Part 6

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_ = 1·44 + 0·252 _R_. +-----------+----------+-------------+----------+-------------+ | | | | |Discrepancies| | Number of | R. | Corrected | F. | between the | |Experiment.|Total load| mean |Calculated|observed and | | | on slide | value of | value of | calculated | | | in lbs. | friction. | friction.| frictions. | +-----------+----------+-------------+----------+-------------+ | 1 | 14 | 4·7 | 5·0 | +0·3 | | 2 | 28 | 8·2 | 8·5 | +0·3 | | 3 | 42 | 12·2 | 12·0 | -0·2 | | 4 | 56 | 15·8 | 15·6 | -0·2 | | 5 | 70 | 19·4 | 19·1 | -0·3 | | 6 | 84 | 23·0 | 22·6 | -0·4 | | 7 | 98 | 25·8 | 26·1 | +0·3 | | 8 | 112 | 29·3 | 29·7 | +0·4 | +-----------+----------+-------------+----------+-------------+

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_ = 0·288 _R_ are shown in Table V., where the greatest difference is 0·7 lb.

TABLE V.—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_ = 0·288 _R_ +-----------+----------+---------+----------+-------------+ | | | | |Discrepancies| | Number of | R. |Corrected| F. | between the | |Experiment.|Total load| mean |Calculated| observed and| | |on slide |value of |value of | calculated | | | in lbs. |friction.| friction.| frictions. | +-----------+----------+---------+----------+-------------+ | 1 | 14 | 4·7 | 4·0 | -0·7 | | 2 | 28 | 8·2 | 8·1 | -0·1 | | 3 | 42 | 12·2 | 12·1 | -0·1 | | 4 | 56 | 15·8 | 16·1 | +0·3 | +-----------+----------+---------+----------+-------------+

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 + 0·266 _R_. 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.

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_ = 0·9 + 0·266 _R_. +-----------+----------+---------+----------+-------------+ | | | | |Discrepancies| | Number of | R. |Corrected| F. | between the | |Experiment.|Total load| mean |Calculated| observed and| | |on slide |value of |value of | calculated | | | in lbs. |friction.| friction.| frictions. | +-----------+----------+---------+----------+-------------+ | 1 | 14 | 4·7 | 4·6 | -0·1 | | 2 | 28 | 8·2 | 8·3 | +0·1 | | 3 | 42 | 12·2 | 12·1 | -0·1 | | 4 | 56 | 15·8 | 15·8 | 0·0 | +-----------+----------+---------+----------+-------------+

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 B C 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 B C; 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.

A smooth plane of pine 72" × 11" carries a loaded slide of pine 9" × 9"; one end of the plane is gradually elevated until the slide starts off. +-----------+-------------+-----------+-----------+-----------+ | Number of |Total load on| Angle of | Angle of |Mean values| |Experiment.| the slide |elevation. |elevation. | of the | | | in lbs. |1st Series.|2nd Series.| angles. | +-----------+-------------+-----------+-----------+-----------+ | 1 | 14 | 19°·5 | —— | 19°·5 | | 2 | 56 | 20°·1 | 17°·2 | 18°·6 | | 3 | 112 | 20°·3 | 18°·9 | 19°·6 | +-----------+-------------+-----------+-----------+-----------+

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.

A smooth plane of pine 72" × 11" carries a loaded slide of pine 9" × 9"; one end of the plane is gradually elevated until the slide, having received a start, moves off uniformly. +-----------+-----------------+------------+ |Number of | Total load on | Angle of | |Experiment.|the slide in lbs.|inclination.| +-----------+-----------------+------------+ | 1 | 14 | 14°·3 | | 2 | 56 | 13°·0 | | 3 | 112 | 13°·0 | +-----------+-----------------+------------+

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.

LECTURE VI. _THE PULLEY._

Introduction.—Friction between a Rope and an Iron Bar.—The use of the Pulley.—Large and Small Pulleys.—The Law of Friction in the Pulley.—Wheels.—Energy.

INTRODUCTION.

151. The pulley forms a good introduction to the important subject of the mechanical powers. But before entering on the discussions of the next few chapters, it will be necessary for us to explain what is meant in mechanics by “work,” and by “energy,” which is the capacity for performing work, and we shall therefore include a short outline of this subject in the present lecture.

152. The pulley is a machine which is employed for the purpose of changing the direction of a force. We frequently wish to apply a force in a different direction from that in which it is convenient to exert it, and the pulley enables us to do so. We are not now speaking of these arrangements for increasing power in which pulleys play an important part; these will be considered in the next lecture: we at present refer only to change of direction. In fact, as we shall shortly see, some force is even wasted when the single fixed pulley is used, so that this machine certainly cannot be called a mechanical power.

153. The occasions upon which a single fixed pulley is used are numerous and familiar. Let us suppose a sack of corn has to be elevated from the lower to one of the upper stories of a building. It may of course be raised by a man who carries it, but he has to lift his own weight in addition to that of the sack, and therefore the quantity of exertion used is greater than absolutely necessary. But supposing there be a pulley at the top of the building over which a rope passes; then, if a man attach one end of the rope to the sack and pull the other, he raises the sack without raising his own weight. The pulley has thus provided the means by which the downward force has been changed in direction to an upward force.

154. The weights, ropes, and pulleys which are used in our windows for counterpoising the weight of the sash afford a very familiar instance of how a pulley changes the direction of a force. Here the downward force of the weight is changed by means of the pulley into an upward force, which nearly counterbalances the weight of the sash.

FRICTION BETWEEN A ROPE AND AN IRON BAR.

155. Every one is familiar with the ordinary form of the pulley; it consists of a wheel capable of turning freely on its axle, and it has a groove in its circumference in which the rope lies. But why is it necessary to give the pulley this form? Why could not the direction of the rope be changed by simply passing it over a bar, as well as by the more complicated pulley? We shall best answer this question by actually trying the experiment, which we can do by means of the apparatus of Fig. 34 (see page 90). In this are shown two iron studs, G, H, 0"·6 diameter, and about 8" apart; over these passes a rope, which has a hook at each end. If I suspend a weight of 14 lbs. from one hook A, and pull the hook B, I can by exerting sufficient force raise the weight on A, but with this arrangement I am conscious of having to exert a very much larger force than would have been necessary to raise 14 lbs. by merely lifting it.

156. In order to study the question exactly, we shall ascertain what weight suspended from the hook B will suffice to raise A. I find that in order to raise 14 lbs. on A no less than 47 lbs. is necessary on B, consequently there is an enormous loss of force: more than two-thirds of the force which is exerted is expended uselessly. If instead of the 14 lbs. weight I substitute any other weight, I find the same result, viz. that more than three times its amount is necessary to raise it by means of the rope passing over the studs. If a labourer, in raising a sack, were to pass a rope over two bars such as these, then for every stone the sack weighed he would have to exert a force of more than three stones, and there would be a very extravagant loss of power.

157. Whence arises this loss? The rope in moving slides over the surface of the iron studs. Although these are quite smooth and polished, yet when there is a strain on the rope it presses closely upon them, and there is a certain amount of force necessary to make the rope slide along the iron. In other words, when I am trying to raise up 14 lbs. with this contrivance, I not only have its weight opposed to me, but also another force due to the sliding of the rope on the iron: this force is due to friction. Were it not for friction, a force of 14 lbs. on one hook would exactly balance 14 lbs. on the other, and the slightest addition to either weight would make it descend and raise the other. If, then, we are obliged to change the direction of a force, we must devise some means of doing so which does not require so great a sacrifice as the arrangement we have just used.

THE USE OF THE PULLEY.

158. We shall next inquire how it is that we are enabled to obviate friction by means of a pulley. It is evident we must provide an arrangement in which the rope shall not be required to slide upon an iron surface. This end is attained by the pulley, of which we may take I, Fig. 34, as an example. This represents a cast iron wheel 14" in diameter, with a V-shaped groove in its circumference to receive the rope: this wheel turns on a ⅝ inch wrought iron axle, which is well oiled. The rope used is about 0"·25 in diameter.

159. From the hooks E, F at each end of the rope a 14 lb. weight is suspended. These equal weights balance each other. According to our former experiment with the studs, it would be necessary for me to treble the weight on one of these hooks in order to raise the other, but now I find that an additional 0·5 lb. placed on either hook causes it to descend and make the other ascend. This is a great improvement; 0·5 lb. now accomplishes what 33 lbs. was before required for. We have avoided a great deal of friction, but we have not got rid of it altogether, for 0·25 lb. is incompetent, when added to either weight, to make that weight descend.

160. To what is the improvement due? When the weight descends the rope does not slide upon the wheel, but it causes the wheel to revolve with it, consequently there is little or no friction at the circumference of the pulley; the friction is transferred to the axle. We still have some resistance to overcome, but for smooth oiled iron axles the friction is very small, hence the advantage of the pulley.

There is in every pulley a small loss of power from the force expended in bending the rope; this need not concern us at present, for with the pliable plaited rope that we have employed the effect is inappreciable, but with large strong ropes the loss becomes of importance. The amount of loss by using different kinds of ropes has been determined by careful experiments.

LARGE AND SMALL PULLEYS.

161. There is often a considerable advantage obtained by using large rather than small pulleys. The amount of force necessary to overcome friction varies inversely as the size of the pulley. We shall demonstrate this by actual experiment with the apparatus of Fig. 34. A small pulley K is attached to the large pulley I; they are in fact one piece, and turn together on the same axle. Hence if we first determine the friction with the rope over the large pulley, and then with the rope over the small pulley, any difference can only be due to the difference in size, as all the other circumstances are the same.

162. In making the experiments we must attend to the following point. The pulleys and the socket on which they are mounted weigh several pounds, and consequently there is friction on the axle arising from the weight of the pulleys, quite independently of any weights that may be placed on the hooks. We must then, if possible, evade the friction of the pulley itself, so that the amount of friction which is observed will be entirely due to the weights raised. This can be easily done. The rope and hooks being on the large pulley I, I find that 0·16 lb. attached to one of the hooks, E, is sufficient to overcome the friction of the pulley, and to make that hook descend and raise F. If therefore we leave 0·16 lb. on E, we may consider the friction due to the weight of the pulley, rope, and hooks as neutralized.

163. I now place a stone weight on each of the hooks E and F. The amount necessary to make the hook E and its load descend is 0·28 lb. This does not of course include the weight of 0·16 lb. already referred to. We see therefore that with the large pulley the amount of friction to be overcome in raising one stone is 0·28 lb.