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Carving



• R_sc = the side-cut radius of the ski. This is determined by Rsc=LTOT2/2000*(S+H-2*W).In simple terms, this is radius of taper in the skis in the center. Skis are made in the manner to allow for canter, as well as increase turning radius. (unknown, Side-Cut Radius Calculator 2015)
• Carving- form of turning and stopping achieved by turning the tips of the skis away from the direction of velocity, achieving either a quick turn or when pressure is applied appropriately, allowing the skier to come to a quick stop, even along a steep incline.
• Skidding- achieved when carving is non-optimal, which we will soon cover. Skidding occurs when the angle between the ski and the force applied to the top of the ski via the skier’s boot is less than 90o.
• R_t= The turning radius of the skis. Rt=Rsccos(φ) where φ is the angle of tilt between the ski and the snow (Jentshura, Physics of Skiing: The Ideal Carving Equation and its Applications 2008). By this definition, we see that turning radius is directly proportional to the side cut radius of skis. Additionally, the longer the total length of the skis, the greater the turning radius. In this same thread, we see that the skinnier either the nose or tail of the ski is, the greater the turning radius. More importantly, as φ approaches 0, Rt=Rsc. This is where canter comes into play.
• Canter: The amount of flex allowed in the center of the ski bring the center closer to the ground whilst bowing the tips or via reverse canter, in which the center is elevated and the tips are pressed harder into the ground. Based on the previous definition of turning radius, we see that the greater the surface area of the ski touching the snow, a skier can achieve tighter turns and less skidding, which leads to faster runs.
I will point you to the following graphic:


We will use this graphic as a means of convention to explain the mathematical processes at hand and to draw our own meaningful conclusions from the physics. For context, velocity v is pointed out of the screen toward the reader through center of mass G. P and Q are the points of contact for the inside and outside ski, respectively. S is the distance between P and Q. L the length of line segment PG. Ѳ is the angle between the snow and the center of mass G. N1 & N2 are defined by the normal force exerted on points P and Q, while F1 & F2 are force acting in the direction of turn x which also takes into account centripetal acceleration ac=v2/Rt. (Physics Problems, Physics Of Skiing 2020)
We will now go through a hypothetical experiment to examine the equations at hand. Let us assume that you, the reader, are a beginner and are using this research paper to draw conclusions about what skis you need and how to use them. Let us assume you are an average sized female, 160cm tall and 77kg. This would mean you have a shoulder width of about roughly 35cm or .35m. We can also assume at this point because you are a beginner that as you ride your skis you will assume an athletic position with feet and skis shoulder width apart. As you develop more as a skier, this length decreases, but for now this, and hopefully no wider, is the stance you are going to take. Now, to calculate mass of the system, as opposed to your mass alone, we will add 11 kilos of weight to accommodate skis, boots, thermal clothing, snow suit, poles, helmet, and goggles. We now have all elements necessary to analyze our first big question: How steep of a run can you go down and not skid or slip?


Steepness or piste of runs is defined by angle α. Traditionally, piste and difficulty are defined into 3 categories: green circles, blue squares, and black diamonds. The pitch of these runs varies from resort to resort, but on average, we can expect greens to be no more than a 25% grade, blues to be no more than a 40% grade, and blacks to be no less than a 40% grade. For contrast of just how steep that really is, “A 45-degree pitch is equivalent to a 100-percent grade, and both mean that a run descends one vertical foot for each horizontal foot. "In perspective, a very steep highway-pass road is approximately 7 percent or about 4 degrees," according to the Highlands Extreme Guide trail map. My carpenter friends who also teach skiing like to tell their students that the incline of a standard house stairs is 30-35 degrees, which seems mild enough on the way to breakfast-but not when you're looking down the Nose of the Headwall at Squaw.” (Editors, How Steep Is Steep? 2018).
Now that we have this information, lets introduce our equations of carving.
By the law of conservation of energy, and assuming that the center of mass G is unmoving in the y direction, we can assume the following:

Where N1 and N2 are the normal forces acting on your skis and α is the measurement, in degrees, piste. The following equation is the measurement of force in the x direction and flows as follows:

Finally, to understand centripetal acceleration more fully, we have:

To solve the following system of equations for α, we will assume all pressure is being applied to the inside leg, which is most commonly done by beginners such as yourself. This implies that N2=F2=0. In addition, Ѳ=83o, L=.84m, N1=810N, v2=6m/s, Rt=12.124m, β=60o. The following mathematically logic flows from this:



Finally, let's talk about taking flight