I thank all the hosts and users for permitting my attempt to reach, or at least to approach scientific soundness. I have been obsessed with in Newton's Shell Theorem recently and have perceived some observed glitches that I will indicate.
The shell theorem as stated by Isaac Newton for the case of a test mass outside the surface of a thin spherical shell goes as follows: "The shell really does behave as if all the mass of the shell was concentrated at the center of the shell." I he concluded at the risk of leaping off the shoulders of some now long gone giants I am compelled to proceed.
Of all the objections to the theorem it is to me best shown how the shell integral leading to the result that the force on a test particle located a distance d from the shell center. The expression for the total force is the familiar, F = GmM/d^2. It is this expression that is referred to as "proving Newton's Shell Theorem". I am unable to discard the law of gravitation as constrained by the inverse distance squared law. To me the law says, for objects of equal mass, located at different distances from M, the object nearest m contributes a greater share of the total force on m than the object located further away. The references I have reviewed have all integrated the expression by summing the forces on rings centered on the r-axis (the m-M axis)with the plane of the circle perpendicular to the r-axis, and then adding the forces on rings from a position closest to m then in a direction retreating away down the r-axis until all rings were summed.
Bear with me on the following. Take any arbitrary ring and project it away from m to a position that reflects the ring as a mirror image of the closest ring to m. Then, draw a horizontal line linking any arbitrary dM on the closest ring with its paired mirror image dM. We are going to sum the forces associated with each mirror image pair and voila, the dM closest to m contributes a greater force on m than the paired mate on the ring farther away from m. When doing this is it not obvious that when all dM in the nearest segment of the sphere now segmented into 1/2 shell segments for the purpose of modifying the order of calculating the forces. When all dM forces in the near ½ shell segment have been counted, then all the dM forces in the farthest ½ shell have been counted.'
This method allows one to determine the exact location of the total of the combined paired forces. Briefly, finding the "gravity center" goes as follows:
Fc is the force from the dM in the closest ½ shell segment, Ff the paired mirror image force. Fc = GmdM/s^2]cos(alpha) where the cosine term effectively projects the calculated dF onto the r-axis. Fc = GmMcos(alpha)/r^2 . For just a single ring cos(��) is a constant for the forward ring as the integration sum the forces on the rings. Ff has a cos(alpha') projection onto the r-axis also. Here ��alpha' < alpha.
Assume the test mass m = 1 and G = 1 , and M = 10 units of mass and assume the dM are each .1 and the shell dm mass is .1. Assume these dM are the first to be counted and both are on the r-axis. If Fc is located 10 units from m and Ff located 14 units from m then the forces are Fc = (1)(1)(.1)/10^2 = 10^-3. Ff = .1/14^2, or Ff = .1/196 =.0005. The total of these "point mass forces" is Fc + Ff = .001 + .0005 = .0015. Now calculate where the force is sensed to originate by m. So (.1 + .1)/ r^2 = .0015. Solving for r^2 gives us .2/r^2 = .0015. r^2 = .2/.0015 = 11.47 – the COF (center of force) of this two object system is offset from the COM in the direction of m. The COM is located at 12 unit distances.
This one calculation can be generalized to the unambiguous result that all calculations will place the COF in the 1/2 shell nearest m . The point is made, if not a bit crudely.
Another point slightly different is recognizing that Newton's shell theorem was postulated before he did all the math and physics and when the integrated force expression turned out to be, F = GmM/d^2, all of Isaac's Groupies sighed when he said [or someone said," the shell really does behave as if the mass were all concentrated at the shell center." But look at the expression. It proves nothing about concentrated mass, the development of the integral form of the expression makes no reference to the concentration of mass or its location. Isaac, however, had previously theorized that that all of this before cranking out the integral: was it luck or skill? Whatever the choice, Isaac's Theorem is seriously flawed. If it had been me, I would have claimed a state of drug addiction, or that I was severely juiced and that I had, "blown it for the m moment., but that me and my chaplain had worked out all the difficulties".
The expression only says, however, "The force on m by a shell of mass M located at d." – the expression does not locate the force center!!!! Sure, the shell was located at d, but all paired force centers, dare we call them the "centers of gravity", are all off set from the COM along the r- axis in the direction of m.
I appreciate anyone who has made it this far.
I know about how we all have benefitted by standing on the shoulders of Giants, but the one I was propped up on decayed suddenly - I dropped a few ignominious feet- no surprise there, after all the dude had been dead for some 250 years, he was just dried dust and his structural integrity was found to be embarrassingly lacking, not his embarrassment , he was very, very dead, and the dead don't blush. It only took me 40 years from first exposure to the theorem before I took a long hard look.
Geistkiesel
