Hi,

> > What does the Earth's motion around the sun have to do with this?"
>
> *Speed*. If a body such as the Moon travels a greater distance than the Sun in the same time of one year, which goes faster?

But what possible relevance does this have? Speed and distance are relative to some arbitrary frame and origin. I'm talking about the earth-moon tidal system, so I am considering a rotating frame of reference, centered on the barycenter of the Earth-moon system, rotating with the angular velocity of the moon's (current) revolution. I am ignoring the effect of the sun, as it is very small at this distance.

> > I don't care how the moon came to be where it is, I am modelling the
> > system as it is now and extrapolating that motion into the far future.
>
> OK, now do it backward and tell me what you find.

Interesting question. I haven't tried that.
Still isn't relevant for this particular discussion, though. I'll take a look at it and get back to you.

> > "Center of gravity is an ill-defined concept. I assume you mean center of mass."
>
> Absolutely not! The center of mass (or momentum) is the point around which a mass or system rotates.

Um, first, the center of mass is not the same thing as the center of momentum. The center of mass is a point, while the center of momentum is a frame in which the net momentum of a system is zero. Second, a mass or system does not necessarily rotate around its center of mass. That is not the definition of a center of mass. Take that 4x4 you mentioned by the end and spin it around your head: it's rotating, but not around its center of mass, nor around the center of mass of you and the board together.

The definition of the center of mass is the integral of the density times the displacement over the integral of the density (i.e. the total mass). While it is true that in the absence of external forces (like your hand), a body will rotate around its center of mass, that is not part of the definition of center of mass. Rather, it is a consequence of the definition.

Similarly, the net torque on a system is the integral of the cross product of the displacement of each mass element and the force vector on each mass element. Clearly, there must be a non-zero torque on the earth-moon system, because that integral will be negative and large over the bulge ahead of and nearer the moon, and positive and small over the bulge on the far side of the earth. (While "r" takes on equivalent values in both directions, the gravitational force goes as 1/d^2, where d is the distance to the moon from the volume element. So the mass closer to the moon feels a stronger gravitational force. Since the magnitude of r cross F goes as the sine of the angle between the vectors, the sign of the torque will clearly be opposite between the two bulges, since r flips direction but F does not. Finally, if we define a right handed coordinate system such that the Earth's current rotational velocity is positive, then the bulge nearer the moon will feel a negative torque by the right hand rule, and the far bulge will therefore feel a positive torque, and the net torque will be negative. QED.)

Since Torque = dL/dt = I dOmega/dt (where I is the moment of Inertia tensor for the earth, in this case, I_{ij} = Volume integral of density times the difference between the displacement squared times a delta function in ij and the quantity x_i times x_j). Those integrals are clearly going to be positive for the coordinate system defined above, so dOmega/dt, the change in the angular velocity of the earth, has to be negative: i.e., it's slowing down. QED.

> In my book the center of gravity is described as follows. "The moment about any axis produced by the weight of the body."

That sentence is poorly worded. It is not clear if the word "produced" refers to the axis or the moment. It's also scientifically not very useful because "weight" is always defined in terms of an external gravitational field, so whatever this "center of gravity" is, it's not an intrinsic property of the object, as it will change given the external circumstances. The center of mass will only change in so far as it is a vector, and thus is defined relative to some arbitrary origin.

> But in large and distant bodies it is appearent that even though the mass is the
> same the gravitational and inertial workings are different...Gravity works on
> weight, inertia works on mass.

Okay, I think I see why we're talking past each other. You seem to think that a graviational force will not accelerate a mass in the same way as, say, an electromagnetic force (which is what force it would be if you pushed on it with your hand). If you think that, you're out of synch with all of physics. The equivalence of gravitational and inertial mass, while not proven analytically, has been measured to be consistant to some phenomenal degree of accuracy (I forget how many, but it's many many decimal places) and there's never been any evidence that they're different (your claims to the contrary I find frankly baffling). This is what allows us to say "F=ma=GMm/r^2" and is the basis for my saying that dOmega/dt must be negative. If you don't accept that, then we can't move any further in this conversation, because everything I'm saying is based on this premise.

You still haven't explained how you think the moon can move away from the earth without slowing down. In the light of Kepler's Third Law, this would seem impossible, regardless of what the earth is doing.

Don