Part 5

PROPERTY OF THE CENTRE OF GRAVITY IN A REVOLVING WHEEL.

107. There are other curious consequences which follow from the properties of the centre of gravity, and we shall conclude by illustrating one of the most remarkable, which is at the same time of the utmost importance in machinery.

108. It is generally necessary that a machine should work as steadily as possible, and that undue vibration and shaking of the framework should be avoided: this is particularly the case when any parts of the machine rotate with great velocity, as, if these be heavy, inconvenient vibration will be produced when the proper adjustments are not made. The connection between this and the centre of gravity will be understood by reference to the apparatus represented in the accompanying figure (Fig. 30). We have here an arrangement consisting of a large cog wheel C working into a small one B, whereby, when the handle H is turned, a velocity of rotation can be given to the iron disk D, which weighs 14 lbs, and is 18" in diameter. This disk being uniform, and being attached to the axis at its centre, it follows that its centre of gravity is also the centre of rotation. The wheels are attached to a stand, which, though massive, is still unconnected with the floor. By turning the handle I can rotate the disk very rapidly, even as much as twelve times in a second. Still the stand remains quite steady, and even the shutter bell attached to it at E is silent.

109. Through one of the holes in the disk D I fasten a small iron bolt and a few washers, altogether weighing about 1 lb.; that is, only one-fourteenth of the weight of the disk. When I turn the handle slowly, the machine works as smoothly as before; but as I increase the speed up to one revolution every two seconds, the bell begins to ring violently, and when I increase it still more, the stand quite shakes about on the floor. What is the reason of this? By adding the bolt, I slightly altered the position of the centre of gravity of the disk, but I made no change of the axis about which the disk rotated, and consequently the disk was not on this occasion turning round its centre of gravity: this it was which caused the vibration. It is absolutely necessary that the centre of gravity of any heavy piece, rotating rapidly about an axis, should lie in the axis of rotation. The amount of vibration produced by a high velocity may be very considerable, even when a very small mass is the originating cause.

110. In order that the machine may work smoothly again, it is not necessary to remove the bolt from the hole. If by any means I bring back the centre of gravity to the axis, the same end will be attained. This is very simply effected by placing a second bolt of the same size at the opposite side of the disk, the two being at equal distances from the axis; on turning the handle, the machine is seen to work as smoothly as it did in the first instance.

111. The most common rotating pieces in machines are wheels of various kinds, and in these the centre of gravity is evidently identical with the centre of rotation; but if from any cause a wheel, which is to turn rapidly, has an extra weight attached to one part, this weight must be counterpoised by one or more on other portions of the wheel, in order to keep the centre of gravity of the whole in its proper place. Thus it is that the driving wheels of a locomotive are always weighted so as to counteract the effect of the crank and restore the centre of gravity to the axis of rotation. The cause of the vibration will be understood after the lecture on centrifugal force (Lect. XVII.).

LECTURE V. _THE FORCE OF FRICTION._

The Nature of Friction.—The Mode of Experimenting.—Friction is proportional to the pressure.—A more accurate form of the Law.—The Coefficient varies with the weights used.—The Angle of Friction.—Another Law of Friction.—Concluding Remarks.

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.

Smooth horizontal surface of pine 72" × 11"; slide also of pine 9" × 9"; grain crosswise; slide is not started; force acting on slide is gradually increased until motion commences. +-----------+-------------+---------------+---------------+-------+ | Number of |Load on slide|Force necessary|Force necessary| Mean | |Experiment.| in lbs., |to move slide. |to move slide. |values.| | | including | | | | | | weight | 1st Series. | 2nd Series. | | | | of slide. | | | | +-----------+-------------+---------------+---------------+-------+ | 1 | 14 | 5 | 8 | 6·5 | | 2 | 28 | 15 | 16 | 15·5 | | 3 | 42 | 20 | 15 | 17·5 | | 4 | 56 | 29 | 24 | 26·5 | | 5 | 70 | 33 | 31 | 32·0 | | 6 | 84 | 43 | 33 | 38·0 | | 7 | 98 | 42 | 38 | 40·0 | | 8 | 112 | 50 | 33 | 41·5 | +-----------+-------------+---------------+---------------+-------+

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.

Smooth horizontal surface of pine 72" × 11"; slide also of pine 9" × 9"; grain crosswise; slide started; force applied is sufficient to maintain uniform motion of the slide. +-----------+-------------+--------------+--------------+-------+ | |Load on slide| Force | Force | | | Number of | in lbs., | necessary | necessary | Mean | |Experiment.| including | to maintain | to maintain |values.| | | weight of | motion. | motion. | | | | slide. | 1st Series. | 2nd Series. | | +-----------+-------------+--------------+--------------+-------+ | 1 | 14 | 4·9 | 4·9 | 4·9 | | 2 | 28 | 8·5 | 8·6 | 8·5 | | 3 | 42 | 12·6 | 12·4 | 12·5 | | 4 | 56 | 16·3 | 16·2 | 16·2 | | 5 | 70 | 19·7 | 20·0 | 19·8 | | 6 | 84 | 23·7 | 23·0 | 23·4 | | 7 | 98 | 26·5 | 26·1 | 26·3 | | 8 | 112 | 29·7 | 29·9 | 29·8 | +-----------+-------------+--------------+--------------+-------+

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.

Friction of pine upon pine; the mean values of the friction given in Table II. (corrected for the friction of the pulley) compared with the formula _F_ = 0·27 _R_. +-----------+----------+----------+----------+---------------+ | Number of | R. |Corrected | F. | Discrepancies | |Experiment.|Total load|mean value|Calculated| between the | | | on slide | of | value of | observed and | | | in lbs. | friction.| friction.| calculated | | | | | | frictions. | +-----------+----------+----------+----------+---------------+ | 1 | 14 | 4·7 | 3·8 | -0·9 | | 2 | 28 | 8·2 | 7·6 | -0·6 | | 3 | 42 | 12·2 | 11·3 | -0·9 | | 4 | 56 | 15·8 | 15·1 | -0·7 | | 5 | 70 | 19·4 | 18·9 | -0·5 | | 6 | 84 | 23·0 | 22·7 | -0·3 | | 7 | 98 | 25·8 | 26·5 | +0·7 | | 8 | 112 | 29·3 | 30·2 | +0·9 | +-----------+----------+----------+----------+---------------+

133. Hence the law _F_ = 0·27 _R_ 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_ + _y R_ 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 + 0·252 _R_.

This formula is compared with the experiments in Table IV.

TABLE IV.—FRICTION.