What Dropping a Fixed Center of Mass Means
But what else do we lose, when we abandon the Center of Mass (CM) concept?
According to the original Center of Mass concept, claims were made about the behavior of rotating non-spherical rigid objects:
(1) An elongated object (such as a hammer) will spin around its Center of Mass (CM), and this CM will move in space according to the other Newtonian Laws. For instance,
(a) The CM of a spinning hammer flying through space in a zero-gravity field will travel in a straight line, along the axis of direction, as if the mass were concentrated at this point.
(b) Similarly, the CM of such an object will follow a parabola trajectory if travelling laterally in a uniform gravity field (such as small objects near the earth's surface).
These claims were not held to be mere 'approximations' in the sense of a failure of the CM concept. Deviations from these predictions were presumed accounted for by things like wind resistance, small changes in the background gravity-field, or the influence of other external forces (electromagnetic etc.).
But now we tell you outright, the CM concept and its calculation is false:
(1) On the one hand, there IS NO fixed "Center of Mass" for non-spherical objects, which would locate the EPM. This is because the EPM of a non-spherical object changes with its geometric orientation to the test-mass. It can only be determined when the test-mass is fixed relative to the non-spherical object.
(2) On the other hand, the force and the EPM are nonetheless fixed and easily calculated for any chosen position of test-mass. Newton's equation works quite well, and the effects of many rigid objects can be known.
Force for a Barbell Reconsidered
For many common situations, the EPM is not difficult to determine at all. We simply revert back to Newton's original formula, and his Sphere Theorem.
Take a simple case of a rigid barbell of two balls (say depleted uranium), connected by a shaft of negligible mass (say structured aluminium honeycomb).
We can develop simple formulas for its gravitational force on a test-mass, for instance along its axis:
We can also easily calculate the static force of a barbell perpendicular in relation to our test-mass:
And finally we can calculate the instantaneous force of a barbell rotating end over end at a stationary distance from our test-mass, using Newton's formula, his Sphere Theorem, which states that each ball of the barbell will act as if its mass was concentrated at the Geometric Center, and the Law of Superposition and Vector Addition:
(1) A barbell spinning in the plane perpendicular to the axis of our test-mass, will have a constant gravitational force, because the forces remain balanced and unchanging, although individual lateral forces rotate while they continue cancelling. This result is trivial, and would suggest that a frisbie or horizontal flying saucer should nonetheless behave as a point-mass in relation to the earth.
(2) Most importantly, a barbell spinning in any other orientation to the axis will have a constantly changing gravitational force, both in magnitude and direction.
Nazaroo's Magic Egg
Remarkably, this varying force is not a simple sine-wave! Due to unbalanced differences in force between near and far ends of the barbell, we get a distorted time/position graph of the Equivalent Point Mass (EPM):
This EPM, tracing the effective force upon the stationary test-mass, traces out an Egg-Pattern in the same plane as the rotation (x/y axis), with the narrow end toward the test-mass. Several important features need to be observed here:
(1) The spinning barbell exerts a varying periodic force (with a simple harmonic content), and therefore is projecting a WAVE of force upon the test-mass. Note that even with a balanced barbell, the wave is NOT a simple sine-wave.
(2) This varying force shows that the EPM moves in geometric relation to a non-spherical object depending upon its orientation to a test-mass, and therefore any oversimplified definition of "Center of Mass" which is fixed relative to an object is falsified.
(3) The base frequency of this Gravity Wave is exactly two times the spin frequency of the object.
Thus at its most basic and accurate level,
(1) Newtonian Theory predicts Gravity Waves. Not only this, but
(2) Newtonian Theory also predicts a 720 degree virtual 'spin' for elongated objects, such as electron-pairs.
Newtonian Theory also suggests that these effects will only appear significant at distances where the radius is in the same order of magnitude as the distance between objects (e.g., molecular levels, and near-collision distances between celestial spheres.)
These waves are stronger when objects are in relatively close proximity relative to the diameter of the objects.