The Death of the Centre of Mass Theorem.

[James Jackson]

So I take it you accept that the concept of the radian as a unit of length is a completely imaginary construction.

 

[krab]

(2) Naturally, the CM for the other half will be the same distance away from the geometric centre of the sphere in the opposite direction away from the test-particle. The total force will be the vector addition of the two halves. But by inspection this is impossible. The increase in force for one half the mass now located closer cannot balance the decrease for the other half, because while the distances are equal, the forces have changed by an unequal amount. Gravity is an inverse exponential force.

Hurkyl gave a poor example with the pencil thing. In the context here, namely quantities and formulas, "exponential" is unambiguously incorrect. It's as bad as saying that a constant speed requires a constant force. A notion that was common knowledge before Newton. But the fact that it was common usage did not make it correct. "Exponential" means that is you increment the independent variable, the effect grows by a constant factor. Hence, without deaths, populations grow exponentially; under constant interest, your savings grow exponentially; etc. Gravity neither grows nor diminishes in this fashion.

 

[quantumdude]

I don't know why you are having trouble with the fact that the radian is defined as a length in physical space describing a distance along an arc or circle's perimeter.

He's not having trouble with the idea of radians. You are. A radian is not a distance. And if you think it is, then please tell us how long a radian is.

You yourself were forced to use the word length to describe it.

Good grief. Just because I have to use the word "distance" to define average speed as the total distance traveled divided by the total elapsed time, it doesn't imply that a speed is a distance.

How you expect to disprove any theorem without having the basics down, I'll never know.

 

[Rogue Physicist]

So I take it you accept that the concept of the radian as a unit of length is a completely imaginary construction.

This is the only reply worth responding to.
Of course radian is essentially a measurement of angle.
It is formally defined by the length along a unit circle in Euclidean space which is proportionate to the angle of concern. In the days of post-Euclidean geometry, the fact that the metric is involved is even more important, and I have no hesitation using radians as the unit of measurement for lengths in problems which are essentially tied to proportions of a circle's relative size. It may have been extreme to actually call the distances 'radians' rather than 'units of radius' or 'radii'.

If we are done nitpicking, we could move on to admitting that it is not trivial that the two main methods (Centre of Mass and Sphere Theorem) of generalization and simplification of gravitational calculations in Newtonian gravitational theory are inaccurate except when d >> r for discrete mass distributions.

In many cases, when the shape of a rigid body, its density distribution and its orientation is known, you can use calculus to find the contribution to the gravitational field of a given massive body. This is very limiting.

When you move to non-rigid bodies (systems of particles), you will I hope agree that the Conservation of Momentum (which is claimed to be even more fundamental than Newton's gravity theory) requires one to again rely upon the axioms and hidden assumptions of the Centre of Mass 'Concept' as defined for general systems of particles. In this case the requirement of 'rigidity' is not required or even assumed.

And this is why all this is important and where the facts lead: The Centre of Mass concept is necessary for the Conservation of Momentum, but in Newtonian Mechanics, the Conservation of Momentum is a circular and poorly defined concept based upon misguided and sloppy descriptions.

The Classical definition and description of the Conservation of Momentum is wrong because it is based upon the misunderstanding and misuse of vectors and their mis-application in Euclidean space.

 

[Doc Al]

What nonsense!