Some what strange that you are arguing against the center of mass model and as an example you use the center of mass. I would wonder if you actually studied the orbital behaviour of your dumbell system what result you would get. You will find that the orbital period is still the same. Your error comes from the assumption that the center of mass has a uniform distribution of mass around it, this is not true for hemispheres.
The Death of the Centre of Mass Theorem
- Thread starter Rogue Physicist
- Start date
I was not suggesting that one actually launch satellites to measure gravitational constant, G. I first stated in my post that there there are better astronomical ways to determine G, but I don't know the details. Consequently, I invented a proceedure that could be used to determine the value of G without the near Earth measurements you think might be corrupted by the lack of perfect uniformity in the Earth's mass distribution.Rogue Physicist said:
BTW the shape of the Earth's gravity field is known very well - exactly how well is a clasified secret. It is described in a tesserial harmonics expansion, where the lowest order term, the spherical component, is dominate. Thus it is not necessary to make any assumption that Earth's gravity is inverse square drop off from the center of the Earth when measuring G. One can use the very well known (almost exactly correct) expression for the shape of Earth's gravity field. Thus you, perhaps in ignorance of the known facts, have set up a strawman to knock down. I think that the coeficients of the higher order tesserial harmonic terms in the expresion for the Earth's gravity field are functions of the date (at least of the month) as the seasonal shift of ice between the N & S hemisphere is easily observed in highly precise measurements of the Earth's gravity field.
One will never know the value of G exactly. There will always be some small error. It is unreasonable to demand it be known free of small error. More than 40 years ago, while attending the annual "spring meeting" of Am. Physics Soc. in Washington DC, I was given a credit card size "Pocket table of the fundamental constants" by the Addison -Wesley book Co. I still have it. The value of G on it is 6.670x 10^-8 dyne cm^2 gm^-2. (error estimate then was 0.005 + or -) I bet now that NASA et al have sent satelittes arround distant planets that the accuracy of G is known to better than one part in a million. How accurate do you want it to be?
I am not looking at your quote about the "Pioneer Anomaly" as I write, but know it is extremely small, and only observable because High precision of radar ranging (or comunication round trip delays) which can measure even very great distances to accuracy of less than 100feet. I don't know where Pioneer is now, but lets guess it is at least twice the distance to Pluto, say 60AU away from sun or earth (your chioce) 1AU is 93,000,000 miles. 60x 5280 is 316,800. Thus a conservative estimate of the distance to Pineer is 29,462,000,000,000 feet, call it 30x10^12. Hence a reasonable, probably conservative, estimate for the error inherent in meassuren the location of Pioneer is one part in 300,000,000,000. Considering (1) all the unknown mass in small rocks in the Ort cloud, which are closer to sun than Pioneer, not to mention the many unknown small comets, asteroids, inside the orbit of Pluto or the residual "dust" which is closer to the sun than Pioneer and (2) the high precision with which Pioneer's location can be measured, it is not surprizing that it is not exactly where computations, using G, and only perhaps 10,000 of the known largest masses predict it to be. Because there is mass closer to the sun, not included in the calculation using G, it is not strange that Pioneer is very slightly closer to the sun than its position, predicted by incomplete computations.
The the surprize would be if there were no "Pioneer Anomaly"!!!! Only someone very ignorant of the facts would argue that the small "Pioneer Anomaly" is proof that the means used to evaluate the value of G is wrong, needs correction, etc. I think your problem is not "that you know too much" but that you know too little and are prone to make assertions that you can not support, such as your claim that the external gravitational field of a uniformly dense sphere if calculated by considering it as the sum of the gravity field of two hemispheres (only a conceptual division, for calculation, not any change in the actual sphere) gives a different value for the external field than the calculation which assumes that all the sphere's mass is at the center of the sphere. I an others have asked you to show this calculation failure or admit you were wrong. Thus far you have done neither.
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Rogue Physicist
Registered Senior Member
The thread is about the failure of the Centre of Mass Theorem.blindman said:
It is not at all hypocritical or suspicious to use the Centre of Mass Theorem in two different ways, which according to vector addition should give the same basic result. This is a simple test of the accuracy which can then be quantified.
I am not claiming the result of the test is meaningful in the sense that it is an accurate way to calculate forces. On the contrary I am using the calculation according to the basic premise of the theorem to show that it is indeed not just inaccurate, but ambiguous and self-contradictory, and these are extra problems.
The 'period' is a time question, and falls under the category of Newtonian Dynamics. Here we are just talking about instantaneous forces and conditions, gravity and electrostatics.blindman said:
However, the truth is a spinning and obiting barbell does *not* act like a point-mass. When Newtonian Mechanics is done properly, it also predicts gravitational waves, which means this is not a test of General Relativity either.
Billy T said:
What I think you are saying here is that if you do the solution by integration either way, you get the same answer. I have never challenged the fact that if you integrate either all at once,(whole sphere) or integrate in two sub-steps (two halves and add) you get the same result. That claim would indeed be absurd, since integration is a process of adding the sections of the sphere together (in infinitesimal pieces).
I will certainly deny that I ever asserted anything that stupid, and I think you are being disengeniuous here. What I did assert was that the Centre of Mass theorem leads to different answers depending upon how you bisect the whole sphere, and I have indeed shown that repeatedly, using the barbell model, which is simply a dual of the Centre of Mass theorem and its consequences.
You have made this claim twice, while at the same time failing to understand the barbell model (or pretending to fail to understand it). At this point, if you really can't grasp the barbell model and why it is an exact consequence of the Centre of Mass theorem, I suggest you leave it there and try again later. Repeating your inaccurate claims will now be embarrassing for you, if you have nothing more to add.
To sum up, my argument is not about two different methods of integration giving different answers, which is not expected and would seem to me to contradict the fundamental theory and premises of integration. I have no quarrel with modern integration techniques.
The quarrel is about the theoretical foundation (premises) of the Centre of Mass theorem and its consequences. In the text-book presentations the CMT is admitted to be an approximation, so in this sense it is a kind of anti-climax. However, what is not adequately covered in physics courses is the details: that it is not simply a straightforward error-term creeping into the calculation, but rather a critical ambiguity in the method for calculating the CMT, which results, not in a quantified error, but a wild set of self-contradictory results.
Why is it a classified secret? Like techniques for separating Uranium 235/238, the gravity field of the earth is indeed rocket science. Presumably if 'crazy people' got their hands on this 'critically important' data, they could accurately launch giant missiles at New York. What can one say, except that military interference with science is brainless and ineffective. I think you have inadvertantly revealed a little something about the problem, and the resistance to examining these ideas in such detail.
I luckily, am under no restraint via government or military contract or agreement. I am quite happy to explain any scientific fact or principle to anyone who asks me. The only restraint I am under is an ethical one, in that I am not interested in helping idiots harm themselves and others for no rational purpose. That would of course include the military of any government, who are all too prone to causing grievous bodily harm to innocent civilians. Of course if someone presented to me a solid ethical rationale for building and detonating an atom bomb, and asked me to help them separate the U235 I'd be happy to help them and explain the latest and most effective techniques: because nobody is paying me not to, and no one has asked me not to. Money talks. Bullshit walks.
You suggested satellites. I quoted the article *not* to show that estimates of G were inaccurate as unit coordinators for solar-system distances, but to show that nothing is simple, and the greatest minds of our age are not satisfied with current models and explanations of actual data.
Of far more importance, is the anomaly in the Inverse Square Law regarding galaxy rotation speeds. This is NOT small, and I am not surprised, but certainly a little disappointed that you skipped over this fact entirely, especially as you claim to be an astrophysicist of sorts, and an expert in (Newtonian?) gravity in the context of planets, stars and Dark Visitors. If I reacted in the same way you have, I could also be a little disengenious and say *you* have made meaningless assertions and have avoided responding to the real arguments here...
Now I think we are bordering on insults. What is the need for this? Everything I have said has been as clear as possible and carefully explained with logic, diagrams, and examples. Misunderstandings are sometimes unavoidable, even by brilliant minds. But once they are sorted out we should be able to move on like adults.
This at least is honest and commendable.
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I an glad you agree that by intgration the total force from two joined hemispheres is same as integration over the sphere, but not sure you think it is also the same (for external ponts) as assuming all the mass were concentrated at the point center of the sphere.Rogue Physicist said:
It is obvious that if you use an approximation (The Center of mass theorem) and compare the results computed inaccruately to accurate results there will be a difference. For example 1/(1+x) can be approximated as 1-x if x< < 1, but if x=0.5 the exact resuult is 2/3 and the approximate result is 0.5
You admit in (2) above that the central mass theorem is only and approximation and is a very bad approximation when d, the distance from the center of mass is only slightly greater than the sphere radius, r.
It appears to me, that this thread has no more merit than one which compares the exact 1/(1+x) with the approximation 1-x.
Why did it get started? It is pointless.
Rogue Physicist
Registered Senior Member
How can a scientific discussion be pointless, if a dozen people participated, and there were 300 viewings of its contents? At least a few people have come away with a better understanding of gravity and Newtonian mechanics, even if they don't understand everything discussed, or agree with either of us on various points.
If education is pointless, or clarifying ideas is pointless, then this thread is pointless.
If education is pointless, or clarifying ideas is pointless, then this thread is pointless.
This "education" is no better than talking about 1-x as an approximation for 1/(1+x) which is more valid when 1 > > x.Rogue Physicist said:
The central mass theorem is an approximation, just as this algebra is. As you noted, it is more valid when d > > r . So what?
Rogue Physicist
Registered Senior Member
Looks like a valid and worthwhile result to me, in the context of proving that Newtonian Gravitation predicts gravity waves, and so the existance of gravity waves fails as a test of General Relativity versus Newtonian Gravitational theory.
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Rogue Physicist said:
No it is not a test of accuracy because you are changing the shape (mass distribution) of the system
The center of mass theorem only applys to objects with an even mass distribution. It does not apply to a hemispheres or any other non spherical object. Your dumbbell system represents an elongated peanut shaped earth and of course you will have less force because you have effectively moved the mass away from the orbiting point.
If the orbiting point is moved around to the poles the force acting on the object is much greater then the center mass solution. The are also 2 circles around the sphere/dumbbell solutions that have equal force.
Once again the center of mass system assumes that the object has an even mass distribution and is a sphere. Changing the center of mass to two points changes the mass distribution of the system and leads to the results you have.
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Rogue Physicist
Registered Senior Member
blindman said:
Once again someone has blurred several different concepts together. You have confused the Centre of Mass Theorem (CMT) with the Sphere Theorem (ST).
Please note the names, similarities and the differences between these notions:
There are four different concepts here that are commonly confused, because they are not taught properly in high-school and 1st-2nd year elementary physics courses. There simply isn't time, and later in specialized courses there isn't room:
(1) The Centre of Mass. This concept is used most often in classical mechanics problems. It has a clear unambiguous mathematical definition, but has no physical counterpart or concrete meaning, aside from being a convenient way of managing a system of particles. It is the arithmetic mean of position for each particle in a system weighted by each particle's mass. (for charges weighting is not necessary, so there is no corresponding concept in electrostatics.)
(2) The Geometric Centre. A similar idea, mostly applicable to regular shapes having some form of symmetry, and meant to represent the mean position of a volume in space.
(3) The Centre of Gravity. A point with behavioral properties that help to describe the conditions of balance of a rigid body under the influence of a uniform (or zero) gravitational field. Essentially the opposing axis through a balancing point on a rigid object must pass through this internal point.
(4) The Equivalent Point-Mass. The position in space one must place a point-mass of the exact same mass as the system to be replaced in order to exert the exact same force in magnitude and direction upon another external test-mass.
----------------------------------------------------
and There are Three different Theorems:
(1) The Sphere Theorem (Newtonian Gravity & Classical Electrostatics)
A hollow sphere of uniform density and negligible thickness acts as if it were a point-mass with all the mass concentrated at the Geometric Centre to objects outside it, and has no gravitational field (net force zero) inside. However, it cannot shield gravitational or other forces imposed from outside.
Also, from this the gravitational field of a solid sphere of radially symmetrical density can be constructed (inside and out).
(2) The Centre of Mass Theorem (Newtonian Gravity)
An Object (rigid) or system of particles (freely floating in space) acts as if the mass were concentrated at the Centre of Mass, a well-defined arithmetic average of the position of the particles weighted by their mass. For purposes of calculating external forces the momentum of the particles can also be averaged, and the Centre of Mass treated as the intertial frame or reference location point of the system.
(3) The Centre of Gravity (Classical Mechanics)
A rigid object acts as if all its mass were concentrated at the Centre of Mass, and this point will move in space according to Newton's Laws of Force (F = mA and Action-Reaction).
A brief discussion of the similarities and differences is in order.
(a) For regular shapes of uniform density, or symmetric arrangements of discrete components in a system, the Centre of Mass often coincides with the Geometric Centre, making it easy to locate without having to average all the particles in a system.
(b) The Centre of Mass is used as the common or best estimate of the location of the rest frame for rigid bodies.
(c) The Centre of Mass is useful for studying rotation, since a system of particles (rigid or not) which is free of external forces or in a uniform field will rotate about this point if it contains any stored rotational energy. This is due to the Conservation of momentum and energy.
(d)The Centre of Mass is expected to follow trajectories as if all the mass were concentrated at that point in a rigid body (ignoring things like wind resistance etc.) For example, a spinning hammer may appear to move irregularly, but if the Centre of Mass is followed, this point will move in a straight line through space or trace a parabola if thrown near the earth's surface, obeying Newton's laws of motion for point-masses.
(e) The Centre of Mass is acknowledged to be only an approximation when dealing with gravitational and electrostatic fields. This is because systems that don't have uniform geometric symmetry don't have distributions of mass that exert uniform forces on arbitrarily located external bodies, even when they have uniform densities.
(f) A Key point is that the Equivalent Point Mass location (EPM) is only definable from the viewpoint of a test-mass with a fixed and defined location. If the test-mass is moved, the EPM also moves, when the object under consideration is not uniform in density or radially symmetric in all directions.
(g) An important concept to come away with here is that an irregular object looks different to all other parties and locations in space, even if those objects are point-masses, and so the EPM is in a different location for each observer under the influence of the gravitational force from the object. In general, there is no unique EPM for an object.
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Rogue Physicist said:
Wrong. Irregular shaped object and objects with Irregular mass distribution should not use the The Center of Mass Theorem. Especially at low orbits, the NEAR space craft that orbited the near Earth asteroid 433 Eros and landed in January 1999, did not use the center of mass theorem in calculations of the orbit. They created a gravity map of the object and used that in their calculation to slowly decrease the orbit and eventually land.
The GRACE mission provided a high resolution gravity map of earth, for very precise orbit requirements, example the LAGEOS satellite.
The Center of Mass Theorem is only a generalization and I'm sure even Newton understood this. Irregular objects will have a varied gravity field, changing the center of mass into two points changes the distribution of mass and will of course change the force acting on an orbiting body.
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Rogue Physicist
Registered Senior Member
You are right. What NASA scientists did was only vaguely related to Centre of Mass.
However, interestingly NASA and other scientists currently still use the inverse-square law as a base for all calculations and integrations. Yet it fails miserably in explaining the rotational speeds of the outside arms of galaxies:
Whether you know it or not, you are actually agreeing with me here. I also don't think it is an accurate or reliable method. Yet it is still found in physics textbooks, which is why it's worth examining.
The galactic rotation is not explained, but the current best guess is the presence of dark matter. Orbits in a galaxy can not be computed via a center of mass solution because orbits are inside the mass. If I theoretically constructed two gigantic spheres inside the earth creating a space for an object to orbit, the smaller the orbital radius the lower the orbital speed, which is directly contradicting the center of mass model.
From the smallest, to the largest distance measured, gravity the value of G remains the same. Some say that at very small scales it changes, and others at very large, but until it is measured it is nothing more then guess work.
From the smallest, to the largest distance measured, gravity the value of G remains the same. Some say that at very small scales it changes, and others at very large, but until it is measured it is nothing more then guess work.
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It is "Centre of Mass Approximation" not the "Centre of Mass Theorem." You yourself stated it correctly in a prior post where you included the essential "when d >> r "Rogue Physicist said:
Your title: "Death of the Center of Mass Theorem" is not only pretensious, but impossible in that "Centre of Mass Theorem" was never alive, did not exist, etc. Again it is an approximation, useful if d >> r where "d" is the distance to the field point where the gravitational force is to be calculated and "r" is a typical dimension of the source of the gravity.
With equal sense I could start a thread called: Death of the Near Unity Theorem" where what I was in fact talking about is the Near Unity Approximation:
1/(1+x) is approximately equal to (1-x), if x is near unity, I.e. if 1 >> x.
This thread would make as much sense, be as educational, etc. as this one.
You can not set up a strawman by claiming that an "approximation" is a "theorem" then show the theorem is "false" - This is what you have done, or at least are trying to do, but I won't let you get away with is strawman slight of hand - I want you to honest.
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Rogue Physicist: I have not read every post to this thread, so I apologize to any who has mentioned this error in your analysis. You posted the following as the beginning of your analysis.
Since one of your initial statements is erroneous, your conclusion is invalid (or certainly not applicable to the CM theorem).
There is a valid proof that a hollow spherical shell of constant density acts as a point mass on external objects when considering gravitational effects. This leads to the conclusion that a sphere whose density is a function of distance from the center also acts as a point mass. Note that this is very approximately correct for many planets, ordinary stars, and other astronomical objects. It is an excellent approximation for non-rotating objects.
A similar claim can be disproved for all (I think) other shapes. The theorem is not valid for spheres whose density is not radially symmetric.
It can be shown that the CM theorem is an excellent approximation for any objects separated by large distances, with the approximation improving with increasing distance.
- (1) The Centre of Mass theorem makes the claim that an object acts as if all its mass is concentrated at the Centre of Mass, a uniquely defined fixed point for any system of particles rigid or not (a macro-sized object).
Since one of your initial statements is erroneous, your conclusion is invalid (or certainly not applicable to the CM theorem).
There is a valid proof that a hollow spherical shell of constant density acts as a point mass on external objects when considering gravitational effects. This leads to the conclusion that a sphere whose density is a function of distance from the center also acts as a point mass. Note that this is very approximately correct for many planets, ordinary stars, and other astronomical objects. It is an excellent approximation for non-rotating objects.
A similar claim can be disproved for all (I think) other shapes. The theorem is not valid for spheres whose density is not radially symmetric.
It can be shown that the CM theorem is an excellent approximation for any objects separated by large distances, with the approximation improving with increasing distance.
Rogue Physicist
Registered Senior Member
This is cleverly worded, but misleads the reader. There *are* such physicists, including most physics-textbook writers, reviewers, and referees. And this is how most students learn physics:
Just a few of the many recent examples of inaccurate statements you can find, from the first two textbooks that came to hand. Since most textbooks are full of such inaccuracies, it is silly to deny it, or pretend that I am attacking 'straw dogs'.
Apology accepted: I simply ask you re-read the posts to see that no one is disputing that the 'Sphere Theorem' is trivially true, for astonomical objects whose separation is typically >>10,000 times their radius. And this is about the Centre of Mass concept, not the Sphere Theorem: post redundant statements in the other thread. If you're attacking windmills, stay on topic.
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Rogue physicist:
Nothing wrong with that. It doesn't say the effect of the mass on other objects is as if all the mass were concentrated at the centre of mass (although for large separations that is true). It talks only about the effect of net external forces on the centre of mass, and what it says is exactly right.
Again, a totally correct statement. Note the "provided..." statement, which is the important bit.
There's nothing inaccurate there.
Nothing wrong with that. It doesn't say the effect of the mass on other objects is as if all the mass were concentrated at the centre of mass (although for large separations that is true). It talks only about the effect of net external forces on the centre of mass, and what it says is exactly right.
Again, a totally correct statement. Note the "provided..." statement, which is the important bit.
There's nothing inaccurate there.
geistkiesel
Valued Senior Member
Get a cannon barrel, separate the critical mass into halves (plus or minus a bit from 'half'). Shoot one of the masses to the other, implode a magnetic field around the now accumulated cricical mass just right,you need only that temperature that will assure an exponential rise in U238 disintegration rate, and poof, "its evaporation time."RoguePhysicist said:
Where do I stand wrt your understanding of it all as hinted at here?
and this one also. RP, I'm in no hurry, so, please, take your time.
Geistkiesel
geistkiesel
Valued Senior Member
Where do I stand wrt your understanding of it all as hinted athere?RoguePhysicist said:
and this one also. I'm in no hurry, so take your time.
geistkiesel
Valued Senior Member
RoguePhysicist said:
RP,
I mean, re my previous post, is there any application wrt your gravity model and the two links? They both, the links at least, have an anti-SRT flavor to them. As corrections to SRT do you see, intuit even, any implications of the links extending beyond the mere SRT corrections?
Thank you RP.
Geistkiesel
I mean, re my previous post, is there any application wrt your gravity model and the two links? They both, the links at least, have an anti-SRT flavor to them. As corrections to SRT do you see, intuit even, any implications of the links extending beyond the mere SRT corrections?
Thank you RP.
Geistkiesel