resistance will have its maximum value (μQ). If P is greater than μQ motion will take place, but the moving force will be less than P, since, although when motion has commenced the frictional resistance is often no longer equal to μQ, yet friction still acts as a force tending to prevent motion. Since the coefficient of friction is independent of the surface of contact, it follows that for a given value of Q the frictional resistance (F) is also independent of the extent of the surface of contact. If A is the area of this surface, then the pressure per unit area is Q/A, and the frictional resistance per unit area is μQ/A. If, while Q remains the same, a is reduced to A', then the pressure per unit area is increased to Q/A', and the frictional resistance per unit area is increased to μQ/A'. Hence the frictional resistance per unit area varies directly as the pressure per unit area. 97*. Limiting Angle.-When motion is just about to commence, and hence P is equal to μQ, the body is in a state of equilibrium under three from E draw EF parallel to P, and hence at right Then, by the triangle of angles to DE, to represent P in magnitude. forces ($72), the reaction which, together with the forces P and Q, maintains the body C in equilibrium, must be represented in magnitude and direction by the line FD. Therefore the angle FDE is equal to the angle between the reaction CR' and the normal. Since DE is equal to Q and EF to P, which is equal to μQ, we have tan : = This angle 4, which represents the greatest angle the line of action of the reaction can make with the normal to the surface of contact, is called the limiting angle. R F N If a force is applied to C along such a direction as FC (Fig. 73), making an angle of with the normal, then if is less than the limiting angle, motion of C will not take place, however great the value of this force. The reason is that we may resolve the force into two components, one parallel to the surface, which tends to produce motion and is resisted by the friction, and the other, which acts along the normal, produces a contact pressure. If F is the force, the component parallel to the surface A FIG. 73. B is F sin, and the component parallel to the normal is F cos . If motion is just about to take place, and we neglect the weight of the body, then But μtan F sin uF cosy, μ=tan . where is the limiting angle. Hence if is less than motion will not take place. 98*. Angle of Repose.-If a body G (Fig. 74) of mass m is placed on an inclined plane AB, then, if there were no friction between G and the plane, the only forces acting would be the weight, which is a force of mg acting vertically downwards and the reaction of the plane GR acting at right angles to AB. As these forces are not in the same straight line, the body would move down the incline. If, however, there is friction between G and the surface of the plane, the friction will tend to prevent motion, and till the plane has a certain slope the body will remain at rest. Το find the maximum inclination (4) of the plane A FIG. 74. B C to the horizontal we resolve the force mg into a component parallel to BA, which tends to produce motion, and a component normal to BA, which acts as the contact pressure. In the triangle DGE, the angle EGD is equal to 4, and ED is parallel to AB. Hence the component of mg parallel to BA is mg sin, and the component perpendicular to BA is mg cos 4. If motion is just about to commence, mg sin umg cos & Hence if is greater than the limiting angle, motion takes place. The maximum inclination to the horizontal of the plane which is possible without the body sliding is called the angle of repose. Thus the angle of repose is equal to the limiting angle, and the coefficient of friction is equal to the tangent of either of these angles. H 99*. Kinetic Friction between Solids.-As mentioned in § 96, after slipping has commenced the friction continues as a force tending to prevent motion, but the magnitude of the friction is in general less than it is just before slipping commences. It is found by experiment that, as long as the speed of the motion is not too great, the frictional resistance is proportional to the total pressure between the two solids, and independent of the velocity. If is the total normal pressure between the solids and F is the frictional resistance, then F=vQ, where is called the coefficient of kinetic friction. Hence if a force P parallel to the plane surface AB (Fig. 72) act on a body C of mass m, then Q=mg and F=vmg. Since the frictional resistance opposes the motion, the resultant force which is available for changing the motion of the body is P-F or P-vmg. The acceleration (a) produced by this force is given by P-F P If there had been no friction the acceleration would have been P/m, so that the effect of friction is equivalent to a negative acceleration of vg units. Of course, if P is less than vmg, the body if in movement will gradually come to rest, and then it will require a force greater than μg to start motion again. In the following table, some of the values of the coefficient of kinetic friction obtained by Morin are given. This table shows in a very marked manner how the friction between solids is reduced by the presence of a layer of lubricant. If the motion is extremely rapid, and the surfaces in contact are sufficiently large, it is possible to use air as a lubricant, and under these circumstances the friction is enormously reduced. 100*. Rolling Friction.-When a wheel or cylinder rolls on a plane surface, there is produced at the point of contact a resistance to the motion which is generally said to be due to rolling friction. This resistance is not a true friction in the sense of the word used in previous pages, since there is no relative motion of the two surfaces at the points of contact, hence there is no slipping. NA H E GL F Suppose that a cylinder EF (Fig. 75) rolls on a horizontal plane AB, and a light string is passed over the cylinder, the tensions P and Q in the two portions of this string being adjusted so that the cylinder, when started, continues to move with a uniform speed, rolling along from A towards B. Since the motion is uniform, it follows that the forces acting on the cylinder are in equilibrium. These forces are the weight w of the cylinder acting vertically downwards. through the axis G, the forces P and Q, A and the reaction of the plane AB. Now it is found experimentally that, if the motion in the direction from A to B is to be uniform, Q must be greater than P. The resultant of the parallel forces Q and P will therefore be a force nearer Q than P (§ 69). Let HK B K W M FIG. 75. ← be this resultant. The resultant of this force HK and the weight of the cylinder must lie somewhere between them, say along LM. If then the forces are in equilibrium, the reaction of the plane must act along LN. In other words the reaction of the plane does not act through the point C, where in the figure the cylinder touches the plane. This apparent impossibility is explainable if we suppose that rolling friction is really due to the fact that the plane becomes deformed and a small ridge is "rolled" up in front of the cylinder, or that the cylinder itself becomes flattened. The former of these effects can be clearly seen if a wheel is rolled on a sheet of india-rubber; for, as shown in is forced up into a small ridge before the wheel. illustrated in the case of a pneumatic bicycle tyre. the resistance to motion in the case of rolling is very much smaller than Fig. 76, the rubber that in the case of sliding. Thus Coulomb found that in the case of a cylinder of lignum-vitæ, 16 centimetres in diameter, when loaded with 1000 pounds, the resistance to rolling amounted to 6 pounds, while with the same load the resistance to sliding would have amounted to at least 200 pounds. Whenever it is possible, it is therefore advantageous to substitute rolling for sliding, if the frictional resistance to motion is to be reduced. Thus, in the case of a carriage, the sliding friction which occurs in a sledge is replaced by rolling friction between the tyre of the wheel and the ground. In the modern bicycle even the sliding friction of a wheel upon its axle is, as far as possible, replaced by rolling friction in the ball-bearing, where a number of hard steel balls are placed, so that the hub of the wheel rolls on them, and they roll on the axle itself. 101. Loss of Available Energy due to Friction.-Since in every case friction acts as a force tending to check the motion, whenever any displacement actually takes place work will have to be done against the frictional resistance. The energy which is necessary to perform this work is converted into heat, and this heat gradually becomes diffused amongst neighbouring bodies, and so the energy is no longer available for doing work. The frictional resistance always opposes motion, so that if we change the direction of motion the direction of the frictional resistance also changes, so that it is impossible to utilise this force to increase the motion of a body or to do work, but work has always to be done against it. It is therefore hardly correct, in view of the definition of force given in § 59, to call the frictional resistance a force. Since, however, it always acts as if it were a force opposing motion, it is convenient so to regard it. 102. Friction-Dynamometer.-One of the applications of friction is to employ it to measure the power or rate of doing work of a machine, such as a steam-engine. A form of friction-dynamometer for this purpose is shown in Fig. 77. A pulley A with a flat edge is fixed to the shaft of the engine, and a strap BCD, on the inside of which blocks of wood are usually fixed, rests on the edge of this pulley. One end of the strap is attached to a spring balance E, by means of which the tension acting on this end of the strap can be measured, while a tension P, caused by a weight suspended on the other end, serves to keep the strap tight. The |