Philosophy 167: Class 6 - Part 12 - Galileo's Key Postulate- Relating Vertical Fall to Fall Along Inclines.
Smith, George E. (George Edwin), 1938-
Now he wants to test it and he's got the problem, the problem I just demonstrated by dropping a coin. It's too damn fast. So he gets the clever idea of doing an incline plane, but if he's going to do an incline plane and he's reaching a conclusion about vertical fall, what's the basis for drawing that conclusion?
And what he does is put forward then an assertion, calling it the postulate of Galileo is still. It's not called a postulate in the text. Stillman adds nice big bold letters above it. The same speed is acquired for many given height, whether in direct fall or along an incline plain.
It is remove air resistance, drop an object from a height then an object down an incline plain, when they arrive at the bottom they will have exactly the same speed which also means that each juncture along the way at the same height, they will have the same speed.
What's so nice about inclined planes, you can make the time last a long long time by having a shallow incline plane. Okay so that's the element of genius here. We're gonna do it on an incline plane, but to justify it we need that claim. How do you justify that claim?
For those who don't know, this gets generalized by Huygens into any path whatsoever, not an incline plane. So I tend to refer to is as the pathwise independence principle. It is one of the most fundamental principles of all of mechanics. It ends up being the definition of a conservative field.
One that for which this is true but that's beside the point. This is an extraordinary proposal by Galileo and it's not clear how you test it, but I'll talk about that in a moment. In the first edition, he proposed this is correct, but with the following sort of argument.
Take a pendulum, it's called an intercepted pendulum. You will see this again. Drop a pendulum from c, and intercept it at different points along the length. It always returns to the same height no matter where you intercept it. And if the pendulum is heavy enough, that's a pretty good result.
Heavy enough for air resistance effects to be minimal. Trouble is how's that related to fall case. Now the answer's gonna be because whatever momentum you get in fall, you get just enough, that exact momentum to get you back to that same height. So if it's getting back to the same height regardless of where you intercept, it's saying that therefore regardless of what happens at the interception point, you're gonna get the same speed, and the same speed's gonna go back to the same height.
Now reverse the argument and have it descend rather than ascend. It's not a terribly good argument. In the posthumous addition Vivani added this argument which doesn't involve motion at all. What it tells you is, look at what weight h is needed to balance the weight g as a function of the angle ca and fed theta, etc.
And it's in Steven, it's a well known result in Steven, and it is indeed the proportionality here goes as the sine of the angle. So the weight needed here, very, very large weight can be held around a pulley by a small weight at the appropriate angle. And now you're gonna be having to transfer this to the case of what's happening with gravity when it's stationary, is you've added a certain number of momenta there, but because it's stationary, they don't express themselves til you let it fall.
And once you let it fall, the same momenta show up. And that's evidence for the thing at the top. In fact, in the second addition there's actually a proof, of sorts, offered for this. It's not Galileos' proof. It became a pretty celebrated principal. It's crucial to Galileo. Galileo had two major proteges in his lifetime, Viviani and Torricelli.
Torricelli, we will talk about a good deal next week and continuing. Torricelli's a very important person. Torricelli's the inventor of the barometer, for example. But he did other things. This is a translation by Judy Nelson and me of a portion. You can see what it is on natural gravitational motion in descent and projectile motion.
It's a beautiful book following up Galileo's two new sciences, and completing, it follows up third and fourth day. It completes shortcomings in the Galileo Lacuna, things missing it completes. It's 1644, Galileo died two years before it. But I'm not gonna read the whole thing. I'm gonna read the opening paragraph just to drive home how important this principle is.
Galileo, when about to discuss naturally accelerated motion, puts forward a principle that he himself thinks not yet clear. As long as he strives to establish it by the not fully precise experiment of the pendulum, which is that the stages of velocity of the same moving object when a massed over differently incline planes are equal when the elevations of the same planes are equal.
From this claim, hangs as it were, his whole doctrine of both accelerated and projectile motion. If anyone has doubts about the principle, he will not have, at all, secure knowledge of the things that follow from it. I know that Galileo, in his last years of his life, tried to demonstrate that supposition.
But because his own argument with his book on motion had not been published, that's the second edition. We have brought forth these few statements on the movements of weights to be fixed at the beginning of our little book so that it may appear that Galileo's supposition can be demonstrated.
And in deed at once, by that theorem that he himself selected has demonstrated for mechanics in this second part of his sixth proposition on accelerated motion. To wit, the momenta of equal weights over plains, unequally inclined are to each other as the perpendiculars of equal parts of the same plains.
This is mean. That is this is the sines of the angles and inclination of the plates. And here's what he does. He puts forward a principle, that's now known as Torricelli's Principle, that two weights joined together can not move of themselves, that is fall, unless their common center of gravity descends.
Okay? That's the claim. It's called Toricelli's Principle, and what he does I'm not showing you. I'm not gonna read it out loud, I put it in small print. But what he does is to show Galileo's path wise independence principle. If it's false, contradicts Torricelli's principle, so he does a reductio to, in effect, establishing it's true by assuming it's false, showing it violates the Torricelli principle.
Huygens gives a proof at the early on with almost the same introduction, Galileo supposed this really important principle, I'm gonna give better proof. And gives us a somewhat superior proof to this using a variant on Torricelli's Principle, but you need some sort of principle to do this. And by the way, for those who are worried about the physics of this, this is an energy principle.
And path-wise independence is an energy principle. It's in effect saying the energy acquired is the same regardless of the path. But that's of course wildly anachronistic. The term energy as we use it is from the 1840s. Long way away.