Motor Daddy, this side discussion has nothing to do with the thread topic. If it is really important to you start another thread and see if anyone joins you. I am done!
It was a question of veracity. The good professor is quoted making a statement that a fraudster would have made. I don't see how. All you need is classical mechanics to solve classical mechanics. I think you missed my point. Nothing is gained by this exercise. Newton's Law of Universal Gravitation gives the same result. You don't need the Hamiltonian to get r = x-X. All you need is a coordinate system. It's vector mechanics in it's simplest incarnation. There I'm going to have to disagree. Although you are now illustrating the point that the skilled posters are scoffing at: one of the common mistakes of inexperienced posters is that they fail to recognize that the first step in a problem in dynamics is to establish the inertial reference system. OnlyMe pointed this out a while back, as did several other folks. I payed out the rope to see if any of the unskilled skeptics could deduce this for themselves. That won't be enough. You have to get agreement on the model. That includes three important decisions [1] what is the inertial reference frame [2] given[1], is this a 1- or 2-body problem and [3] what are the requirements and/or limits of accuracy for this problem. I was hoping you would tell us that. I'm trying to be helpful. One can lead a horse to water, but not make it drink. Obviously there are two bodies. The question is how many free bodies there are. And that is based on the outcome of decision [1]. That's debatable, but it belongs to decision [3] if at all. When I say "debatable" I mean it's a debate between skilled and unskilled posters. The skilled folks will either all agree, or else they will need a few exchanges to sort out any assumptions, and/or to refresh any rusty skills. The unskilled folks will tend to gnash and rend garments, particularly the religious ones who are trolling on the cool. Once we decide [1], [2] and [3] the answer you're seeking should be a walk in the park -- for the posters with the basic skills.
I feel these are important points to agree on. 1) The inertial reference frame is the frame before the object is dropped above the Earth. The center of gravity of the combined object+Earth system is our reference point, and remains so throughout. Galileo made the forgivable mistake of assuming that he stayed in this frame when he dropped the object, but this is of course not true because the Earth began falling towards the object by obeying Newton's Third Law. 2) This is obviously a 2-body problem. 3) I think we can all agree that this has no practical use for dropping small objects to the Earth. This is a theoretical exercise, and the point of the OP is to show that, TECHNICALLY, Galileo was mistaken. He could also have made the claim that light travels with an infinite speed and would have been forgiven because, for all practical purposes (particularly at that point in history), he would have been correct. This thread is clearly being pedantic and technical. If anyone is offended because we're talking about absurdly minuscule differences in drop times then you don't understand the point of the exercise.
I can agree with that. To add to that, if you are offended that we are talking about absurdly miniscule differences, then you don't appreciate how small we are in this infinite volume of space we live in.
I made two points. The first point is that what RJBeery is talking about doesn't actually contradict anything Gallileo actually said because, in effect, RJBeery has shifted the goal-posts, he's changed the thought experiment from that which Gallileo {as I understand it} originally stated to something else. The second point I made is that even if RJBeery is right the result is physically meaningless for anything he has proposed that's even remotely relevant to the original thought experiment {talk of dropping black holes on earth, for example}. It's physically meaningless because of issues around accuracy and exactness, and it's physically meaningless because the motion of the earth is less than the size of the subatomic particles that make the earth up.
One thing I want to make clear, because it seems to be confusing people: Hamiltonian dynamics are classical physics. They are not relativity, and they are not quantum mechanics [although the Hamiltonian does play a similar, important role in quantum]. Hamiltonian dynamics are just a formalism to derive a system's classical equations of motion from its kinetic and potential energies, which is what I did. It sounds like you think that because the ball and the rest of the Earth are parts of the same larger object, they should be treated differently from two objects that are "really" separate from each other. The problem is, there is no room in physics for subjective judgments like what "really" counts as a separate object. So a golf ball 10 ft up counts as part of the Earth? What about a golf ball 10000 ft up? What about a golf ball in deep orbit around the Earth? Does it matter whether the orbit is geosynchronous? There is no objective break point; all we can do is say that we want to examine the interaction between the ball and the rest of the Earth, so we should treat the two as separate objects. And the accelerations you gave definitely violate Newton's third law; saying that I am "messing up a simple problem" is not actually a counter-argument. See above: Hamiltonian mechanics IS classical mechanics, just a different formalism from the one most people learn in intro physics. I thought it more appropriate for the problem at hand because it starts with the energy equation, which everyone [I thought] would agree on. For heaven's sake. I didn't use the Hamiltonian to derive r=x-X. I used it to transform the energy equation (H=\frac{P^2}{2M}+\frac{p^2}{2m}+\frac{GMm}{r}) into the equation of motion (\frac{dv_{rel}}{dt}=\frac{G[M+m]}{r^2}). I'll address the question of the inertial frame below, but I want to make one thing very clear: if you agree that (H) gives the energy of the system, my expression for relative acceleration necessarily follows, based entirely on classical mechanics. I see RJ already answered this bit, but I'll do the same. 1. The physics should play out the same in any inertial frame, but the easiest to use is probably one with its origin at the system's center of mass. 2. The Earth and the ball can both move with respect to the center of mass and each other, so there are two free bodies. 3. The effect we're discussing is small enough to be just a technicality, so its accuracy is limited to a result of either "the effect is zero" or "the effect is nonzero." The arguments given by RJ, myself, and others show that the effect is nonzero. The actual numeric values of the fall times will be affected far more by factors like the topography of the Earth's surface than by this effect, so they are not useful beyond showing a nonzero difference. I thought so too, which is why I jumped in; an uncharacteristic number of the usually "skilled" posters seemed to be getting on the wrong side of this debate.
The weight of a single feather is sufficient to move an elevator full of bowling balls to the top floor, provided the elevator's counterweight is precisely balanced enough, and the bearings are made frictionless (actually possible with superconducting bearings). I doubt the process "uses up" any significant gravitational energy in any permanent manner, no matter how many times the elevator moves them. Of course, the weight of the elevator cables need to be taken into account after the first move. This is a wonderful thread (even if just a tiny bit off topic). I regret that we never had time to have this extended discussion out in physics classes, but I also understand why now. Finally, Galileo didn't need to be 'exactly' correct in order to advance an idea a great deal closer to reality than Aristotle's. If Aristotle was to be believed, a large heavy stone could be made to fall more slowly simply by tethering it to smaller, lighter ones that fall slower.
It will be interesting to note, how many people on this thread agree to the fact that earth and ball (as in OP) will have different acceleration in free body analysis without violating Newton's any law, as shown in my calculations ?? Those who disagree, along with Fednish48 and RJbeery, should google search the "Pseudo Force" and try to understand the same beyond wiki.... I am quite surprised at the complete lack of understanding of basic Newtonian mechanics as shown by many members, and these very members appear to be quite at ease with Hamiltonian ?? what a farce.... This is science forum guys and you are letting a Tom come, and declare Galileo wrong with some silly and outright incorrect math/Physics. This is not a kind of implicit admission that yeah, may be, poor Galli was technically wrong..My foot...This is a kind of admission that we are ignorant, we could not collectively prove this Tom wrong. As some one said in this thread.. I can only bring water near to the horse mouth, but I cannot make him drink....
This is a good point. Galileo's contribution was debunking the idea that an object's acceleration toward the Earth was a function of its mass, which is without doubt. Inertial mass = gravitational mass. I frankly don't know if Galileo ever actually said "FALL TIMES TO EARTH ARE INDEPENDENT OF MASS", which strictly differs, but this statement is simply false.
I'm not "some Tom", and I only consider you ignorant if you blind yourself to facts and reason. This issue has clearly become a litmus test for exposing Physics intuition and the ability to digest new ideas. I'm quite comfortable having rpenner, Fednis, and, most importantly, the truth on my side.
You have to choose the origin. Specify a place. Before you can decide anything else you have to decide the physical origin of the coordinate system. While you're thinking this through you should ask yourself where Galileo's origin was. Let's get past step [1] before we jump to any conclusions. You misunderstand. Whether it's a 1- or 2-body problem depends on the choice of the origin of the reference frame. If you get a different result by choosing a different origin than him, then all bets are off. He didn't make any claims per se, other than what he could measure to the best of his ability. I think all of the skilled posters are ahead of the game here. You won't get too much agreement on that. Hamiltonian math/physics is considered one grade of evolution more modern than classical physics. ' The Hamiltonian is not covered in freshman physics because it's not necessary to teach students how to solve classical mechanics problems. That's getting closer to admitting that it adds nothing. For that we just state early on that we are going to apply conservation of energy. That's the universal everyone will agree on. That was what I saw in your post. You ended back at the Law from whence you started, with one of the radii expressed as the difference in x-coordinates. The equation of motion follows directly from the Law, as I mentioned previously. I don't agree that this added anything , that's all. No RJ hasn't yet answered the question. All motion is relative. Once you change the reference frame, you alter the definition of the motion. That's all this boils down to. Only because you chose to place the inertial reference frame at the c.m. This is an alteration of the experiment attributed to Galileo. Therefore the results will be different. There are different kinds of misunderstandings in play. The main one is the confusion over switching reference frames, first cited by [I think] OnlyMe. Several posters I know for sure are aware of the confusion. The ones that aren't aware of it may be lead to that water, but they probably will refuse to drink. You mean the difference between the two problems is nonzero. In this case it may be useful for teaching Galilean relativity. Maybe the horses will drink and the dry throats will be quenched.
Uhh, I literally said that our physical origin is the center of gravity of the Earth+object system. Then I literally said that this was Galileo's technical error. Since we are talking about reality, and simultaneously discussing Galileo's error, this is undeniably a 2-body problem.
RPenner, I'm a bit frustrated with and disappointed in you. I know for a fact that you understand and technically agree with this thread's premise, yet you give the doubters just enough ambiguity in your answers to pull what they need to support their misconceptions and drag this thing on and on. Again I challenge you: If you aren't going to verify the mathematical analysis please supply the wording of the "right mental model" which would make Galileo TECHNICALLY right in regards to the following claim: FALL TIMES TO EARTH ARE INDEPENDENT OF MASS.
There are as many reference frames as you can think of. But modelling the problem requires choosing one. However, it takes skill to understand the implications of that choice. Since the choice of reference frame is arbitrary, how does one err in choosing one frame or the other? We are talking about a specific kind of test of gravity on Earth. You're putting the conclusion before the fact. State the facts and you'll get a lot more bang for the buck. Only once you've chosen a coordinate system. I gathered from Fednis48 that you are setting the origin of your system at the c.g. That will invalidate any results you get, unless you take your results and convert them to Galileo's coordinates. Until then, there is no basis for comparing results, since relative quantities can not be taken from one frame of reference to another without a coordinate transformation.
Aqueous Id, I've chosen one. I declared what it was. It's one which remains inertial during the entire experiment. Please note that even if you attempt to declare gravity is a "pseudo-force" like some folks are doing, the Earth pressing UP against your feet with a greater force after the object is dropped is a legitimate one.
When stating a problem of this sort, always give all of your assumptions up front, esp. the placement of the origin from which all vectors are measured. Okay, now we are at a crossroads. You have left the readers to understand that you are in a different coordinate system than Galileo. The skilled readers are waiting for you to show the transformation that you need in order to compare your results against Galileo's. I sense that you don't understand me, so I'll do my best to clarify this for you if it's unclear. None of what you just said makes much sense to me. I see some vague reference to GR, which is moot, and perhaps something about Newton's 3rd Law, which I assume you acknowledge. But none of this is at issue. All you are actually attacking is Galileo's choice of reference frame. Once you've figured out what the coordinate transform is to get from your system to his, you will be able to compare your numbers to his, and then you can reach the kind of conclusion a skilled reader would reach.
Galileo did not have a valid reference frame because it was not inertial during the experiment. You're starting to sound like a bit of a blowhard.
Note, no one knows if Galileo was even involved in this, it's only a legend. The reference frame the experiment uses is Earth-based. So by definition it's inertial: In Newtonian physics and special relativity, an inertial frame of reference [or Galilean reference frame] is a frame of reference in which Newton's first law of motion applies: an object moves at a constant velocity unless acted on by an external force. http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Inertial_frame_of_reference.html I'm just posting facts relevant to the discussion. Feel free to trump them with better facts.
