Jonathan Wong wrote:
OK I have something now for the real meat of the linear/spiral discussion. If I need to be clued into what started this attempted discussion (I may have missed it) then please let me know. But so far as I know, these comments still are relevant. Although, it isn't much improvement over what Dave wrote. At any rate this describes the fact that we can volitionally control our force output across time, as well as the fact that our body plan may "prefer" or influence that temporallyvarying force output. Mike, this may all be silly until we get into your ramifications you alluded to.
I. What is meant by "linear?"
The term linear refers to the rate of change of a function across a domain. It means that the rate of change is constant. I take it we are talking about the direction of force produced. Of course at any one moment in time, the force is going in a straight line just like at any one value on the xaxis, a single, defined rate of change (derivative; slope of the function) exists for y=x as well as for y=sin(x). But it doesn't mean y=sin(x) is linear because that discrete value for rate of change varies across the x axis, unlike for y=x.
So when one says the force produced is linear and not something to be described in other ways (spiral? torsional?), what exactly is being argued? That there is no change in direction of the force across time? Of course no one could argue that, because we can change direction at will. So, we need to nail that down.
II. The rolling ball question.
Here is a series of scenerios that might be very illuminating to discuss.
1. A ball rolls across a table. The ball produces a vertical, downward, obviously linear force. The point of application of the force shifts along the table as the ball rolls. Simple.
2. A sticky ball of the same mass rolls across the table. I know there will be additional heat terms and a deceleration, but this is still a bit mysterious to me. The downward weight is the same, but does the net downward force change? (The "upstroke" on the trailing side pulling upward via the adhesive? Or I guess that pull is tangential so it is purely in the horizontal decelerating direction? Never mind, #3 and #4 below are the fun part!)
3. The same sticky ball rolls while deflating as it goes along. Now it is getting interesting. The deflation action trades internal pressure for upward pull, and there is significant reduction in downward force upon the table. The center of mass of the ball is in fact going through a braked fall while the ball rolls.
4. Now imagine situation #3 without the flat table. We can have a curved table, with sections of curvature that match the ball's radius of curvature. The table can be poorly secured to walls or foor. The 'table' surface can in some spots be a pretty severe incline, like close to vertical it's ok b/c the ball is sticky. Also now, the ball can breathe, not just deflate. So what happens? The table is going to get pretty worked by the ball's rotating, expanding/contracting grip. Will the forces be linear? I think the table will get pressed and pulled (linear forces?) as well as torqued.
So what is the point? When I hold my bokken, it is like the table, and I am the ball (the floor is another table). I am a pretty cool ball though because of things like hands. As I expand and contract, sink and rise, my tissues coil about as we discussed, being the nature of the human body. That bokken experiences some twisting forces that are fun to use on humans too. So how do we fit this into the conceptual framework of linear forces?
