoles” that center on Lagrangian points L4
(upper, on central vertical line) and L5 (lower) merge to form a
“horseshoe”. This contour
plot was generated from a function of the falling rate difference of lighter and
heavier bodies and their relative positions. It shows the shape of the
“orbits”
of Lagrange’s Trojan asteroids relative to the Lagrangian points L4 and L5
of the other two bodies (the main body, e.g. the Sun, on the left on the central
horizontal line; the secondary, e.g. Jupiter, on the right).
IN A NUTSHELL: by Newton’s own theory
of gravity:
If
Galileo had held a 1 kg mass in one hand and a 2 kg in the other, 1 meter
apart, and dropped them simultaneously from the top of the Tower of Pisa, at the instant of release
the lighter body and the Earth would have accelerated together (i.e. Earth
reference frame) faster by
approximately m/sec2
(ignoring, of course, all the
usual: “gravitational anomalies”, wind/viscosity, buoyancy, electro-magnetic
effects, etc.)
The lighter body falls faster!
It
falls faster since it accelerates/falls toward the heavier mass faster than the
heavier mass falls toward it; the 2 bodies form an isosceles triangle with the
Earth, so their unequal accelerations toward each other have unequal components
in the direction of the center of the Earth; i.e. the lighter body has
the larger such
component, so it and the Earth fall together faster (yes, even taking into
account the greater acceleration of the Earth toward the heavier body). When such bodies have an
extended period of time to fall, as they do in orbit, the difference can show
up quite visibly, as it does with the Lagrange’s Trojan asteroids; the bodies behave as though the approximation of “fall at the same
rate” were incorrect, which it is if extrapolated sufficiently
far.
We can note the irony of the situation: since the lighter body falls faster,
Galileo and Aristotle were both wrong.
In any case, Newton’s... oversight is so egregious that it can be referred to as:
By way of explanation, there is an old and somewhat
satirical chess aphorism:
When
a beginner gives away his queen, it’s a blunder.
When a grandmaster gives away his queen, it’s an… oversight.
The use of humor is intended as mnemonic and as a teaching device, and to
some extent as comic relief. Shakespeare is a constant reminder of how important
that last is. Here the ellipsis… has been added to the old aphorism to further
all those.
While perhaps less important theoretically than e.g. the
“infinitesimal” advance in the perihelion of the orbit of Mercury
— apparently accounted for by Einstein’s relativity but not by
Newton’s gravity
— the “infinitesimal” difference in the falling rates of
lighter and heavier bodies is important in its own way (not considering whether
it itself bears on relativity).
Or rather more to the point, the fact that this
both theoretical and actual falling rate difference has been
overlooked for
well over 300 years is crucially important, to the psychology of science, to the
philosophy of science (i.e. the love of wisdom as it pertains to science), and
to science itself and its practice in that it is a clear example of how science can fail, in
practice, on
its own terms, and with regard to fundamental results, for centuries on end,
without scientific notice let alone public acknowledgement.
The importance can be judged somewhat by the
response of some (luckily not all) leading scientists to this formal
falsification of Galileo’s fundamental result: anger. Some scientists’ first
reaction is overt anger, anger that science is being questioned, especially such accepted and fundamental results. One, at first
quite angry (“you shouldn’t be
questioning...”), later admitted, after it
was insisted that going over the equations was in order, that there was indeed a
difference in the falling rates, but he dismissed it contemptuously as
“too
small to be important”. When it
was pointed out that the advance in the perihelion of the orbit of Mercury was
also very small, but was considered scientifically important none-the-less...
end of communication.
Another reaction was that of a high school science teacher whose students had shown her
the equations that they found on the PAIAS web site. We exchanged several
e-mails discussing the falling rate difference, until she finally pleaded:
“but is it all right if I tell my students that lighter and heavier bodies
do fall at the same rate?” When I reiterated that although they fall at
different rates, they do fall at approximately the same rate
for our usual purposes here on Earth, that satisfied her enough that our
correspondence ended.
These sorts of reactions should give us all pause
—
scientists and non-scientists alike. We are all currently in need of several good doses of:
“... when
wrong, to be put right.”
(The historical basis for
this common and quite anti-scientific anger of scientists is explored somewhat
in this article’s sub-section
Newton’s…
Oversight: How?! Why?!
of the section
SCIENCE’S
PERENNIAL… OVERSIGHTS. A more philosophically oriented article on...
Oversights is under construction, based on the section just referred to.)
PICTURES and
PLOTS
     
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6a
Figure 6b
Figure 5 is same as the contour plot seen near the top of page. See also
Link Bar at left, above.
Copyright © 1995 - 2002,
Michael Hugh Knowles. All rights reserved.
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