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2 In stationary systems, friction manifests itself as a force equal and opposite to the shear force applied to the interface. Thus, as in Fig. 2, if a small force S is applied, a friction force P will be generated, equal and opposite to S, so that the surfaces remain at rest. P can take on any magnitude up to a limiting value F, and can therefore prevent sliding whenever S is less than F. If the shear force S exceeds F, slipping occurs. During sliding, the friction force remains approximately equal to F (though often it is smaller by about 20%) and always acts in a direction oppos- ing the relative motion. The friction force is proportional to the normal force L, and the constant of proportionality is defined as the friction coefficient f. This is expressed by the equation F = fLF = fL. The "laws" of friction are essentially statements about the friction coefficient, and have been worked out over the past five centuries by a number of distinguished engineers and scientists, among them Leonardo da Vinci and Charles de Coulomb. These laws, which are approximately, but never per- fectly, obeyed by typical sliding systems are as follows: (1) The friction coefficient is independent of the normal force. This is another way of saying that the friction force is proportional to the normal force. (2) The friction coefficient is independent of the sliding velocity. However, as was stated above, the friction for surfaces at rest is often about 20% greater than for the same system when sliding. Such systems are often spoken of as hav- ing larger static friction than kinetic friction. (3) The friction co- efficient is independent of the apparent area of contact. Thus, flat surfaces and surfaces with contacting asperities but of the same projected area give the same friction coefficient. (4) The friction coefficient is independent of surface roughness. Actually, the friction coefficient is primarily a property of the contacting materials and the contaminants or lubricants at the interface. Closely related to the friction coefficient is the concept of friction angle. The friction angle θf is the largest angle rela- tive to the horizontal at which a surface may be tilted, so that an object that is placed on the surface does not slide down. Leonhard Euler was the first to show that tan θf is equal to the friction coefficient. Mechanism A number of processes occur at the interface between two solids that tend to inhibit sliding or to use up energy during sliding. All these contribute to friction. Over the last three cen- turies, there has been much discussion over which one of these processes is the most important and can thus be considered the main cause of friction. Until about 1940 it was considered that surface roughness was the main cause of friction, and that work had to be done during sliding to lift one sliding surface over the high spots on the other one. Modern work has largely discounted the importance of surface roughness: first, from the standpoint of theoretical considerations, because the work done in sliding up a high spot on a surface is largely recovered on the down side, and second, because experimental testing shows that for most sliding systems friction coefficients are largely independent of the roughness of the surfaces. Very smooth surfaces, like cleaved mica, which is smooth to within one atomic diameter, give friction at least as great as that of ordinary surfaces. For reasons that are not clear, most people find it very hard to accept the fact that smooth surfaces give as much friction as rough ones. Classical friction laws have far outlived a variety of attempts to explain them on a fundamental basis. Surface roughness was ruled out definitively in the mid-1950s as a possible mechanism for most friction. Molecular adhesion, though, remained a strong possibility, due in large part to studies which found that friction, although independent of apparent macro- scopic contact area, is in fact proportional to the true contact area. That is, the microscopic irregularities of the surfaces touch and push into one another, and the area of these contacting re- gions is directly proportional to the friction force. Subsequent experiments explored the possibility that friction arose from sufficiently strong bonding at the true contact points so as to produce continual tearing away of tiny fragments of mate- rial. This explanation failed, however, to predict experimental observation. Indeed, it was proved incorrect in the 1970s, when a "surface force apparatus" developed for atomic-scale friction measurements found clear evidence for friction in the total absence of wear. The surface force apparatus consists of two perfectly flat, cleaved mica surfaces which are separated by Friction (continued) + ward ' s science Fig. 2: The forces acting on a book resting on a flat surface when a sheer force S is applied. The friction from P is equal to S (up to a limiting value F), while L, the normal force, is equal to the weight W of the book.