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When I was in 9th grade, I learned that Aristotle was responsible for holding back physics for centuries because he said that heavier objects fall faster than lighter objects. Finally, in the 16th century Galileo disproved this theory by dropping two balls of different masses from the Leaning Tower of Pisa showing that they both fell at the same speed.

And when I took physics in 12th grade, I learned that Newton's Law of Gravitation explains the results of Galileo's experiment, showing that the acceleration of an object near the earth's surface is always the same $g=GM/R^2=9.80 m/s^2$, where $G$ is the gravitational constant, $R$ is the distance of the object to the center of the earth, and $M$ is the mass of the earth.

This all seemed to conclusively disprove Aristotle's theory that heavier objects fall faster than lighter objects. However, this line of argument neglects to consider Newton's Third Law, which implies that the falling object forces the earth to move at acceleration proportional to the mass of the falling object. And this will cause the distance between the falling object and the center of the earth to decrease faster for heavier falling objects, implying that the heavier objects do in fact fall faster than lighter objects.

So my question is why was I taught that Aristotle was completely wrong when his prediction seems to be totally in agreement with Newtonian mechanics?

Added: If you don't believe me, just check out the differential equations obtained from Newton's Law of Gravitation:

$MR''=GmM/|R-r|^2$ and $mr''=-GmM/|r-R|^2$,

where $m$ is the mass of the object, $M$ is the mass of the earth, $R$ is the position of the earth, $r$ is the position of the object. Making it simpler, we get:

$R''=Gm/|R-r|^2$ and $r''=-GM/|r-R|^2$.

When $m$ is large, $R''$ is large, implying that $|R'-r'|$ becomes large faster than when $m$ is small, implying that the object will eventually move faster towards the earth when $m$ is large than when $m$ is small.

See a similar question here: https://physics.stackexchange.com/questions/3534/dont-heavier-objects-actually-fall-faster-because-they-exert-their-own-gravity

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    $\begingroup$ First of all, it is wrong to "read" ancient theory "with insight" ... Aristotle (and ancient "natural philosophers") never formulated a precise law (with some mathematical formalism available) expressing the relation between speed and time, nor made "predictions" regarding the behaviour of bodies in motion. This is the main reason why it is quite useless trying to compare those theories with "modern" (i.e.based on mathematical formulation of relations capable of numerical predictions) ones. Note : the "Leaning Tower" experiment has (quite certainly) never been performed by Galileo. $\endgroup$ – Mauro ALLEGRANZA Jan 14 '15 at 15:17
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    $\begingroup$ For sure A did not think at gravity in the same way as Newton did ! :) In my opinio, speaking of "right" or "wrong" in historical context is not very useful. Since Galileo, we "live" in a world were physical science is "made of" mathematical laws and theory capable of numerical prediction and it is hard to think in a non-math way. Galileo uses the same math of Arstotle's time : Eudoxus/Euclide's theory of proportion and he was able to find and test the correct law for free falling bodies. 1/2 $\endgroup$ – Mauro ALLEGRANZA Jan 14 '15 at 16:36
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    $\begingroup$ @CraigFeinstein You've got everything completely mixed up. Newton's laws are easily used to demonstrate that gravitational acceleration and gravitational acceleration only (unless one considers noninertial coordinates) is independent of mass: $F_g=\frac{GMm}{r^2}=ma\implies \frac{GM}{r^2}=a$, i.e. the acceleration of the object only depends on the other object (with mass $M$) which it is attracted by in Newtonian mechanics. It is obvious that Aristotle was wrong. In this light, it is clear that your question rests on a misconception about the laws of physics. $\endgroup$ – Danu Jan 14 '15 at 16:47
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    $\begingroup$ @CraigFeinstein Well, yes, but that's generally ignored when dealing with Newtonian physics problems. $\endgroup$ – HDE 226868 Jan 14 '15 at 21:31
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    $\begingroup$ A very similar question can be found here on Physics. Your question seems to be along these lines, not historical ones. And yes, it's true, but the reason you're not taught it - which appears to be what you're asking, which would also make it off-topic - is that it's generally unnecessary in most calculations. $\endgroup$ – HDE 226868 Jan 14 '15 at 22:09

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I'll try with some calculations : please, check it and the formulae used ...

A solid ball with a mass $m$ of $1$ kg falls (with the usual approxiamtions : no drag, etc.) with an acceleration $a$ that is about $10 \ m/sec^2$.

This means that falling from a tower $80$ meters heigh, it will touch ground after $4$ sec, with a final velocity of about $40 \ m/sec$.

You are right : in the same time, the Earth will "fall towards" the ball, attracted by the same gravitational force.

The mass $M$ of the Earth is about $6 \times 10^{24} \ kg$.

The acceleration $A$ of the Earth that is proportional to $a$ as the reciprocal of the masses; i.e. :

$A = a \times m/M = 10/(6 \times 10 ^{24}) \approx 2 \times 10 ^{-24} \ m/sec^2$.

After $4$ seconds, the Earth reachs a velocity of fall of $8 \times 10 ^{-24} \ m/sec$ and it has traversed a space $s \approx 16 \times 10 ^{-24} \ m$.

This means that, due to the reciprocal attraction, the two bodies will touch each other in slightly less than $4 \ sec$ and that the "real" space traversed by the falling ball with respect to the Earth is about :

$80$ meters minus the space traversed by the "falling" Earth during the short time of the fall.

It can be useful to recall that :

the size of atoms is measured in picometers : trillionths ($10 ^{-12}$) of a meter.


If the ball has a mass $m'$ of $1000 \ kg$, the force with which it "pulls" the Earth will be $1000$ times greater, producing an acceleration $A' \approx 2 \times 10 ^{-21} \ m/sec^2$.

This implies that in this second case the space traversed by the Earth duing the fall of the heavier ball will be $s' \approx 16 \times 10 ^{-21} \ m$.


According to Aristotle (Physiscs, Book VII) the "law" of dynamics is :

"if a power $\alpha$ moves a body $\beta$ during time $\delta$ for a distance $\gamma$, then an equal power $\alpha$ will move a body half of $\beta$ along a distance twice as $\gamma$ in the same time".

In an anachronistic way, we can say :

$F \propto V$.

Thus, if we apply this law to "our model" of free fall, with the weight of the body as the force, we have that - assuming that after an initial short time of acceleration the falling body will reach a constant "terminal velocity" - in the second case the acquired speed $v'$must be $1000$ times the first one : $v$.

This means that after $4$ seconds the heavier ball has traversed a space : $s'= v' \times t$, i.e. $s' = 1000 \times v \times t = 1000 \times s$, where $s$ is the space traversed by the lighter body after a fall of $4$ seconds.


Conclusion

As you can see the "same factor" : $1000$ acts in a completely different way in the two models.

Due to the huge mass of the Earth, for bodies of "normal" size (meaning body of our daily life experience) it has no experimentally verifiable effect on the behaviour of different bodies falling due to the gravitational force.

In the Aristotelian model, that factor has an evident experimentally verifiable effect on the behaviour of different bodies falling due to the "tendency towards the centre".

This is exactly where is the "conceptual" difference : to perform some sort of experimental test.

Thus, the answer to :

Was Aristotle really wrong about gravity?

is a definitive YES if we try to answer the question from the point of view of modern science, a point of view that is not that of aristotelian natural philosophy.

If instead we want to compare two different qualitative "worldviews", things are different (see at least the philosophical debate involving : Thomas Kuhn, The Incommensurability of Scientific Theories, Scientific Revolutions, Historicist Theories of Scientific Rationality, Imre Lakatos and Paul Feyerabend).


Notes

i) In our computations we made approximations; approximation is a modern concept.

Without precise mathematical laws there are no possible approximations.

Aristotle's "natural laws" are not approximations in the modern sense; they are qualitative description of facts, like the one made by the same Aristotle on botany and zoology (whih were impressively accurate, by the way).

ii) Writing the "aristotelian equation" we make an "historical mistake" : he never thinked in terms of mathematical formuale.

Please, note that the mathematics used by Galileo in his analysis of the free falling body problem was only the theory of proportions of Eudoxus/Euclid, that was already available in Aristotle's time.

Thus, the "tools" available to ancient natural philosophers were more or less the same compared to the ones available to Galileo (not so with Newton ...).



Postscriptum

The above "experiment" was alredy discussed by Galileo, in advance of the correct formulation of the law of universal gravitation. See :

SALV. We infer therefore that large and small bodies move with the same speed provided they are of the same specific gravity.

SIMP. Your discussion is really admirable; yet I do not find it easy to believe that a bird-shot falls as swiftly as a cannon ball.

SALV. Why not say a grain of sand as rapidly as a grindstone? But, Simplicio, I trust you will not follow the example of many others who divert the discussion from its main intent and fasten upon some statement of mine which lacks a hair’s-breadth of the truth and, under this hair, hide the fault of another which is as big as a ship’s cable. Aristotle says that “an iron ball of one hundred pounds falling from a height of one hundred cubits reaches the ground before a one-pound ball has fallen a single cubit.” I say that they arrive at the same time. You find, on making the experiment, that the larger outstrips the smaller by two finger-breadths, that is, when the larger has reached the ground, the other is short of it by two finger-breadths; now you would not hide behind these two fingers the ninety-nine cubits of Aristotle, nor would you mention my small error and at the same time pass over in silence his very large one. Aristotle declares that bodies of different weights, in the same medium, travel (in so far as their motion depends upon gravity) with speeds which are proportional to their weights; this he illustrates by use of bodies in which it is possible to perceive the pure and unadulterated effect of gravity, eliminating other considerations, for example, figure as being of small importance, influences which are greatly dependent upon the medium which modifies the single effect of gravity alone.Thus we observe that gold, the densest of all substances, when beaten out into a very thin leaf, goes floating through the air; the same thing happens with stone when ground into a very fine powder.

But if you wish to maintain the general proposition you will have to show that the same ratio of speeds is preserved in the case of all heavy bodies, and that a stone of twenty pounds moves ten times as rapidly as one of two; but I claim that this is false and that, if they fall from a height of fifty or a hundred cubits, they will reach the earth at the same moment.

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    $\begingroup$ Wow, not much movement of the earth. $\endgroup$ – Craig Feinstein Jan 15 '15 at 20:49
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In short, you were taught that Aristotle was wrong because he was wrong. He didn't make a prediction, he made an observation about rock and feather, and then sloppily generalized it to all objects without a second thought. The subtle effects you are describing weren't even noticable in his time, but that a feather falls slower because it is much more affected by air resistance, would have been obvious to sailors, or anyone who dealt with winds, even then. Already in antiquity John Philoponus pointed out that if one corrects for that the sole basis for Aristotle's conclusion disappears: "But this is completely erroneous, and our view may be completely corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights, one many times heavier than the other you will see that the ratio of the times required for the motion does not depend [solely] on the weights, but that the difference in time is very small."

But you were taught that Aristotle was "holding back science" not because he was just wrong about falling bodies. As Philoponus pointed out, it was merely an illustration of a general attitude, unfortunately adopted by many after him, that facts about nature can be reasoned out of their heads with spotty and misconstrued observations, if any at all. To be fair, Aristotle's contribution wasn't all negative, he gave first systematic descriptions in what now became established natural sciences, and tried to organize and structure what was known about the world in his time. But his method of inquiry was wrong headed, and it took a lot of time and effort to overcome it.

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Yes, Aristotle was wrong about gravity. But I think it is unfair to say “that Aristotle was responsible for holding back physics for centuries”. The ones who held back physics for centuries were the late-antique and mediaeval (Christian, Muslim and Jewish) so-called philosophers who transformed Aristotelianism into an ossified dogmatic doctrine. Aristotle himself was always willing to change his mind and to consider alternative explanations.

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No. Aristotle was not necessarily wrong. This is in substance Carlo Rovelli's view in Aristotle’s Physics: a Physicist’s Look. As the abstracts announces it

Aristotelian physics is a correct and non-intuitive approximation of Newtonian physics in the suitable domain (motion in fluids), in the same technical sense in which Newton theory is an approximation of Einstein’s theory.

If one agrees that falling occurs in a fluid, then it is not different from 'sinking'. Heavier bodies sink faster. (Buoyancy which can play a crucial role is also due to gravity).

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  • $\begingroup$ This is a really nice article! $\endgroup$ – Arnold Neumaier May 8 '18 at 18:02
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That Aristotle (and you with your example) are wrong is proved by the following simple argument: imagine two bricks of equal mass. Each of them falls with certain acceleration. Now glue them together and let them fall. According to Aristotle two bricks will fall faster than each brick separately. It is evident that this is absurd: what difference does it make whether the bricks are glued together or not?

I suppose this argument is due to Galileo, but I am not 100% sure.

Actually Aristotle's "physics" says that two bricks will fall twice faster than one brick.

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    $\begingroup$ This argument is due to Galileo, but unfortunately it is circular. Assuming that gluing makes no difference already presupposes that Aristotle is wrong. philsci-archive.pitt.edu/2524/1/… To Aristotle glued bricks become a single item, and behave as such with its "natural place" and weight that determines the rate of approaching that place. Two unglued bricks are two separate items. $\endgroup$ – Conifold Jan 16 '15 at 1:33
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    $\begingroup$ It seems intuitive to us today that force field acts the same way on both, but that's because we absorbed Newtonian forces and fields. To Aristotle there is no reason why the "natural pull" should not distinguish between objects of different nature. Even without "natural motions" it is conceivable that reaction forces introduced between the bricks by gluing can alter how they move. There really is no way to reason it out without observations :) $\endgroup$ – Conifold Jan 16 '15 at 1:34
  • $\begingroup$ I read once (and I don't remember where any more) that this argument actually predates Galileo by perhaps 100-200 years. But nothing was really made of it at the time. What Galileo did was provide mathematical models for simple physical situations. That was the real breakthrough. It's one of the cornerstones of the "scientific revolution". $\endgroup$ – Carl Offner Sep 11 '16 at 1:14
  • $\begingroup$ @Conifold: Alexandre's (Galilei's) argument is not circular. We do not assume that gluing makes no differencem, but we prove it by gluing only in our imagination. $\endgroup$ – Otto May 29 '17 at 13:07
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You are right that a bigger object will reach the ground in a miscroscopically smaller time if dropped independently. Note however that if dropped at the same time, this will still not be the case, as the earth will be attracted to the force of gravity created by both of those objects.

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  • $\begingroup$ Welcome to HSM, rickmcn1986. Because of the nature of this particular Stack Exchange site, we want references with all our answers. In addition, it seems that this is in the same vein as the other answers. Can you spruce this up in some way? Thanks. $\endgroup$ – HDE 226868 Apr 10 '15 at 22:39
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Aristotle concluded in his law of motion that the speed of an object depends on the viscosity of the medium it is in. In keeping with this line of thinking, since a perfect vacuum has zero viscosity, the speed of a falling object should approach infinity, as viscosity approaches zero. Galileo in his incline plane experiment identified the role of gravity, explaining it to be Aristotle's attractive force pulling toward the " natural place". In addition, Galileo's Leaning Tower of Pisa experiment addressed the question: do falling masses of different sizes fall at the same speed? He concluded that they will fall at the same speed. Newton corrected Galileo's conclusion with his own gravitational theory which states that the the force of gravity exert the same acceleration on objects regardless of size; an idea that has gained acceptance over the years. Just as Aristotle law of gravitation, ignoring the role of mathematics available to him at the time, was was the first approximation of the role of gravity on falling objects, Galileo, utilizing the same math that was available to Aristotle, came up with a second approximation of the role of gravity. Newton with more advance mathematics came up with his law of gravitation, a third approximation to the law of gravitational motion; and of course, Einstein made a radical departure from all existing theories of the time with his upgrade.

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Actually Carlo Rovelli considers the situation ignoring any viscosity/friction effects and points out that a body still falls slower in a denser medium.

Consider a mass of specific gravity 2 falling through water. The net gravitation force is mg/2 but the mass is still m, hence the acceleration is halved.

Of course the specific gravity of air is 0.0013 so that effect is negligible in air. Aristotle thought that in a zero dense medium the mass would fall infinitely fast and he was obviously wrong.

"if a thing moves through the thickest medium such and such distance in such and such time, it moves through the void with a speed beyond any ratio". (Physics 215b)

People who think Aristotle's treatment was purely qualitative should try reading him. Look at Physics 215b and 216a in particular. There's plenty of calculation there. It seems to me that most of his problems arise from not understanding the concept of zero - hardly surprising since it wasn't introduced into maths for another thousand years (by the Indians). Maybe his horror of the void comes down to the same issue.

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Actually, if we are talking about Earth, there is another sense in which Aristotle was (accidentally) right, Earth has an atmosphere, and an atmosphere exerts a drag force on falling objects. It is not difficult to calculate terminal velocity and if we assume drag is proportional to velocity (R = kv), at terminal velocity acceleration and hence resultant force is zero, mg - kv = 0, kv=mg, v=mg/k so if the objects are roughly the same size and shape so k is constant, terminal velocity is proportional to mass! Interesting.

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Physicists now tend to think in terms of effective theories; that is a theory which is accurate in a certain energy regime but fails in another. Thus, for example we get super-gravity as a low energy effective (ie approximate) theory of string theory.

It's not quite fair to compare Newton and Aristotle after all a span of two millenia separates them. This is a hefty amount of time by any measure; and the earlier work should be evaluated in its own terms, in the context of its time, as well as it's influence.

Julian Barbour in his History of Dynamics clearly points out that Aristotles critical approach to the problem of change, space, time & motion was a clear precursor to later thinking about motion; and I think, if I recall rightly, that he said that Newtons laws were in some sense, in an embryonic form in his works.

It was the later infatuation with Aristotle beginning in the Renaissance after Averroes rediscovery of Aristotles works and his commentaries on them; and also the close association of the Church with Aristotle and then then the rise of anti-clericalism in Europe after the bourgeois revolutions that our modern dislike of Aristotle stems from; the popular literature is littered with disdain for his works; this is not an accurate view of his importance; and nor, to be honest of the Greek work on dynamics of which Aristotle is the most prominent representative.

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It's not that Aristotle himself who held back scientific thought; after all, he himself was not backward in critiquing Plato, Parmenides or Democritus when he felt their ideas hadn't sufficient justification. He was not over-awed by them, perhaps due to the closeness in time to these giant figures, after all, he himself was a student of Plato.

Had the early to late medivals understood this aspect of his critical activity and took it upon themselves then perhaps scientific thought might have been recovered more rapidly; instead, they found themselves over-awed by their achievements and on the whole, could do no more than tinker with them, and it took the time to the early moderns before general scientific culture in Europe had risen to a sufficient height and had become fluid and vigorous that they could begin to develop where the Ancient world had stopped.

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Aristotle was considering the max speed of an object being pulled through a resistant media - eg: an ox pulling a cart, or a feather falling through air.

It is obvious to any idiot that two 1 lb weights falling side by side hit the ground at the same time as 1 2lb weight formed by gluing the two together.

It is far less obvious how long it will take an identical shape and size weighing twice as much while falling through a resistant media. It will certainly be faster, but since resistance is NOT directly proportionate to speed, it will not be twice as fast.

Still, give Aristotle some credit, he was NOT taking about objects falling in a vacuum.

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  • $\begingroup$ Please consider specific historical evidence when answering. "Obvious to any idiot" is generally not a useful basis for putting forward a response. On the other hand, "obvious in view of idea x given in book y" might be a more useful contribution (provided, of course, that book y was e.g. known to the considered author in particular, or generally known in the relevant community and time). $\endgroup$ – terry-s Mar 14 '18 at 15:23

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