Okay, thank you for reading through my last posts.
I think there may still be an issue of language here,
in regard to your point (12) , where you introduce "right-angle".
I think however I can cut to the chase with a new diagram:
This diagram asks the pertinent question,
namely where will the resultant final vector end up?
Could it be that by 'luck' or by the order of the universe,
that after all is said and done, the final vector turns out to
align at least with the Cone-Axis, since we know it won't align
with the original GC axis as posed by the Center of Mass idea.
In fact, there is strong appeal in this argument, on the same
Euclidean basis that we concluded that the Cone-Axis was fixed
by the angle of the plane from the GC line.
We may also (although it will be a leap for some) "by inspection"
conclude that WHATEVER answer the Gravity-Law may finally produce,
the PROPORTION of the Vectors for the Near and Far halves will be constant.
This intuitive eureka is probably quite right (although it may not hold
for just ANY hypothetical Gravity Law).
If this is so, we naturally hold out hope that the final Vector may indeed
be locked onto the Cone Axis.
However, I strongly suggest you look now TWICE at the diagram above:
The following additional observations can be made:
(1) The lower force component Vector will always be BELOW the GC line.
This is a natural result of ALL the mass for the lower half being below the line.
(2) This constraint will require the UPPER Force Vector to be a MINIMUM
magnitude and length (for any expected direction), to pull the final
Vector into line with the moving Cone Axis, as the ellipse is tilted or
elongated.
I say this comfortably, but hoping to avoid doing a calculation,
that "by inspection" this cannot be likely for all angles of the ellipse.
In the diagram above, for instance, the Near-force is 3x the magnitude
of the Far-force Vector, so that the Final vector is aligned with the Cone Axis.
Is this a plausible ratio of magnitudes for the diagram before us?
To change the magnitudes to something more sensible, you must
also change the direction of the component vectors to something less plausible...
The reason we ask this question, namely can we align the forces with the Cone Axis,
is because although this isn't sufficient to save Newton's argument,
(it still has other gaping holes), it would be a minimum necessary requirement,
for the analogy to hold.
And it would be a remarkable result in itself, which would go far beyond
proving the Parallel Planes Theorem.
I think there may still be an issue of language here,
in regard to your point (12) , where you introduce "right-angle".
I think however I can cut to the chase with a new diagram:
This diagram asks the pertinent question,
namely where will the resultant final vector end up?
Could it be that by 'luck' or by the order of the universe,
that after all is said and done, the final vector turns out to
align at least with the Cone-Axis, since we know it won't align
with the original GC axis as posed by the Center of Mass idea.
In fact, there is strong appeal in this argument, on the same
Euclidean basis that we concluded that the Cone-Axis was fixed
by the angle of the plane from the GC line.
We may also (although it will be a leap for some) "by inspection"
conclude that WHATEVER answer the Gravity-Law may finally produce,
the PROPORTION of the Vectors for the Near and Far halves will be constant.
This intuitive eureka is probably quite right (although it may not hold
for just ANY hypothetical Gravity Law).
If this is so, we naturally hold out hope that the final Vector may indeed
be locked onto the Cone Axis.
| Reminder: We have shown before that this is NOT a necessary condition for the Parallel Planes / Opposing Cones Law, since wherever the final Vector goes, there is a complimentary tilt on the other cone that keeps the two vectors on a single line. |
However, I strongly suggest you look now TWICE at the diagram above:
The following additional observations can be made:
(1) The lower force component Vector will always be BELOW the GC line.
This is a natural result of ALL the mass for the lower half being below the line.
| It also incidentally shows the failure of the Center of Mass idea, since even when the ellipse (or circle) is perpendicular to the GC line, the component vectors will not be on the line, and the force will be proportionately weakened, AND the Equivalent Point Mass (EPM) will actually have to be placed on the GC-line FURTHER AWAY than the plane of the disk/ellipse, from the test-mass. This is nonetheless a natural result, since spreading the mass over a plane has the effect of moving the mass further away from the test-mass, and naturally weakens the gravitational force. |
(2) This constraint will require the UPPER Force Vector to be a MINIMUM
magnitude and length (for any expected direction), to pull the final
Vector into line with the moving Cone Axis, as the ellipse is tilted or
elongated.
I say this comfortably, but hoping to avoid doing a calculation,
that "by inspection" this cannot be likely for all angles of the ellipse.
In the diagram above, for instance, the Near-force is 3x the magnitude
of the Far-force Vector, so that the Final vector is aligned with the Cone Axis.
Is this a plausible ratio of magnitudes for the diagram before us?
To change the magnitudes to something more sensible, you must
also change the direction of the component vectors to something less plausible...
The reason we ask this question, namely can we align the forces with the Cone Axis,
is because although this isn't sufficient to save Newton's argument,
(it still has other gaping holes), it would be a minimum necessary requirement,
for the analogy to hold.
And it would be a remarkable result in itself, which would go far beyond
proving the Parallel Planes Theorem.
-------------
Here's another couple of diagrams to assist in visualization:
Remember that it was Newton who composed the original problem,
and used opposing Cones with aligned Cone-axes.
Irrespective of the shape, or regularity of the cones themselves,
(Newton chose circular cones, but elliptical cones could also be used,
in a similar thought experiment),
the final opposing force Vectors must fulfill TWO conditions:
(1) They must be aligned in opposite directions along a single axis.
(2) They must be equal in magnitude.
Now consider the simpler case of parallel planes cut by Newton's
opposing cones, aligned on a single axis:
In the parallel planes case,
the the opposing forces must still share a single axis,
and be equal in magnitude, but that axis can pass through
the test-particle in any direction.
(any direction that is, along the vertical line of symmetry of the ellipses).
In this case, since the forces for each ellipse are complimentary
in the geometrical sense, and equal force components will also align
on a shared axis opposing each other, it is trivial that all the forces
cancel, and there is no NET force on the particle.
The only constraint imposed here is that the planes be parallel,
which forces the alignment of the component Vectors,
and vertical symmetry down the middle of the array on
the plane through both axes.
Now look at the REAL case that Newton proposed, where
the ellipses could actually tilt in different directions and are
NOT simply intersections of parallel planes;
Here a very significant extra constraint must be imposed,
SINCE THE OPPOSING COMPONENT VECTORS ARE NOT EQUAL
in either Magnitude or Direction:
Now they must be coordinated such that however they were acquired,
the RESULTANT FINAL Force Vectors remain on a single shared line,
and equal in magnitude.
To fulfill Newton's original conditions, we must still have opposing cones,
and they must share an axis of symmetry, even if they are not circular
(i.e., radially symmetrical). We assume circular cones for now,
since we need not arbitrarily change the original problem.
In the diagram, it can be seen clearly now that
the Geometric Center line, must now be BENT, into two separate lines,
in order to maintain the condition of opposing cones.
The opposing Ellipse also slides downward, to keep it aligned with
the opposing cone. (we keep the original left GC line and particle fixed,
as per previous convention).
This makes the new GC lines of no use for balancing opposing forces,
and we must look to other lines that can remain straight while
passing through the test particle.
By simple inspection, the only line with the required symmetry
and potential to function as a balance of forces is the CONE axis.
We must presume then intuitively that the component forces for
each half-ellipse must coordinate such that the final Vector is the Cone axis.
Furthermore, we may further reason by inspection that since the
new location of the Cone Axis relative to the ellipses, pierces them
off-center from the original GC of each ellipse, the NEW division
into "portions" of each ellipse must fall on this piercing-point.
This is where your reasoning presumably comes in.
That is, on the one hand, it is trivial that dividing the ellipses
down the middle vertically gives symmetry and equal forces,
due to symmetry.
But it is NOT trivial to show that the "horizontal" division
of the ellipses above and below the Cone Axis must have equal forces,
which is actually REQUIRED if the new orientation is to balance.
Because the new PLANE aligned with the Cone Axis chops the
ellipses into unequal pieces, it must now be shown that their
distances and distribution of mass perfectly balance, which is
equivalent to saying that the force from each of the FOUR 'portions'
are equal in magnitude and direction.
This new strict condition must hold true for every complimentary pair
of angled ellipses.
Essentially we are describing the condition that the Force from
every such ellipse fall upon the Cone Axis at all times.
----------
Remember that it was Newton who composed the original problem,
and used opposing Cones with aligned Cone-axes.
Irrespective of the shape, or regularity of the cones themselves,
(Newton chose circular cones, but elliptical cones could also be used,
in a similar thought experiment),
the final opposing force Vectors must fulfill TWO conditions:
(1) They must be aligned in opposite directions along a single axis.
(2) They must be equal in magnitude.
Now consider the simpler case of parallel planes cut by Newton's
opposing cones, aligned on a single axis:
In the parallel planes case,
the the opposing forces must still share a single axis,
and be equal in magnitude, but that axis can pass through
the test-particle in any direction.
(any direction that is, along the vertical line of symmetry of the ellipses).
In this case, since the forces for each ellipse are complimentary
in the geometrical sense, and equal force components will also align
on a shared axis opposing each other, it is trivial that all the forces
cancel, and there is no NET force on the particle.
The only constraint imposed here is that the planes be parallel,
which forces the alignment of the component Vectors,
and vertical symmetry down the middle of the array on
the plane through both axes.
Now look at the REAL case that Newton proposed, where
the ellipses could actually tilt in different directions and are
NOT simply intersections of parallel planes;
Here a very significant extra constraint must be imposed,
SINCE THE OPPOSING COMPONENT VECTORS ARE NOT EQUAL
in either Magnitude or Direction:
Now they must be coordinated such that however they were acquired,
the RESULTANT FINAL Force Vectors remain on a single shared line,
and equal in magnitude.
| That is, removing the constraint that the component vectors be equal and opposite, imposes another two constraints: (1) that they nonetheless combine to align on a single axis shared by each side. (2) that this line pass through the test particle. There is already an unspoken constraint that ALL the final force lines under discussion fall upon the vertical plane passing through the GC line and the Cone Axis, since this is the plane of bilateral symmetry for the system. |
To fulfill Newton's original conditions, we must still have opposing cones,
and they must share an axis of symmetry, even if they are not circular
(i.e., radially symmetrical). We assume circular cones for now,
since we need not arbitrarily change the original problem.
In the diagram, it can be seen clearly now that
the Geometric Center line, must now be BENT, into two separate lines,
in order to maintain the condition of opposing cones.
The opposing Ellipse also slides downward, to keep it aligned with
the opposing cone. (we keep the original left GC line and particle fixed,
as per previous convention).
This makes the new GC lines of no use for balancing opposing forces,
and we must look to other lines that can remain straight while
passing through the test particle.
By simple inspection, the only line with the required symmetry
and potential to function as a balance of forces is the CONE axis.
We must presume then intuitively that the component forces for
each half-ellipse must coordinate such that the final Vector is the Cone axis.
Furthermore, we may further reason by inspection that since the
new location of the Cone Axis relative to the ellipses, pierces them
off-center from the original GC of each ellipse, the NEW division
into "portions" of each ellipse must fall on this piercing-point.
This is where your reasoning presumably comes in.
That is, on the one hand, it is trivial that dividing the ellipses
down the middle vertically gives symmetry and equal forces,
due to symmetry.
But it is NOT trivial to show that the "horizontal" division
of the ellipses above and below the Cone Axis must have equal forces,
which is actually REQUIRED if the new orientation is to balance.
Because the new PLANE aligned with the Cone Axis chops the
ellipses into unequal pieces, it must now be shown that their
distances and distribution of mass perfectly balance, which is
equivalent to saying that the force from each of the FOUR 'portions'
are equal in magnitude and direction.
This new strict condition must hold true for every complimentary pair
of angled ellipses.
Essentially we are describing the condition that the Force from
every such ellipse fall upon the Cone Axis at all times.
----------
We should now be ready to trivially disprove the Cone-Axis idea:
If the forces on each side of the new division of the ellipse balance,
and result in a force along the axis, then the two portions are intechangeable.
That is, the left and right side of the diagram below should be equivalent:
But by inspection, this is unlikely...
At this point, the idea that the smaller amount of mass on the left,
NEARER to the test mass, can balance out the larger mass on the right,
farther away, must be abandoned.
Its not about mere proportions anymore.
We leave this proof to the student.
If the forces on each side of the new division of the ellipse balance,
and result in a force along the axis, then the two portions are intechangeable.
That is, the left and right side of the diagram below should be equivalent:
But by inspection, this is unlikely...
At this point, the idea that the smaller amount of mass on the left,
NEARER to the test mass, can balance out the larger mass on the right,
farther away, must be abandoned.
Its not about mere proportions anymore.
We leave this proof to the student.
---------
You should also have a second look at these three points you made:
(9) I think is where you have have gotten a bit muddled again:
I would not assert that the near-half EPM is above the Cone Axis.
That would be yet to be determined.
What we can assert from the initial argument is that the
near-half EPM is above the GC axis.
Whether it coincides with, lags, or surpasses the moving Cone Axis,
is the open question.
I would suggest from the diagrams that although we have been
exaggerating the directions of the component Vectors for visualization
purposes, in fact they are less extreme than the tilt of the Cone Axis
away from the GC axis.
You rightly note that the lower component vector tilts downward at
less of an angle than the upward tilt of the upper component vector,
but we should be talking at this stage in terms of the GC axis, not the
Cone axis, which is also moving as the ellipse is tilted.
Your (11) will stand, provided we are talking about the angle in relation
to the GC axis, not the relation to the cone axis, which is unknown
from the discussion to that point.
9) Under the premise (which I do not contest) that the near half EPM is above the cone axis, the direction of its force vector to the test particle will be at a finite angle to the cone axis. 10) Similarly, the far half EPM is below the cone axis, and thus its force vector will be directed at an angle to the cone axis. But since it is below the cone axis, the angle will veer off the cone axis on the opposite side from the near side–to-cone axis angle. 11) Geometry dictates that the magnitude of the far side-to-cone axis angle will be smaller than the near side-to-cone axis angle |
(9) I think is where you have have gotten a bit muddled again:
I would not assert that the near-half EPM is above the Cone Axis.
That would be yet to be determined.
What we can assert from the initial argument is that the
near-half EPM is above the GC axis.
Whether it coincides with, lags, or surpasses the moving Cone Axis,
is the open question.
I would suggest from the diagrams that although we have been
exaggerating the directions of the component Vectors for visualization
purposes, in fact they are less extreme than the tilt of the Cone Axis
away from the GC axis.
You rightly note that the lower component vector tilts downward at
less of an angle than the upward tilt of the upper component vector,
but we should be talking at this stage in terms of the GC axis, not the
Cone axis, which is also moving as the ellipse is tilted.
Your (11) will stand, provided we are talking about the angle in relation
to the GC axis, not the relation to the cone axis, which is unknown
from the discussion to that point.
Quote:
Originally Posted by DavisBJ
I am going to shift gears a little
bit. I contend that in broad scope, we are asking what is the net
gravitational force on a randomly placed particle within the shell. To
do that, we will have to consider every bit of the shell, and ask
ourselves what is the vector total of every bit of the shell, no matter
how we chose to chop the shell up into manageable pieces. For now, the
“forward” and “backward” elliptical pieces will do, as long as we
ultimately account for the whole shell.
|
Please remember that the modern calculus solution was never in dispute,
as a mathematical structure.
In fact, we went over the mathematical solution to the problem via calculus,
and did the complete process including every stage of integration.
Finding fault with the calculus itself was not our thrust in any way.
As a physicist however, I objected originally to the calculus solution to
to the problem, not as a mathematical techique asserting something
about mathematical entities, but rather as an application to a physical
situation.
My objections there involve in particular the spatial distribution of
mass, which is "quantized" or rather non-smooth at the scale of
atomic particles; i.e., the use of the continuum to solve this problem
fails directly as a result of the clumped distribution of mass as particles.
This quantization of mass distribution results in near-proximity
imbalances and anomalies in the gravitational field, that prevent
the results of the Sphere Theorem from being relevant and applicable
to a physical situation of that type.
The problem there at distances and sizes within range of molecular
structures is perfectly plain, namely that there in fact is no continuum,
and so no balance of forces is possible in spherical structures like
Carbon micro-spheres and 'Bucky-ball' type structures.
The Sphere Theorem fails however, as we've observed, on several levels,
particularly involving misunderstandings regarding the Center of Mass
techniques and reasoning based on an incomplete comprehension of
those approximations.
Thus, other anomalies and inaccuracies are expected, and naturally,
other measures must be taken to bring the early ideas of Newton
in regard to application of gravitational formulas in line with a
properly self-consistent and coherent gravity theory with applications.
Quote:
| Now we look at the tilted ellipse that we have discussed at length.
We conclude that no way does that piece result in a force aligned with
the cone axis that outlines the ellipse. Fine, it is what it is. We have
(in abstract) computed the magnitude of the force that ellipse has on
the test particle, along with its direction. Part of job done. Get out a
tally sheet out, and record “ellipse 1 results = qq Newtons of force
directed in such and such a specific direction. There is no restriction on which way I next turn the cone from the test particle, as long as I faithfully follow our procedure of vectorially adding up the forces due to the “near side” and “far side” halves of the enclosed ellipse. Whatever that answer comes out to be, we will add that to the tally sheet and then move on. So, to make sure we don’t leave bothersome gaps, I am going to rotate the cone so the cone axis just skims the edge of the far half of the ellipse we just finished with. Now, we consider the new ellipse, and its two halves. The new ellipse far half is now even farther away than the far half of the prior ellipse. And the near half … the near half … the near half is mostly the old far half of the prior ellipse. But we have already accounted for the effect that has on the test particle, it’s already a major part of the figure on our force tally sheet. We can compute the force contribution of this overlap of the old and the new ellipse in the net force from the first ellipse, or the second, but we don’t get to double dip. That overlapping segment of the wall contributes one force vector, and where do you want it counted – as part of the first ellipse net force, or as part of the second ellipse net force? |
Your idea of overlapping ellipses is interesting.
I'm not sure what advantage it is going to procure
in an analysis of the problem, but please go ahead:
In particular, can you sketch a few diagrams of what you are intending?
You can use an online drawing program and link to it if you want to.
--------
Actually one of the things I'd like to do before abandoning this thread
to the other trolls, is to make what should be an obvious observation:
DavisBJ, whoever he may be, appears to me to be 'legit',
in the sense that whether or not he is a physicist of some branch
or other, he appears at least to be an engineer.
I acknowledge this from his plain pattern of behaviour:
(1) Although not all engineers/physicists are equal, especially in areas
outside of their expertise and interest, all engineers that I've met,
don't let a matter of science rest until they understand it, and typically
they will try and solve the problem, albeit in their own way.
This is precisely what DavisBJ has done.
And on that note, I have to confess that his first pass at explaining
his own calculations are not at all clear (I didn't expect them to be),
and he has not provided diagrams. This is unfortunate, for it is also
unclear to me (and I am at least very familiar with the problem),
whether he has indeed got a proof, and even whether in his own way,
he has come around to acknowledging that there in fact is some pull
on a particle inside a uniform hollow sphere.
It would really appreciated by me (although that is not essential),
if I could get him to clarify just what he has seen in his own calculations,
because at least others, for instance Stripe and many readers,
will want to know what side of the question he comes down on,
and will want to also see why.
We can put DavisBJ's actions in direct contrast to those of gcthomas,
who simply assumes he understands the questions, makes no analysis
of his own, shows no interest in the problem other than to contradict
the claim without proofs or mathematical evidence, only seeks to
argue a smokescreen about Newton's Principia in Latin, and finally,
simply asserts my discussion is wrong and the 'status quo' as he
thinks he understands it is 'right'. This kind of behaviour is completely
unscientific, unconvincing to the readers he attempts to divert, and
rather blatantly shows he is the laziest 'scientist' around, or simply a fake.
(2) DavisBJactually did struggle with unfamiliar material, making some
mistakes, and asking questions, and clarifying his own understanding of
what the thread is about. He was polite and actually did read the stuff,
spent a lot of effort trying to understand the arguments, and was very
interested and concerned about the conclusions and claims.
Again this is precisely what another scientist or technician would do,
as opposed to the behaviour of others. His actual success in his
endeavour is not so relevant, because of course there are all kinds of
engineers and scientists with specialist training and holes in their skillset.
Once again DavisBJ comes across as convincing, whereas gcthomas
fails and faceplants himself again.
(3) DavisBJ showed the typical confidence of an engineer/physicist,
utterly convinced he would have no real issue solving the problem,
and answering for himself any lingering questions. Also typically, it
naturally turned out to be harder and more complex than he anticipated,
but that hardly phased him and he pressed ahead to his own solution,
applying such tools as he found himself with.
Again, a point awarded for realism and authenticity. Its hard not to
believe that DavisBJ is a scientist of some kind. He is utterly consistent
in his method, activity, and attitudes. Maybe a bit heavier on the
'beer-drinking engineer' side of the scale than the 'tea-totaling physicist'
side, but more than in the ballpark.
Contrast that again with gcthomas, who can't make the effort to
crawl out of his armchair far enough to reach for a pencil or even
an online calculator, and try to get it, yet has ample energy for
troll-like contradicting, mockery, and insults.
I give 3 out of 3 for DavisBJ as being the more authentic scientist/engineer.
The peanut gallery crowd is self-evident too,
but I feel obligated to give recognition where due,
even if DavisBJ isn't on our team.
Actually one of the things I'd like to do before abandoning this thread
to the other trolls, is to make what should be an obvious observation:
DavisBJ, whoever he may be, appears to me to be 'legit',
in the sense that whether or not he is a physicist of some branch
or other, he appears at least to be an engineer.
I acknowledge this from his plain pattern of behaviour:
(1) Although not all engineers/physicists are equal, especially in areas
outside of their expertise and interest, all engineers that I've met,
don't let a matter of science rest until they understand it, and typically
they will try and solve the problem, albeit in their own way.
This is precisely what DavisBJ has done.
And on that note, I have to confess that his first pass at explaining
his own calculations are not at all clear (I didn't expect them to be),
and he has not provided diagrams. This is unfortunate, for it is also
unclear to me (and I am at least very familiar with the problem),
whether he has indeed got a proof, and even whether in his own way,
he has come around to acknowledging that there in fact is some pull
on a particle inside a uniform hollow sphere.
It would really appreciated by me (although that is not essential),
if I could get him to clarify just what he has seen in his own calculations,
because at least others, for instance Stripe and many readers,
will want to know what side of the question he comes down on,
and will want to also see why.
We can put DavisBJ's actions in direct contrast to those of gcthomas,
who simply assumes he understands the questions, makes no analysis
of his own, shows no interest in the problem other than to contradict
the claim without proofs or mathematical evidence, only seeks to
argue a smokescreen about Newton's Principia in Latin, and finally,
simply asserts my discussion is wrong and the 'status quo' as he
thinks he understands it is 'right'. This kind of behaviour is completely
unscientific, unconvincing to the readers he attempts to divert, and
rather blatantly shows he is the laziest 'scientist' around, or simply a fake.
(2) DavisBJactually did struggle with unfamiliar material, making some
mistakes, and asking questions, and clarifying his own understanding of
what the thread is about. He was polite and actually did read the stuff,
spent a lot of effort trying to understand the arguments, and was very
interested and concerned about the conclusions and claims.
Again this is precisely what another scientist or technician would do,
as opposed to the behaviour of others. His actual success in his
endeavour is not so relevant, because of course there are all kinds of
engineers and scientists with specialist training and holes in their skillset.
Once again DavisBJ comes across as convincing, whereas gcthomas
fails and faceplants himself again.
(3) DavisBJ showed the typical confidence of an engineer/physicist,
utterly convinced he would have no real issue solving the problem,
and answering for himself any lingering questions. Also typically, it
naturally turned out to be harder and more complex than he anticipated,
but that hardly phased him and he pressed ahead to his own solution,
applying such tools as he found himself with.
Again, a point awarded for realism and authenticity. Its hard not to
believe that DavisBJ is a scientist of some kind. He is utterly consistent
in his method, activity, and attitudes. Maybe a bit heavier on the
'beer-drinking engineer' side of the scale than the 'tea-totaling physicist'
side, but more than in the ballpark.
Contrast that again with gcthomas, who can't make the effort to
crawl out of his armchair far enough to reach for a pencil or even
an online calculator, and try to get it, yet has ample energy for
troll-like contradicting, mockery, and insults.
I give 3 out of 3 for DavisBJ as being the more authentic scientist/engineer.
The peanut gallery crowd is self-evident too,
but I feel obligated to give recognition where due,
even if DavisBJ isn't on our team.
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