Gliding for Gold: The Physics of Winter Sports

Technical Notes

T E C H N I C A L N O T E S These technical notes provide derivations for some of the equations that are graphed in the main text and back up certain claims that are made in the text. The level of math is variable—some of it is high school physics and math, and some is college level—and it is always condensed, to save space. However, if you want to reconstruct the derivations, there is enough development presented here so that you can fill in the gaps. NOTE 1. FRICTION AND DRAG Here I present the equations that we need elsewhere in this book for winter sports physics calculations. Sliding friction involves two solid objects in contact, moving with respect to one another. A typical classroom example is that of a block sliding down an incline, as shown in figure N1. If the incline angle, a, is carefully chosen, the block will not accelerate down the slope nor will it stick; it will slide down at a constant speed. In such a case there can be no net force acting upon the block, and so the forces shown in figure N1 must exactly balance. These forces are the force due to gravity (acting straight down), the force due to sliding friction (acting in the direction opposite to the block velocity), and the force N of the incline acting upon the block. This ‘‘normal’’ force points upward at an angle that is perpendicular to the slope. (The incline must exert a force on the block because the block does not fall through the incline, despite gravity.) The equation describing sliding friction is Fs = mN, (N1.1) where m is the coe≈cient of kinetic friction and N is the normal force (which, for the case of fig. N1, is given by N = mg cos a, where m is block mass and g is the acceleration due to gravity). The inclined plane can be used to determine the 150 TECHNICAL NOTES
 





   Figure N1. Sliding friction. A block sliding down an incline is subject to three forces (solid arrows). The gravitational force resolves into two components (dashed arrows) that are parallel and perpendicular to the incline; these components must equal in magnitude the normal force and the sliding friction force. kinetic friction coe≈cient: because there is no net force acting on the block, you can see (by balancing the forces of fig. N1) that m = tan a. For lubricated friction, describing the physical situation that is illustrated in figure 1.6b, the friction force is expressed as FL = mvAv h , (N1.2) where A is the slider area, v is slider speed, h is the thickness (assumed small) of the lubricating layer of liquid, and mv is the coe≈cient of dynamic viscosity of the liquid. Note that lubricated friction depends upon di√erent parameters than sliding friction; in particular, it depends upon slider speed. Aerodynamic drag is the friction force exerted upon a moving object (such as the puck of fig. 1.6c) by the air through which it travels. The magnitude of this drag force is FD = QcDrAv2 . (N1.3) Here, A is projectile area (this time it is the cross-sectional area, as seen from the front), r is air density, v is projectile speed, and cD is drag coe≈cient. Note that drag force depends upon the square of speed. To describe all the forms of friction that act upon a winter sports athlete who is TECHNICAL NOTES 151 rapidly traversing a section of snow or ice, we will need either equation (N1.1) or equation (N1.2), plus equation (N1.3). Sometimes, if the athlete’s speed is very low, we will not need (N1.3) because drag force will be small enough to be negligible. Sometimes we can ignore lubricated friction, equation (N1.2), because it is small in comparison to sliding friction. For many of our calculations we will have to take into consideration both sliding friction and aerodynamic drag. NOTE 2. THE PUSH-OFF To calculate the e≈ciency of a skater who pushes o√ from a standing start, consider the geometry of his skating motion, shown in figure 2.4b. I will make the simplifying assumption that friction is negligibly small for a skate moving along its length over ice. This assumption is reasonable for short steps such as occur when accelerating, though for longer, gliding...