Fric3. force F k and the equation (4.2) may be used. The sense of F k is opposite


 Nigel Jacobs
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1 4. FRICTION 4.1 Laws of friction. We know from experience that when two bodies tend to slide on each other a resisting force appears at their surface of contact which opposes their relative motion. The laws of dry friction are best understood by the following experiment. A block of weight W is placed on a horizontal surface (Fig.4.1a). Suppose, now, that a horizontal force P is applied to the block (Fig.4.1b). If P is small, the block will not move; some other horizontal force must therefore exist, which balances P. This other force is the staticfriction force F, which is actually the resultant of a great number of forces acting over the entire surface of contact between bodies. The nature of these forces is not known exactly, but it is generally assumed that these forces are due to the irregularities of the surface in contact and also, to a certain extent, to molecular attraction. A detail examination of the nature of friction is complex physicomechanical problem lying beyond the scope of theoretical mechanics. If the force P is increased, the friction force F also increases, continuing to oppose P, until its magnitude reaches a certain maximum value F m (Fig.4.1c). If P is further increased, the friction force cannot balance it any more and the block starts sliding. As soon as the block has been set in motion, the magnitude of F drops from F m to a lower value F k. From then on, the block keeps sliding with increasing velocity while the friction force, denoted by F k and called kineticfriction force, remains approximately constant. Experimental evidence shows that the maximum value of staticfriction force F m is proportional to the normal component N of the reaction of the surface, i.e. F m = µ s N (4.1) where µ s is a constant called the coefficient of static friction. Similarly, the magnitude F k of the kineticfriction force may be put in the form F k = µ k N (4.2) where µ k is a constant called the coefficient of kinetic friction. Fric1
2 The coefficients of friction µ s and µ k do not depend upon the area of the surfaces in contact. Both coefficients, however, depend strongly on the nature of the surfaces in contact. Since they also depend upon the exact condition of the surfaces, their value is seldom known with an accuracy greater than 5%. Approximate values of coefficients of static friction are given in Table 4.1 for various dry surfaces. The corresponding values of the coefficient of kinetic friction would be about 25% smaller. Table 4.1 Approximate values of coefficients of static friction for dry surfaces Metal on metal 0,15 0,35 Metal on wood 0,20 0,60 Metal on leather 0,30 0,60 Wood on wood 0,30 0,70 Rubber on concrete 0,60 0,90 Steel on ice ~0,03 From the description given above, it appears that four different situations may occur when a rigid body is in contact with a horizontal surface: a) The forces applied to the body do not tend to move it along the surface of contact; there is no friction force (Fig.4.2a); b) The applied forces tend to move the body along the surface of contact but are not large enough to set it in motion. The friction force F which has developed may be found by solving the equations of equilibrium for the body. Since there is no evidence that the maximum value of staticfriction force has been reached, the equation (4.1) cannot be used to determine the friction force F (Fig.4.2b); If the magnitude F of the friction force is smaller than its maximum value F m, i.e. F < F m = µ s N, we say that the friction is not developed Fric2
3 c) The applied forces are such that the body is just about to slide. We say that motion is impending. The friction force F has reached its maximum value F m, and, together with the normal force N, balances the applied forces. Both the equation of equilibrium and the equation (4.1) may be used. We also note that the friction force has a sense opposite to the sense of impending motion (Fig.4.2c); d) The body is sliding under the action of the applied forces, and the equations of equilibrium do not apply any more. However, F is now equal to the kineticfriction force F k and the equation (4.2) may be used. The sense of F k is opposite to the sense of motion (Fig.4.2d). Let us resume the above statements in a form of three Coulomb laws (1781!): At the condition of impending motion and once motion has begun, it is possible for given dry surfaces to relate the frictional and normal components of force so that: 1. The total amount of friction which can be developed is independent of the magnitude of the area of contact. 2. The total friction force which can be developed is proportional to the normal force transmitted across the surface of contact. 3. For low velocities, the total amount of friction which can be developed is practically independent of velocity. However, it is less than the frictional force corresponding to impending motion. Remarks: 1. Very often we limit our modelling task to either static or dynamic problem only. In other words we don t need to create a model for static and dynamic purposes simultaneously. In those cases a friction force as well as a coefficient of friction may be denoted simpler without lower indices, i.e. F and µ. The latter simplified symbols will be used in what follows. 2. In a majority of Polish handbooks the letter T is used for denoting the friction force (from the word tarcie). Fric3
4 4.2 Angle of friction. It is sometimes found convenient to replace the normal force N and the friction force F by their resultant R. Let us consider again a block of weight W resting on a horizontal plane surface. If no horizontal force is applied to the block, the resultant R reduces to the normal N (Fig.4.3a). However, if the applied force P has a horizontal component P x which tends to move the block, the force R will have a horizontal component F and, will form a certain angle with the vertical (Fig.4.3b). If P x is increased until motion becomes impending, the angle between R and the vertical grows and reaches a maximum value (Fig.4.3c). This value is called angle of friction and denoted by φ. From Fig.4.3c, we note that tan φ = F m /N = µn/n. Thus finally tan φ = µ (4.3) 4.3 Problems involving dry friction. Problems involving dry friction are found in many engineering applications. Some deal with simple situations particularly when a single rigid body has only one point (or one plane) of contact with the ground or another body (Fig.4.4a). There is also a simple situation when a single body has more than one point of contact but an impending or actual motion in one point of contact results in the same in all others contact points (Fig.4.4b,c). This means that fully developed friction appears simultaneously in all contact points. There are, however, more complicated situations when: a) at the stage of modeling we have to select from an infinite number of possible contact points only one or two actual points of contact (Fig.4.5a); b) a single rigid body has two points of contact with other bodies but the impending or actual motion may appear at one or two points and this is not known in advance (Fig.4.5b) c) a system of rigid bodies has more than one point of contact but the motion of one member of a system does not necessarily affect the motion of others members (Fig.4.5c) Also, a number of common machines and mechanisms may be analyzed by applying the laws of dry friction. These include wedges, screws, journal and Fric4
5 thrust bearings, and belt transmissions. They can be studied in more advanced texts and some of them will be considered in the final chapter of this lecture entitled Selected Applications. There is another problem caused by friction. It appears when a freebody diagram is to be drawn. As long as friction is not involved for bodies in contact an arbitrary sense may be given to all unknown reactions. This is not the case when friction has to be taken into account (is involved). Now, both components of a total reaction force, i.e. normal and tangential reaction must be drawn with an appropriate (correct) sense. A relatively simple situation occurs when two bodies A and B are in contact and tend to slide on each other (Fig.4.6a). The friction forces exerted, respectively, by A on B and by B on A are equal and opposite (Newton s third law). It is important, in drawing the freebody diagram of one of the bodies, to include the appropriate friction force with its correct sense. The following rule should then be applied: the sense of the friction force acting on A is opposite to that of the motion (or impending motion) of A as observed from B (Fig.4.6b). The sense of the friction force acting on B is determined in a similar way (Fig.4.6c). Note that the motion of A as observed from B is a relative motion. A more difficult situation is presented in Fig.4.7a. Now a rotating disc is put at a rough surface. In order to determine the correct sense of friction force in a phase when the disc touches the ground, we need to establish the relative velocity of the touch point of the disc with respect (relative) to the ground. The actual friction force acts in a direction opposite to this velocity (see Fig.4.7b). In fact, it is the friction force which initiates and then accelerates the translational motion of the disc. 4.4 Rolling resistance wheel friction. The wheel is one of the most important inventions of our civilization. Its use makes it possible to move heavy loads with relatively little effort. Because the point of the wheel in contact with the ground at any given instant has no relative motion with respect to the ground, the wheel Fric5
6 eliminates the large friction forces, which would arise if the load were in direct contact with the ground. In practice, however, there exists some resistance to the motion of the wheel. This resistance is mostly due to the fact that wheel and ground deform, with the result that contact between wheel and ground takes place, not at a single point, but over a certain area. This resistance is known as the rolling friction or rolling resistance. The phenomenon of rolling resistance and the law of rolling friction can be best understood by the following experiment. Consider a wheel of weight W resting on a rough horizontal surface (Fig.4.8a). Under the load W, both wheel and the ground deform slightly, causing the contact between wheel and the ground to take place over a certain area. Suppose, now, that a small horizontal force P is applied at center of wheel (Fig.4.8b). Experimental evidence shows that the resultant R of the forces exerted by the ground on the wheel over this area is applied at a point B, which is not located directly under the center O of the wheel, but slightly in front of it. Let the force P increases such that the wheel rolls at constant speed. To balance the moment of W about B and to keep the wheel rolling at constant speed, the force P must satisfy the following equation Pr = Wb (4.4) where r is the radius of wheel and b is the horizontal distance between O and B. The distance b is commonly called the coefficient of rolling resistance, which has the dimension of length. For instance b = 0,5 mm for a wheel rolling on a rail, both of which are made of mild steel. The value of b depends on the material of the bodies and is determined experimentally. The ratio b/r for most materials is much less than the coefficient of static friction µ s. That is why in mechanisms rolling parts (wheels, rollers, ball bearings, etc.) are preferred to sliding parts. Fric6
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