Ski Coaching

There is some discussion as to whether or not skis CAN be skied truly parallel, whilst making arcs.

IF they can, then the circles, the segments of which are being described in the snow by the skis, MUST be concentric. If those circles are not concentric then the skis can only be parallel at one infinitessimally small point. At two points on the circles they describe, the lines they are drawing will cross. This means the lines are approaching and diverging from one another. Clearly then, they are not parallel. The skis would then cross, as would the skier's legs and the result is not pleasant to contemplate.

If you wish to test this you can, very simply and without complex physics being involved. The skis can only describe circles of either the same or different radii.

Example 1: Same radius. If the two circles being described are the same radius, then either they will be concentric or not. If concentric there will only be one line drawn in the snow and since there are two skis that is not possible. So - if the radii of the two circles are the same, then the circles must be centred on different points.

Their circumferences cannot then be parallel. All you need in order to prove this to yourself is to take a set of compasses, (or in their absence a circular saucer or even a coffee cup), and draw two circles of matching radius, centred on two different, albeit nearby points. You will see that the circles cross one another twice. Therefore their circumferences are not parallel.

Example 2: Now consider two circles of slightly different radii. The only circumstance in which their circumferences can be parallel is if they are concentric.

IF ( and it may be big "IF" ) truly parallel arcs can be described by two skis acting simultaneously, then whatever the physical influences at work are, the inner ski must be reacting differently to the outer ski. Skis can only bend, tilt (with or without torsional distortion), or pivot.

Bending and tilting combined can lead to "carving". Pivoting leads to skidding, as does torsional distortion.

I have no idea whether true, absolutely exact parallel skiing is possible, but it seems  clear to me, that if it is, it can only be done with arcs which are part of concentric circles.

So - either it is possible and the inner and outer skis are being separately influenced by the external forces; or the external forces are equalised between the two skis, and while "parallel" skiing may APPEAR to be happening, that is a mistaken perception resulting from not being able to watch and measure sufficient of the circumference of the circles being described.

Bob Trueman

My fellow coach Dave Tapley reported to me that one or two skiing blogs have recently been filled with discussion about the true nature of "parallel" skiing and the perplexing questions that arise once you start thinking in depth about it.

The discussion hinges on whether or not "parallel" skiing is actually possible. Dave quoted his own observation that when you are "carving" perfectly you can inspect your skis' tracks and they look to be perfectly parallel. However, they are not drawing the same radius arcs (part-circles).

So, After some thought I wrote to a pupil and friend of mine,  physicist Tony York. Here is the e-mail like what I wrote.

Let's say we have a skier effecting an arc, a perfectly "carved" arc - an arc during which both skis slide perfectly (no skid) -and let's say that his skis are parallel to one another all the way round that arc.

For this to happen, the inner ski must perforce travel a shorter distance than the outer ski.  For this to happen without skidding, the inner ski must either, tilt more, or bend more, or a combination of both. Were this not to be the case, they would necessarily be describing segments of arcs of non-concentric circles.

To bend more it would need to be receiving greater centripetal force, which we know would be very unstable for the skier, so optimally no more than 50% of the force should be being resisted by the inner ski. Unless - I wonder - being nearer to the circle's centre it inevitably receives more force ? ?

Even in this scenario, the inner ski must be tilted slightly more than the outer ski, or it would skid. This is because were it to be tilted to the same degree it would be describing a circle of the same diameter as the outer, but in a different location - they would not be concentric; and if you draw this out on a piece of paper it becomes obvious that the two circles must cross (twice) which thereby denies the "parallel" requirement of this experiment.

Now, there is plenty enough bio-mechanical movement in the hips and ankles to permit this variation, but here a little confusion arises in my mind ( which is rather unusual  -  because usually there is a lot of confusion in my mind; I must do this again!).

There will be one aggregate centre of mass for the skier, supported against the centripetal force by two platforms.  Here then is where my confusion arises.

Where, precisely is the centripetal force's own centre of origin? Or is this a daft question?  Is there, for example, just one centre of centripetal force, or since there are two platforms, are there also two centres of this force? After considering this I feel there must be two, because each ski (platform) is resisting a force, and I feel that this necessitates having two forces, coming from two slightly different directions. This being the case, then there are two reasons for the inner ski to tilt more - 1) in order to present a platform at 90 degrees to the force, and 2) in order to enable the ski to slide perfectly around a circle of smaller radius.

But if this is so, then if you followed the directional lines of these forces (or this force) from whence do they emanate? Is it for example on the snow's surface? Or precisely at the interface between the platform and supporting surface? Or - does it emanate from somewhere else, underground? And if so, how far away/down?

I think it must be at the interface only, which is where the force and the resistance meet. Am I right? After all ( I conjecture) unless there is resistance, there will be no centripetal force - in effect they are one and the same???? Without the one, you cannot have the other.

Bob

PS - It's just occurred to me that the bend in the ski is created at least in part by a force from ahead of it, acting on the shovel through a couple between the shovel and the ski's centre. The shorter the radius of the circle being followed for any given tangential speed, wouldn't the force be inevitably greater? So might we not get more bend anyway even though the skier's mass was being equally distributed between the two skis?

Tony, after considerable cogitation answered as follows, and I'm very grateful to him.

OK, (he said) here are my thoughts so far:
 
Since the skis are going round curves of different radii, and are therefore travelling at different speeds, it is mathematically easier to say they are both moving with the same angular velocity (ie they would both take the same time to complete a full circle).  The expression for the force is then mw²r (m is mass, w is angular velocity, r is radius).  Because r is greater for the outside ski, there will be more force, which is what the skier needs, in order to be stable.
 
So far so good - but then how do the skis provide this force?  If the outer one is producing more of the centripetal force, and they are both at the same angle, it will bend more, making it impossible for both skis to be "carving", as the inner one is following a tighter curve.  If the inner one is tilted more, perhaps it could be describing a tighter arc, but be bent less, consistent with it producing less force.  I should stop now while I'm ahead, but I have a horrible feeling that if you look at a still photo of a racer in a turn, the outside ski is tilted more! (Yes, but you'll usually see that the inner ski is all but "floating" and is not actually carving, even though that's what they would like. Bob)
 
The bending of the ski is a result of the snow pushing against it, but that won't be simple either.  Even in the simplest imaginable scenario of the same force from the snow against each cm of the ski, the front of the ski will have more bending moment, as it is longer than the tail.  Whether this leads to more actual bending depends on the stiffness of the ski, which varies along the ski in a very complex manner, I would imagine.
 
As implicit in last para, as far as the ski is concerned the force comes from the snow immediately in contact with it, but that snow is in turn supported by the snow beneath it, which is in turn supported by the ground beneath it.  This is of course why the skier sinks deeper into powder before there is enough force generated to support him/her.
 
I don't think the idea of a "centre of centripetal force" is useful.  The vector sum of all the forces from both skis must pass through the centre of mass of the skier and be directed towards the centre of the circle in which he/she is travelling.  One also needs to be careful in talking about reaction forces.  This vector sum is effectively a single force acting on the skier.  There is no sense in which the skier is in equilibrium; he/she is being continually accelerated towards the centre of the circle.
 
God knows how ski designers do the business, & God knows how any of us can actually get the skis to do what we want (sometimes).  I should probably stick to making furniture or high energy nuclear physics; that would be simpler. 

I am very grateful to Tony for his observations, and if anyone wants to join in, then please do so; it won't make anybody's skiing any better, but it keeps the old grey matter from atrophying any more quickly than is necessary! And it sure as Hell beats "doing turns!"

Bob Valentine Trueman