In Defense of Inaccuracy

Ethan at Starts With a Bang busts two Galileo myths.

1) That Galileo actually dropped weights off the Leaning Tower of Pisa. He almost certainly didn't. Like the story of George Washington and the cherry tree, it's an instructive parable not at variance with the character of the man - but not an event that actually happened.

2) That the experiment would have worked even if Galileo had done it. It wouldn't have. Air resistance would mean that otherwise identical objects of the same mass wouldn't have hit the ground at the same time.

Those are the two points that Ethan makes. They're both quite correct.

Nonetheless, ol' Galileo's (fictitious) Pisa experiment deserves a lot of credit. At the dawn of the 17th century our knowledge of physics was terrible. Aristotle's millennium-old ideas were still quite dominant, and though Aristotle was a brilliant man who at least got the ball rolling (or falling), he was in fact very wrong about a great many things in physics. Gravity was one of those things - Aristotle thought that the acceleration of a falling object was proportional to its mass. A lump of rock weighing 10 mina should fall ten times faster than a 1 mina rock.

But it doesn't. Not even close. Sure if you carried out the experiment the odds are the 10 mina rock would make it to the ground a little before the 1 mina rock. But would it make it down more than three* times faster? Not a chance. Though the experiment doesn't quite match Galileo's "they fall at the same rate" theory, it matches his theory a lot closer than Aristotle's "they fall in proportion to their mass" theory. What's more, Galileo actually covered air resistance in his theory. He predicted that in a vacuum the experiment actually would result in the rocks hitting at exactly the same time. Though Galileo never lived to see the moon landing or even a vacuum pump, he was right. In a vacuum we really do observe the objects falling at the same rate. So even if the Pisa experiment didn't quite illustrate the point perfectly, it still would have served the purpose of eliminating Aristotle and supporting Galileo just fine.

Fig 1: If you drop this very heavy statue of Aristotle out of a plane, it will still only fall at 9.8 m/s^2.

As usual our knowledge has improved over the centuries but the general concept keeps repeating itself. Many times experiments are done at the bleeding edge of experimental and theoretical capability. Sometimes the data is not clear enough to tell you what is happening, but it's plenty good enough to tell you what isn't happening. Almost every field of physics has experienced this, so to pick an example at random we might take Bose-Einstein condensation. The theory predicted that very cold helium-4 would undergo a phase transition to a quantum condensate at a particular temperature. When it was first tested it was found that helium-4 did possess very unusual properties, but not quite at the temperature predicted and not quite the properties that the theory predicted. But did it prove that Bose and Einstein were right that bosons didn't conduct business as usual at low temperatures? Absolutely. Later on corrections along the lines of air resistance in mechanics were made to the original theory and now theory and experiment match very well. The experiment taught us something important and thoroughly smashed the old ideas even if it wasn't the exact first-shot perfection we might have preferred.

In concluding the whole matter: should you ever find yourself at the top of that perilously leaning tower in Pisa holding two unequal weights, don't worry that there's a small difference in the times they reach the ground below. Even with the slight variance with the naive theory of uniform acceleration the experiment is still more than enough to shatter the Aristotelian physics into another Greek ruin.

*Three because distance covered in uniform acceleration is proportional to the time of fall squared. Inverting, the time is proportional to the square root of the distance. Thus ten times the acceleration reduces the time required to cover a given distance by a factor of the square root of 10, or a little above 3.

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Somebody should look.

I remember performing an undergraduate vibrations lab experiment in which we were given a mathematical model for the vibration of two masses embedded in an aluminum cantilever beam (one at the end and the other near the middle). In its derivation, the model assumed two virtually independent superimposed vibrating systems (one that did not take into account the continuity of the slope of the flexing beam). I managed to derive a more suitable mathematical model which predicted the results much better than the given model. I served as TA for that class the following year.

I wonder if educators should reinforce the idea that there may be results that you expect but not results that you are "supposed to" get and that refining the existing model is possible. Coming up with that new model was a bit arduous but I wonder how difficult it would be to construct suitable experiments for students in which they can reasonably come up with a better model on their own with their existing skills.

By Tim Gaede (not verified) on 09 Jun 2009 #permalink

After running a quick computer simulation, it is seems that very massive iron balls would separate vertically by only a few inches by the time they hit the ground at the base of the Tower of Pisa. That may be suitable as a public demonstration rather than an experiment. What Galileo actually did was roll objects down inclined planes. He was able to establish that for an object released at rest, the distance traveled was proportional to the square of time. Unfortunately, he couldn't quite integrate the concepts force, mass and acceleration the way Newton would eventually do late in the 17th century.

Here is a demonstration by an Apollo 15 astronaut. Of course, it was not under the best laboratory conditions but it was a delightful tribute.

You may also want to read this very brief sobering account made popular by Digg of college student perceptions of gravity.

By Tim Gaede (not verified) on 09 Jun 2009 #permalink

John Wilkins,

Could you then say that both Aristotle and Galileo recognized that heavy objects fall faster-but-not-that-much-faster than light ones? But Aristotle attributed the "faster" part to gravity and the "not that much faster" part to air resistance, while Galileo did the reverse?

By Anton Mates (not verified) on 09 Jun 2009 #permalink

Aristotle certainly knew that heavier objects fell faster than lighter ones, and he properly ascribed that to resistance from the medium through which it fell. As he did not recognise a vacuum, he would have thought this to be true in all cases.

I have been through the Physics and On The Heavens, which are often cited by those claiming that Aristotle said this or that, and there's no passages I can find that say otherwise. "Heavy" objects (those composed of earth or water) will fall to their proper place in a straight line unless acted upon; I can only surmise that he meant this to be constant for the same materials, acted upon by the medium (water or air).

Sorry to come late to this, but this might still be interesting:
The argument that falling speed cannot depend on mass can be found by thought:
Imagine two blocks weighing one unit each. Let them fall side by side so that they touch. Measure their falling speed.
Now glue the two together. Nothing significant changes to the first case, so they cannot suddenly fall faster.
During my physics study, I was told that Galileo actually used this kind of argument, but I never looked it up.