|Originally posted by Ric B:
...PM, I don't have the math or computer skills you have, but this diagram still seems lacking to me. If I were to draw what I see in my minds eye, I would have opposing arrows, drawn perpendicular to the direction of travel,(our inertia), representing centripetal force and the resistance of the snow. They would show the amount of force and opposing resistance being created by their size drawn, and this would be in direction relation to the amount of change in direction of our CoM that is happening and the acceleration/deacceleration of going into and out of the fall line. And yes, I intentionaly left out centrifugal.
You bring up a good point - one that is absolutely critical to an understanding of how dynamical situations are analyzed: In the good old F = ma equation, if you know where an object (the skier) is at every point in time, you also know exactly the acceleration that the object is undergoing, because "a" is just the second derivative of the (known) position as a function of time.
Earlier in this thread, someone suggested that skiers making a sine-wave path down the hill would be an interesting case. Once this assumption was made, the rest of the acceleration analysis is a slam dunk. Basically, we just turn the mathematical crank, and immediately can determine the acceleration (due to motion in our reference frame) that is acting on the skier at every point in time. Add in the acceleration due to gravity because the slope angle is not zero, and you have an exact value for the net acceleration of the object at every point along its path. Multiply these values by the mass, "m", and you have the net force acting on the body at every point along its path.
The previous paragraph was a long-winded version of Bob's response to the same question. My diagrams (exactly as do his), show the resultant, or net acceleration of the body, which is essentially the same as showing the net force acting on the body. I did not show the components that went into each of my "resultant" vectors because, as Bob suggested, I wanted to emphasize the flow and continuity and so, wanted to show the resultant vector at a lot of points. Including the two vectors that add together to make each resultant in my diagram would have hopelessly crowded the drawing. Besides, Bob's drawing (with fewer total number of points) already shows the components quite well.
The above analysis makes absolutely no attempt to say how this "net force" arises from the interactions between ski and the snow, just that IF the skier is doing what we said he is doing (ie, making sine-waves), then the net force is precisely and inarguably known.
On the other hand, RicB brought up a very interesting and important question, namely, the other side of the F=ma equation - How do we decompose this net force into the various physical mechanisms that actually create it. These mechanisms include the snow drag that was mentioned, but also include aerodynamic drag, forces exerted by the snow perpendicular to the base and edges, etc..
I didn't attempt to decompose the net force into the various physical interactions between the ski/skier and his environment which create these forces for a couple of important reasons:
1) It wasn't necessary for what I wanted to show (ie, which is exactly what BobB wanted to show by his illustration), namely, how the overall net force varies through a turn.
2) To decompose the net force into the various ski-snow interaction mechanisms, one would have to make a whole bunch of weak assumptions, none of which are fundamental or readily verifiable, and all of which depend in complicated ways on parameters applicable only to very specific skiing situations including pressure on the snow surface, type of snow surface, edge angle (if skidding when edged), angle of attack, speed, etc.
Put differently, I believe my diagram is complete as is (and shows what it is supposed to illustrate), but there is another, completely different diagram with lots of arrows on it that is not shown at all. The only relation between the two diagrams is that for corresponding points on each diagram, the vector sum of all the components in one diagram must be identical to the vector sum in the other diagram. In other words, the left side of F=ma has to equal the right side of F=ma at each point. What RicB suggested in his post (eg, adding drag and other vectors to the diagram I showed) is essentially putting these non-acceleration arrows in the wrong diagram. They go in the vector diagram for the other side of the equation (that I decided not to show).
I hope this further elaboration helped.
As I get time over the next few days, I am going to try to develop this model a bit further to specifically answer questions about the main subject of this thread, namely, the motion of the skier's center of mass across his skis.
Tom / PM[ November 09, 2003, 12:04 AM: Message edited by: PhysicsMan ]