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Cross-Country Skiing

Cross-country skiing or nordic skiing is a sport that involves sliding over snow on long, slender “boards.” Unlike downhill skiing it is performed in varying terrain: Uphill, downhill and flat. Because weight is more of a concern and stability less so than downhill skiing, the skis tend to be much narrower.

Forward motion on the flats and uphills is achieved in 1 of 2 techniques: classic and skate. In classic skiing, part of the ski's base is roughed up or a hydrocarbon material is applied to it to increase friction with the snow. Movement is achieved by pushing off on one ski, while swinging the other ski and leg forward and then letting the momentum transfer into glide. This is then repeated on the other side. For a video click this link.

Forward movement in skate skiing is achieved by placing the ski diagonal to the direction of and then pushing out perpendicular to the direction of motion. The vector of this force has components in both the forward and lateral directions. The result of this is a motion forward and across to the other ski, where the same action is performed again. For a video click this link.

In both techniques poles are used to supplement the forward force, but for the purpose of this website they will be ignored.

While on a good day it may seem like it takes no effort to get a pair of skis to glide the truth is that even on the flats and downhills there is always a force that will be opposing the direction of motion. This force is composed of 3 major parts: gravity, air resistance and friction.

FTotal = Fg  + Far + Ff

On a set course gravity isn’t something  that can be changed and air resistance isn’t affected much by the skis. When looking at the skis the force of friction is the only changeable force and so it is a major concern when preparing and manufacturing skis.

To get a better picture of this we can split the force of friction into 4 parts. Electrostatic, dirt, capillary and dry. While the electrostatic and capillary forces aren’t actually frictional forces it makes the picture simpler to define them as such.

Ff = Felec + Fdirt + Fcap + Fdry

The electrostatic force comes from the electric properties of snow and the ski base. Electrons are transferred from one to the other creating an electric field. This electric field does not want to change and so as the ski moves forward there is an electric force opposing this movement. The force is proportional to the charge developed on the ski base and the snow and the distance between them.  Where q is the charge and r is the distance between the charges.

Felec=(kq1q2)/r^2 k=8.99E9

The friction from dirt is actually a form of dry friction. It occurs because a ski trail has more than just snow on it. There are particles of dust, wax and other debris from the environment. This dirt can get stuck to the ski base from both the electric field and the wax. The dirt then drags on the snow and creates friction.

The interaction of the ski base and snow crystals creates dry friction. The sharp points of the snow crystals dig into the base of the ski and break off or drag, creating friction. This can be modeled with the equation.

Fdry=ukFn

Where uk is the friction coefficient, which is given depending on the characteristics of the snow and ski base. Fn is the normal force, which is equal to the weight being applied perpendicular to the ski. An interesting note about this equation is that surface area is absent, meaning that friction does not increase or decrease with surface area. This will be important when we talk about ski flex.

Capillary force is due to the stickiness of water. Water molecules are very polar, meaning the electrons tend to group on one side of the molecule while the nuclei are on the other. This polarity cause the molecules to attract and stick to each other. It is the same reason why it is easier to separate pieces of paper when you lick your fingers. As a ski glides across the snow the other forms of friction create heat, which melts a very thin layer of water under the ski. This layer of water is attracted to itself and the ski base, creating drag against the direction of motion.

Fighting Friction

When building and preparing a ski, minimizing friction is one the most important aspects to consider. There are three major parts of the ski that can affect this friction: Flex, structure and wax. Generally they are in that order for the influence they have. Flex is the most important while wax is the least.