Disproving
Newton
Part
3
The
Death of the Center of Mass Theorem
by Nazaroo
Newton’s Centre of Mass Theorem is shown to be self-contradictory
(With a New Introduction)
Last Updated: June 19, 2005
Synopsis
This paper examines the Centre of Mass Theorem ( CMT ), originated by Newton, as it is currently formulated and presented in physics textbooks around the world today.
Specifically, after examining the details of the theorem, we show:
1) That CMT is incapable of properly handling external forces exerted upon a system, because its method discards crucial information regarding those forces.In summary, the Centre of Mass Theorem is essentially incorrect, and flawed as a theoretical basis for understanding the physical phenomena it is expected to account for. Teachers and students are better off avoiding this theorem entirely and waiting for a better theoretical overview or presentation to be offered.
2) That CMT cannot be patched, modified or extended simply and easily, because its flaws are deeply entrenched and are systemic to its derivation and formulation.
3) That CMT is self-contradictory and mathematically incoherent. Its theoretical foundations are misguided and confused.
4) No sensible or useful reformulation of CMT is really possible, and its only value is historical. It cannot be retained as an integral part of modern Newtonian mechanics.
5) That the few valid components contained in CMT should be left to stand alone under restricted circumstances, such as the Conservation of Linear Momentum ( CLM ), or else be absorbed into other valid theorems and methods, such as those covering Angular Momentum and Equilibrium of forces.
6) That CMT should be abandoned as a teaching tool, since it misleads people regarding both its meaning and value for Newtonian mechanics.
7) That CMT cannot be used either as a theoretical foundation or even an auxiliary support for the Sphere Theorem ( ST ) in regard to Newtonian gravitation.
Table of Contents
Introduction
1. Quick Review: Newtonian Gravity
2. Gravity made Practical
3. The Background and Purpose of the Centre of Mass Theorem
4. The Centre of Mass Concept ( CM )
5. Special Properties of the Centre of Mass
6. The Centre of Mass and the Geometric Centre
7. The Generic Nature of the Centre of Mass
8. Practical Methods of Calculating the CM
9. Derivation of the Centre of Mass
The Basic Theorem of the Centre of Mass ( CMT )
1. A Typical Formulation
2. The Scope and Claims of CMT
3. Interpretation of CMT
4. Counterexamples contradicting CMT
5. Realistic Restrictions for CMT
6. Re-interpretation of CMT ( ‘no free lunch’ theorem )
Analysis and Criticism of CMT
1. Flaw A: Failure to distinguish between types of systems
2. Flaw B: Non Uniqueness of the CM
3. CMT : Blow by Blow Critique
Demonstration of the Failure of CMT
1. Analyzing Newton’s Balls
2. CMT contradicts NTF
3. CMT contradicts Itself
4. The 'Success' of CMT: A Cosmological Coincidence
Introduction
1. Quick Review: Newtonian Gravity
In brief, Newton viewed gravity as a force of attraction between bodies. Matter is made of basic particles, such as protons and electrons. Each particle attracts every other, and so there is a small force between every pair of particles.
These forces are independent, in the sense that they do not interfere with each other as sound waves or light waves can. That is, unlike waves that vary over time, gravity is a static force that only varies with distance. Gravitational forces pass through one another in a ghostlike fashion, only acting on actual particles.
Gravity apparently reaches infinite distances and penetrates everything. Interposing a third mass between two objects does not diminish or deflect the force. In fact it has no effect at all, except to add two more paired forces. The gravitational force between each pair of objects is dependent upon their own mass and distance alone. The total net force is just the superimposing and adding up of all these forces between pairs.
Every object exerts the same total force 100 light-years away as it does next door (!):
Although gravity weakens as distance squared, this really just reflects a fixed amount of force spreading out over an increasing spherical surface area. It can be represented by a fixed number of lines of force spreading out like spokes of a wheel. This essentially lossless spread could even be called the Conservation of Gravity.
Even more remarkably, gravity seems able to draw upon infinite amounts of energy to apply itself. One object exerts the same force upon one other object, or an infinite number of them, as necessary. Gravity doesn’t run down like a clock, or drain like a battery. The force doesn’t become defocused, diluted, or randomized as it stretches across space. These basic discoveries are mystifying. Nor is there any obvious mechanism or medium for the actual transmission of the force. This led Newton to view gravity as a spooky, instantaneous action at a distance, and led Einstein to interpret it as a geometric property of space-time itself.
2. Gravity made Practical
Although mysterious in terms of ordinary notions of cause and effect, at least gravitational forces are easy to handle in principle. We don’t need to worry about extra interactions or side-effects. For any given pair of particles, Newton’s basic formula could be used to calculate the force. With more particles we just add all the gravitational forces together like we do other forces as vectors to get the total net force on any one particle or object.
Easy in principle however becomes impractical with billions of particles per handful of matter. Useful formulas must apply to large groups of particles at a time. Newton addressed this problem with two claims:
(1) The Sphere Theorem: Uniform spheres act as if they were point-masses, and so the simple formula for point-masses can be used ‘as is’ with celestial bodies. One assumes the mass is concentrated at the geometric centre (GC), and distances are measured centre-to-centre between spheres.These two key theorems are the basis for practical calculations and predictions for the motion of large masses under Newtonian gravity.
(2) The Centre Of Mass Theorem: Similarly, any object’s centre of mass ( a fixed point of balance inside it1) also moves according to the laws of motion, as if all the mass were concentrated there. External forces can be summed and applied to the centre of mass (CM) to predict its new position etc.2
______________________________________________
1. An object’s centre of
mass ( the weighted average of position of all particles ) can be
found by measuring weight, volume, composition, density & shape,
without descending to the atomic level. The
geometric centre ( GC ) of an object is determined by its shape,
i.e., distribution of volume and extension in space. The centre of
mass is determined by the distribution of mass. If the system is
symmetrical in both shape and distribution of mass, its centre of
mass coincides with its geometric centre. These two are not the
same as the centre of gravity, which is only relevant in a uniform
field. Nor are they the same as an equivalent point-mass position,
which is what we have to calculate.
2. That is, the centre of
mass traces out a parabola or moves in a straight line, even if the
object appears to move eccentrically.
_______________________________________________3. The Background and Purpose of the Centre of Mass Theorem
To assess the impact of a flaw in the Centre of Mass Theorem ( CMT ), we need to review its function. Newton’s theoretical foundation rests upon atoms exerting mutual forces of attraction upon one another. That is, his gravity formula actually only applies to particles. This would be hopelessly impractical unless we could economically deal with large groups of particles at once.
The whole purpose of CMT is to enable us to do so. CMT then is a key part of Newton’s complete theory. Without CMT or some other practical technique any gravity theory is just a curiosity. Newton’s spooky notion of invisible, instantaneous action-at-a-distance would probably have been branded as mere magical thinking long ago, if not for the spectacular success bought for it by CMT. 3
That is, people accepted Newton’s theory, because his formulas and methods actually worked, or seemed to. Even those today who reject Newton’s theory presumably don’t object to engineers continuing to use Newton’s techniques ( i.e. CMT ) to build bridges and to calculate rocket trajectories.
It is hard to deny that CMT is still presented as a currently valid theorem, and that most scientists assume the consistency and validity of its derivation and formulation, at least within the sphere of Newtonian mechanics.4 Now and then CMT is called an ‘approximation’, although no quantification is offered. The basic position given on CMT is that it is accurate. 5
_______________________________________
3. Even with
the success of Newton’s techniques, many brilliant minds questioned
his theoretical foundations. Newton’s success regarding celestial
bodies relied upon the Sphere Theorem ( ST
) but this appears to be just a special case of CMT.
4. Angular Momentum in
particular has been modified to accommodate Relativity, but this is
really only relevant for particle physics. For ordinary objects
Newtonian versions of angular momentum are expected to be valid and
applicable to gravitational problems also.
5.
For
instance see Elements
of Newtonian Mechanics
Knudsen/Hjorth, Springer, 2000 pg. 195
The Current State of Affairs as to Gravitational Theories
The only viable alternative is General Relativity ( GR ). But GR is still in its infancy and in its current form is simply impractical. For most purposes Newton’s CMT is all we have. Even those who prefer GR as a theoretical foundation still use CMT for ordinary calculations. This is no surprise, since Einstein designed GR to mimic or reduce to CMT in ordinary situations. For these cases GR then offers no increased accuracy or practical advantage, nor does it make any alternate predictions. Apparently CMT (with a few corrections for relativistic effects when necessary) is all we really need. So Newton’s method ( CMT ) is used for accuracy and convenience, while Einstein’s theory ( GR ) is preferred as a theoretical foundation for other reasons, such as compatibility with Special Relativity.
Other Related Theorems and Methods
To complete the picture presented so far, we must mention the Angular Momentum and Equilibrium theorems. These handle rotation and the application of forces to rigid bodies, and they also handle balance under uniform gravitational fields. Both theorems make use of the Centre of Mass concept, ( CM ) but do not directly rely upon the Centre of Mass Theorem. ( CMT ). One might argue that most engineers rely upon these other theorems more heavily than they rely upon CMT .
Perceived Agenda for Gravitational Theories and Techniques
Naturally, Newton’s theoretical foundation ( NTF ) is expected to fade away because it can’t explain new discoveries. It wasn’t written to explain them, and would need ongoing revision to do so. In hindsight, NTF appears not to be a real explanation at all, but simply the ad hoc concept Newton needed to justify CMT , in the same way he used ‘Absolute Space’ to justify inertial frames and simplify the laws of motion. That is, NTF really depends upon CMT for its existence, not vise versa. Both NTF and Absolute Space were necessary for Newton, but are no longer so for us. If this is so, we can discard NTF and keep CMT as one removes the scaffold once the house is built. If CMT doesn’t require NTF, or they were found incompatible in some unforeseen way, we could even now choose CMT over NTF on more than empirical grounds.
In a happy world, so it would be. Newton needs CMT , and we need CMT, but we don’t need Newton’s NTF. It could be left as a learning tool, or retired to a history shelf, and CMT would be retained in the honourable service of GR.
Probable Outcome of a New Analysis
This however appears not to be the future of the Centre of Mass theorem. The reader should discover as we did that CMT has fatal flaws, if they can find the patience to step carefully through the arguments below. Is this an exaggeration? Surely people will go on using CMT as they always have, in the same manner as they did after GR. Not exactly.
GR was clearly revolutionary, overturning theoretical foundations. Yet, for ordinary mechanics, life went on pretty much undisturbed: In this case, the theoretical modifications are obviously far more modest. Yet something very different has happened qualitatively. The function of CMT has inevitably changed, because it can no longer be trusted. Its results, even its successes, require careful reinterpretation. CMT must now be applied more cautiously, keeping an eye out for much more than just relativistic effects. Many results and theorems will need reformulating in cases where it is clear that errors may become significant.
4. The Centre of Mass Concept ( CM )
The centre of mass ( CM ) is not a vague concept in Newtonian mechanics. It is a unique point related to a system, and is located within the volume of space that encompasses the system. A system in this context is a group of particles, each having a definite mass and position at some instant in time.6 A system need not be a rigid body, and systems can overlap and contain one another. However, the instantaneous position of the CM for a system is always defined in principle, because a reliable method is given for calculating it. The CM is called the ‘weighted average of position’7 of the particles, and each coordinate is defined by a calculation averaging the coordinates and masses of the particles in the system:
This is simply an arithmetic mean of position, weighted by mass.
This procedure always gives an unambiguous location, which is fixed relative to the rest frame of the system it belongs to. This relation can be called an ‘internal position’ to the system.
___________________________________________
7' e.g.
University
Physics
7th
Ed. Sears/Zemansky/Young, Addison Wesley 1987 pg. 197
____________________________________________
5. Special Properties of the Centre of Mass
The CM cannot change without a change in the distribution of mass.So the CM is always a fixed position for a rigid object, which is defined as a system with a fixed relative distribution of mass. This allows us to relax the simultaneity restriction for rigid objects, and we can track the motion of the CM as easily as other parts of a system without having to constantly recalculate it.
It is trivially true and desirable that a change of coordinate system, or scaling of all the masses and/or distances by some factor (i.e., a change of units) has no effect on the internal position of the CM. For instance, since the CM is only a point, altering the orientation of a system has no effect on the CM.
For the CM to hold its position relative to other systems however, the system must be rotated only on an axis through the CM itself. It is also true that changing the orientation, distance, mass or size of a system relative to a second system, or changing the second system’s orientation to the first, also has no effect on the internal location of the CM. But this latter case is not a mere change of coordinates, scales or units.
We expect the CM to be unaffected by trivial transforms like rotations, scaling and reflections. However, a system can actually be squashed flat, exploded, or stretched symmetrically in any number of dimensions without affecting the CM. One can even add or subtract mass, particles, combinations, whole systems, provided the effect is symmetrical.
Conversely however, even extreme changes in the distribution of mass don’t necessarily affect the position of the CM.Any number of systems could average out to the same CM. Although dependent upon distribution, the CM is a one-way trapdoor, preserving no information whatsoever as to the actual distribution of mass. Neither its location nor its motion can really tell us anything more than the general point of balance within a system.
6. The Centre of Mass and the Geometric Centre
The CM is closely related to the geometric centre ( GC ), because for systems of point-masses, the GC can be similarly defined as the ‘unweighted’ average of position, which simply means the ‘equally weighted’ average.
When dealing with systems of point-masses, the CM and GC will be the same as long as all the points have equal mass.Because of this, the CM and GC can share both topological and geometric properties. For instance, if the system’s point-masses sit on a straight line, the CM is also on that line. If they lie on a plane, the CM is also on the plane. Like the CM, the GC is also indifferent to orientation, is a fixed internal point in a rigid object, and lacks unique association to a specific point-mass distribution.
Where the mass can be considered continuous because of the quantity of point-masses, all that is really required is uniform density for the CM and GC to coincide.
If the mass is distributed symmetrically around either the CM or GC, then it is so around both, and they are identical. For many symmetrical objects, such as spheres, disks, cylinders, and regular polygons, the CM and GC are again the same, as long as the objects have merely a radially symmetrical gradient.
7. The Generic Nature of the Centre of Mass
Above, we referred to different systems having the same CM. This is only meaningful with an independent way of defining the position of a system, which the GC provides. The GC gives an alternate means of defining position, and helps to locate the CM, but does little else for us. The CM is the more relevant concept for gravitational forces, and for rotation as well.
Using the CM and the GC alone only allows us to distinguish two kinds of systems topologically, those in which they coincide, and those in which they don’t. For example, take any two non-symmetrical systems, each of which has non-coinciding CM and GC. Superimposing them with their GC at same location, we can rotate and scale one of them until their CM s also coincide, making them equivalent systems according to CMT. Clearly, even when the GC and CM do not coincide there is little to distinguish between systems, applying these concepts alone.
If the current claims found in CMT actually hold in the physical world, we would expect diverse systems to behave in an identical manner, namely the manner predicted by CMT.
8. Practical Methods of Calculating the CM
Obviously we can’t know the instantaneous position and mass of billions of particles, let alone calculate a weighted average from the data. The whole interest in the CM is to develop practical methods to measure and predict motion of large systems, without having to handle individual point-masses.
Ultimately, people rarely use the formal method of calculating the CM , except perhaps for a system with only a handful of particles or celestial bodies. Otherwise, matter is treated as though its mass were continuously distributed, and cases are limited to rigid symmetrical shapes with simple density patterns. However, the CM is invisible, often sits in empty space between masses in a system, and sometimes can only be found through the GC. If they don’t coincide, the orientation of the system is also needed as well as the GC, in order to find the CM. A combination of geometry and calculus is used along with assumptions about the behaviour of the CM found in explanations of CMT stated below.
In a locally uniform gravitational field, such as near the earth with small objects, a simple method for finding the CM is to exploit symmetry (it must be on an axis of symmetry in an object with a known symmetrical distribution of mass). One can also hang an object from some point two times, and in each case extend the vertical line through the object or along its surface, and mark their intersection.
9. Derivation of the Centre of Mass
Modern treatments derive the CM by first applying Newton’s 2nd law (F = mA) to each particle in a system. Position is substituted for A as a derivative (acceleration is the 2nd derivative of position). Mass is allowed to slip inside the derivative, since it is assumed to be a constant for each particle: 8
Next, an important declaration is made:
‘The total force on the system 9 is the vector sum of all the forces on all the particles.’ 10This is such a remarkable theoretical step that it deserves a comment: The idea is that just as we get the net force on a particle by summing the forces, so we should calculate the final net force upon a system by summing the forces applied to each particle in it, regardless of where in the system they are actually applied.
Thus, the total force of the system is written as the vector sum of the individual forces on each particle. Using the sum of derivatives = derivative of the sum rule, the summation is also slipped inside:
Finally, multiplying by M/M ( M = total mass ) and slipping the 1/M portion inside the derivative as well, puts it all back into the form of Newton’s 2nd law, ( F = MA ) only this time A is clearly seen to be the 2nd derivative of the formula given earlier for the CM, namely the weighted arithmetic mean average:
From Newton’s third law it is noted that all the internal forces between particles within the system come in equal and opposite pairs, and so they cancel leaving zero net force. Only forces from outside the system are unpaired and unbalanced and so represent a leftover net force.
That is, while Ftotal = Fexternal + Finternal , Finternal = 0 and so Ftotal = Fexternal
Thus the final equation Fexternal = Mtotal * AccelerationCM seems to imply that the CM actually obeys Newton’s 2nd law as long as Fexternal is understood to be the vector sum of all the external forces, as we originally defined it in the derivation.
____________________
8. This is a minor fudge. Mass varies slightly as energy is stored or released in chemical bonding or atomic decay.
9. Physics 3rd ed. Wolfson/Pasachoff, Addison Wesley, 1999 pg. 239
10. University
Physics 7th
ed. Sears/Zemansky/Young, Addison Wesley, 1987 pg. 198 etc.
_________________
The Basic Theorem ( CMT )
We are now ready to state clearly what CMT essentially asserts: Every system of particles having mass acts as a single point-mass concentrated at its CM, and external forces act upon the system as if they were all acting upon the CM.
1. A Typical Formulation
1) Suppose we have a suitable means of measuring the position and time of point-masses in any direction and at any distance.
2)We establish a suitable rest-frame by measuring the motion of a sufficiently isolated point-mass and assuming Newton’s first law of motion.
3)We define the CM of a system as the weighted average of position (as above.)
4)The displacement of the CM is then the weighted average of the displacements.
5)The velocity of the CM is then the weighted average of all the velocities.
6)Momentum is defined for particles as mv, and the vector sum of all the particle momenta in an isolated system is shown to be conserved. m1v1 + m2v2 + … = ( a constant vector )
7)This vector sum is identified as a momentum, that of the system. (MVsystem ) = m1v1 + m2v2 + … and we get a law of Conservation of Momentum .
8)The total mass M is defined as the arithmetic sum of all the masses.
9)The velocity of the system is defined and isolated by dividing both sides of the momentum equation by M.
10)The velocity of the system then turns out to be the velocity of the CM. (!)
11)The total mass M is then assigned to the CM.
12)The momentum of the CM is then the momentum of the system.
13)The law of conservation of Momentum is applied to the CM.
14)The CM appears to obey the first law of motion when the system is isolated, and it has uniform motion, even if parts of the system have complex motion. For instance, if the system is rotating, it rotates around the CM.
15)The system appears to behave as though it were a single point-mass located at the CM, with all its mass concentrated there. This assumption is extended to non-isolated systems, under acceleration from the effects of external forces.
16)Newton’s second law (F = mA) is applied to the system as a whole (as above).
17)Just as the net force upon an individual point-mass is the vector sum of all the forces, the net total force on the system/CM is assumed to be the simple vector sum of all the external forces. (In this case, the forces are those upon the individual particles at different locations however.)
18)The acceleration of the CM is calculated by dividing the total force on the system by the total mass ( by rearranging the equation to A = F / M ).
19)It turns out that with force so defined, the CM is the position found on the right side of the equation following Newton’s 2nd law (on paper), and vise versa.
20)Therefore, the CM acts as a point-mass of the same mass as the system, and all external forces can be simply applied there (as a vector sum). We can either predict the CM motion from known forces, or the forces from the CM motion.
2. The Scope and Claims of CMT
The scope and generality of CMT is as follows: It applies to all systems of point-masses, rigid or not, isolated or not. It is not dependent upon ‘uniform fields’ or special cases. This is often explicitly stated in physics texts, for instance:
‘ ( in contrast to the centre of gravity concept, ) …the centre of mass, conversely, is defined independently of any gravitational effect.’ 11
At this point the following claims are typically made:
‘(The CM ) does obey Newton’s law…We have defined the CM so that we can apply the 2nd law to the entire system rather than each … particle.12 …As far as its overall motion is concerned, a complex system acts as though all its mass were concentrated at the CM…we defined the CM so that its motion obey(s)…F = MA. ’ 13Physical support for CMT is usually offered in the form of examples such as non-uniformly weighted hammers or wrenches. When rotating, they revolve around their CM. The CM can be marked on the object and observed to move in a straight line as the object spins through space or slides across a table.
‘When a (system) of particles is acted on by external forces, the CM moves just as though all the mass were concentrated at that point and it were acted on by a resultant force equal to the (vector) sum of the external forces on the system’14
‘The ( CM ) moves as if the entire mass were concentrated in that point, and all external forces act there.’15
‘The ( CM ) of the system moves as if all the mass of the system were concentrated at that point. … the system moves as if the resultant external force were applied to a single particle of (equal) mass … located at the centre of mass.’16
3. Interpretation of CMT
One can search through the most recent physics texts, yet not find any qualification on the claims for CMT. The word ‘approximation’ does not appear there, although liberally used in other contexts. The one time we have found it used in relation to CMT is as follows:
‘Newton’s second law guides the motion of the CM just as it guides the motion of a single mass point. This is the justification for the often used approximation of an extended body as a mass-point.’ 17This is hardly a statement that CMT is an ‘approximation’ in any significant or quantifiable sense. If anything, CMT is once again asserted to be accurate. There is no hint of when CMT might prove inaccurate, or by how much. There is no mechanism, no example offered, no acknowledgment of any failure of CMT.
4. Counterexamples contradicting CMT
Ordinary physics students can be forgiven for thinking that here at last is some promise of a general method for dealing with systems and forces that is practical, reliable, and easy to use. They will not stay misled for long however, once they experience problems involving rotation and torque.
Note that if one applies equal and opposite force vectors sideways to each end of a simple rod, it spins (about its CM in fact! ). But if we were to follow CMT as presented, and simply add them vectorally and apply the resultant zero vector to the CM of the rod, we would never predict rotation or discover the rules of torque.
This isn’t a new discovery, obviously. But we would do well to discover the nature of the failure of CMT in cases such as this, so that we can modify its scope and its interpretation to reflect reality. One simple way to do this would be to just include theories of torque and equilibrium within CMT as a basic modification.
This would be unsatisfactory for several reasons. For one thing, it assumes the validity of CMT in other cases not involving torque. But we have already seen that CMT has been poorly defined as to scope and interpretation, and there may be other flaws in CMT.
5. Realistic Restrictions for CMT
CMT only deals with linear (translational) motion.How did CMT fail in the previous example? Because in fact it has no mechanism at all to deal with rotation and associated forces. It cannot, because by nature a point-mass has no orientation, nor any mechanism to connect to, store, or even measure relative rotation. Rotation can be thought of as balanced acceleration. CMT ‘handles’ balanced acceleration by canceling it out and ignoring it. One way to modify CMT to handle rotation would be to convert angular momentum into a kind of mass vector. In this case however, mass becomes a variable, and some of the mathematical steps in CMT obviously become invalid.
CMT only deals with vectors passing through a CM axisRigid bodies can be thought of as machines able to convert linear motion into rotational motion. As such, they are unsuitable for a method using a simple summation of vectors regardless of application point. Arbitrary forces often produce a mixture of translational and rotational motion. One solution would limit external forces under consideration to those applied along an axis passing through the CM. Then at least one point-mass in the system needs to be on this axis to receive the force. We could also handle cases having a secondary mechanism that distributes and delivers force in a parallel direction, such as a flat plunger.
In other special cases, CMT could handle balanced forces acting on parts of a system. For instance, one could simultaneously apply two vectors having appropriate directions and proportionate magnitude relative to a CM axis. This would work like a uniform gravitational field, where forces trivially move the system as if the sum of forces were applied to the CM.
Otherwise, CMT can only handle forces (or fields) acting symmetrically upon the system, around the axis of net direction through the CM.These restrictions limit forces and combinations of forces to those that produce pure linear (translational) motion without rotation. The careful expansion beyond uniform fields allows for the kind of spatial distortion of a system that doesn’t affect the relative location of the CM on any axis of symmetry.
6. Re-interpretation of CMT ( ‘no free lunch’ theorem )
With a sober assessment of the restraints upon CMT , we can now carefully reinterpret its claims.
‘The ( CM ) moves as if the entire mass were concentrated in that point, and all external forces act there.’18 And,These statements are true (and only true) if by ‘resultant external force’ we mean the result after we filter all those external forces through a technique which calculates and discards all torque and rotation effects. This technique would leave us with a net linear force related the system as a whole through the CM.
‘… the system moves as if the resultant external force were applied to a single particle of (equal) mass … located at the centre of mass.’19
External forces act on the CM indirectly since they are altered by mechanisms unique to each particular system, and their effects rely upon things like rigidity, interconnection of parts, and distribution of mass. There is no universal or simple method to convert real external forces into linear motion of the system or CM.
‘the CM moves just as though …it were acted on by a resultant force equal to the (vector) sum of the external forces on the system’20This statement is blatantly false. In fact, if a force is applied to a point on a rigid body so that its directional axis does not pierce the CM, it produces a torque, making a variable amount of its magnitude available for application to the overall linear motion of the system. There is no way of knowing the net contribution of the force to linear (translational) motion without calculating and subtracting the torque.
The example above is even more damning for CMT than the last. The ratio of translational to rotational motion which vector force F gives to the rod depends upon the distance R from the CM. The vector itself gives no indication of linear motion, and a sum of such vectors can’t either.21
Analysis and Criticism of CMT
We will skip general criticisms of Newtonian mechanics regarding relativistic effects. Our concern is whether CMT is consistent with the rest of Newton or even self-consistent. Granted that it appears to give good results in many instances, we want to know why, and if it fails or can fail, where and why it fails.
1. Flaw A: Failure to distinguish between types of systems
When CMT leaves the definition of ‘system’ open, allowing both arbitrary groups of point-masses in space, and also rigid bodies, it clearly fails:
The vector sum approach of dealing with external forces is only valid If we have a system of truly independent point-masses. That is, if we have a purely classical system of elastic particles having no significant gravitational, electromagnetic or other forces connecting them. In this case, each particle is truly isolated and has a definite uniform velocity, which cannot change without a collision and an exchange of energy, preserving the law of conservation of momentum.
External forces then act upon the system by direct application of a force vector to an individual point-mass, say through a collision. In this case, the definition of the velocity of the system as the weighted average of the individual velocities will naturally and correctly reflect the application of a force vector to any particle of the system. This force vector, although only physically affecting one particle, will be correctly assimilated into the average position, velocity, and momentum of the system.
But what systems conform to this classical billiard-ball style model? A bag of billiard balls released into deep space. Each exerts negligible gravitational and electrostatic force upon the others. We can even correctly handle uniform gravitational fields, since each ball receives a force proportional to its mass independent of position, and the force vectors can be simply summed. Adding the individual masses gives a total mass independent of position as well, but that is fine. Thus, if the particles are spread apart far enough so that they act independently, we can use CMT safely, and the CM will reflect the system’s velocity and momentum.
But this is precisely not the type of system we want to apply CMT to! All large masses of interest consist of closely grouped particles which are bound together by strong electromagnetic and gravitational forces. These lumps of actual matter form a class of rigid or ‘semi-rigid’ bodies in which the particles interact strongly through fields, and are capable of converting linear forces into rotation and storing energy through angular momentum.
Thus a group of (non rotating) billiard balls is a legitimate ‘system’ for CMT, but an individual billiard ball is not. Real billiard balls and other objects can rotate..
Since a raw vector sum of external forces on the system cannot distinguish between rotational or translational results, it cannot reliably represent the translational motion of any rigid or semi-rigid system of closely interacting particles. The vector sum cannot properly reflect the motion of the system or the CM, and is a meaningless value. To work, any version of CMT would require pre-treatment of external forces via torque handling techniques, and a consideration of the effects of rotation on the inertia of a system.
An easy fix is not possible for the current version of CMT however, because the vector sum is already deeply imbedded in its structure and formulation! The current version of CMT is clearly and fatally flawed. The next question is where exactly, and can it be fixed?
2. Flaw B: Non Uniqueness of the CM
Suppose we try to extend CMT to non-rotating rigid bodies. In this case, we would still need the CM concept in order to apply force without causing rotation. Thus the CM itself is retained in the theory of torque. We would also have to restrict external forces to those essentially applied along an axis of direction through the CM, but this generalizes the concept from a point to an axis.
The body would then move according to the laws of motion. However, now so does every particle in the rigid body, and nothing distinguishes the CM. In fact, part of the interpretation in the Centre of Gravity Theorem ( CGT ) predicts that all the particles in a system have the same velocity and parabolic trajectory, even when separated in the air, excluding any extra motion imparted in separation. The function of the CM is then reduced to an aid in applying force, but has no physical uniqueness in a rigid system , and no other physical meaning.
What is now left in our hands, after all these exceptions, constraints and limits? We had to abandon the summation of external forces, except for ‘systems’ of free non-interacting particles. We had to abandon rotating and rotatable systems to methods devised to handle torque. We had to abandon applying forces except uniformly or on a special axis, according to laws of equilibrium and balance. We had to abandon uniqueness and other overstated claims about the CM. The only substantive item left is a basic conservation of momentum law.
The theorem of Conservation of (linear) Momentum:An understated, (and possibly unintended) sober assessment of these results is found in Knudsen:
‘When the external force is zero, the total momentum P (the vector sum of the individual momenta) remains constant.’22
And, ‘the velocity of the CM is the same before and after (an interaction) in which the total linear momentum is conserved.’23
‘The two theorems, the conservation of (linear) momentum for a closed (isolated) system, and ( CMT ) have the same physical content.’24Note particularly the necessary word ‘closed’, i.e., completely isolated with no external forces. What is implied for systems with external forces? Not much.
Where does this leave us?
It leaves us with the perhaps daunting task of explaining the successes of CMT, removing the contradictions in it, and reformulating it in a manner that saves the phenomena, gives us a more solid theoretical foundation. Also desirable is increased accuracy and reliability, some practical methods, and some new and interesting predictions regarding Newtonian Kinematics and Dynamics. CMT itself now lies in shreds before us. The consequences for the Sphere Theorem ( ST ) which is a special case of CMT will be considered later.
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3. CMT : Blow by Blow Critique
Referring to the boxed and numbered steps in the formulation:
On 3) : The choice of weighted arithmetic mean could be challenged. It is the ‘first moment’ or centroid. This comes out of applications involving balance in a uniform field or equilibrium under rotation. Because of the physical properties of levers, torque is proportional to m x d. But field forces are proportional to m / d2 or q / d2 and other ways of weighting particle position could easily be proposed.
Take a non-uniform hollow sphere: At one point inside the gravitational field is zero ( ZG ) . This is not the GC or the CM. For obvious reasons, one may prefer this as the fundamental location of the sphere. The theoretical justification for the CM therefore is weak for both rigid and non-rigid systems.
Any supplementary argument drawing on rotation or equilibrium is irrelevant if we restrict CMT to systems only having linear motion. That the CM can actually sit in space between particles should remind us that it is not a real physical entity.
On 5) : Having the velocity etc. of the system dependent on the same definition gives consistency and beauty. But the potential circularity from interdependence throughout is far more important and worrisome. A solid physical basis for every observable, not relying upon the properties of numbers, vectors and algebraic systems, is desirable.On 6) & 7) : Momentum is a powerful and deep concept, and total momentum of a system is also clearly a useful number, but are they fundamental physical entities? Does adding bound vectors together give us another bound vector, or an unbound abstraction? We don’t need the vector sum of individual momenta to be a ‘real’ vector that can be decomposed into ‘Mass’ and ‘Velocity’, in order to have Conservation of Momentum. We could just as well have Conservation of a Sum of Momenta.
On 8) : The total mass may be the most questionable entity of all, although on the surface it looks unassailable. Yet can we just add the masses together, when they are really spread all over space? This is actually one of the very things that CMT is supposed to demonstrate, not assume.On 10) - 13): velocity CM = momentumsystem / Mtotal Is this miraculous confirmation of a theory? … or just an illusion predetermined by bad definitions and pulled off by taking abstractions too literally and allowing nonsensical operations?
On 14) & 15): The characterization of the CM seems extreme and overextended.25 But does the behaviour of the CM at rest or in uniform motion really tell us anything about how the system will accelerate under external forces? Should a baboon observe a sleeping lion to determine its temper before poking it?
On 16) - 19): This result is probably the fatal flaw in CMT and seems to have been completely misunderstood. Force applied from outside to systems as simple as a rigid stick have variable effects on linear motion, and the effects are not reflected in vector sums. We know that just adding the forces gives ludicrous results. The fact that combining such a sum with another suspect entity, total mass , results in a ‘solution’ for Newton’s 2nd law ought to make us jump out of our skins. - Not in the direction of making magical claims for the CM, but in the direction of questioning the applicability of the 2nd Law to abstract non-physical entities, and toward abandoning the current formulation of CMT.Generally, force vectors are a restricted type of vector. They are not ‘free vectors’, such as displacement vectors. They are linked to a location and time, or to a particle, and so they are called ‘bound vectors’. Normally, they can only be added vectorally (i.e., by parallelogram law) if they are bound to the same location and time, or the same particle. The directional sense of a vector often only has physical meaning when assigned to point to or from a location or source of the force.
The resultant vector is also bound to a particle or location in the same way and it also can only act upon the system through actual physical mechanisms.
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25. Theorists
seem to have stopped just short of giving it intelligence and a will
of its own! Not bad for an abstract point-mass whose location in
real space is actually empty.
Demonstrating CMT Failure
1. Analyzing Newton’s Balls
First, let's take a pair of equal point-masses joined by a rod of negligible mass (a barbell), and call it System A. The centre of mass ( CM ) for system A is at the geometrical centre (GC). For clarity we draw our point-masses as a small balls. To get the total force on another test-mass, (System B), aligned on the same x-axis we just add the forces from each end of the barbell using Newton’s formula. We don't even need the Sphere Theorem or vectors with the point-masses are all in a row in a single dimension.
For clarity we normalize the equations by making the masses of each system 1 unit, and choose distance units to make the gravitational constant G = 1 as well. This lets us drop the Gmamb part of Newton's formula. Now the formulas simply become 1/ d2 for each part, the d in this case is the distance between the test-mass (system B) and each each end of the barbell. The new equation for the force between the systems is restated as
In this form we can keep the system to system distance constant while spreading our balls. We could normalize d as well, ( i.e., d = 1 ) but we leave it in to show that the formula reduces to the inverse square law. We construct a simple triangle in the diagram to identify Theta in the equation. By inspection the angle Theta in the diagram is always less than 45o because the right triangle is formed using the distance d , the distance between the geometrical centre ( GC ) of system A and the test-mass.
The force depends upon the spread in system A and becomes Newton’s formula when the two point-masses merge as r = 0 and Theta = 0 o.We use this formula for the force of system A to calculate the position of a singular equivalent point-mass ( EPM ). That is, we wish to replace system A with a single point-mass having exactly the same mass as system A. We need to place it so that it exerts the very same force on B. By inspection we know that this location will be somewhere along the axis of alignment. We only need to find the distance from test-mass B.
Since we want a new distance ( dnew ), giving the same force, and the EPM is just a point-mass, we simply rearrange Newton’s point-mass formula ( NGF ) to solve for distance, and substitute our new equation for force into it. The Gravitational Constant and the masses cancel and drop out, leaving a general formula for the distance from the test-mass, based on Theta alone. Remember, Theta is just a convenient way to express the spread of the balls in system A:
Keeping the centre-to-centre distance d constant, we spread our balls. Naturally, the EPM distance ( dnew ) decreases as the spread and the force increases. The position of the EPM drifts closer to test-mass B , catching up with the inner point-mass by the time it reaches test-mass B.
This is precisely as it should be. Gravity is an inverse-square law. It is non-linear (exponential) by nature. The increase in force as one point-mass approaches test-mass B is not balanced by a complimentary decrease from the retreating point-mass, because the rate of change slows down farther away. This would be true of any non-linear distance law, and is not remarkable.
…Except that the system does not behave ‘as though all its mass was concentrated at the CM’ at all! At least as to magnitude of force, it behaves as though its mass were located at an entirely different position. Rotating system A will change the direction of force as well. This will be generally true of any object with a mass distributed in space in a non-spherically symmetric manner.
It is important to grasp the quality of the variation here: It is not a small error of approximation, or even adjustable with a simple factor. The force can fluctuate almost infinitely in any direction < 90o. It is virtually unpredictable without exact information as to both the distribution of mass within systems and the distances between the systems.
2. CMT contradicts NTF
It is important at this point to carefully explain what actually went wrong here. We calculated the correct location for an equivalent point-mass the hard way, by actually computing and adding the (two) individual forces acting on the system from outside. It is not the location predicted by CMT. That is, CMT contradicts results using Newton’s Gravity Formula for point-masses ( NGF ), which is the core of NTF. But CMT was supposed to be a shortcut for NGF, to save us having to use it.
Is NGF at fault, or CMT? In fact, the fault is clearly with CMT. It simply provides no mechanism to accommodate the vast fluctuations in force possible with even the simplest gravitational objects. The failure of CMT is built in. Creating a weighted average of position just throws away critical information as to distribution of mass right from the start. This self-mutilation fatally cripples CMT. It simply can’t do what is being asked of it, and so CMT invariably produces force vectors which are incorrect in magnitude and/or direction. It is no surprise that CMT is internally inconsistent. As a result, CMT not only contradicts NGF, but also ST , as we will see below. Since ST is just a special case of CMT, it simply contradicts itself!
Although we’d prefer CMT to NGF, we can’t. Abandoning NGF wouldn’t save CMT in any case. There is no way to relax the absolute requirement that force vectors sum in a coherent and self-consistent way. Couldn’t CMT be valid for some other unknown reason? No. CMT itself is an invalid method for calculating external gravitational forces and predicting motion, because it wipes out critical information.
3. CMT contradicts Itself
A simple example shows that CMT and ST contradict one another. Take a solid uniform half-sphere, and calculate its centre of mass ( CM ). For our purposes, it is irrelevant where it actually is, but by inspection it must be somewhere on the axis of symmetry. We can mark it with a dot . The important thing is that the CM is definitely a fixed point inside the object, since it is the weighted average of all the atoms making it up, and its a rigid solid.
Placing two half-spheres together to make a whole sphere puts the CM for each half at a fixed, equal opposing distance from the GC of the whole sphere. By symmetry, the CM for the whole sphere is at the GC itself.
Now add a test-mass. This is a precise parallel to the case above with the balls. The forces upon each half should add up to the force upon the whole sphere. Remember, this is just a static case: there is no motion involved. CMT should enable us to replace the halves with point-masses at their respective CM s , and add the vector forces. If CMT is self-consistent, an EPM for the halves should be at the same location as the CM of the whole sphere. But we already know this is not the case. Each half contributes unequally to the force, because of its position. The resultant force gives an EPM location that is closer to the test-mass than the CM of the sphere. Clearly CMT can’t deliver the goods.
Similarly, One could split up the sphere horizontally: Again treating the halves separately and using the Centre of Mass Theorem diverges from the result predicted by the Sphere Theorem:
According to CMT, any half-sphere, spinning or not, and regardless of orientation, will follow the laws of motion as if its mass were concentrated at that point. This must entail applying vectors to the CM and using actual masses for practical calculation. For such purposes, one is to treat the half-sphere as a point-mass of exactly the same mass located at the CM. That is, if CMT has any value, it must mean that we can use it along with NGF to calculate the external force between objects, and predict the resulting motion. This is precisely how Newton uses it to calculate and predict projectile motion under earth gravity. There, the gravitational field is virtually uniform for small objects near the surface.
This is not a trivial point, nor can one get around it by insisting that the centre of mass concept is only meaningful as a centre of gravity and ‘only applicable in a uniform field’. The gravity field in the isolation of deep space would presumably be the most uniform field of all. There, the mass of the universe (the distant backdrop of stars) is randomly distributed but essentially spread evenly in all directions.
If you can use CMT out in space, and also next to a huge mass like the earth, you must be able to use it pretty well everywhere. But if we can use it almost anywhere, what are the real restrictions as to when and how we can use it? CMT itself offers no help or explanation for such internal contradictions as plainly exist.
4. The 'Success' of CMT: A Cosmological Coincidence
As it turns out, both Newton’s and our good fortune has more to do with circumstantial factors than universal physical principles. The apparent success of CMT is just that: an appearance created by special circumstance. The vast distances of space dwarf the actual size of stars so much that those spherical bodies are essentially point-masses, and ST while trivially true is also untestable as such. On the other hand, gravity is so weak a force that it is virtually immeasurable between even adjacent objects at the normal human scale of experience. Earth gravity originates at an average distance roughly the radius of the earth, again dwarfing the size of ordinary objects.
Thus the mechanism for CMT ‘s apparent success is really just a set of special circumstances and cases. The root of the matter is simply that for most objects considered, the mass is sufficiently concentrated and point-like for any errors in CMT to pass undetected under the umbrella of imprecision of measurement.
CMT and GR
But does this sufficiently explain the success and accuracy of CMT ? In fact, no it does not. There remains another aspect of CMT not yet discussed, and a hidden mechanism to explain it. Basing CMT on NTF alone is a failure, because any simple version of CMT fails a compatibility test with NGF, the heart of NTF, as well as failing the basic self-consistency test.
Looking back at examples given to justify CMT, one notices that rotation is always involved, although not necessarily gravity. This indicates there is real physical content behind the mistaken current formulation. If our only real knowledge of the CM comes through rotation or interaction with a uniform gravitational field, we should look more closely here for a better foundation than CMT can offer for both phenomena in relation to the CM.
But a new formulation of CMT is necessary and desirable, to account for rotation, moments of inertia and angular momentum. That is, paradoxically, while rotation effects are the downfall of CMT as currently formulated, and must be excluded from it, rotation is probably the only justification for any continued use of the CM concept! Basing some new version of CMT on GR or a revised version of Newtonian Mechanics offers promise.
Hidden Penalties and Dangers
Our good practical good fortune has its flip side in our theoretical misfortune and the potential dangers in assuming the validity of the concepts packaged in CMT. It seems reasonable to address these issues.
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