Part 7

164. Let us now perform precisely the same experiment with the small pulley. I transfer the same rope and hooks to K, and I find that 0·16 lb. is not now sufficient to overcome the friction of the pulley, but I add on weights until C will just descend, which occurs when the load reaches 0·95 lb. This weight is to be left on C as a counterpoise, for the reasons already pointed out. I place a stone weight on C and another on D, and you see that C will descend when it receives an additional load of 1·35 lbs.; this is therefore the amount of friction to be overcome when a stone weight is raised over the pulley K.

165. Let us compare these results with the dimensions of the pulleys. The proper way to measure the effective circumference of a pulley when carrying a certain rope is to measure the length of that rope which will just embrace it. The length measured in this way will of course depend to a certain extent upon the size of the rope. I find that the circumferences of the two pulleys are 43"·0 and 9"·5. The ratio of these is 4·5; the corresponding resistances from friction we have seen to be 0·28 lb. and 1·35 lbs. The larger of these quantities is 4·8 times the smaller. This number is very close to 4·5; we must not, as already explained, expect perfect accuracy in experiments in friction. In the present case the agreement is within the ¹/₁₆th of the whole, and we may regard it as a proof of the law that _the friction of a pulley is inversely proportional to its circumference_.

166. It is easy to see the reason why friction should diminish when the size of the pulley is increased. The friction acts at the circumference of the axle about which the wheel turns; it is there present as a force tending to retard motion. Now the larger the wheel the greater will be the distance from the axis at which the force acts which overcomes the friction, and therefore the less need be the magnitude of the force. You will perhaps understand this better after the principle of the lever has been discussed.

167. We may deduce from these considerations the practical maxim that large pulleys are economical of power. This rule is well known to engineers; large pulleys should be used, not only for diminishing friction, but also to avoid loss of power by excessive bending of the rope. A rope is bent gradually around the circumference of a large pulley with far less force than is necessary to accommodate it to a smaller pulley: the rope also is apt to become injured by excessive bending. In coal pits the trucks laden with coal are hoisted to the surface by means of wire ropes which pass from the pit over a pulley into the engine-house: this pulley is of very large dimensions, for the reasons we have pointed out.

THE LAW OF FRICTION IN THE PULLEY.

168. I have here a wooden pulley 3"·5 in diameter; the hole is lined with brass, and the pulley turns very freely on an iron spindle. I place the rope and hooks upon the groove. Brass rubbing on iron has but little friction, and when 7 lbs. is placed on each hook, 0·5 lb. added to either will make it descend and raise up the other. Let 14 lbs. be placed on each hook, 0·5 lb. is no longer sufficient; 1 lb. is required: hence when the weight is doubled the friction is also doubled. Repeating the experiment with 21 lbs. and 28 lbs. on each side, the corresponding weights necessary to overcome friction are 1·5 lbs. and 2 lbs. In the four experiments the weights used are in the proportion 1, 2, 3, 4; and the forces necessary to overcome friction, 0·5 lb., 1 lb., 1·5 lbs., and 2 lbs., are in the same proportion. Hence the friction is proportional to the load.

WHEELS.

169. The wheel is one of the most simple and effective devices for overcoming friction. A sleigh is an admirable vehicle on a smooth surface such as ice, but it is totally unadapted for use on common roads; the reason being that the amount of friction between the sleigh and the road is so great that to move the sleigh the horse would have to exert a force which would be very great compared with the load he was drawing. But a vehicle properly mounted on wheels moves with the greatest ease along the road, for the circumference of the wheel does not slide, and consequently there is no friction between the wheel and the road; the wheel however turns on its axle, therefore there is sliding, and consequently friction, at the axle, but the axle and the wheel are properly fitted to each other, and the surfaces are lubricated with oil, so that the friction is extremely small.

170. With large wheels the amount of friction on the axle is less than with small wheels; other advantages of large wheels are that they do not sink much into depressions in the roads, and that they have also an increased facility in surmounting the innumerable small obstacles from which even the best road is not free.

171. When it is desired to make a pulley turn with extremely small friction, its axle, instead of revolving in fixed bearings, is mounted upon what are called friction wheels. A set of friction wheels is shown in the apparatus of Fig. 66: as the axle revolves, the friction between the axles and the wheels causes the latter to turn round with a comparatively slow motion; thus all the friction is transferred to the axles of the four friction wheels; these revolve in their bearings with extreme slowness, and consequently the pulley is but little affected by friction. The amount of friction in a pulley so mounted may be understood from the following experiment. A silk cord is placed on the pulley, and 1 lb. weight is attached to each of its ends: these of course balance. A number of fine wire hooks, each weighing 0·001 lb., are prepared, and it is found that when a weight of 0·004 lb. is attached to either side it is sufficient to overcome friction and set the weights in motion.

ENERGY.

172. In connection with the subject of friction, and also as introductory to the mechanical powers, the notion of “work,” or as it is more properly called “energy,” is of great importance. The meaning of this word as employed in mechanics will require a little consideration.

173. In ordinary language, whatever a man does that can cause fatigue, whether of body or mind, is called work. In mechanics, we mean by energy that particular kind of work which is directly or indirectly equivalent to raising weights.

174. Suppose a weight is lying on the floor and a stool is standing beside it: if a man raise the weight and place it upon the stool, the exertion that he expends is energy in the sense in which the word is used in mechanics. The amount of exertion necessary to place the weight upon the stool depends upon two things, the magnitude of the weight and the height of the stool. It is clear that both these things must be taken into account, for although we know the weight which is raised, we cannot tell the amount of exertion that will be required until we know the height through which it is to be raised; and if we know the height, we cannot appreciate the quantity of exertion until we know the weight.

175. The following plan has been adopted for expressing quantities of energy. The small amount of exertion necessary to raise 1 lb. avoirdupois through one British foot is taken as a standard, compared with which all other quantities of energy are estimated. This quantity of exertion is called in mechanics the unit of energy, and sometimes also the “foot-pound.”

176. If a weight of 1 lb. has to be raised through a height of 2 feet, or a weight of 2 lbs. through a height of 1 foot, it will be necessary to expend twice as much energy as would have raised a weight of 1 lb. through 1 foot, that is, 2 foot-pounds.

If a weight of 5 lbs. had to be raised from the floor up to a stool 3 feet high, how many units of energy would be required? To raise 5 lbs. through 1 foot requires 5 foot-pounds, and the process must be again repeated twice before the weight arrive at the top of the stool. For the whole operation 15 foot-pounds will therefore be necessary.

If 100 lbs. be raised through 20 feet, 100 foot-pounds of energy is required for the first foot, the same for the second, third, &c., up to the twentieth, making a total of 2,000 foot-pounds.

Here is a practical question for the sake of illustration. Which would it be preferable to hoist, by a rope passing over a single fixed pulley, a trunk weighing 40 lbs. to a height of 20 feet, or a trunk weighing 50 lbs. to a height of 15 feet? We shall find how much energy would be necessary in each case: 40 times 20 is 800; therefore in the first case the energy would be 800 foot-pounds. But 50 times 15 is 750; therefore the amount of work, in the second case, is only 750 lbs. Hence it is less exertion to carry 50 lbs. up 15 feet than 40 lbs. up 20 feet.

177. The rate of working of every source of energy, whether it lie in the muscles of men or other animals, in water-wheels, steam-engines, or other prime movers, is to be measured by the number of foot-pounds produced in the unit of time.

The power of a steam-engine is defined by its equivalent in horse-power. For example, it is meant that a steam-engine of 3 horse-power, could, when working for an hour, do as much work as 3 horses could do when working for the same time. The power of a horse is, however, an uncertain quantity, differing in different animals and not quite uniform in the same individual; accordingly the selection of this measure for the efficiency of the steam-engine is inconvenient. We replace it by a convenient standard horse-power, which is, however, a good deal larger than that continuously obtainable from any ordinary horse. A one horse-power steam-engine is capable of accomplishing 33,000 foot-pounds per minute.

178. We shall illustrate the numerical calculation of horse-power by an example: if a mine be 1,000 feet deep, how much water per minute would a 50 horse-power engine be capable of raising to the surface? The engine would yield 50 × 33,000 units of work per minute, but the weight has to be raised 1,000 feet, consequently the number of pounds of water raised per minute is

50 × 33,000 ———————————— = 1,650. 1,000

179. We shall apply the principle of work to the consideration of the pulley already described (p. 90). In order to raise the weight of 14 lbs., it is necessary that the rope to which the power is applied should be pulled downwards by a force of 15 lbs., the extra pound being on account of the friction. To fix our ideas, we shall suppose the 14 lbs. to be raised 1 foot; to lift this load directly, without the intervention of the pulley, 14 foot-pounds would be necessary, but when it is raised by means of the pulley, 15, foot-pounds are necessary. Hence there is an absolute loss of ¹/₁₅th of the energy when the pulley is used. If a steam-engine of 1 horse-power were employed in raising weights by a rope passing over a pulley similar to that on which we have experimented, only ¹⁴/₁₅ths of the work would be usefully employed; but we find

33,000 × 14 = 30,800. ———— 15

The engine would therefore perform 30,800 foot-pounds of useful work per minute.

180. The effect of friction on a pulley, or on any other machine, is always to waste energy. To perform a piece of work directly requires a certain number of foot-pounds, while to do it by a machine requires more, on account of the loss by friction. This may at first sight appear somewhat paradoxical, as it is well known that, by levers, pulleys, &c., an enormous mechanical advantage may be gained. This subject will be fully explained in the next and following lectures, which relate to the mechanical powers.

181. We shall conclude with a few observations on a point of the greatest importance. We have seen a case where 15 foot-pounds of energy only accomplished 14 foot-pounds of work, and thus 1 foot-pound appeared to be lost. We say that this was expended upon the friction; but what is the friction? The axle is gradually worn away by rubbing in its bearings, and, if it be not properly oiled, it becomes heated. The amount of energy that seems to disappear is partly expended in grinding down the axle, and is partly transformed into heat; it is thus not really lost, but unfortunately assumes a form which we do not require and in which it is rather injurious than otherwise. Indeed we know that energy cannot be destroyed, however it may be transformed; if it disappear in one shape, it is only to reappear in another. A so-called loss of energy by friction only means a diversion of a part of the work to some purpose other than that which we wish to accomplish. It has long been known that matter is indestructible: it is now equally certain that the same may be asserted of energy.

LECTURE VII. _THE PULLEY-BLOCK._

Introduction.—The Single Moveable Pulley.—The Three-sheave Pulley-block.—The Differential Pulley-block.—The Epicycloidal Pulley-block.

INTRODUCTION.

182. In the first lecture I showed how a large weight could be raised by a smaller weight, and I stated that this subject would again occupy our attention. I now fulfil this promise. The questions to be discussed involve the most advantageous methods of employing a small force to overcome a greater. Here is a subject of practical importance. A man of average strength cannot raise more than a hundredweight without great exertion, yet the weights which it is necessary to lift and move about often weigh many hundredweights, or even many tons. It is not always practicable to employ numerous hands for the purpose, nor is a steam-engine or other great source of power at all times available. But what are called the mechanical powers enable the forces at our disposal to be greatly increased. One man, by their aid, can exert as much force as several could without such assistance; and when they are employed to augment the power of several men or of a steam-engine, gigantic weights, amounting sometimes to hundreds of tons, can be managed with facility.

183. In the various arts we find innumerable cases where great resistances have to be overcome; we also find a corresponding number and variety of devices contrived by human skill to conquer them. The girders of an iron bridge have to be lifted up to their piers; the boilers and engines of an ocean steamer have to be placed in position; a great casting has to be raised from its mould; a railway locomotive has to be placed on the deck of a vessel for transit; a weighty anchor has to be lifted from the bottom of the sea; an iron plate has to be rolled or cut or punched: for all of these cases suitable arrangements must be devised in order that the requisite power may be obtained.

184. We know but little of the means which the ancients employed in raising the vast stones of those buildings which travellers in the East have described to us. It is sometimes thought that a large number of men could have transported these stones without the aid of appliances which we would now use for a similar purpose. But it is more likely that some of the mechanical powers were used, as, with a multitude of men, it is difficult to ensure the proper application of their united strength. In Easter Island, hundreds of miles distant from civilised land, and now inhabited by savages, vast idols of stone have been found in the hills which must have been raised by human labour. It is useless to speculate on the extinct race by whom this work was achieved, or on the means they employed.

185. The mechanical powers are usually enumerated as follows:—The pulley, the lever, the wheel and axle, the wedge, the inclined plane, the screw. These different powers are so frequently used in combination that the distinctions cannot be always maintained. The classification will, however, suffice to give a general notion of the subject.

186. Many of the most valuable mechanical powers are machines in which ropes or chains play an important part. Pulleys are usually employed wherever it is necessary to change the direction of a rope or chain which is transmitting power. In the present lecture we shall examine the most important mechanical powers that are produced by the combination of pulleys.

THE SINGLE MOVEABLE PULLEY.

187. We commence with the most simple case, that of the single moveable pulley (Fig. 35). The rope is firmly secured at one end A; it then passes down under the moveable pulley B, and upwards over a fixed pulley. To the free end, which depends from the fixed pulley, the power is applied while the load to be raised is suspended from the moveable pulley. We shall first study the relation between the power and the load in a simple way, and then we shall describe a few exact experiments.

188. When the load is raised the moveable pulley itself must of course be also raised, and a part of the power is expended for this purpose. But we can eliminate the weight of the moveable pulley, so far as our calculations are concerned, by first attaching to the power end of the rope a sufficient weight to lift up the moveable pulley when not carrying a load. The weight necessary for doing this is found by trial to be a little over 1·5 lbs. So that when a load is being raised we must reduce the apparent power by 1·5 lbs. to obtain the power really effective.

189. Let us suspend 14 lbs. from the load hook at B, and ascertain what power will raise the load. We leave the weight of the moveable pulley and 1·5 lbs. of the power at C out of consideration. I then find by experiment that 7 lbs. of effective power is not sufficient to raise the load, but if one pound more be added, the power descends, and the load is raised. Here, then, is a remarkable result; a weight of 8 lbs. has overcome 14 lbs. In this we have the first application of the mechanical powers to increase our available forces.

190. Let us examine the reason of this mechanical advantage. If the load be raised one foot, it is plain that the power must descend two feet: for in order to raise the load the two parts of the rope descending to the moveable pulley must each be shortened one foot, and this can only be done by the power descending two feet. Hence when the load of 14 lbs. is lifted by the machine, for every foot it is raised the power must descend two feet: this simple point leads to a conception of the greatest importance, on which depends the efficiency of the pulley. In the study of the mechanical powers it is essential to examine the number of feet through which the power must act in order to raise the load one foot: this number we shall always call the _velocity ratio_.

191. To raise 14 lbs. one foot requires 14 foot-pounds of energy. Hence, were there no such thing as friction, 7 lbs. on the power hook would be sufficient to raise the load; because 7 lbs. descending through two feet yields 14 foot-pounds. But there is a loss of energy on account of friction, and a power of 7 lbs. is not sufficient: 8 lbs. are necessary. Eight lbs. in descending two feet performs 16 foot-pounds; of these only 14 are utilised on the load, the remainder being the quantity of energy that has been diverted by friction. We learn, then, that in the moveable pulley the quantity of _energy_ employed is really greater than that which would lift the weight directly, but that the actual _force_ which has to be exerted is less.

192. Suppose that 28 lbs. be placed on the load hook, a few trials assure us that a power of 16 lbs. (but not less) will be sufficient for motion; that is to say, when the load is doubled, we find, as we might have expected, that the power must be doubled also. It is easily seen that the loss of energy by friction then amounts to 4 foot-pounds. We thus verify, in the case of the moveable pulley, the approximate law that the _friction is proportional to the load_.

193. By means of a moveable pulley a man is able to raise a weight nearly double as great as he could lift directly. From a series of careful experiments it has been found that when a man is employed in the particular exertion necessary for raising weights over a pulley, he is able to work most efficiently when the pull he is required to make is about 40 lbs. A man could, of course, exert greater force than this, but in an ordinary day’s work he is able to perform more foot-pounds when the pull is 40 lbs. than when it is larger or smaller. If therefore the weights to be lifted amount to about 80 lbs., energy may really be economized by the use of the single moveable pulley, although by so doing a greater quantity of energy would be actually expended than would have been necessary to raise the weights directly.

194. Some experiments on larger loads have been tried with the moveable pulley we have just described; the results are recorded in Table IX.

TABLE IX.—SINGLE MOVEABLE PULLEY.

Moveable pulley of cast iron 3"·25 diameter, groove 0"·6 wide, wrought iron axle 0"·6 diameter; fixed pulley of cast iron 5" diameter, groove 0"·4 wide, wrought iron axle 0"·6 diameter, axles oiled; flexible plaited rope 0"·25 diameter; velocity ratio 2, mechanical efficiency 1·8, useful effect 90 per cent.; formula _P_ = 2·21 + 0·5453 _R_. +-----------+---------+----------+----------+------------------+ | | | | P. | Discrepancies | | Number of | R. | Observed |Calculated| between observed | |Experiment.| Load | power | power | and calculated | | | in lbs. | in lbs. | in lbs. | powers. | +-----------+---------+----------+----------+------------------+ | 1 | 28 | 17·5 | 17·5 | 0·0 | | 2 | 57 | 33·5 | 33·3 | - 0·2 | | 3 | 85 | 48·5 | 48·6 | + 0·1 | | 4 | 113 | 64·0 | 63·8 | - 0·2 | | 5 | 142 | 80·0 | 79·6 | - 0·4 | | 6 | 170 | 94·5 | 94·9 | + 0·4 | | 7 | 198 | 110·5 | 110·2 | - 0·3 | | 8 | 226 | 125·5 | 125·5 | 0·0 | +-----------+---------+----------+----------+------------------+

The dimensions of the pulleys are precisely stated because, for pulleys of different construction, the numerical coefficients would not necessarily be the same. An attentive study of this table will, however, show the general character of the relation between the power and the load in all arrangements of this class.

The table consists of five columns. The first contains merely the numbers of the experiments for convenience of reference. In the second column, headed _R_, the loads, expressed in pounds, which are raised in each experiment, are given; that is, the weight attached to the hook, not including the weight of the lower pulley. The weight of this pulley is not included in the stated loads. In the third column the powers are recorded, which were found to be sufficient to raise the corresponding loads in the second column. Thus, in experiment 7, it is found that a power of 110·5 lbs. will be sufficient to raise a load of 198 lbs. The third column has thus been determined by gradually increasing the power until motion begins.