..."The shell really does behave as if all the mass of the shell was concentrated at the center of the shell."..
This becomes true only at the sufficient distance from shell, doesn't it? Anyway, I can't see your exact problem here...
Originally Posted By: geistkiesel
..."The shell really does behave as if all the mass of the shell was concentrated at the center of the shell."...
This becomes true only at the sufficient distance from shell, doesn't it? Anyway, I can't see your exact problem here...
No, Newton used the model in earth-moon calculations, even though the physical extension of both orbs is readily apparent.
I assume you are familiar with the rings on the shell locating the dM masses that are the source of the gravitational force of the thin shell mass M acting on m. After all the dM on the ring are calculated the ring moves over the surface of the sphere until the total surface of the sphere and all dM have been calculated using, F = GmM/x^2 . For the shell center located at a distance d from m have been calculated, that is after the integral has been evaluated the resulting expression for the total force on m is, F = GmM/d^2. This expression is the proof that, "the shell really does behave as if all the mass on the shell was concentrated at the shell center, the COM."
The expression is describing a force and it says nothing about the location of the force. The expression says, describes the "force of gravity by a shell of mass M with the shell center (COM) a distance d from m". Does this remind you of any proof?
There is nothing in the structure of the mathematics alluding to the location of any force, to concentration of mass. The mathematics provided a mechanism to calculate the total force on m by M. There is no calculation of the location of any combined forces.
Take three masses in a line where two of the [point] masses are a considerable distance to the third test mass. Using the force expression and calculating the forces on m singularly to get a total force, this sum can then be used to calculate the 'apparent' location of the mass contributing force on m. The force expression constrains the forces such that the closer of the two equal masses contributes a greater share of the force on m than does the more distant mass. The test mass m sees only one force. Making the calculation will result in the combined force's location off set from the COM of the two distant masses in the direction of m.
This is what the shell theorem requires as a correction. Simply calculate the forces from mirror image pairs of dM with a mirror image ring located symmetrically in the nearest ½ shell segment and the other ring in the more distant ½ shell segment. By inspection the nearest ½ shell will contribute a greater share of the total force on m than the more distant 1/2 shell segment. The 'center of force' for the shell, just call it the 'center of gravity', AKA the cg.
While the science industry continues to glorify itself from the lofty height reached when standing on the shoulders of the 'Giants', if for no other reason than this slight intrinsic error in force calculation amounts to a couple of percentage points, the complete ignorance of believing that the mass may be concentrated at the COM of the shell is witness to an aching void in lack of completeness and just plain seemed be rather ignorant regarding what should be obvious - referring to the closest dM to m contributes a greater share of the total force on m than the dM located more distant from m.
These are the two corrections for which I demand my money back.
Other somewhat irksome flaws include, 1. Treatment of the m-M system using vector analysis. If two persons hold the ends of equal length ropes where both ropes are tied to an object between the rope holders. The men pull with equal force of 50 kgf in opposite directions. Vector rules conclude the forces sum to zero, when I say the forces sum to 100 kgf. 2. There never been an observation of a stellar mass in the form of a spherical this shell, yet a universal law of gravitation is recognized and faithfully enforced, yet the law is founded in a mathematical abstraction n with absolutely no common characteristics justifying even pretending the expression has scientific value. 3. The forces outside the sphere having been calculated, now we enter the wicked witch of the west's house and are told that we could float inside the sphere without having to flap our arms – one absurd physical impossibility is now offered as justification to make real yet another event seen only by Alice when she fell down the hole.
These nit picking items are said to follow logically – scientifically – from the "proof " of the shell theorem. Why do so many commentators use the expression that the 'zero force' result felt on an object in side the shell is 'not immediately intuitive", or that it 'strains the sense of physics'.
Zephir, are you convinced, not by my incessant whining, but by the math and the physics? I just remember, a UC Berkeley PhD in Physics first lectured me on the shell theorem, but that was in another lifetime.
Where do I go to get my money back?
