Disproving Newton Pt2: (c) Solid Sphere (cont.)

(C)   Force for a Solid Sphere

CONTENTS

1. Force for a Uniform Disk 

2. Graphing the Force for a Disk 

3. Force for a Solid Sphere ( Disk Method )

4. Formula and Graph for Solid Sphere ( Disk Method )

5. Force for a Solid Sphere ( Shell Method ) 

6. Integral and Graph for Solid Sphere ( Shell Method ) 

7. Summary of Part 2 so far: 

1.  Force for a Uniform Disk

The case of a solid thin uniform disk perpendicular to a test-mass is a standard physics problem, but it is usually presented and solved in an inconvenient form.  The formula describes a scenario where disk density and distance are held fixed, while the disk radius is varied by adding mass in the form of more outer rings, or else the distance is varied.
Our  interest is in what happens when a given mass is spread out into a disk while distance and mass is held constant.  We need a form which shows the actual factor diluting Newton’s original formula, as the point-mass is spread out into a thin uniform disk.
Typical Form:

 



2.  Graphing the Force for a Disk

A look at the basic formula shows it isn’t valid for negative values of distance d     ( x-coordinate for the test-mass ), because of the ( 1 – cos ) structure.  So we have to tweak it a bit by manipulating the sign using the Absolute Value function.  …Voila!
The radius is set to   r = 1  so that distance units are in radians of the disk.






Note the vertical asymptote at the origin.  Physically, the forces should balance there, and the net force should be zero.  Again, the mathematics does not perfectly reflect the expected physical situation.



3.  Force for a Solid Sphere  ( Disk Method )

    We can use an approach similar to the hollow sphere, however, this time the mass will vary for each slice.  Now, the constant will be disk density. Again as before,  r2  is defined using the Pythagorean Theorem, and we can automatically handle near  and far  slices by using +ve and -ve values for i :

4.  Formula and Graph  for Solid Sphere  ( Disk Method )

The following formula settles around  n > 400, at this scale of graph.  The original formula is only valid in the domain  D > 1  and appears the same as Newton’s   1/ d2   prediction.   Manipulating signs with the Absolute Value function extends the formula to values of  D < 1 and allows negative x-coordinates for the test-mass as well.

Comments:  When  -1 < D < 1 , then during the summation, there will always be a point where D = i / N.  Here the disk is at the same location as the test-mass, and the force should be zero1 , but our sign patch will have an undefined denominator.  With large N, this single disk should have an insignificant mass and can be ignored.
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6.   zero in the x- direction and balanced in the y and z directions, although the physical meaning of this will be discussed later on.

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5.  Force for a Solid Sphere ( Shell Method )

For the shell method, we divide the sphere up into concentric hollow spheres or shells, each having a mass proportional to its area.  From this perspective, the total mass of the solid sphere is the sum of the masses of each shell.  The force from each shell will be weighted by its fraction of the total mass.  It is important to maintain consistency in the method of calculating the masses.  For this we waive ordinary volume formulas for the sphere and resort to the following formulas:

 The shell surface area would normally be 4r 2, but since the 4 in the denominator and numerator of the shell mass formula cancels out, it can be dropped.  Integrating the AreaTotal formula gives simply 1/ 3, simplifying further:


We generalize our integrated formula for a hollow sphere to accommodate any radius, and take the constants to the outside:

6.  Integral and Graph for Solid Sphere  ( Shell Method )

 A graph (below) of the formula using the summation is without surprises, converging well at this resolution with  N > 400.  The integral can also be set up as follows:  ( By inspection,  there will be a discontinuity at certain values of D and x  due to denominator divide-by-zero cases, as with other formulas. )




7.  Summary of Part 2 so far:

We have completed our detailed look at the essential mathematical content of the Sphere Theorem  ( ST ).  We used only simple algebra and trigonometry to allow almost any reader to follow the steps.  Integration was only brought in at the end for completeness, to provide simple, exact and labor-saving formulas, although algebraic methods can be used to achieve any desired degree of accuracy.    

We remind the reader at this point that contrary  to the Centre of Mass Theorem, (CMT)  which is conceded to be an ‘approximation’ however ill-defined, the Sphere Theorem ( ST ) is supposed to be an exact  theorem, and working in conjunction with electrostatic formulas, is believed to be accurate down to distances of 10 –13 cm. 2 

(See Berkeley Physics Course: Mechanics Vol 1, 2nd Ed. Pg 270 for example.)
We have shown that the hollow sphere formula suffers from a discontinuity, an undefined singularity when the test-mass is at the surface.  This exactly parallels the singularity for the disk formula, and also Newton’s original formula for point-masses.  By inspection, it is obvious that these anomalies are simply occasional divide-by-zero errors in the formulas, and have no actual physical meaning.  For certain values, the formulas are undefined and nonsensical, and are simply invalid. 

At least according to classical notions of reality, the force acting on a particle can be zero or unknown, but not ‘undefined’.  On this basis, we assume that a test-mass arriving at the centre of the earth doesn’t experience ‘infinite’ gravity but rather a balance of forces resulting in zero net force.  Similarly, a test-mass piercing a hollow sphere doesn’t undergo infinite forces in both directions, but more likely experiences a temporary balance of forces, again resulting in an instantaneous force of zero.  

It is a common practice to take an otherwise continuous function with only one problem-point and use it for all other values, substituting a value by hand at that point, rather than just discard it.  We have no problem with this pragmatic procedure.  It serves to remind us however not to take the mathematical formulas too literally, as to their physical truth claims. 
It should be clear that the formulas for the solid sphere also have these discontinuities hidden inside them, since they are built out of the hollow sphere and solid disk formulas.   This is especially true of the integral and integrated versions of the solid sphere formulas, even though the discontinuities appear to vanish. 

The real  problems with the integrated versions of  the sphere formulas will be shown in the next section.  The danger involved in assuming the integrals have the same accuracy and validity as the summation formulas upon which they are based will then become all too apparent.