## Philosophy 167: Class 10 - Part 4 - Surface Gravity: Two Theory-Mediated Measurements, Via Conical and Cycloidal Pendulums.

Smith, George E. (George Edwin), 1938-

2014-11-04

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As I said, in October of 1659, he got compulsively preoccupied with measuring the strength of surface gravity. Starting out with Mersenne's method. Which was to take a 90 degree pendulum and try to time the dropping of a ball simultaneously with the swing of the pendulum. And he concluded, after messing with that for a week or so, it could not be made precise.

So, then he started asking himself how can I do something precisely. And he comes up with two methods of measurement. I'll show you the details later. The first is a constant height conical pendulum where he derives the Law of the Conical Pendulum I stated in our terms on the right.

Namely the period of one revolution is two pi times the square root of the height of the pendulum divided by our acceleration of gravity. He of course was working with the distance of fall in one second, which is half of our g, so his formula is on the left.

Now if you set up a device to do this and you can time it and it stays a nice, constant height, conical pendulum, I'll show you later how we did that. You can measure the distance of fall in one second by measuring the height of the conical pendulum and the period.

You then infer the acceleration of gravity, fair enough? This is an indirect theory-mediated measurement, it's one of the first really striking. Highly indirect theory mediated measurements. He got it, he measured, as you'll see in just a moment. Got a very, very good value. Decided he needed to cross check it and started looking at what would be a simple pendulum.

He realized the conical pendulum, as it declines in arc. Gets closer and closer to a simple small arc circular pendulum. So he goes back to this question of what's the isochronal path. Comes up with the solution I just described to you, the cycloid, discovers that what he needs in the way of cheeks that the string would be bend on is the same cycloid.

Derives the law of the cycloidal pendulum. And once again, he can measure the distance of fall in one second not by dropping anything but by measuring the length of the pendulum and the period of it. Okay? So he's got two different ways of measuring it, and needless to say, as I'll show you in just a moment, they agree.

Okay? So he's got two theory mediated measurements that revolve completely different, not completely, almost completely different theory and they produce the same results. So, here are his results as of 1659 you see, I'm just doing the history of them and showing you the modern value on the far right.

We had left off with Riccioli doing 15 Roman feet, which I've now concluded is 935 centimeters per second, per second. Huygens' first measurement is15.6 Greenwich feet. But he rounded off. Had he not rounded off, it would have been the same as his first measurement with the cycloidal pendulum.

15 Roman feet, seven and a half inches. Once he had that he further realized that in a cycloidal pendulum if the arc is small enough, you're approximating a circular pendulum so closely that you don't really have to set up the cycloid to measure it. You can just do a very small arc circular pendulum, keep the arc low, and measure gravity that way.

The idea became, I'll lay it out in just a moment, the idea became all right, we're gonna keep changing the length of a pendulum until it beats one second. I'll show you how we would do that. Then that length for the one second pendulum will let us infer what the distance the fall is in the first second, or hour of acceleration of gravity.

So that became a standard measurement. That is, not only he did it people all over the place started doing it because he gave them a way of doing it that wasn't that complicated. And that number became, that's a Paris number. The standard Paris number, the length of that is three feet and eight.

Have to do this right. Three feet, eight lines. A line is a 12th of an inch. And one half line. So it's 440.5 12ths of an inch is the length of a one second pendulum. It's a standard number all through this period for Paris. And if you look, that's 980.7, 980.9.

Our current value is 980 0.97 and that happens to be for Paris with an altitude adjustment that he didn't make. So in effect, he's measured surface gravity to four significant figures. Within one digit or two digit in the fourth significant figure in 1659, two different ways. Okay, and established that measure for the public so that now people can go around and compare it elsewhere.

Fair enough? That's a breakthrough. I'll have to give you all the theory behind this, but that's all done between October 1659, and Christmas 1659. And, of course, he starts telling people, when he visits he shows them how to do it, etc., ways you will see. What ensues is people start doing this measurement all over the place.

And the really interesting one is John Rochet in Cayenne It's done in roughly 1670. He was in Cayenne for about a four year period. You'll learn about that in the next class. He discovered that near the equator the seconds pendulum has to be made a little bit shorter.

Than in Paris. How much shorter? One and a quarter lines. How much is that? 2.8 millimeters. Now 2.8 millimeters is about the distance between my fingers. In a three foot length. That's how sensitive the measurement was. You could determine the gravity at the equator is definitely less than gravity at the at Paris.

And after that, people start measuring it all over the place, usually not very carefully. The problem in this measurement, if the arc gets too large, you don't get a good number, as I was about to show you there. But this variation of surface gravity becomes the most important single piece of evidence for Newtonian universal gravity.

Okay, so it's a very big deal. Very big deal to Huygens' too. Okay, because he decided the reason the gravity is less at the equator is the Earth is rotating. In fact at one point he says I'm going to prove the Earth is rotating. Because I'll show you that the amount of gravity is less in the equator exactly corresponds to the centrifugal force from the Earth's rotation.