When writing my previous post in this thread, because of a lack of time, I referred to terms like "local, instantaneous vertical", "edging angle", and "critical edge angle" and their relation to this angulation / banking discussion, but didn’t have time to fill in the details (let alone get into the whole centrifugal / centripetal definition thing).
Lets imagine a skier standing on one ski, traversing across the hill in a straight line on hardpack. No matter what contortions he puts his body through (e.g., angulation, counter, etc.), unless his center of mass is directly above the uphill edge of his downhill ski (as viewed from the rear), he will fall over – no ifs, ands or buts about it. This is the meaning of “being in balance” for a static skier. In everything that follows, I assume that the skier is always in balance.
If this traversing skier is in balance and fairly upright (i.e., not in an angulated or banana-shaped position (as viewed from the rear)), the bottom of his skis will be close to parallel with the surface of the earth. As he goes across the hill, the compression of the snow under his skis will generate a little shelf in the snow for him to stand on. In this case, the little snow shelf will also be close to parallel to the surface of the earth. This will happen no matter how steep the hill he is on (within reason).
On the other hand, if the skier is again in balance (i.e., his CM is directly above his edge), but has assumed an angulated position, the little snow shelf that he is actually standing on will no longer be parallel to the surface of the earth – the part of it furthest into the hill will actually be slightly lower than the part just under the surface of the snow. Under the weight of the skier, this sideways ramp angle makes the ski want to cut into the hill even more, and ensures that the ski won’t sideslip.
For the in-balance, but upright skier, should it happen that the little snow shelf that he is standing on tilts slightly outward instead of inward, then, the weight of the skier will tend to drive the ski off of this shelf, and ensure that the ski begins to sideslip.
This is THE fundamental phenomena that makes angulation important.
In technical terms, first draw a line between the center of mass of this hypothetical one-legged skier and the ski edge in contact with the snow. Next, draw a line perpendicular to the ski passing through its edge. The angle between these two lines is the “critical edge angle”. If it’s one way (positive), your skis get driven into the shelf, and don’t sideslip. If it’s the other way (negative), they slide off the shelf and you slip.
It is important to realize that this angle is NOT the normal edge angle that people generally talk about in skiing. The normal edge angle is simply the angle between the base of the ski and the snow. In contrast to the critical edge angle, the normal edge angle depends on the angle of the hill you are on, and might only vary by a few degrees out of many between a skidding and non-skidding situation.
When the skier is in a turn, centrifugal force is added to the situation, but nothing fundamentally different is happening with respect to the issue of critical edge angle and skidding. At any point in a turn, the instantaneous value and direction of the centrifugal force is added to the force of gravity acting on the skier. This can be thought of as a single resultant net force acting on the skier. It acts in a direction that is not the same as either gravity or centrifugal force alone, and it varies in magnitude and direction depending on where the skier is in the turn, how fast he is going, etc.. This direction is the “local, instantaneous vertical” direction, and is the single most important thing to a skier trying to stay in balance.
Again, draw a line from the skier’s center of mass to the ski edge in contact with the snow, and draw another line representing the direction of this net force through the skier’s center of mass. Unless the two lines are exactly coincident, the net force will make the skier fall to either the inside or outside of the turn. This is the technical meaning of being “in-balance” in a turn, and is exactly analogous to being in-balance while not turning, except that centrifugal force has been added to the mix.
For a skier in a turn, his instantaneous vertical direction is always inclined at some angle with respect to the static vertical direction of a stationary observer. Our hypothetical one-legged skier always must lean over in his turns at this angle to stay in balance, whether or not he is simultaneously employing angulation (e.g., to adjust skidding). I have always heard the angle (previous paragraph) the CM-edge line makes with respect to earth vertical called the angle of bank (in analogy to aircraft).
Bob Barnes posted a diagram related to this phenomena a few weeks ago in another thread. In that thread, the issue was forces in the plane of the hill, so that diagram was “from above”. In the present case of dynamic balance and the concept of the “instantaneous local vertical”, this can be seen more clearly in a side view. Maybe Bob’s graphic arts skills can be persuaded to be brought to bear on this topic as well.
Hope this helps,
Tom / PM
PS – As they say in textbooks, “I will leave it to the interested reader” to derive the details of a real, two-legged skier in a wide stance (instead of our hypothetical one-legged skier). There are no new physical phenomena going on in this case, but it can be a royal pain to describe everything in words.
PS#2 - Looking over earlier posts in this thread, I sensed the feeling that banking was something you could decide to do or not do in a turn. For a single ski (or snowboard) turn this is impossible - you've got to bank to stay in balance. OTOH, for two-legged turns, this is not the case, but I would point out that you immediately run into a problem in defining banking, and have to think about things like unequal leg extension, the geometry of which, as I said above, gets awfully complicated.
[This message has been edited by PhysicsMan (edited September 10, 2001).]</FONT>