Help me buy men's skis...

[jack97]

Originally Posted by Marc

As far as the forces involved in a turn go, height will affect the placement of the center of mass of the skier, and where the skier can move his CM to his advantage to balance the forces induced by the angular acceleration of a turn. Remember, any time you change velocity, you accelerate, so when you arc through a turn, you're experiencing acceleration.

The force placed on the skier through a turn of like radii is shown by Newton's 2nd law, force is equal to mass times acceleration. For the same skier making a turn of the same radius, the mass of the skier is accelerated by the same amount, so the skier with the greater mass will experience a greater force, which he will exert back on the snow through his skies (3rd law).

Since it is force on the ski that causes it to deform (deflect), it is only the mass of the skier and the radius of your turn that affects the force on the ski. In general, a longer ski of the same model is not only scaled up to provide a bigger spring for the bigger force, but also gives more surface edge area that the heavier skier needs to maintain the no slip edge condition.

Hmm interesting conversation.... from the top of my head, for the same weight, a skier height does come into play. The com is higher but when he/she moves laterally to make a turn, the axis of rotation is further away from the ski. IIRC, the angular forces generated would be greater b/c of the difference in com location, again assume same weight, same speed going into turn, same snow conditions and so on. Thus it would be relatively easier to deform (reverse camber) the ski due to the extra force generated.

[Marc]

Originally Posted by jack97

Hmm interesting conversation.... from the top of my head, for the same weight, a skier height does come into play. The com is higher but when he/she moves laterally to make a turn, the axis of rotation is further away from the ski. IIRC, the angular forces generated would be greater b/c of the difference in com location, again assume same weight, same speed going into turn, same snow conditions and so on. Thus it would be relatively easier to deform (reverse camber) the ski due to the extra force generated.

Well, the problem is a lot more detailed then my inital response to Greg would lead on. Height does play a factor in the placement of a skier's CM, which certainly has the potential to affect the force on the ski, but in my estimation, does not make enough of a difference (or any at all) to matter when choosing the length of a ski. I'll draw it out so it's easier to discuss...

[Marc]

Above we have a crude drawing of the forces acting on a skier, let's say, mid way through a carved turn.

The blue angled line represents the outside ski, inside ski not shown for simplicity's sake and the fact that much more force is typically applied to the outside ski during a turn.

CM1 represents center mass for a tall skier, CM2 represent center mass from a shorter skier, both of the same mass. The blue line marked AOR is the axis of rotation of the carved turn (approximating a circular radius if viewed from above).

Now the forces- the horizontal green arrow represents the centripetal force applied by the snow on the skier accelerating him through the turn. The vertical green arrow represents the reaction force of the ground pushing against the skiers weight, which is the same in both instances. The two black vertical arrows represent the force of gravity acting on the skiers' respective center masses. The horizontal black arrows represent the reaction force, "centrifugal" force, to the centripetal acceleration, which act on the skiers' respective center masses. The black angled force pointing on the ski is the result of the vector sum of the forces acting on the skier's CM and applied at this point. This is the force that loads the ski.

Now for torques acting on the skier, which must balance, otherwise the skier goes tumbling off tangential to the turn. All moment arms are represented by the double ended red arrows. Negative (counterclockwise) torque is generated by the centrifugal force acting on a moment arm that has a length equal to the height of the skier's CM from the ground. This torque is balanced by the skier bringing his CM closer to the AOR, and creating a moment arm on which the weight of the skier acts (lateral distance from the dashed red line to the CM of the skier). This generates postivie torque, to balance the negative.

The last element of the equation - the limit of adhesion of the ski. This is generated by the frictional force of the ski digging into the snow, which is proportional to the weight of the skier, which in this case, is the same.

Now let's assume they are tracking through a turn of the same radius (meaning ski to AOR). They both lean over to get the ski on edge to generate frictional force and to generate that positive torque. The centripetal force generated is proportional to lateral distance from CM to AOR. It first glance, it looks like the taller skier can keep his mass closer to the AOR and thereforce reduce the centripital force generated and therefore the force on the ski.

However, if you'll look at the diagram, there's an imbalance on torque on the taller skier. As he leans over moving his CM closer to the AOR, the centripital force is less, however it is acting on a larger moment arm due to his height. The shorter skier generates more centripital force, however this has a smaller moment arm to act on due to the height of the skier. So the two "centripital torques" are close to equal (not everything scales linearly, but pretty close). This means the taller skier of the same weight, by leaning over to generate less centripetal force ends up giving more moment arm to his weight, which means he is generating more positive torque than the shorter skier. To compensate, and to keep from falling over, the taller skier must bend his legs, and reduce that moment arm. To balance, he then moves his center of mass to where the shorter skier's is, and thus the applied force on the ski is still about the same.

This would only increase or decrease significantly if you changed the weight of the skier.

[Greg]

My head just exploded. I'm outta this one...

[severine]

OMG, it's just like talking to my cousin. Who was an engineer (although he recently switched professions to becoming a patent lawyer). I could totally see him diagram his argument in a discussion.

[Greg]

Marc - I have to know. Is all this stuff off the top of your head, or do you have Wikipedia or something fired up in another browser? If the former, then I think it's quite possible you are an alien.

[Marc]

Originally Posted by Greg

Marc - I have to know. Is all this stuff off the top of your head, or do you have Wikipedia or something fired up in another browser? If the former, then I think it's quite possible you are an alien.

Nope, that's all in my head. It's easy, just identify and balances the forces and torques

[MRGisevil]

Originally Posted by Marc

Nope, that's all in my head. It's easy, just identify and balances the forces and torques

I work with Engineers all day. I can vouch for all of that being as common knowledge to him as shoe tieing is to the rest of us.

[Marc]

Originally Posted by MRGisevil

I work with Engineers all day. I can vouch for all of that being as common knowledge to him as shoe tieing is to the rest of us.

Oh man, and you shoulda seen the last time I tried to tie my shoes on my own. What a disaster.

[jack97]

Originally Posted by Marc

Now for torques acting on the skier, which must balance, otherwise the skier goes tumbling off tangential to the turn.

I have thinks about this one, I would agree that there is a critical angle that the ski edge has to take else one would tumble over. I think there are more parameters to this.

Originally Posted by Marc

However, if you'll look at the diagram, there's an imbalance on torque on the taller skier. As he leans over moving his CM closer to the AOR, the centripital force is less, however it is acting on a larger moment arm due to his height.

Hmm, ok I got it, the centripital force would be lower for the taller skier since there is less mass at the radius of rotation. And it would be roughly the same for the tall and short skier.

But I beleive that the conservation of angular momentum does favors the taller skier.... still have to think one through.