No, not a straw man argument. You continually refer how you think that the axe handle makes more sense to reference because it's not an invisible line. I'm saying that the axle makes more sense because its where the axe is actually balancing and therefore where it wants to center its lateral rotation. Because it contains the CoG, it's also the line along which it will balance fore/aft when held 90° to the horizontal.
Here's about how much an axe "inherently wants to move like that", according to conventional physics:
The simplified example with a 4-pound axe (with 36 inch handle) held vertically, with the center of gravity one inch in front of the vertical, results in only 0.33 foot-pounds more torque to hold it in this position, when compared with the other example with the CoG directly overhead. The extent that the axe "inherently wants to move" to reduce this 0.33 foot-pound torque is much less than the extent that the axe "wants to move" downward due to gravity (4 pounds in hand, compared to the equivalent of 4 ounces held level a foot away from the hand), and yet the axeman still holds up those 4 pounds, just like he/she will still hold back those relatively minor 0.33 foot-pounds.
It would want to inherently move, but the purpose of mentioning this was not that the force had a major impact on the axe in fore-aft pivot, but rather that it represented starting the axe in your example from a starting position that did not properly constrain the effects of moving the CoG. This is much like when people make the old flat vs. convex edge argument by saying that a convex is stronger because it fits outside the bounds of a flat V when the three points are kept the same...the reason it is stronger is actually because in the process of that manipulation the edge angle has been made thicker, and so for the basis of comparing geometry the edge angle needs to be held constant. In this case, the starting position of the CoG should always be in the starting position that results in zero torque in 2D profile.
Big mistake you're making here, the calculations I did are for movement in 2 dimensions, as mentioned earlier (and presumably obvious). This was spurred on by your claims about "desire to tilt forward or backward", which happens in 2 dimensions. I showed a way that these affects could actually be quantified, to give a fuller understanding of what's going on. (I thought you'd be appreciative.
) "Wobble" and inherent "wobbliness" occur in 3 dimensions, with some other causes unrelated to these calculations.
And that's fine, but it doesn't change the fact that the behavior occurs--it only quantifies it. There's value in that, but the more fundamental matter is that those forces and behaviors are present at all. You had said:
And I think you're reading too much into the CoG's properties. An axe does not inherently want to move like that; instead, the axeman moves it where he wants to move it, and the CoG location influences the resulting torque that he/she feels in hand.
It's true that if you limit motion to only fore/aft without permitting lateral rotation then only the fore/aft torque and the handle matter significantly, but...axes don't move like that. In cases where the axle sits forward of the handle the user has to be
trying to constrain the input force on the handle to the 2D plane of the stroke because the greater the distance of the handle at the grip point from the axle the greater the leverage any lateral motion at that point will have on the axle, making it easier to cause it to wobble if there's any deviation
Again, your discussion included some dubious-sounding (and changing) claims about "desire to tilt forward or backward", which is the reason I did these calculations, to provide an improved way to make those determinations. Again, I thought you would be appreciative.
My actual claims aren't changing. I keep trying to explain it in ways that you're able to digest it because it seems to me that you don't fully get it yet. Again, this is something that's conceptually and practically very simple to see and feel--but it's hard to put it into words that are suitably effective at communicating that information. I appreciate the fact that you went through the effort, but it has only a minimal impact on the core concept. The desire to tilt forward or backward when the CoG is changed while keeping the edge position fixed in space is only significant from the standpoint that it shows that the tool is starting from a position that is not neutral like it was before, and that it would need to have its pivot corrected in order to keep that torque you're discussing equal to before at the start of the stroke.
I'm honestly surprised that you are holding on so tightly to your theories and methods. Remember, I'm just addressing the 2-D aspects of the motion. With "real physical" axes, the forces or torques required to hold an axe straight out horizontally are clearly much greater than the forces or torques required to hold it overhead (with the center of gravity off the vertical by a couple inches), and yet the axeman is able to achieve both these "feats" (despite those "inherent desires" of the axe). This is in agreement with the calculation results. Yet, you keep insisting that the axe "inherently wants to" move back those couple inches to bring the center of gravity back to vertical, as if this were such a major force relative to all the other forces in play.
This is a major flaw in your theories, regardless of your attempts to minimize these issues.
I'm "clinging" to it because they're not flaws in my understanding of how the tool wishes to behave--I've rather apparently failed to communicate it in a way where you understand what I'm trying to get at. The reason why it's important that it want to move in that way to bring itself into balanced suspension is that it represents a shift in the axle, which is the axis around which the tool wants to rotate. This means that how the handle is oriented relative to the axle has changed and there may now be more or less potential for wobble as a result.
I strongly disagree. As I said earlier, I think it points out some major (but fixable) flaws in your theories. Mostly about reading too much into the concepts related to Center of Gravity, especially regarding motion in 2 dimensions.
See above. The importance is less so about the quantity of the force as much as what the presence of that force is indicating. Granted that the greater the force, the more change has been enacted, but the fact that it is a much smaller force than gravity acting on the axe when horizontal is inconsequential to its significance. It's mostly about what impact that's had on the axle position relative to the handle and if the altitude of a second grip point has been increased or decreased as a result, and/or if the handle has changed from following to leading the CoG (though this isn't likely under most circumstances.)
