Part 8

195. From an examination of the columns showing the power and the load, we see that the power always amounts to more than half the load. The excess is partly due to a small portion of the power (about 1·5 lbs.) being employed in raising the lower block, and partly to friction. For example, in experiment 7, if there had been no friction and if the lower block were without weight, a power of 99 lbs. would have been sufficient; but, owing to the presence of these disturbing causes, 110·5 lbs. are necessary: of this amount 1·5 lbs. is due to the weight of the pulley, 10 lbs. is the force of friction, and the remaining 99 lbs. raises the load.

196. By a calculation based on this table we have ascertained a certain relation between the power and the load; they are connected by the formula which may be enunciated as follows:

The power is found by multiplying the weight of the load into 0·5453, and adding 2·21 to the product. Calling _P_ the power and _R_ the load, we may express the relation thus: _P_ = 2·21 + 0·5453 _R_. For example, in experiment 5, the product of 142 and 0·5453 is 77·43, to which, when 2·21 is added, we find for _P_ 79·64, very nearly the same as 80 lbs., the observed value of the power.

In the fourth column the values of _P_ calculated by means of this formula are given, and in the last we exhibit the discrepancies between the observed and the calculated values for the sake of comparison. It will be seen that the discrepancy in no case amounts to 0·5 lb., consequently the formula expresses the experiments very well. The mode of deducing it is given in the Appendix.

197. The quantity 2·21 is partly that portion of the power expended in overcoming the weight of the moveable pulley, and partly arises from friction.

198. We can readily calculate from the formula how much power will be required to raise a given weight; for example, suppose 200 lbs. be attached to the moveable pulley, we find that 111 lbs. must be applied as the power. But in order to raise 200 lbs. one foot, the power exerted must act over two feet; hence the number of foot-pounds required is 2 × 111 = 222. The quantity of energy that is lost is 22 foot-pounds. Out of every 222 foot-pounds applied, 200 are usefully employed; that is to say, about 90 per cent. of the applied energy is utilized, while the remaining 10 per cent. is lost by friction.

THE THREE-SHEAVE PULLEY-BLOCK.

199. The next arrangement we shall employ is a pair of pulley-blocks S T, Fig. 35, each containing three sheaves, as the small pulleys are termed. A rope is fastened to the upper block, S; it then passes down to the lower block T under one sheave, up again to the upper block and over a sheave, and so on, as shown in the figure. To the end of the rope from the last of the upper sheaves the power H is applied, and the load G is suspended from the hook attached to the lower block. When the rope is pulled, it gradually raises the lower block; and to raise the load one foot, each of the six parts of the rope from the upper block to the lower block must be shortened one foot, and therefore the power must have pulled out six feet of rope. Hence, for every foot that the load is raised the power must have acted through six feet; that is to say, the _velocity ratio_ is 6.

200. If there were no friction, the power would only be one-sixth of the load. This follows at once from the principles already explained. Suppose the load be 60 lbs., then to raise it one foot would require 60 foot-pounds; and the power must therefore exert 60 foot-pounds; but the power moves over six feet, therefore a power of 10 lbs. would be sufficient. Owing, however, to friction, some energy is lost, and we must have recourse to experiment in order to test the real efficiency of the machine. The single moveable pulley nearly doubled our power; we shall prove that the three-sheave pulley-block will quadruple it. In this case we deal with larger weights, with reference to which we may leave the weight of the lower block out of consideration.

201. Let us first attach 1 cwt. to the load hook; we find that 29 lbs. on the power hook is the smallest weight that can produce motion: this is only 1 lb. more than one-quarter of the load raised. If 2 cwt. be the load, we find that 56 lbs. will just raise it: this time the power is exactly one-quarter of the load. The experiment has been tried of placing 4 cwt. on the hook; it is then found that 109 lbs. will raise it, which is only 3 lbs. short of 1 cwt. These experiments demonstrate that for a three-sheave pulley-block of this construction we may safely apply the rule, that _the power is one-quarter of the load_.

202. We are thus enabled to see how much of our exertion in raising weights must be expended in merely overcoming friction, and how much may be utilized. Suppose for example that we have to raise a weight of 100 lbs. one foot by means of the pulley-block; the power we must apply is 25 lbs., and six feet of rope must be drawn out from between the pulleys: therefore the power exerts 150 foot-pounds of energy. Of these only 100 foot-pounds are usefully employed, and thus 50 foot-pounds, one-third of the whole, have been expended on friction. Here we see that notwithstanding a small force overcomes a large one, there is an actual loss of energy in the machine. The real advantage of course is that by the pulley-block I can raise a greater weight than I could move without assistance, but I do not create energy; I merely modify it, and lose by the process.

203. The result of another series of experiments made with this pair of pulley-blocks is given in Table X.

TABLE X.—THREE-SHEAVE PULLEY-BLOCKS.

Sheaves cast iron 2"·5 diameter; plaited rope 0"·25 diameter; velocity ratio 6; mechanical advantage 4; useful effect 67 per cent.; formula _P_ = 2·36 + 0·238 _R_. +-----------+---------+----------+----------+------------------+ | | | | P. | Discrepancies | | Number of | R. | Observed |Calculated| between observed | |Experiment.| Load | power | power | and calculated | | | in lbs. | in lbs. | in lbs. | powers. | +-----------+---------+----------+----------+------------------+ | 1 | 57 | 15·5 | 15·9 | + 0·4 | | 2 | 114 | 29·5 | 29·5 | 0·0 | | 3 | 171 | 43·5 | 43·1 | - 0·4 | | 4 | 228 | 56·0 | 56·6 | + 0·6 | | 5 | 281 | 70·0 | 69·2 | - 0·8 | | 6 | 338 | 83·0 | 82·8 | - 0·2 | | 7 | 395 | 97·0 | 96·4 | - 0·6 | | 8 | 452 | 109·0 | 109·9 | + 0·9 | +-----------+---------+----------+----------+------------------+

204. This table contains five columns; the weights raised (shown in the second column) range up to somewhat over 4 cwt. The observed values of the power are given in the third column; each of these is generally about one-quarter of the corresponding value of the load. There is, however, a more accurate rule for finding the power; it is as follows.

205. To find the power necessary to raise a given load, multiply the loads in lbs. by 0·238, and add 2·36 lbs. to the product. We may express the rule by the formula _P_ = 2·36 + 0·238 _R_.

206. To find the power which would raise 228 lbs.; the product of 228 and 0·238 is 54·26; adding 2·36, we find 56·6 lbs. for the power required; the actual observed power is 56 lbs., so that the rule is accurate to within about half a pound. In the fourth column will be found the values of _P_ calculated by means of this rule. In the fifth column, the discrepancies between the observed and the calculated values of the powers are given, and it will be seen that the difference in no case reaches 1 lb. Of course it will be understood that this formula is only reliable for loads which lie between those employed in the first and last of the experiments. We can calculate the power for any load between 57 lbs. and 452 lbs., but for loads much larger than 452 or less than 57 it would probably be better to use the simple fourth of the load rather than the power computed by the formula.

207. I will next perform an experiment with the three-sheave pulley-block, which will give an insight into the exact amount of friction without calculation by the help of the velocity ratio. We first counterpoise the weight of the lower block by attaching weights to the power. It is found that about 1·6 lbs. is sufficient for this purpose. I attach a 56 lb. weight as a load, and find that 13·1 lbs. is sufficient power for motion. This amount is partly composed of the force necessary to raise the load if there were no friction, and the rest is due to the friction. I next gradually remove the power weights: when I have taken off a pound, you see the power and the load balance each other; but when I have reduced the power so low as 5·5 lbs. (not including the counterpoise for the lower block), the load is just able to overhaul the power, and run down. We have therefore proved that a power of 13·1 lbs. or greater raises 56 lbs., that any power between 13·1 lbs. and 5·5 lbs. balances 56 lbs., and that any power less than 5·5 lbs. is raised by 56 lbs.

When the power is raised, the force of friction, together with the power, must be overcome by the load. Let us call _X_ the real power that would be necessary to balance 56 lbs. in a perfectly frictionless machine, and _Y_ the force of friction. We shall be able to determine _X_ and _Y_ by the experiments just performed. When the load is raised a power equal to _X_ + _Y_ must be applied, and therefore _X_ + _Y_ = 13·1. On the other hand, when the power is raised, the force _X_ is just sufficient to overcome both the friction _Y_ and the weight 5·5; therefore _X_ = _Y_ + 5·5.

Solving this pair of equations, we find that _X_ = 9·3 and _Y_ = 3·8. Hence we infer that the power in the frictionless machine would be 9·3; but this is exactly what would have been deduced from the velocity ratio, for 56 ÷ 6 = 9·3 lbs. In this result we find a perfect accordance between theory and experiment.

THE DIFFERENTIAL PULLEY-BLOCK.

208. By increasing the number of sheaves in a pair of pulley-blocks the power may be increased; but the length of rope (or chain) requisite for several sheaves becomes a practical inconvenience. There are also other reasons which make the differential pulley-block, which we shall now consider, more convenient for many purposes than the common pulley blocks when a considerable augmentation of power is required.

209. The principle of the differential pulley is very ancient, and in modern times it has been embodied in a machine of practical utility. The object is to secure, that while the power moves over a considerable distance, the load shall only be raised a short distance. When this has been attained, we then know by the principle of energy that we have gained a mechanical advantage.

210. Let us consider the means by which this is effected in that ingenious contrivance, Weston’s differential pulley-block. The principle of this machine will be understood from Fig. 36 and Fig. 37.

It consists of three parts,—an upper pulley-block, a moveable pulley, and an endless chain. We shall briefly describe them. The upper block P is furnished with a hook for attachment to a support. The sheave it contains resembles two sheaves, one a little smaller than the other, fastened together: they are in fact one piece. The grooves are provided with ridges, adapted to prevent the chain from slipping. The lower pulley Q consists of one sheave, which is also furnished with a groove; it carries a hook, to which the load is attached. The endless chain performs a part that will be understood from the sketch of the principle in Fig. 36. The chain passes from the hand at A up to L over the larger groove in the upper pulley, then downwards at B, under the lower pulley, up again at C, over the smaller groove in the upper pulley at A, and then back again by D to the hand at A. When the hand pulls the chain downwards, the two grooves of the upper pulley begin to turn together in the direction shown by the arrows on the chain. The large groove is therefore winding up the chain, while the smaller groove is lowering.

211. In the pulley which has been employed in the experiments to be described, the effective circumference of the large groove is found to be 11"·84, while that of the small groove is 10"·36. When the upper pulley has made one revolution, the large groove must have drawn up 11"·84 of chain, since the chain cannot slip on account of the ridges; but in the same time the small groove has lowered 10"·36 of chain: hence when the upper pulley has revolved once, the chain between the two must have been shortened by the difference between 11"·84 and 10"·36, that is by 1"·48; but this can only have taken place by raising the moveable pulley through half 1"·48, that is, through a space 0"·74. The power has then acted through 11"·84, and has raised the resistance 0"·74. The power has therefore moved through a space 16 times greater than that through which the load moves. In fact, it is easy to verify by actual trial that the power must be moved through 16 feet in order that the load may be raised 1 foot. We express this by saying that the _velocity ratio_ is 16.

212. By applying power to the chain at D proceeding from the smaller groove, the chain is lowered by the large groove faster than it is raised by the small one, and the lower pulley descends. The load is thus raised or lowered by simply pulling one chain A or the other D.

213. We shall next consider the mechanical efficiency of the differential pulley-block. The block (Fig. 37) which we shall use is intended to be worked by one man, and will raise any weight not exceeding a quarter of a ton.

We have already learned that with this block the power must act through sixteen feet for the load to be raised one foot. Hence, were it not for friction, the power need only be the sixteenth part of the load. A few trials will show us that the real efficiency is not so large, and that in fact more than half the work exerted is merely expended upon overcoming friction. This will lead afterwards to a result of considerable practical importance.

214. Placing upon the load hook a weight of 200 lbs., I find that 38 lbs. attached to a hook fastened on the power chain is sufficient to raise the load; that is to say, the power is about one-sixth of the load. If I make the load 400 lbs. I find the requisite power to be 64 lbs., which is only about 3 lbs. less than one-sixth of 400 lbs. We may safely adopt the practical rule, that with this differential pulley-block a man would be able to raise a weight six times as great as he could raise without such assistance.

215. A series of experiments carefully tried with different loads have given the results shown in Table XI.

TABLE XI.—THE DIFFERENTIAL PULLEY-BLOCK.

Circumference of large groove 11"·84, of small groove 10"·36; velocity ratio 16; mechanical efficiency 6·07; useful effect 38 per cent.; formula _P_ = 3·87 + 0·1508 _R_. +-----------+---------+----------+----------+---------------+ | | | | P. | Difference | | Number of | R. | Observed |Calculated|of the observed| |Experiment.| Load | power | power | and calculated| | | in lbs. | in lbs. | in lbs. | values. | +-----------+---------+----------+----------+---------------+ | 1 | 56 | 10 | 12·3 | + 2·3 | | 2 | 112 | 20 | 20·8 | + 0·8 | | 3 | 168 | 31 | 29·2 | - 1·8 | | 4 | 224 | 38 | 37·7 | - 0·3 | | 5 | 280 | 48 | 46·1 | - 1·9 | | 6 | 336 | 54 | 54·6 | + 0·6 | | 7 | 392 | 64 | 63·1 | - 0·9 | | 8 | 448 | 72 | 71·5 | - 0·5 | | 9 | 504 | 80 | 80·0 | 0·0 | | 10 | 560 | 86 | 88·4 | + 2·4 | +-----------+---------+----------+----------+---------------+

The first column contains the numbers of the experiments, the second the weights raised, the third the observed values of the corresponding powers. From these the following rule for finding the power has been obtained:—

216. To find the power, multiply the load by 0·1508, and add 3·87 lbs. to the product; this rule may be expressed by the formula _P_ = 3·87 + 0·1508 _R_. (See Appendix.)

217. The calculated values of the powers are given in the fourth column, and the differences between the observed and calculated values in the last column. The differences do not in any case amount to 2·5 lbs., and considering that the loads raised are up to a quarter of a ton, the formula represents the experiments with satisfactory precision.

218. Suppose for example 280 lbs. is to be raised; the product of 280 and 0·1508 is 42·22, to which, when 3·87 is added, we find 46·09 to be the requisite power. The mechanical efficiency found by dividing 46·09 into 280 is 6·07.

219. To raise 280 lbs. one foot 280 foot-pounds of energy would be necessary, but in the differential pulley-block 46·09 lbs. must be exerted for a distance of 16 feet in order to accomplish this object. The product of 46·09 and 16 is 737·4. Hence the differential pulley-block requires 737·4 foot-pounds of energy to be applied in order to yield 280 useful foot-pounds; but 280 is only 38 per cent. of 737·4, and therefore with a load of 280 lbs. only 38 per cent. of the energy applied to a differential pulley-block is utilized. In general, we may state that not more than about 40 per cent. is profitably used, and that the remainder is expended in overcoming friction.

220. It is a remarkable and useful property of the differential pulley, that a weight which has been hoisted will remain suspended when the hand is removed, even though the chain be not secured in any manner. The pulleys we have previously considered do not possess this convenient property. The weight raised by the three-sheave pulley-block, for example, will run down unless the free end of the rope be properly secured. The difference in this respect between these two mechanical powers is not a consequence of any special mechanism; it is simply caused by the excessive friction in the differential pulley-block.

221. The reason why the load does not run down in the differential pulley may be thus explained. Let us suppose that a weight of 400 lbs. is to be raised one foot by the differential pulley-block; 400 units of work are necessary, and therefore 1,000 units of work must be applied to the power chain to produce the 400 units (since only 40 per cent. is utilized). The friction will thus have consumed 600 units of work when the load has been raised one foot. If the power-weight be removed, the pressure supported by the upper pulley-block is diminished. In fact, since the power-weight is about ¹/₆th of the load, the pressure on the axle when the power-weight has been removed is only ⁶/₇ths of its previous value. The friction is nearly proportional to that pressure: hence when the power has been removed the friction on the upper axle is ⁶/₇ths of its previous value, while the friction on the lower pulley remains unaltered.

We may therefore assume that the total friction is at least ⁶/₇ths of what it was before the power-weight was removed. Will friction allow the load to descend? 600 foot-pounds of work were required to overcome the friction in the ascent: at least ⁶/₇ × 600 = 514 foot-pounds would be necessary to overcome friction in the descent. But where is this energy to come from? The load in its descent could only yield 400 units, and thus descent by the mere weight of the load is impossible. To enable the load to descend we have actually to aid the movement by pulling the chain D (Figs. 36 and 37), which proceeds from the small groove in the upper pulley.

222. The principle which we have here established extends to other mechanical powers, and may be stated generally. _Whenever more than half the applied energy is consumed by friction, the load will remain without running down when the machine is left free._

THE EPICYCLOIDAL PULLEY-BLOCK.

223. We shall conclude this lecture with some experiments upon a useful mechanical power introduced by Mr. Eade under the name of the epicycloidal pulley-block. It is shown in Fig. 33, and also in Fig. 49. In this machine there are two chains: one a slight endless chain to which the power is applied; the other a stout chain which has a hook at each end, from either of which the load may be suspended. Each of these chains passes over a sheave in the block: these sheaves are connected by an ingenious piece of mechanism which we need not here describe. This mechanism is so contrived that, when the power causes the sheave to revolve over which the slight chain passes, the sheave which carries the large chain is also made to revolve, but very slowly.

224. By actual trial it is ascertained that the power must be exerted through twelve feet and a half in order to raise the load one foot; the velocity ratio of the machine is therefore 12·5.

225. If the machine were frictionless, its mechanical efficiency would be of course equal to its velocity ratio; owing to the presence of friction the mechanical efficiency is less than the velocity ratio, and it will be necessary to make experiments to determine the exact value. I attach to the load hook a weight of 280 lbs., and insert a few small hooks into the links of the power chain in order to receive weights: 56 lbs. is sufficient to produce motion, hence the mechanical efficiency is 5. Had there been no friction a power of 56 lbs. would have been capable of overcoming a load of 12·5 × 56 = 700 lbs. Thus 700 units of energy must be applied to the machine in order to perform 280 units of work. In other words, only 40 per cent. of the applied energy is utilized.

226. An extended series of experiments upon the epicycloidal pulley-block is recorded in Table XII.

TABLE XII.—THE EPICYCLOIDAL PULLEY-BLOCK.

Size adapted for lifting weights up to 5 cwt.; velocity ratio 12·5; mechanical efficiency 5; useful effect 40 per cent.; calculated formula _P_ = 5·8 + 0·185 _R_. +-----------+---------+----------+----------+---------------+ | | | | P. | Difference | | Number of | R. | Observed |Calculated|of the observed| |Experiment.| Load | power | power | and calculated| | | in lbs. | in lbs. | in lbs. | values. | +-----------+---------+----------+----------+---------------+ | 1 | 56 | 15 | 16·2 | + 1·2 | | 2 | 112 | 27 | 26·5 | - 0·5 | | 3 | 168 | 40 | 36·9 | - 3·1 | | 4 | 224 | 47 | 47·2 | + 0·2 | | 5 | 280 | 56 | 57·6 | + 1·6 | | 6 | 336 | 66 | 68·0 | + 2·0 | | 7 | 392 | 78 | 78·3 | + 0·3 | | 8 | 448 | 88 | 88·6 | + 0·6 | | 9 | 504 | 100 | 99·0 | - 1·0 | | 10 | 560 | 110 | 109·4 | - 0·6 | +-----------+---------+----------+----------+---------------+