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- Thread starter Rogue Physicist
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- #52

krab

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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.Rogue Physicist said:(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.

- #53

Tom Mattson

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He's not having trouble with the idea of radians.Rogue Physicist said:I don't know why you are having trouble with the fact that the radian is defined as alengthin physical space describing adistancealong an arc or circle's perimeter.

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.You yourself were forced to use the wordlengthto describe it.

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

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This is the only reply worth responding to.James Jackson said:

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

In many cases, when the shape of a

When you move to non-rigid bodies (systems of particles), you will I hope agree that the

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 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.

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Doc Al

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