Philosophy 167: Class 10 - Part 11 - Converging Theory-Mediated Measurements of Surface Gravity: Using Cycloidal and Conical Pendulums to Confirm Different Principles of Motion.
Smith, George E. (George Edwin), 1938-
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All right. I said earlier that Huygens had measured, in 1659, measured surface gravity with a constant height conical pendulum. We don't know how he managed to do that. But Joella Yoder found in his notebooks the drawing of the following clock. And so the assumption is this is what he did.
He took a clock, took the device for a clock, and built a constant height conical pendulum. Now, I hope you appreciate the problem. I've got a conical pendulum. What's resistance gonna do to it? It's gonna slow it down. It does not affect the period. It just slows it down.
When it slows it down, it falls a little, because gravity is now higher than the centrifugal force. Right? So, it's gonna tend to come down. I want it to stay a constant height. How do I make a constant height? I have to shorten the cord. This is classic, it has a name in engineering.
It's called a servo mechanism. He's got a chain here hanging with enough weight that, as this slows up, the tension in the string drops, and as a result, this weight pulls it up to keep it at constant height. And he can figure out how much weight he needs to do that from that statics problem I showed you before.
Okay. So if you think about it, this is just a beautiful device to maintain a constant heights pendulum. And now, you can do it, you can find out what the length is, you know, gut the whole clock, you can see how this clock's running versus another one versus the stars.
So that's his first measurement. That's in 1659. And it's Joella's guess, and I take Joella's word quite seriously here. It's Joella's guess that this is the device by which he got the numbers. Okay, in the Horologium Oscillatorium he does something much neater. This is his diagram, but I've already told you if you can keep an object on a paraboloid, then you're home free, because it's gonna have the same period of revolution regardless of where it is on the paraboloid.
So he comes along and puts just the right cheek, I'll explain how he calculates just the right cheek, to keep this bob always on the surface of a paraboloid. And built a clock like that, and describes it in Horologium Oscillatorium, telling you he got the same distance of fall in the first second, the fifteen Paris feet, one point one inches.
Now you may think a clock like this can't work, but in 1995, when I was at the 300th anniversary of Huygens' death, they had operating in the Boerhaave Museum a replica of this clock. About this big, the little ball spun fast as hell, and it was clearly keeping beautiful time, and when I was back there, three or four years later I tried to buy that clock, and they couldn't find it anywhere.
I spent the better part with two people in a warehouse trying to find that clock, and we couldn't find it anywhere. I was ready to spend $1,000 for that clock at drop of a hat. But it's just a beautiful little device, a conical pendulum clock. And it's totally self-regulating, thanks to this cheek.
A really beautiful piece of work. So we've got two different conical pendulum measurements. One in 1659 announced, and one in 1673 in Horilogium itself. This diagram, this is what's in Horilogium. But that's the one in his notebooks when he was figuring out how to do this. I'll come back to that cheek in a little bit.
Independently of that, we have the measurement I told you about before, from the cycloidal pendulum. It's from four Galilean principles, we'll get to them in a moment, using the height as a proxy for V square. You deduce cycloid is the isochronal path, cycloidal cheeks make the pendulum path the same cycloid, the law of the cycloidal pendulum I gave you before.
The same law holds for small arc circular pendulum. And where the bound is about four point five degrees. And if you do that, you get exactly the same measurement, 15 Paris feet, one point one inches for the fall of gravity with two different ways. Okay, those who had Philosophy of Science with me, and Cory was reminding me, I've pointed out that J.J. Thomson measured the mass to charge ratio of the electron two different ways.
And I pointed out there, Huygens had done exactly that in 1659, two complementary measurements. But I want to show you what that means, and this is gonna take a moment. I'm gonna do each one separately, the one on the left is the cycloidal pendulum. If the Galilean principles of pathwise independence return to height in no effective weight.
If those three principles hold, then gravity is uniformly accelerated, if and only if the cycloidal and small arc circular pendulum measures of distance to fall in the first second are stable. Now I have to tell you what I mean by stable, it's my technical term. I can do a pendulum that's this long.
I can do a pendulum that's this long. I can do a pendulum that's very, very long. I should get exactly the same measure of distance of fall in the first second as I vary it. That's what it is to be stable. I'm in effect showing that the constant of proportionality in the law is a constant by varying everything else, varying circumstances, doing whatever I do to vary, and have it constantly come up the same amount.
If it comes up the same amount in all of those, then I've got confirmation that it's a uniform gravity. It has uniform acceleration. Now we know, if I varied it enough, if I got the string five miles long the inverse square effect would start happening, and etc. But that's the point.
The logic here is this is a test of uniform gravity. It's being stable. And those who studied the JJ Thompson with me, you change the electrode materials, you change the gas, you keep getting this same value. Over here we have different principles. They're not the Galilean principles whatsoever.
Forces proportional to weight times the distance of departure from uniform straight line motion over time squared. That's essentially Newton's second law of motion. Okay? And if that holds, then I get gravity is uniform acceleration if and only if the conical and paraboloidal pendulum measurements fall in the first second are stable.
I can use big conical pendulums. I can use small conical pendulums. I can use the constant height conical pendulum. I can use the paraboloidal. I should keep getting the same number all the time. Okay? Now look when you put the two together. Okay they're both giving you the same value.
That's what it means to be converted. They both give you stable values. They give you the same value. That's evidence for the stuff in blue. Okay in particular, if you think these things on the left are reasonably well established, you've got very strong evidence for the assumption on the right.
In fact, I'll now make this statement. The strongest evidence for Newton's first two laws of motion at the time they were published were these measurements. The fact that the conical pendulum measurement, which is strongly dependent on the first two laws of motion gives you exactly the same measure as the cycloidal and small arc pendulums, which are, at most, weakly dependent on the first two laws of motion, cuz they're kinematic, okay?
That's a new form of evidence. Complementary measures of the same fundamental quantity, both theory mediated, but using different theories can be used to support the theories. The famous example now, I've already mentioned it a couple weeks ago, I think it was a couple weeks ago when we were talking about Mersenne and Richie Oley.
There are seven or eight measurements of the fine structure constant being done in QED today. One of them is very strongly dependent on QED. It gives a result to nine significant figures for the constant. Another is weakly dependent on QED, the theory. It gives the same nine significant figures.
And the conclusion, I'm now almost quoting the published conclusion, this is remarkably strong evidence for quantum electrodynamics. Okay? It's a form of evidence, being able to use the theory to measure quantities, but now do complementary measurements, using different parts of the theory. And weakly, parts of the theory that are weak, parts of the theory that are strong.
Huygens never seemed to appreciate this slide. Let me assure you, Newton appreciated it totally. Huygens' view seemed to be uniform acceleration, and these other principles were already established. Why worry about it? Let's just go out and do the measurement and cross check. That the good thing about the cross check is you eliminate sources of systematic error if they agree.
Cuz the systematic errors and conical pendulums and cycloidal pendulums are very different. I should explain that, by the way. I didn't, and it's not in my notes. Error resistance on a pendulum. The effect tends to cancel. As you come down a pendulum the air resistance is gonna delay the point where it reaches maximum velocity.
As you go up, air resistance is gonna hasten the point where you reach zero. The two effects cancel out. Air resistance is a third order, not a second order, effect on pendulums. On a conical pendulum, particularly if you have it with the paraboloid, air resistance is irrelevant, has no effect on anything.
Cuz it's perpendicular to the measurements. Okay, the equilibrium has to be between the other two forces. The fact that there's a third force coming this way is totally irrelevant. So this is a case where we're getting around the Galilean problem of handling air resistance. And they new this, they knew perfectly well that the air resistance affects tend to cancel out in the simple pendulum, and they have no effect on a conical pendulum.
Huygens in particular knew. The note the last paragraph of the reading for today on part five announces it, among other places. All right. Do people see this? This is a whole new form of evidence. It's from Newton forward, it's been very dominant, doing very precise measurements.