I've decided to give you one more shot at this, with diagrams.
The ideas may be a bit slippery, but they only require a grasp
of Euclidean geometry:
Consider the following NEW diagram, this time with ellipses:
Here we have for argument's sake AN ELLIPSE, rotated on an axis through
its Geometric Center (GC), dividing the equally distributed mass on either side
of the line.
Intuitively we know that the near side, being closer, will exert more
gravitational force than the far side.
So far, so good. This means that the DIRECTION OF FORCE,
must necessarily tilt away from the GC axis, and toward the top (upward)
in the diagram, which is basically a SIDE VIEW.
However, as the ellipse is tilted, the Cone of intersection with a plane
coinciding with the flat ellipse (of uniform density) tilts, distorts, and shrinks.
(The max excursion of the cone will be when the elliptical disk is perpendicular).
So now lets look at the diagram with the Cone of Intersection superimposed:
Immediately we should notice that the CONE axis, has drifted away from the GC Axis.
This corresponds to your previous observation that a larger part of the ellipse
is now BELOW the half-way point of the cone.
Put another way, as the Cone tilts to accommodate the rotating ellipse,
its Center Axis, tilts upward. Remember that the Cone Axis of Symmetry,
being always centered in the cone, will be equal angles from every side.
From the side view, when the cone is projected flat, the Cone Axis will
always BISECT the angle between the upper and lower edge of the cone.
We already know that relative to the GC axis, the direction of force also tilts
away upward from it. The question is, will it follow exactly the Cone Axis?
There are a few observations from Euclidean geometry that will help here.
(1) Regardless of the distance of an ellipse intersecting the cone
(which is now held constant for the thought experiment),
the angle of the Cone Axis to the GC Axis will stay the same.
That is, by the law of Euclidean proportion, (regardless of how big we draw),
the angles and position of each axis stay the same. The only thing
that would move would be the size and position of the ellipse, but NOT
its own angle re: the GC axis and the test particle.
(2) Regardless of the size and distance of the ellipse,
the position of the Cone axis and the PROPORTION of the Ellipse on
either side of the Cone Axis will stay the same, again by the Law of
Euclidean proportions.
As long as the ellipse stays at the same angle, we can simply increase
its size and move it to the left, or decrease its size and move it to the
right. The Cone Axis and cone don't change, and neither does the GC Axis.
(3) The force itself is not dependent upon the geometry.
That is, only its direction is dependent, dependent upon the angle of the Ellipse.
The actual force could be defined by any Law, not just inverse square,
and yet the result should be the same:
(4) The DIRECTION of the force (only) IS dependent on the geometry.
That is, once the actual direction is determined, by whatever law,
the ANGLE of the LINE OF FORCE will not change. It will be strictly
dependent upon the actual law, when we calculate the force for each
side of the ellipse. This is done by dividing the ellipse as usual,
exactly in half for easy calculation, and assuming the force is
strictly dependent upon distance/direction for an equivalent point-particle.
That is, even if the 'Center of Mass' idea strictly fails as a generalization
for shapes that are not radially symmetric in proximity, it still enables us to ballpark:
The approximate Center of Mass (CM) for each half will be located
somewhere near the physical geometric average of the positions of
all the components of each half (i.e., somewhere near the GC of each half).
Making the diagram larger or smaller won't change the angle of
direction for the combined forces for each half. This is a simple
"by inspection" line of reasoning based on the Euclidean Law of proportions
once again.
The final question then is:
(A) Does the real direction of force track the Cone Axis?
Because if it does, we could make a much larger generalization
than that of the parallel planes law. In fact, it would mean that
even the ANGLE of an ellipse didn't matter. it would act as if
it were a point particle at the point where the Cone Axis pierced it.
This would in essence mean that the force was only dependent upon density of mass.
In this case, the 'tilt' of an ellipse wouldn't matter for the purposes
of balancing the forces between two opposing ellipses. Their effective
Center of Mass would always be on the CONE axis, and the force would
always be along that axis, balanced in direction at least, if not strength.
It would follow that the forces would only depend upon the density value.
The problem of course is that since the force is ALWAYS (also) dependent
upon the DENSITY VALUE, the direction of force for a given configuration
is not fixed, even though the CONE axis IS fixed.
This means that the Direction of Force CANNOT track the Cone Axis,
and we already know that it can't track the GC axis, so ...
Both the Center of Mass (approximation) and the argument of Newton
regarding the balancing of opposing 'cone intersections' on the sphere
surface are false, and the Sphere Theorem must fail.
The ideas may be a bit slippery, but they only require a grasp
of Euclidean geometry:
Consider the following NEW diagram, this time with ellipses:
Here we have for argument's sake AN ELLIPSE, rotated on an axis through
its Geometric Center (GC), dividing the equally distributed mass on either side
of the line.
Intuitively we know that the near side, being closer, will exert more
gravitational force than the far side.
So far, so good. This means that the DIRECTION OF FORCE,
must necessarily tilt away from the GC axis, and toward the top (upward)
in the diagram, which is basically a SIDE VIEW.
However, as the ellipse is tilted, the Cone of intersection with a plane
coinciding with the flat ellipse (of uniform density) tilts, distorts, and shrinks.
(The max excursion of the cone will be when the elliptical disk is perpendicular).
So now lets look at the diagram with the Cone of Intersection superimposed:
Immediately we should notice that the CONE axis, has drifted away from the GC Axis.
This corresponds to your previous observation that a larger part of the ellipse
is now BELOW the half-way point of the cone.
Put another way, as the Cone tilts to accommodate the rotating ellipse,
its Center Axis, tilts upward. Remember that the Cone Axis of Symmetry,
being always centered in the cone, will be equal angles from every side.
From the side view, when the cone is projected flat, the Cone Axis will
always BISECT the angle between the upper and lower edge of the cone.
We already know that relative to the GC axis, the direction of force also tilts
away upward from it. The question is, will it follow exactly the Cone Axis?
There are a few observations from Euclidean geometry that will help here.
(1) Regardless of the distance of an ellipse intersecting the cone
(which is now held constant for the thought experiment),
the angle of the Cone Axis to the GC Axis will stay the same.
That is, by the law of Euclidean proportion, (regardless of how big we draw),
the angles and position of each axis stay the same. The only thing
that would move would be the size and position of the ellipse, but NOT
its own angle re: the GC axis and the test particle.
(2) Regardless of the size and distance of the ellipse,
the position of the Cone axis and the PROPORTION of the Ellipse on
either side of the Cone Axis will stay the same, again by the Law of
Euclidean proportions.
As long as the ellipse stays at the same angle, we can simply increase
its size and move it to the left, or decrease its size and move it to the
right. The Cone Axis and cone don't change, and neither does the GC Axis.
(3) The force itself is not dependent upon the geometry.
That is, only its direction is dependent, dependent upon the angle of the Ellipse.
The actual force could be defined by any Law, not just inverse square,
and yet the result should be the same:
(4) The DIRECTION of the force (only) IS dependent on the geometry.
That is, once the actual direction is determined, by whatever law,
the ANGLE of the LINE OF FORCE will not change. It will be strictly
dependent upon the actual law, when we calculate the force for each
side of the ellipse. This is done by dividing the ellipse as usual,
exactly in half for easy calculation, and assuming the force is
strictly dependent upon distance/direction for an equivalent point-particle.
That is, even if the 'Center of Mass' idea strictly fails as a generalization
for shapes that are not radially symmetric in proximity, it still enables us to ballpark:
The approximate Center of Mass (CM) for each half will be located
somewhere near the physical geometric average of the positions of
all the components of each half (i.e., somewhere near the GC of each half).
Making the diagram larger or smaller won't change the angle of
direction for the combined forces for each half. This is a simple
"by inspection" line of reasoning based on the Euclidean Law of proportions
once again.
The final question then is:
(A) Does the real direction of force track the Cone Axis?
Because if it does, we could make a much larger generalization
than that of the parallel planes law. In fact, it would mean that
even the ANGLE of an ellipse didn't matter. it would act as if
it were a point particle at the point where the Cone Axis pierced it.
This would in essence mean that the force was only dependent upon density of mass.
In this case, the 'tilt' of an ellipse wouldn't matter for the purposes
of balancing the forces between two opposing ellipses. Their effective
Center of Mass would always be on the CONE axis, and the force would
always be along that axis, balanced in direction at least, if not strength.
It would follow that the forces would only depend upon the density value.
The problem of course is that since the force is ALWAYS (also) dependent
upon the DENSITY VALUE, the direction of force for a given configuration
is not fixed, even though the CONE axis IS fixed.
This means that the Direction of Force CANNOT track the Cone Axis,
and we already know that it can't track the GC axis, so ...
Both the Center of Mass (approximation) and the argument of Newton
regarding the balancing of opposing 'cone intersections' on the sphere
surface are false, and the Sphere Theorem must fail.
No comments:
Post a Comment