Newton's Shell Theorem –Bad mathematics - Bad physics Take three mass point objects m1 = m2 = m3 = 1 unit mass, G=1 unit gravitation constant, and using init distances the force of attraction between m1 and m3 separated by 10 unit distance is calculated using the universal law of gravity expression, F = Gm1m2/r^2 (minus sign omitted). F12 = (1)(1)(1)/10^2. Similarly, the force generated between m1 and m3 separated by 12 unit distance F13 = 1/144. When the masses are arranged along a common axis the total force of m2 and m3 on m1 is F123 = 1/100 + 1/144 = (1)(1)(1 + 1)/r^2 . Rearranging the terms, 2/r^2 = 244/(100)(144), or, r^2 = (2)(14400)/244 or R^2 = 118.03. The result r = 10.86 and, 10 < r < 11, where 11 is the location of the m1m2 system center of mass (COM). The combined forces' COM is located a distance 11 from m1. However, the center of mass-force (CMF) is located at 10.86, which is off set from the COM in the direction of m1. The total forces of the mass of a this spherical shell is calculated manually (see above) using mirrored image pairs of masses on the shell where one membermass of each pair is in opposite hemispheres, one closest to m1, one farthest from m1. Clearly when all forces are calculated, all CMFs of each calculation are located in the nearest sphere segment to m1, contrary to Newton's Shell theorem that without any physical basis, claimed that m1 may consider the mass of the sphere concentrated at the COM of the sphere. The links below are consistent examples of the rote acceptance of a developed shell theorem, referenced as an unchallenged law of physics and cheerfully communicated as rigorously copied scientific gospel, chiseled in stone, as it were, and enjoying immunity from heretical thought or criticism by the innoculated consensus of a solemnly deferential scientific community. http://en.wikipedia.org/wiki/Shell_theorem http://www.physclips.unsw.edu.au/jw/NewtonShell.pdf http://www.absoluteastronomy.com/topics/Shell_theorem From inspection of the sphere and m1 externally located at some point r from the sphere center it should be obvious that using the concept of "inverse distance squared" as a starting point, the mass of M in the hemisphere closest to m1 will contribute a greater share of the total force on m1 than the farthest hemisphere, hence the CMF for the entire mass M is located on the m-M axis off set from the COM in the direction of m. Where, and how, does the Newton Shell theorem (NST) place the CMF at the sphere COM? It doesn't. The question of locating the CMF is not discussed! The NST model begins with a ring on the sphere of differential mass dM oriented perpendicular to m and centered on r. By summing the force for each dM on each ring then integrating over the surface of the sphere, the total force, F = GmM/r^2 results, which says nothing, absolutely nothing, regarding the location of the CMF. The Wikipedia model referenced above states after dF is integrated, that, "The shell really does act as though all the mass is concentrated at the center!" Some commentators call this Shell theorem result, "proof" of what is claimed. The big problem here is that the developed algorithm made no inclusion for determining the location of the CMF. The intuitive assumption that the CMF is located at the COM of the sphere was arbitrarily (instinctively) made (or so this writer has surmised) as clued from the 'r^2' term in the expression, that clearly is an expression for determining the total gravitational force of M on m, only. Another flaw peeking from the expression is seen iwhere calculating the net vector force in the m-M direction as derived by taking the cosine projection of the force onto the "r" [m-M] axis and from this, supposedly, the inference was made that the CMF followed the projection of the force onto the m-M axis – the projection of a force vector is mathematically proper (forces perpendicular to the m-M axis 'cancel, or so we are told), but to include the scalar quantity location in the projection of the CMF is just plain "bad mathematics"; not properly placing the CMF in the nearest hemisphere, by inspection, of the conditions, re m and M, is just plain, "bad physics". Caveat emptor – beware of standing on the shoulders of giants who have been dead for 300 years. PU!Please Register or Log in to view the hidden image!