Monday, April 04, 2005

Physics Conundrum

The other day ( or so )
i was reading on The Internet a Posting that asked The Question:
"What would happen if i were to visit the Interior of a Hollow Planet,
Such as Our own Hollow Earth?"

The answer; Apparently provided by a Graduate Student of Physics,
Postulated that if you were in the very center of The Hollow Sphere,
you would find yourself to be completely Weightless...
( Which seems very Sensible )
But then she went on to Insist that if you were to float a little bit off to one side,
In fact, Regardless of where you were inside the Hollow Sphere,
You would be completely Weightless, since although you would be nearer one side,
Which would exert a greater Gravitational Attraction upon you,
The Area/Volume below you would be offset by the comparatively Greater Area/Volume Above you...!!!???
She goes on to claim that she'd actually 'Worked Out' the 'Math' of this Dilemma,
and found that this is 'In Fact' The Case...!!!
Those of you that have ever 'Worked Anything Out' by means of Mathematical Computations,
Or have heard Anecdotes of 'Scientists' that have 'Worked Out' this or that...
Should be reasonably 'Skeptical' about any such Claim...!!!
To Properly 'Work Something Out' by Mathematical means,
Requires that you have a Complete & Thorough Understanding of The Problem & Be Able to Use all The Necessary Tools to Solve The Problem in A Responsible Manner.
On The Surface; This Solution to The Hollow Earth Problem Just Seems Wrong to Me.
i can easily view myself near one wall of the Sphere and it just seems to me that the Surface beside me would exert an overwhelming Gravitational Effect upon me, While the Surface on The Far Wall would be inconsequential. The surfaces on the sides would be exerting only a partial 'Upward' Pull, while most of their Gravitational Effort would be CounterActed by the Surface Directly Opposite of them.
So i tried working this out on my own...
i got the Appropriate Equation from my HP48 Equation Library
F = Force in Newton's
G = The Gravitational 'Constant'
m1 * m2 = The Two Masses that are being Attracted to One Another
r = The Distance Between the Two Masses
i wasn't sure if i should measure the distance between the two masses from their Centers or from their leading edges, so i tried it both ways.
First Approach:
i wrote a Simple program to calculate a representative number of points around a circle, and Added their Gravitational x & y Components to a figure off center within the Circle.
If the Figure was indeed Weightless, The final result should have yielded a Sum of (0,0).
No 'Net' Pull in either the + or - Directions for Either the x or y Axis.
The Result i got was a net pull toward the nearer surface.
Second Approach:
Then i tried calculating the Area for a large Angled Area both Above and Below a Figure,
With the Figure standing very near to one surface,
With the Vertex of the Angles cutting through the Figure:
See Diagram:
( Taken from my Journal )
See Appendix A for What was on the Other Side of The Page

i think you can just barely make out The Nested Circles,
and The Figure towards the Bottom, and The Angular Slices.
Basically; i was thinking that these Two Areas,
The Larger one at The Top,
And it's CounterWeighting Smaller Region at The Bottom
Should provide a Calculation with The Relevant Proportions that would Show if They are 'In Fact' Counter Acting One Another.
The Outer Circle has a Radius of : 10
The Inner Circle has a Radius of : 7
The Angle of the Slices is 26 Degrees from The Midline, or 52 Degrees all together.
The Area of the Upper 'Mass' was first crudely Calculated at 40.3639, and later Refined to 46.1932.
The Lower 'Mass' was first Estimated at 4.3639 and later Redetermined to be more like 4.5726
The 'Mass' of the Person was first Arbitrarily given as 2 and later .5.
i was attempting to determine a consistent 'Proportionality' for a Sphere, and Not Terribly Concerned with providing all the extremely accurate data for an actual Sphere, made of actual materials and what their actual Gravitational Attractiveness might be...!!!
The Results again Reflected a Decidedly Clear Surplus of Attractiveness towards the Nearer Side!
Third Approach:
Thinking that maybe i was getting Anomalous Results from looking at Areas when this Problem is Really all About Volumes...
i attempted to Calculate the Volume of the Mass for a Sphere with the given Dimensions of the Circle previously used...
Upper Volume = 1207.6654 Cubic Whatever's
Lower Volume = 41.0040 Cubic Whatever's
Person = .5 Cubic Whatever's
For this:
The Force Pulling from The Top was 5.5766x10^-10
The Force Pulling from The Bottom was 6.0801x10^-10
The Gravitational Attraction for 'The Ground' was greater, but not by much...???
All the Numbers given for this Inquiry were pretty small, and i don't know how to interpret the 'Newton' Results...
So The Net Effect of this is--
i will continue to think that if i was in a Hollow Sphere, i would be able to walk around on the Inner Surface,
But i'm not quite as Certain of this as when i Started...!!!
As a result of looking into this; i created a couple of Programs to Calculate various solutions for this problem...!!!
So now i have a Program that will Calculate The Area of a Slice out of a Circle, as Delineated by a Choid that is Defined by any Two Complex Points or One Complex Point and an Angle...
And another Program that will Calculate the Area of Any Convex Polygon as Defined by a list of its Vertices!

Appendix A
[ What was on The Other Side ]

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