THE NATURE OF FRICTION.

112. A discussion of the force of friction is a necessary preliminary to the study of the mechanical powers which we shall presently commence. Friction renders the inquiry into the mechanical powers more difficult than it would be if this force were absent; but its effects are too important to be overlooked.

113. The nature of friction may be understood by Fig. 31, which represents a section of the top of a table of wood or any other substance levelled so that c d is horizontal; on the table rests a block a of wood or any other substance. To a a cord is attached, which, after passing over a pulley p, is stretched by a suspended weight b. If the magnitude of b exceeds a certain limit, then a is pulled along the table and b descends; but if b be smaller than this limit, both a and b remain at rest. When b is not heavy enough to produce motion it is supported by the tension of the cord, which is itself neutralized by the friction produced by a certain coherence between a and the table. Friction is by this experiment proved to be a force, because it prevents the motion of b. Indeed friction is generally manifested as a force by destroying motion, though sometimes indirectly producing it.

114. The true source of the force lies in the inevitable roughness of all known surfaces, no matter how they may have been wrought. The minute asperities on one surface are detained in corresponding hollows in the other, and consequently force must be exerted to make one surface slide upon the other. By care in polishing the surfaces the amount of friction may be diminished; but it can only be decreased to a certain limit, beyond which no amount of polishing seems to produce much difference.

115. The law of friction under different conditions must be inquired into, in order that we may make allowance when its effect is of importance. The discussion of the experiments is sometimes a little difficult, and the truths arrived at are principally numerical, but we shall find that some interesting laws of nature will appear.

THE MODE OF EXPERIMENTING.

116. Friction is present between every pair of surfaces which are in contact: there is friction between two pieces of wood, and between a piece of wood and a piece of iron; and the amount of the force depends upon the character of both surfaces. We shall only experiment upon the friction of wood upon wood, as more will be learned by a careful study of a special case than by a less minute examination of a number of pairs of different substances. 117. The apparatus used is shown in Fig. 32. A plank of pine 6'×11"×2" is planed on its upper surface, levelled by a spirit-level, and firmly secured to the framework at a height of about 4' from the ground. On it is a pine slide 9"×9", the grain of which is crosswise to that of the plank; upon the slide the load a is placed. A rope is attached to the slide, which passes over a very freely mounted cast iron pulley c, 14" diameter, and carries at the other end a hook weighing one pound, from which weights b can be suspended.

118. The mode of experimenting consists in placing a certain load upon a, and then ascertaining what weight applied to b will draw the loaded slide along the plane. As several trials are generally necessary to determine the power, a rope is attached to the back of the slide, and passes over the two pulleys d; this makes it easy for the experimenter, when applying the weights at b, to draw back the slide to the end of the plane by pulling the ring e: this rope is of course left quite slack during the process of the experiment, since the slide must not be retarded. The loads placed upon a during the series of experiments ranged between one stone and eight stone. In the loads stated the weight of the slide itself, which was less than 1 lb., is always included. A variety of small weights were provided for the hook b; they consisted of 0·1, 0·5, 1, 2, 7, and 14 lbs. There is some friction to be overcome in the pulley c, but as the pulley is comparatively large its friction is small, though it was always allowed for.

119. An example of the experiments made is thus described. A weight of 56 lbs. is placed upon the slide, and it is found on trial that 29 lbs. on b (including the weight of the hook itself) is sufficient to start the slide; this weight is placed upon the hook pound by pound, care being taken to make each addition gently.

120. Experiments were made in this way with various weights upon a, and the results are recorded in Table I.

Table I.—Friction.

In the first column a number is given to each experiment for convenience of reference. In the second column the load on the slide is stated in lbs. In the third column is found the force necessary to overcome the friction. The fourth column records a second series of experiments performed in the same manner as the first series; while the last column shows the mean of the two frictions.

121. The first remark to be made upon this table is, that the results do not appear satisfactory or concordant. Thus from 6 and 7 of the 1st series it would appear that the friction of 84 lbs. was 43 lbs., while that of 98 lbs. was 42 lbs., so that here the greater weight appears to have the less friction, which is evidently contrary to the whole tenor of the results, as a glance will show. Moreover the frictions in the 1st and the 2nd series do not agree, being generally greater in the former than in the latter, the discordance being especially noticeable in experiment 8, where the results were 50 lbs. and 33 lbs. In the final column of means these irregularities are lessened, for this column shows that the friction increases with the weight, but it is sufficient to observe that as the difference of the 1st and the 2nd is 9 lbs., and that of the 2nd and the 3rd is only 2 lbs., the discovery of any law from these results is hopeless.

122. But is friction so capricious that it is amenable to no better law than these experiments appear to indicate? We must look a little more closely into the matter. When two pieces of wood have remained in contact and at rest for some time, a second force besides friction resists their separation: the wood is compressible, the surfaces become closely approximated, and the coherence due to this cause must be overcome before motion commences. The initial coherence is uncertain; it depends probably on a multitude of minute circumstances which it is impossible to estimate, and its presence has vitiated the results which we have found so unsatisfactory.

123. We can remove these irregularities by starting the slide at the commencement. This may be conveniently effected by the screw shown at f in Fig. 32; a string attached to its end is fastened to the slide, and by giving the handle of the screw a few turns the slide begins to creep. A body once set in motion will continue to move with the same velocity unless acted upon by a force; hence the weight at b just overcomes the friction when the slide moves uniformly after receiving a start: this velocity was in one case of average speed measured to be 16 inches per minute.

124. Indeed in no case can the slide commence to move unless the force exceed the friction. The amount of this excess is indeterminate. It is certainly greater between wooden surfaces than between less compressible surfaces like those of metals or glass. In the latter case, when the force exceeds the friction by a small amount, the slide starts off with an excessively slow motion; with wood the force must exceed the friction by a larger amount before the slide commences to move, but the motion is then comparatively rapid.

125. If the power be too small, the load either does not continue moving after the start, or it stops irregularly. If the power be too great, the load is drawn with an accelerated velocity. The correct amount is easily recognised by the uniformity of the movement, and even when the slide is heavily laden, a few tenths of a pound on the power hook cause an appreciable difference of velocity.

126. The accuracy with which the friction can be measured may be appreciated by inspecting Table II.

Table II.—Friction.

127. Two series of experiments to determine the power necessary to maintain the motion have been recorded. Thus, in experiment 7, the load on the slide being 98 lbs., it was found that 26·5 lbs. was sufficient to sustain the motion, and a second trial being made independently, the power found was 26·1 lbs.: a mean of the two values, 26·3 lbs., is adopted as being near the truth. The greatest difference between the two series, amounting to 0·7 lb., is found in experiment 6; a third value was therefore obtained for the friction of 84 lbs.: this amounted to 23·5 lbs., which is intermediate between the two former results, and 23·4 lbs., a mean of the three, is adopted as the final result.

128. The close accordance of the experiments in this table shows that the means of the fifth column are probably very near the true values of the friction for the corresponding loads upon the slide.

129. The mean frictions must, however, be slightly diminished before we can assert that they represent only the friction of the wood upon the wood. The pulley over which the rope passes turns round its axle with a small amount of friction, which must also be overcome by the power. The mode of estimating this amount, which in these experiments never exceeds 0·5 lb., need not now be discussed. The corrected values of the friction are shown in the third column of Table III. Thus, for example, the 4·9 lbs. of friction in experiment 1 consists of 4·7, the true friction of the wood, and 0·2, which is the friction of the pulley; and 26·3 of experiment 7 is similarly composed of 25·8 and 0·5. It is the corrected frictions which will be employed in our subsequent calculations.

FRICTION IS PROPORTIONAL TO
THE PRESSURE.

130. Having ascertained the values of the force of friction for eight different weights, we proceed to inquire into the laws which may be founded on our results. It is evident that the friction increases with the load, of which it is always greater than a fourth, and less than a third. It is natural to surmise that the friction (F) is really a constant fraction of the load (R)—in other words, that F=kR, where k is a constant number.

131. To test this supposition we must try to determine k; it may be ascertained by dividing any value of F by the corresponding value of R. If this be done, we shall find that each of the experiments yields a different quotient; the first gives 0·336, and the last 0·262, while the other experiments give results between these extreme values. These numbers are tolerably close together, but there is still sufficient discrepancy to show that it is not strictly true to assert that the friction is proportional to the load. 132. But the law that the friction varies proportionally to the pressure is so approximately true as to be sufficient for most practical purposes, and the question then arises, which of the different values of k shall we adopt? By a method which is described in the Appendix we can determine a value for k which, while it does not represent any one of the experiments precisely, yet represents them collectively better than it is possible for any other value to do. The number thus found is 0·27. It is intermediate between the two values already stated to be extreme. The character of this result is determined by an inspection of Table III.

The fourth column of this table has been calculated from the formula F = 0·27 R. Thus, for example, in experiment 5, the friction of a load of 70 lbs. is 19·4 lbs., and the product of 70 and 0·27 is 18·9, which is 0·5 lb. less than the true amount. In the last column of this table the discrepancies between the observed and the calculated values are recorded, for facility of comparison. It will be observed that the greatest difference is under 1 lb.

Table III.—Friction.

133. Hence the law F=0·27R represents the experiments with tolerable accuracy; and the numerical ratio O·27 is called the coefficient of friction. We may apply this law to ascertain the friction in any case where the load lies between 14 lbs. and 112 lbs.; for example, if the load be 63 lbs., the friction is 63×0·27=17·0.

134. The coefficient of friction would have been slightly different had the grain of the slide been parallel to that of the plank; and it of course varies with the nature of the surfaces. Experimenters have given tables of the coefficients of friction of various substances, wood, stone, metals, &c. The use of these coefficients depends upon the assumption of the ordinary law of friction, namely, that the friction is proportional to the pressure: this law is accurate enough for most purposes, especially when used for loads that lie between the extreme weights employed in calculating the value of the coefficient which is employed.

A MORE ACCURATE LAW OF FRICTION.

135. In making one of our measurements with care, it is unusual to have an error of more than a few tenths of 1 lb. and it is hardly possible that any of the mean frictions we have found should be in error to so great an extent as 0·5 lb. But with the value of the coefficient of friction which is used in Table III., the discrepancies amount sometimes to 0·9 lbs. With any other numerical coefficient than 0·27, the discrepancies would have been even still more serious. As these are too great to be attributed to errors of experiment, we have to infer that the law of friction which has been assumed cannot be strictly true. The signs of the discrepancies indicate that the law gives frictions which for small loads are too small, and for large loads are too great.

136. We are therefore led to inquire whether some other relation between F and R may not represent the experiments with greater fidelity than the common law of friction. If we diminished the coefficient by a small amount, and then added a constant quantity to the product of the coefficient and the load, the effect of this change would be that for small loads the calculated values would be increased, while for large loads they would be diminished. This is the kind of change which we have indicated to be necessary for reconciliation between the observed and calculated values. 137. We therefore infer that a relation of the form F=x+yR will probably express a more correct law, provided we can find x and y. One equation between x and y is obtained by introducing any value of R with the corresponding value of F, and a second equation can be found by taking any other similar pair. From these two equations the values of x and of y may be deduced by elementary algebra, but the best formula will be obtained by combining together all the pairs of corresponding values. For this reason the method described in the Appendix must be used, which, as it is founded on all the experiments, must give a thoroughly representative result. The formula thus determined, is

F=1·44+0·252R.

This formula is compared with the experiments in Table IV.

Table IV.—Friction.

The fourth column contains the calculated values: thus, for example, in experiment 4, where the load is 56 lbs., the calculated value is 1·44+0·252×56=15·6; the difference 0·2 between this and the observed value 15·8 is shown in the last column.

138. It will be noticed that the greatest discrepancy in this column is 0·4 lbs., and that therefore the formula represents the experiments with considerable accuracy. It is undoubtedly nearer the truth than the former law (Art. 132); in fact, the differences are now such as might really belong to errors unavoidable in making the experiments.

139. This formula may be used for calculating the friction for any load between 14 lbs. and 112 lbs. Thus, if the load be 63 lbs., the friction is 1·44+0·252×63=17·3 lbs., which does not differ much from 17·0 lbs., the value found by the more ordinary law. We must, however, be cautious not to apply this formula to weights which do not lie between the limits of the greatest and least weight used in those experiments by which the numerical values in the formula have been determined; for example, to take an extreme case, if R = 0, the formula would indicate that the friction was 1·44, which is evidently absurd; here the formula errs in excess, while if the load were very large it is certain the formula would err in defect.

THE COEFFICIENT VARIES WITH
THE WEIGHTS USED.

140. In a subsequent lecture we shall employ as an inclined plane the plank we have been examining, and we shall require to use the knowledge of its friction which we are now acquiring. The weights which we shall then employ range from 7 lbs. to 56 lbs. Assuming the ordinary law of friction, we have found that 0·27 is the best value of its coefficient when the loads range between 14 lbs. and 112 lbs. Suppose we only consider loads up to 56 lbs., we find that the coefficient 0·288 will best represent the experiments within this range, though for 112 lbs. it would give an error of nearly 3 lbs. The results calculated by the formula F=0·288R are shown in Table V., where the greatest difference is 0·7 lb.

Table V.—Friction.

141. But we can replace the common law of friction by the more accurate law of Art. 137, and the formula computed so as to best harmonise the experiments up to 56 lbs., disregarding the higher loads, is F=0·9+0·266R. This formula is obtained by the method referred to in Art. 137. We find that it represents the experiments better than that used in Table V. Between the limits named, this formula is also more accurate than that of Table IV. It is compared with the experiments in Table VI., and it will be noticed that it represents them with great precision, as no discrepancy exceeds 0·1.

Table VI.—Friction.

THE ANGLE OF FRICTION.

142. There is another mode of examining the action of friction besides that we have been considering. The apparatus for this purpose is shown in Fig. 33, in which bc represents the plank of pine we have already used. It is now mounted so as to be capable of turning about one end b; the end c is suspended from one hook of the chain from the “epicycloidal” pulley-block e. This block is very convenient for the purpose. By its means the inclination of the plank can be adjusted with the greatest nicety, as the raising chain g is held in one hand and the lowering chain f in the other. Another great convenience of this block is that the load does not run down when the lifting chain is left free. The plank is clamped to the hinge about which it turns. The frames by which both the hinge and the block are supported are weighted in order to secure steadiness. The inclination of the plane is easily ascertained by measuring the difference in height of its two ends above the floor, and then making a drawing on the proper scale. The starting-screw D, whose use has been already mentioned, is also fastened to the framework in the position shown in the figure.

143. Suppose the slide a be weighted and placed upon the inclined plane bc; if the end C be only slightly elevated, the slide remains at rest; the reason being that the friction between the slide and the plane neutralizes the force of gravity. But suppose, by means of the pulley-block, c be gradually raised; an elevation is at last reached at which the slide starts off, and runs with an accelerating velocity to the bottom of the plane. The angle of elevation of the plane when this occurs is called the angle of statical friction.

144. The weights with which the slide was laden in these experiments were 14 lbs., 56 lbs., and 112 lbs., and the results are given in Table VII.

We see that a load of 56 lbs. started when the plane reached an inclination of 20°·1 in the first series, and of 17°·2 in the second, the mean value 18°·6 being given in the fifth column. These means for the three different weights agree so closely that we assert the remarkable law that the angle of friction does not depend upon the magnitude of the load.

Table VII.—Angle of Statical Friction.

145. We might, however, proceed differently in determining the angle of friction, by giving the load a start, and ascertaining if the motion will continue. To do so requires the aid of an assistant, who will start the load with the help of the screw, while the elevation of the plane is being slowly increased. The result of these experiments is given in Table VIII.

Table VIII.—Angle of Friction.

We see from this table also that the angle of friction is independent of the load, but the angle is in this case less by 5° or 6° than was found necessary to impart motion when a start was not given.

146. It is commonly stated that the coefficient of friction equals the tangent of the angle of friction, and this can be proved to be true when the ordinary law of friction is assumed. But as we have seen that the law of friction is only approximately correct, we need not expect to find this other law completely verified. 147. When the slide is started, the mean value of the angle of friction is 13°·4. The tangent of this angle is 0·24: this is about 11 per cent. less than the coefficient of friction 0·27, which we have already determined. The mean value of the angle of friction when the slide is not started is 19°·2, and its tangent is 0·35. The experiments of Table I. are, as already pointed out, rather unsatisfactory, but we refer to them here to show that, so far as they go, the coefficient of friction is in no sense equal to the tangent of the angle of friction. If we adopt the mean values given in the last column of Table I., the best coefficient of friction which can be deduced is 0·41. Whether, therefore, the slide be started or not started, the tangent of the angle of friction is smaller than the corresponding coefficient of friction. When the slide is started, the tangent is about 11 per cent. less than the coefficient; and when the slide is not started, it is about 14 per cent. less. There are doubtless many cases in which these differences are sufficiently small to be neglected, and in which, therefore, the law may be received as true.

ANOTHER LAW OF FRICTION.

148. The area of the wooden slide is 9"×9", but we would have found that the friction under a given load was practically the same whatever were the area of the slide, so long as its material remained unaltered. This follows as a consequence of the approximate law that the friction is proportional to the pressure. Suppose that the weight were 100 lbs., and the area of the slide 100 inches, there would then be a pressure of 1 lb. per square inch over the surface of the slide, and therefore the friction to be overcome on each square inch would be 0·27 lb., or for the whole slide 27 lbs. If, however, the slide had only an area of 50 square inches, the load would produce a pressure of 2 lbs., per square inch; the friction would therefore be 2×0·27=0·54 lb. for each square inch, and the total friction would be 50×0·54=27 lbs., the same as before: hence the total friction is independent of the extent of surface. This would remain equally true even though the weight were not, as we have supposed, uniformly distributed over the surface of the slide.

CONCLUDING REMARKS.

149. The importance of friction in mechanics arises from its universal presence. We often recognize it as a destroyer or impeder of motion, as a waster of our energy, and as a source of loss or inconvenience. But, on the other hand, friction is often indirectly the means of producing motion, and of this we have a splendid example in the locomotive engine. The engine being very heavy, the wheels are pressed closely to the rails; there is friction enough to prevent the wheels slipping, consequently when the engines force the wheels to turn round they must roll onwards. The coefficient of friction of wrought iron upon wrought iron is about 0·2. Suppose a locomotive weigh 30 tons, and the share of this weight borne by the driving wheels be 10 tons, the friction between the driving wheels and the rails is 2 tons. This is the greatest force the engine can exert on a level line. A force of 10 lbs. for every ton weight of the train is known to be sufficient to sustain the motion, consequently the engine we have supposed should draw along the level a load of 448 tons.

150. But we need not invoke the steam-engine to show the use of friction. We could not exist without it. In the first place we could not move about, for walking is only possible on account of the friction between the soles of our boots and the ground; nor if we were once in motion could we stop without coming into collision with some other object, or grasping something to hold on by. Objects could only be handled with difficulty, nails would not remain in wood, and screws would be equally useless. Buildings could hardly be erected, nay, even hills and mountains would gradually disappear, and finally dry land would be immersed beneath the level of the sea. Friction is, so far as we are concerned, quite as essential a law of nature as the law of gravitation. We must not seek to evade it in our mechanical discussions because it makes them a little more difficult. Friction obeys laws; its action is not vague or uncertain. When inconvenient it can be diminished, when useful it can be increased; and in our lectures on the mechanical powers, to which we now proceed, we shall have opportunities of describing machines which have been devised in obedience to its laws.