Newton's theory of gravity

[Doctordick]

Some thirty years ago, soon after getting my Ph.D., I received a letter from a niece, my sister's youngest daughter. She was in high school and had been assigned (or chose) a science project consisting of presenting a talk on Newton's theory of gravity. Since I was the most educated person in the family and in physics to boot, she wrote me a letter requesting help in preparing that talk. So I wrote her back what I thought was a high school level presentation of Newton's theory.

When I saw the mess created when my mother-in-law's house burned, I was moved to clean out our attic. They had a lot of useless stuff in the attic which just created a burning pile of junk the fire department almost couldn't put out; in fact, two days after the firemen left we still found smoldering stuff in the bottoms of some piles of absolutely unidentifiable stuff.

At any rate, cleaning out my attic, I ran across my original work sheets for that letter to my niece. What follows here is essentially what I wrote to her.

The whole thing starts with Galilean relativity. Galilean relativity is essentially the idea that the rules of physics are the same in all frames of reference (Euclidean frames are all they had back then). Galileo really didn't worry much about “accelerating frames” so he was essentially talking about what Newton later called “inertial frames”: frames moving at a constant speed. Galileo pointed out that a rock dropped from the crow's nest on a ship hit the same place on the deck when the ship was moving as it hit when the ship was at rest. Essentially the fact that the ship was moving was completely hidden by the fact that the rock (when it started) was moving at exactly the same speed and in exactly the same direction as the ship.

Galileo also provided another deduction of great significance: everything fell at the same speed. People always say that Galileo dropped two objects from the leaning tower, but I have heard that he didn't; he simply used the idea of dropping two rocks from the leaning tower as a thought experiment and no one was able to show him wrong. From my understanding (which of course could be wrong) Galileo proposed the following thought experiment.

At the time, the authorities claimed that heavier objects fell faster. So suppose we drop two rocks from the top of the leaning tower at exactly the same time (one heavy and one light). The heavy one should hit the ground first. So now tie the two together with a string (loose enough so they don't actually touch one another) and drop them again. Since the heavy one falls faster, it must drag the light one by the string; right? Makes sense doesn't it? Ah, but it brings up another problem; the two rocks together weigh more so the combination (when they are tied together) must fall faster than the heavy rock when it is not tied to the light rock. So the existence of the light rock must cause the heavy rock to fall faster. So exactly how does it do this? It “pushes” the heavy rock through the string?

The only rational conclusion is that they fall at the same speed. Putting this together with his principle of relativity (for some reason no one ever refers to Galileo's theory of relativity; there is another paper right there), Galileo came up with the idea that the acceleration of gravity was constant (though I am not sure he referred to it as “gravity”). I don't read Italian and have never read what he actually said.

So now we come to Newton and his take on the principle of relativity and the constant acceleration of gravity. Essentially, what the falling apple pointed out to him was the idea that “the moon might be falling towards the earth” (an idea everyone else considered to be utterly ridiculous). But let us look at the idea carefully. When we fire a gun, the bullet (after it leaves the barrel) simply falls. This fact can be used to figure out exactly how far a bullet will go if the gun is perfectly level and there was no air to slow the bullet down. Just as with Galileo's rock dropped from the crows nest, it will hit exactly the same spot which would be hit by a bullet dropped from a platform moving horizontally at exactly the same speed as the bullet.

So let us put a little math to that. Let us start with a gun a distance “d” above a flat plane. We have an imaginary ship moving at exactly the same velocity and direction as the bullet. Let our imaginary ship consist of two important parts: a “crows nest” who's edge is exactly level with the end of the barrel and a flat deck exactly a distance “d” below that point. From that platform we drop a rock from a point on the crow's nest exactly at the end of the barrel at the very moment the gun is fired. What do we know? We know that both the bullet and the rock will, at exactly the same time, hit exactly the same spot on the deck of our imaginary ship. All we really need to know is, “how long does that take?” That together with the muzzle velocity of the gun and we have the fact that x=vt as the distance the bullet has gone.

Well, to figure that out you need calculus which you either know or you don't; therefore I will just give you the answer (it is quite easy if you know calculus). The distance the bullet drops as a function of its distance from the end of the barrel is given by . If we know how far the bullet has to drop to hit the ground, we know exactly how far it will go.

Now, how about the fact that the earth is a sphere? As you go straight out in the direction the gun is pointed, the distance to the earth is further and further away. Suppose the muzzle velocity of the gun is high enough so that the distance the earth drops is exactly equal to the distance the bullet drops. Well, for that you need trigonometry so I will again just give you the answer: by similar triangles it is quite easy to show that x divided by the radius of the earth is identical to the distance the earth drops divided by half of x.

which implies

So, if the distance the earth drops (or rather a perfect sphere with no atmosphere) when the bullet moves a distance x is exactly the same as the distance the bullet drops, the height of the bullet above the sphere will remain exactly the same. (Don't worry about the acceleration; the force is perpendicular to the motion so the velocity is unchanged.) Putting the two equations above together, , we can easily solve for that specific muzzle velocity. The critical velocity is exactly . Since the radius of the earth is very close to 6.578x10⁶ meters, if the muzzle velocity is exactly equal to 7,906 meters per second, you will shoot yourself in the back of the head. Or another way to view it, if you get out of the way, the bullet will be in orbit and it will stay at that altitude forever. A better measure of the critical velocity is the orbital period so we should look at the case where the circumference (or x) is exactly : i.e., or which is totally equivalent to .

In other words, if the velocity is exactly equal to the velocity which will yield no change in the measured altitude above the surface, the period of the orbit will be given by .

So it is entirely possible that the moon is just falling and the fact that its velocity is perpendicular to the direction to the earths center is just sufficient to yield a constant elevation is a quite possible circumstance. That would imply that the period of the moon (the length of time it takes to complete an orbit around the earth) given that it is at a radius of approximately 3.8x10⁸ meters, would be roughly 10 hours, 55 minutes and 30 seconds. But that is the wrong answers! The correct answer is 27.3 days. So the idea that the moon is just falling towards the earth makes no sense at all.

But wait a moment, if you check it out, the answer is almost exactly one sixtieths of the correct answer! Put this together with the fact that the moon is 60 times as far away from the center of the earth as we are suggests the possibility that T is proportional to r divided by the radious of the earth or “g” (the acceleration of gravity at the moons orbit) is [imath]\left(\frac{6.578^6}{r}\right)^2 times that at the surface of the earth! Gee, that would yield exactly the correct period for the moon's orbit. That makes a lot of sense. It suggests that if we set the value of g to we will get 9.8 meters per second per second at the earths surface and the correct period for the moon. Gee, that is pretty impressive.

So, let's check it. If that is the case, what would we expect the period of the earth around the sun to be? The distance to the sun is approximately 1.5x10¹¹ meters. Putting in the expected value of g we get the period (the length of time it takes the earth to go once around the sun) of about 570 years. That is certainly the wrong answer. So let's dump this whole idea.

But wait a moment. Suppose that the answer is again an incorrect value for g! The value of g is apparently too small by a facto of 3.25x10⁴. Suppose that g is a function of the mass of the originating body. The assumption that the mass of the sun is 3.25x10⁴ times the mass of the earth is not at all an unreasonable assumption. That would make the density of the sun roughly equal to the density of water and the fact that all evidence is that it is made of mostly hydrogen suggests that, if the gravity is considerably greater than the earth, that hydrogen might be sufficiently compressed to be roughly at the same density as water.

Best of all, setting the acceleration of gravity to would yield the correct value of g at the earths surface, the correct period of for the moon and, in addition, the correct length of the year. That is one hell of a lot of support for the idea that they are all just falling toward one another. Furthermore, examination of the implied orbit periods of the rest of the planets in the solar system yield very little reason to disbelieve the theory. In fact, since then most everyone uses that very equation to determine the mass of the various planets.

And that, my dear, is the essence of Newton's theory of gravity. Actually it is pretty obvious if you think about it.

Have fun -- Uncle Dick

[Turtle]

Originally Posted by Doctordick

...
Galileo also provided another deduction of great significance: everything fell at the same speed. People always say that Galileo dropped two objects from the leaning tower, but I have heard that he didn't; he simply used the idea of dropping two rocks from the leaning tower as a thought experiment and no one was able to show him wrong. From my understanding (which of course could be wrong) ...

Have fun -- Uncle Dick

Yes, your understanding is wrong. In fact, Galileo devised an inclined plane down which he rolled his bronze balls to conduct the necessary experiments. Seems he lacked a sufficiently accurate timepiece if he simply dropped his balls.

Here's an in depth description of the experiments and reasoning of Galileo.

Galileo and the Mathematics of Motion: Part I

...Galileo wanted to study gravity -- and how it affected acceleration -- in great detail, but falling objects accelerated too quickly, and the time was too short, to make accurate observations.

Was there any way he could try to slow down the effect of gravity -- to observe the rate of acceleration in slow motion? This is exactly what the inclined plane allowed Galileo to do.
...

PS Galileo's own description: >> http://books.google.com/books?id=cza...esult&resnum=5

[arkain101]

That is a very rich and interesting explanation of Newtons Gravity. The interconnectedness of ideas makes it much more exciting to read than some stuff you get out of a textbook.

I'm sure your niece had a bit of her mind blown when reading that. for the first time.

How did she do on her project?

[Doctordick]

Hi arkain, sorry I missed your post here.

Originally Posted by arkain101

That is a very rich and interesting explanation of Newtons Gravity. The interconnectedness of ideas makes it much more exciting to read than some stuff you get out of a textbook.

I'm sure your niece had a bit of her mind blown when reading that. for the first time.

How did she do on her project?

I have no idea. Though she was the only one of my sister's five children that went to college. Perhaps she thought about things a bit more than her siblings. She became a Certified Professional Accountant (maybe a little interest in math ???; arithmetic anyway ).

Have fun -- Dick

[Doctordick]

Hi Turtle, I had also missed this post.

Originally Posted by Turtle

Yes, your understanding is wrong. In fact, Galileo devised an inclined plane down which he rolled his bronze balls to conduct the necessary experiments. Seems he lacked a sufficiently accurate timepiece if he simply dropped his balls.

I was aware of his inclined plane work but I was talking about an issue one of my ancient professors told me: i.e., that Galileo was the originator of that thought experiment. That is a slightly different issue (which may or may not be true; I have personally found no conformation of it).

Have fun -- Dick

[Erasmus00]

Also, while the above is a reasonable assertion of evidence that could lead to an inverse square law, the evidence that Newton and Hook actually followed to get inverse square was that an inverse square law produces conic section orbits (like Kepler's ellipses).

[Doctordick]

Originally Posted by Erasmus00

Newton and Hook actually followed to get inverse square was that an inverse square law produces conic section orbits (like Kepler's ellipses).

Could be but I wasn't there and that provides no reason for the apple thing. I won't bother responding again as you and I have very different views of the universe and I think talking to you is a waste of time.

Anyone who believes more than ten percent of what they hear or fifty percent of what they read or ninety percent of what they see with their own eyes is gullible. I have chosen to believe nothing; I only see what could be.

Have fun -- Dick

[modest]

Originally Posted by Erasmus00

Also, while the above is a reasonable assertion of evidence that could lead to an inverse square law, the evidence that Newton and Hook actually followed to get inverse square was that an inverse square law produces conic section orbits (like Kepler's ellipses).

Apparently there was some question as to whether Newton or some fella named Christiaan Huygens first found the inverse square law. To make his case, Newton showed his early work preceding the well-known proofs in the Principia by some 20 years where he found the centripetal force and apparently used Kepler's 3rd law from that. I get from:

The birth of a new physics - Google Books

My question, which I know betrays the novice I am, how would you combine Kepler's third and v^2/r to get 1/r^2?

~modest

EDIT: Sorry I got it. I had Kepler's velocity relationship wrong

(from Kepler's third)

[AnssiH]

Originally Posted by Doctordick

And that, my dear, is the essence of Newton's theory of gravity. Actually it is pretty obvious if you think about it.

Haha, and what is really interesting, when you think about the logical steps that led to the theory, how far from obvious it was when thought through the previous paradigm. I.e. people stuck in their intuitive ideas are just not capable of thinking these sorts of things through.

It is fascinating, that when it become apparent that our law of gravity does not explain the velocity of the outer stars in large galaxies (they are moving much faster than the newtonian law predicts ), people were quite reluctant to consider the possibility that our law of gravity is wrong. Much easier to keep the old paradigm and add invisible matter to the soup, in just the places where it is needed to explain the observations. To me that tastes a lot like "adding epicycles" (or postulating figure of eight orbits) to an old model until it explains everything we see.

And the history of the world is just chocked full of fascinating examples like this. Here's an interesting quote from "Saint Augustine", and his argument about why the "bottom of the earth" cannot be inhabited by people;


But as to the fable that there are Antipodes, that is to say, men on the opposite side of the earth, where the sun rises when it sets to us, men who walk with their feet opposite ours, that is on no ground credible. And, indeed, it is not affirmed that this has been learned by historical knowledge, but by scientific conjecture, on the ground that the earth is suspended within the concavity of the sky, and that it has as much room on the one side of it as on the other: hence they say that the part which is beneath must also be inhabited. But they do not remark that, although it be supposed or scientifically demonstrated that the world is of a round and spherical form, yet it does not follow that the other side of the earth is bare of water; nor even, though it be bare, does it immediately follow that it is peopled.

Notice how many concepts from the old paradigm are just mixed in his train of thought, even after he already (apparently) believes earth is in fact spherical.


It is too absurd to say, that some men might have taken ship and traversed the whole wide ocean, and crossed from this side of the world to the other, and that thus even the inhabitants of that distant region are descended from that one first man.


I.e. the bottom half cannot be inhabited by people because all people originated from one place on earth (from Adam, of course), and it has not been possible for anyone to traverse to the bottom half over the ocean that lies in between.

Don't make the mistake that those people were just dumber than people today. It is not a trivial thing to analytically investigate a different paradigm. If you don't recognize the undefendable nature of your beliefs, the different paradigms do sound quite ridiculous. "But if you inquire from those who defend these marvellous fictions, why all things do not fall into that lower part of the heaven, they reply that such is the nature of things, that heavy bodies are borne to the middle, and that they are all joined together towards the middle, as we see spokes in a wheel; but that the bodies which are light, as mist, smoke, and fire, are borne away from the middle, so as to seek the heaven. I am at a loss what to say respecting those who, when they have once erred, consistently persevere in their folly, and defend one vain thing by another."

Even today people argue entirely from within their pre-existing paradigms, and they do not recognize how their argument hinges on undefendable beliefs on very many aspects about reality. The arguments may be more elaborate than those old examples, but if really start analyzing the situation, it is exactly the same.

Furthermore, examination of the implied orbit periods of the rest of the planets in the solar system yield very little reason to disbelieve the theory. In fact, since then most everyone uses that very equation to determine the mass of the various planets.

Yes it is also fascinating that people tend to forget that it was a specific model that allowed us to determine a variable (the mass of a planet), and when they forget that, they start using that determined variable as "experimentally measured and verified fact" as a defense towards the validity of the very model that allowed them to give a number to the variable in the first place. It still is, ladies and gentleman, a pre-existing theory of an atom that allowed us to "probe the atom", i.e. to interpret the results of our experiments.

Of course you can say that there exists dark matter "that does not interact with anything but gravity" (so to speak) and thus "the motion of the stars is the proof of its existence", much like some people readily claim M&M experiment is the proof of relativistic spacetime, and that the redshift of light from faraway galaxies is the proof that "the fabric of space is expanding" (Put this together with "spacetime", and "thus reality began from a singularity" etc.)

It is fascinating that people take the explanations to observations as being proven "ontologically true" by the observations themselves (in the sense that it is "the" way things are), and even though we have done that throughout the history of the world and been mistaken over and over, that belief stays just as firmly in the minds of people today.

I would say it is much more objective to take the theories as an explanation to a given observation, and so explicitly keep in mind that there always exists incredibly many ways to explain anything.

Furthermore it is fascinating to me, that people tend to still use "Occam's razor" to defend their pet explanation, because in their mind it is somehow simpler than another. I have no idea how they became convinced that ontological correctness is measured by the simplicity of a model, and furthermore I have no idea how to measure simplicity. Many people are of the opinion that dark matter is simpler explanation, while other people say modification to gravity is clearly much simpler. That is of course a matter of your perspective. And certainly both can be formulated into equally valid form (predictionwise).

So now tie the two together with a string (loose enough so they don't actually touch one another) and drop them again. Since the heavy one falls faster, it must drag the light one by the string; right? Makes sense doesn't it? Ah, but it brings up another problem; the two rocks together weigh more so the combination (when they are tied together) must fall faster than the heavy rock when it is not tied to the light rock. So the existence of the light rock must cause the heavy rock to fall faster. So exactly how does it do this? It “pushes” the heavy rock through the string?

Heh, that's pretty clever, and it reminds me of another unrelated debate I once had (regarding the philosophy of the mind), which displays nicely the thoughtlessness of people stuck inside their paradigm. This guy had convinced himself that subjective experience is something that everything has got to a degree (panpsychism). It's just the more complex systems that "understand more about nature" (such as our brain), but a rock also has got an experience about the hill it is rolling down, it's just much simpler experience.

So he had a problem with my assertion that subjective experience stands on a model about reality, i.e. a reality needs to be expressed in some form by the configuration of a system (the rock does not hold a definition of "a hill", the defined "objects" can be freely chosen, and the chosen references are quite immaterial), and furthermore one specific definition about subjective experience is that while there are many parts to a system, there is still just a single subjective experience over the "collective effort" of those many parts (which, if you think about it, leads to the so-called "hard problem of consciousness", which upon closer analysis arises from the undefendable assumption that reality is made of "objects" with identity).

So, I led the debate towards the inconsistency of his view, which is upon simple analysis that a rock has got a subjective experience, as does another rock next to it. A lump of clay has got a subjective experience too, and if you halve it, there's 2 different subjective experiences. A pile of sand has got an experience, as has got a pile of rocks (just an order of magnitude more complex subjective experience than the single rock). Tie two piles of rocks together with a rope; voila twice as complex subjective experience arises.

From that point it's easy to move the argument to the fact that in his definition subjective experience occurs to any randomly chosen set of elements and also over the entire universe, and ends up losing all its meaning. The air molecules in a box have got a subjective experience, as does the two halves of the box independently.

At this point we have entered the fact that our definitions of objects are, for all we know, semi-persistent patterns. Any semi-persistent pattern has got subjective experience ("yes" he said). A tornado consists of air molecules; also has got a subjective experience. When you take out the plug from a pool of water and there's that little vortex. Has got subjective experience, of course.

To this day he is of the opinion that the little vortex of water is subjectively aware of its surroundings.

Note that I am not talking about a belief regarding whether that is so or not. I am talking about how his definition of subjective experience lost its meaning with the requirement that it is applied self-coherently on his worldview.

-Anssi

[ldsoftwaresteve]

Doubt is a demon with a white plume.