So I take it you accept that the concept of the radian as a unit of length is a completely imaginary construction.
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.
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.Rogue Physicist said: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.
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 word length to describe it.
This is the only reply worth responding to.James Jackson said:So I take it you accept that the concept of the radian as a unit of length is a completely imaginary construction.