Philosophy 167: Class 6 - Part 7 - Pendulums- Isochronism, and the Problem of Resistance.
Smith, George E. (George Edwin), 1938-
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Now, on his claim about the pendulums. This is rather distressing, I'm just gonna read the whole passage out because it's so striking. So I fell to thinking how one might many times repeat descents from small heights. And accumulate many of those minimal differences of time that might intervene between the arrival of a heavy body at the terminus and that of the light one.
So that added together in this way, they would make up a time not only observable, but easily observable. Problem is, when you drop objects from a small height, it's difficult to get a measure of how different in time they land. Ultimately it's small skip you can see where it is.
Ultimately I took two balls. One of led and one of cork. The former being at least 100 times as heavy as the latter and I attach them to equal thin strings four or five brachia long. Brachia is a forearm roughly two feet. So we're talking here eight to ten feet long tied high above.
Notice the word pendulum isn't being used here because it's not quite there yet. Removed from the vertical these were sent going at the same moment. And following along the circumferences of the circles described by the equal strings that were the radii, they pass the vertical and return by the same path.
Repeating their goings and comings a good hundred times by themselves, they sensibly showed that they heavy one kept time with the light one so well that not in a hundred oscillations, nor in a thousand, does it go ahead in time even by a moment but the two travel with equal pace.
The operation of the medium is also perceived, offering some impediment to the motion. It diminishes the oscillations of the cork much more than those of the lead. But it does not make them more frequent of less so. Indeed, when the arcs passed by the cork were not much more than five or six degrees.
You can picture it slowing down from the resistance effect, and those of the lead were 50 or 60, they were passed over in the same times. That's the claim. And he sure seems to be describing a real experiment. Doesn't he? But you do that experiment there's no way they stay remotely in synchrony with one another.
I just want to pick out one thing and then come back to the main point. Do notice he knew, and every one knew really. The heavier an object, the less the effect on resistance. Okay, so that's in there. The sequence here, is this is first described in a letter of 1602 to one of his mentors at Pisa.
I have to look up the name. It's not a name I know immediately. Del Monte. Then me makes a slightly less detailed. Well significantly less detailed statement in the dialogue. But he certainly suggests that the time of any pendulum is independent of the arc length of a circular pendulum.
And it just flat falls. When I told my daughter, the telescope daughter, the daughter who loved Galileo. When I told him, well I said to initially, cuz I didn't know of the letter from 1602 that he was just lying about doing this experiment. She got very upset with me, and made me do the experiment.
So we actually reproduced the experiment, and she was very discouraged to see sure enough they go out of whack almost immediately depending on how large of an arch. So what I've actually given you I've just worked this out the formulas at the top. You wont see this formula in anything but a very advanced book.
Nobody in the 17th century or the first half of the 18th century could derive this formula it involves an elliptical integral. I don't recall whether it was LeGrange or Euler who first actually solved this problem, but whatever it is, it's an infinite series solution. But you can see the number of full cycles before a 20% discrepancy, if you're running 15 degrees, which is not very much.
It sure isn't a hundred or a thousand, but if you're running 30 degrees, which is the kind of arc he would naturally have done, it's very, very few cycles until they're 20% out of sync. And, now, the question is, you know, at one point I said he lied.
Once I saw the letter, it was actually Nico Bartolome merely finding the letter in the Dibner Institute and showing it to me and say, what do you want to say about this. I had to say he clearly did the experiment, the question is what in the world did he see?
And what Bartolome merely says in his book is, he probably just described the discrepancy to something like some secondary effect he chose to ignore. But that's not the description here. It's possible to set things up in synchrony, if you link them in the right way, they'll feed one another.
So, it could be just a badly designed experiment. We don't have an answer to how he could have thought it would synchronize. Circular arc pendulum were synchronous. The technical term here is isochronous. What path you reach the bottom in the same time, regardless of how far away from the bottom you start.
That's a peculiar concept. You start at one thousandth of an inch away, you start it way up many feet away. You arrive at the bottom in the same time. That became a question out of this. Particularly when Mersin said circular arc pendulums are not isochronis. But he said that and then Galileo repeats it in two new science's so we don't know what's going on.
But they're not a isochronous and it becomes a major question. But it is the question that's the principle question in the book I told you about earlier Huygens book of 1673 he solves this problem. It's a very big deal in the history of clock making. As you, I trust you can immediately see the virtues of it being isochronous.
If clock making because as the art slows down it doesn't effect the timing. And Galileo proposed it for clocks, they would not have been very good. All right with that said let me turn to a different question, now more one of getting at the radical character. His splitting vertical fall into two and motion.
And near the air into separate mechanism. That's really what he's doing. He's saying that natural motion and projectile motion really consists of two separate mechanisms. One mechanism is motion in the absence of air resistance, and the second mechanism is induced by that motion in the medium and takes the form of resistance.
Okay, and the natural question is when is it legitimate to say that something that happens in nature is being composed out of two completely separate mechanisms? And we should approach it as composed. And I give three answers here. One is that even to get at the secondary mechanism, you need a full theory of the primary mechanism.
And that's Newton's approach to trying to get at resistance forces. Is to have such a good account of the primary mechanism that you can then work out what the added effect of air resistance is. That's in the Principia as you'll see. Then the second is you can get results of evidential value.
Only if you remove the confounding effects of the secondary mechanism because they are confounding, they're messing things up. So give a theory and then introduce corrections for the secondary effect. I'll give you an example of that. Mercury in a thermometer, there is a cohesive effect, capillary effect on the tube.
You really want to design it to make corrections for that so it doesn't give you a misleading value. I can give you many examples of that. And the third one which seems to be Galileo's, is no theory of the second mechanism is possible at all. So trying to do the two together is going to prevent you from ever having a theory.
The key is to to separate the two mechanisms and get a totally adequate theory for the principle mechanism. Then take care of the secondary mechanism by various approximations, corrections, engineering devices, etc. And I've sort of already told you that's the state we're in a lot of engineering fields, where certainly that states the strength of materials.
I didn't tell you this, by the way, I should have told you just to whet your curiosity. Solid state physics measures the cohesive energy of a solid. It's an order of magnitude greater than the energy needed to fracture it. It's one of the deep mysteries of solid state deep research problems in solid state physics.
Why are real crystals ten times weaker than solid state physics says they are, the area is called Dislocation Theory. But it's essentially a search for an answer they have not succeeded in getting yet. One last thing, just for people's interest. Tom Kuhn's physics doctoral dissertation was on cohesive energies in solid state physics.
But that's neither here nor there. So, those are the reasons in this Galileo finally tells you in day four that his reason for getting rid of air resistance is the third one. Descartes's gonna say the same thing. He's gonna toss out air resistance as being beyond science, so they're not unique.