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by Robert S. BallApril 24th, 2023

*Experimental Mechanics by Robert S. Ball is part of the HackerNoon Books Series. You can jump to any chapter in this book **here**. THE WHEEL AND AXLE
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*Introduction.—Experiments upon the Wheel and Axle.—Friction upon the Axle.—The Wheel and Barrel.—The Wheel and Pinion.—The Crane.—Conclusion.*

**INTRODUCTION.**

297. The mechanical powers discussed in these lectures may be grouped into two classes,—the first where ropes or chains are used, and the second where ropes or chains are absent. Belonging to that class in which ropes are not employed, we have the screw discussed in the last lecture; and the lever discussed in Lecture VIII.; while among those machines in which ropes or chains form an essential part of the apparatus, the pulley and the wheel and axle hold a prominent place. We have already examined several forms of the pulley, and we now proceed to the not less important subject of the wheel and axle.

298. Where great resistances have to be overcome, but where the distance through which the resistance must be urged is short, the lever or the screw is generally found to be the most appropriate means of [Pg 150]increasing power. When, however, the resistance has to be moved a considerable distance, the aid of the pulley, or the wheel and axle, or sometimes of both combined, is called in. The wheel and axle is the form of mechanical power which is generally used when the distance is considerable through which a weight must be raised, or through which some resistance must be overcome.

Fig. 46.

299. The wheel and axle assumes very many forms corresponding to the [Pg 151]various purposes to which it is applied. The general form of the arrangement will be understood from Fig. 46. It consists of an iron axle b, mounted in bearings, so as to be capable of turning freely; to this axle a rope is fastened, and at the extremity of the rope is a weight d, which is gradually raised as the axle revolves. Attached to the axle, and turning with it, is a wheel a with hooks in its circumference, upon which lies a rope; one end of this rope is attached to the circumference of the wheel, and the other supports a weight e. This latter weight may be called the power, while the weight d suspended from the axle is the load. When the power is sufficiently large, e descends, making the wheel to revolve; the wheel causes the axle to revolve, and thus the rope is wound up and the load d is raised.

300. When compared with the differential pulley as a means of raising a weight, this arrangement appears rather bulky and otherwise inconvenient, but, as we shall presently learn, it is a far more economical means of applying energy. In its practical application, moreover, the arrangement is simplified in various ways, two of which may be mentioned.

301. The capstan is essentially a wheel and axle; the power is not in this case applied by means of a rope, but by direct pressure on the part of the men working it; nor is there actually a wheel employed, for the pressure is applied to what would be the extremities of the spokes of the wheel if a wheel existed.

302. In the ordinary winch, the power of the labourer is directly applied to the handle which moves round in the circumference of a circle.

303. There are innumerable other applications of the principle which are constantly met with, and which can be easily understood with a [Pg 152]little attention. These we shall not stop to describe, but we pass on at once to the important question of the relation between the power and the load.

EXPERIMENTS UPON THE WHEEL AND AXLE.

304. We shall commence a series of experiments upon the wheel a and axle b of Fig. 46. We shall first determine the velocity ratio, and then ascertain the mechanical efficiency by actual experiment. The wheel is of wood; it is about 30" in diameter. The string to which the power is attached is coiled round a series of hooks, placed near the margin of the wheel; the effective circumference is thus a little less than the real circumference. I measure a single coil of the string and find the length to be 88"·5. This length, therefore, we shall adopt for the effective circumference of the wheel. The axle is 0"·75 in diameter, but its effective circumference is larger than the circle of which this length is the diameter.

305. The proper mode of finding the effective circumference of the axle in a case where the rope bears a considerable proportion to the axle is as follows. Attach a weight to the extremity of the rope sufficient to stretch it thoroughly. Make the wheel and axle revolve suppose 20 times, and measure the height through which the weight is lifted; then the one-twentieth part of that height is the effective circumference of the axle. By this means I find the circumference of the axle we are using to be 2"·87.

306. We can now ascertain the velocity ratio in this machine. When the wheel and axle have made one complete revolution the power has been lowered through a distance of 88"·5, and the load has been raised [Pg 153]through 2"·87. This is evident because the wheel and axle are attached together, and therefore each completes one revolution in the same time; hence the ratio of the distance which the power moves over to that through which the load is raised is 88"·5 ÷ 2"·87 = 31 very nearly. We shall therefore suppose the velocity ratio to be 31. Thus this wheel and axle has a far higher velocity ratio than any of the systems of pulleys which we have been considering.

307. Were friction absent the velocity ratio of 31 would necessarily express the mechanical efficiency of this wheel and axle; owing to the presence of friction the real efficiency is less than this—how much less, we must ascertain by experiment. I attach a load of 56 lbs. to the hook which is borne by the rope descending from the axle: this load is shown at d in Fig. 46. I find that a power of 2·6 lbs. applied at e is just sufficient to raise d. We infer from this result that the mechanical efficiency of this machine is 56 ÷ 2·6 = 21·5. I add a second 56 lb. weight to the load, and I find that a power of 5·0 lbs. raises the load of 112 lbs. The mechanical efficiency in this case is 112 ÷ 5·5 = 22·5. We adopt the mean value 22. Hence the mechanical efficiency is reduced by friction from 31 to 22.

308. We may compute from this result the number of units of energy which are utilized out of every 100 units applied. Let us suppose a load of 100 lbs. is to be raised one foot; a force of 100 ÷ 22 = 4·6 lbs. will suffice to raise this load. This force must be exerted through a space of 31', and consequently 31 × 4·6 = 143 units of energy must be expended; of this amount 100 units are usefully employed, and therefore the percentage of energy utilized is 100 ÷ 143 × 100 = 70. It follows that 30 per cent. of the applied energy is consumed in overcoming friction.

309. We can see the reason why the wheel and axle overhauls—that is, [Pg 154]runs down of its own accord—when allowed to do so; it is because less than half the applied energy is expended upon friction.

310. A series of experiments which have been carefully made with this wheel and axle are recorded in Table XVIII.

By the method of the Appendix a relation connecting the power and the load has been determined; it is expressed in the form—

P = 0·204 + 0·0426 R.

311. Thus for example in experiment 5 a load of 84 lbs. was found to be raised by a power of 3·7 lbs. The value calculated by the formula is 0·204 + 0·0426 × 84 = 3·8. The calculated value only differs from the observed value by 0·1 lb., which is shown in the fifth column. It will be seen from this column that the values calculated from the formula represents the experiments with fidelity.[Pg 155]

312. We have deduced the relation between the power and the load from the principle of energy, but we might have obtained it from the principle of the lever. The wheel and axle both revolve about the centre of the axle; we may therefore regard the centre as the fulcrum of a lever, and the points where the cords meet the wheel and axle as the points of application of the power and the load respectively.

313. By the principle of the lever of the first order (Art. 237), the power is to the load in the inverse proportion of the arms; in this case, therefore, the power is to the load in the inverse proportion of the radii of the wheel and the axle. But the circumferences of circles are in proportion to their radii, and therefore the power must be to the load as the circumference of the axle is to the circumference of the wheel.

314. This mode of arriving at the result is a little artificial; it is more natural to deduce the law directly from the principle of energy. In a mechanical power of any complexity it would be difficult to trace exactly the transmission of power from one part to the next, but the principle of energy evades this difficulty; no matter what be the mechanical arrangement, simple or complex, of few parts or of many, we have only to ascertain by trial how many feet the power must traverse in order to raise the load one foot; the number thus obtained is the theoretical efficiency of the machine.

FRICTION UPON THE AXLE.

315. In the wheel and axle upon which we have been experimenting, we have found that about 30 per cent. of the power is consumed by friction. We shall be able to ascertain to what this loss is due, and then in some degree to remove its cause. From the experiments of Art. 165 [Pg 156]we learned that the friction of a small pulley was very much greater than that of a large pulley—in fact, the friction is inversely proportional to the diameter of the pulley. We infer from this that by winding the rope upon a barrel instead of upon the axle, the friction may be diminished.

Fig. 47.

316. We can examine experimentally the effect of friction on the axle by the apparatus of Fig. 47. b is a shaft 0"·75 diameter, about which a rope is coiled several times; the ends of this rope hang down freely, and to each of them hooks e, f [Pg 157]are attached. This shaft revolves in brass bearings, which are oiled. In order to investigate the amount of power lost by winding the rope upon an axle of this size, I shall place a certain weight—suppose 56 lbs.—upon one hook f, and then I shall ascertain what amount of power hung upon the other hook e will be sufficient to raise f. There is here no mechanical advantage, so that the excess of load which e must receive in order to raise f is the true measure of the friction.

317. I add on weights at e until the power reaches 85 lbs., when e descends. We thus see that to raise 56 lbs. an excess of 29 lbs. was necessary to overcome the friction. We may roughly enunciate the result by stating that to raise a load in this way, half as much again is required for the power. This law is verified by suspending 28 lbs. at f, when it is found that a power of 43 lbs. at e is required to lift it. Had the power been 42 lbs., it would have been exactly half as much again as the load.

318. Hence in raising f upon this axle, about one-third of the power which must be applied at the circumference of the axle is wasted. This experiment teaches us where the loss lies in the wheel and axle of Art. 304, and explains how it is that about a third of its efficiency is lost. 85 lbs. was only able to raise two-thirds of its own weight, owing to the friction; and hence we should expect to find, as we actually have found, that the power applied at the circumference of the wheel has an effect which is only two-thirds of its theoretical efficiency.

319. From this experiment we should infer that the proper mode of avoiding the loss by friction is to wind the rope upon a barrel of considerable diameter rather than upon the axle itself. I place upon a similar axle to that on which we have been already experimenting a barrel of about 15" circumference. I coil the rope two or three times [Pg 158]about the barrel, and let the ends hang down as before. I then attach to each end 56 lbs. weight, and I find that 10 lbs. added to either of the weights is sufficient to overcome friction, to make it descend, and raise the other weight. The apparatus is shown in Fig. 47. a is the barrel, c and d are the weights. In this arrangement 10 lbs. is sufficient to overcome the friction which required 29 lbs. when the rope was simply coiled around the axle. In other words, by the barrel the loss by friction is reduced to one-third of its amount.

THE WHEEL AND BARREL.

320. We next place the barrel upon the axis already experimented upon and shown in Fig. 46 at b. The circumference of the wheel is 88"·5; the circumference of the barrel is 14"·9. The proper mode of finding the circumference of the barrel is to suspend a weight from the rope, then raise this weight by making one revolution of the wheel, and the distance through which the weight is raised is the effective circumference of the barrel. The velocity ratio of the wheel and barrel is then found, by dividing 14·9 into 88·5, to be 5·94.

321. The mechanical efficiency of this machine is determined by experiment. I suspend a weight of 56 lbs. from the hook, and apply power to the wheel. I find that 10·1 lbs. is just sufficient to raise the load.

322. The mechanical efficiency is to be found by dividing 10·1 into 56; the quotient thus obtained is 5·54. The mechanical efficiency does not differ much from 5·94, the velocity ratio; and consequently in this machine but little power is expended upon friction.

323. We can ascertain the loss by computing the percentage of applied [Pg 159]energy which is utilized. Let us suppose a weight of 100 lbs. has to be raised one foot: for this purpose a force of 100 ÷ 5·54 = 18·1 lbs. must be applied. This is evident from the definition of the mechanical efficiency; but since the load has to be raised one foot, it is clear from the meaning of the velocity ratio that the power must move over 5'·94: hence the number of units of work to be applied is to be measured by the product of 5·94 and 18·1, that is, by 107·5; in order therefore to accomplish 100 units of work 107·5 units of work must be applied. The percentage of energy usefully employed is 100 ÷ 107·5 × 100 = 93. This is far more than 70, which is the percentage utilized when the axle was used without the barrel (Art. 309).

324. A series of experiments made with care upon the wheel and barrel are recorded in Table XIX.

The formula which represents the experiments with the greatest amount of accuracy is P = 0·5 + 0·169 R. This formula is compared with the experiments, and the column of differences shows that the calculated and the observed values agree very closely. The constant part 0·5 is partly due to the constant friction of the heavy barrel and wheel, and partly, it may be, to small irregularities which have prevented the centre of gravity of the whole mass from being strictly in the axle.

325. Though this machine is more economical of power than the wheel and axle of Art 305, yet it is less powerful; in fact, the mechanical efficiency, 5·54, is only about one-fourth of that of the wheel and axle. It is therefore necessary to inquire whether we cannot devise some method by which to secure the advantages of but little friction, and at the same time have a large mechanical efficiency: this we shall proceed to investigate.

THE WHEEL AND PINION.

326. By means of what are called cog-wheels or toothed wheels, we are enabled to combine two or more wheels and axles together, and thus greatly to increase the power which can be produced by a single wheel and axle. Toothed wheels are used for a great variety of purposes in mechanics; we have already had some illustration of their use during these lectures (Fig. 30). The wheels which we shall employ are those often used in lathes and other small machines; they are what are called 10-pitch wheels,—that is to say, a wheel of this class contains ten times as many teeth in its circumference as there are inches in its diameter. I have here a wheel 20" diameter, and consequently it has 200 teeth; here is another which is 2"·5 diameter, and which consequently [Pg 161]contains 25 teeth. We shall mount these wheels upon two parallel shafts, so that they gear one into the other in the manner shown in Fig. 46: f is the large wheel containing 200 teeth, and g the pinion of 25 teeth. The axles are 0"·75 diameter; around each of them a rope is wound, by which a hook is suspended.

327. A small weight at k is sufficient to raise a much larger weight on the other shaft; but before experimenting on the mechanical efficiency of this arrangement, we shall as usual calculate the velocity ratio. The wheel contains eight times as many teeth as the pinion; it is therefore evident that when the wheel has made one revolution, the pinion will have made eight revolutions, and conversely the pinion must turn round eight times to turn the wheel round once: hence the power which is turning the pinion round must be lowered through eight times the circumference of the axle, while the load is raised through a length equal to one circumference of the axle. We thus find the velocity ratio of the machine to be 8.

328. We determine the mechanical efficiency by trial. Attaching a load of 56 lbs. to the axle of the large wheel, it is observed that a power of 13·7 lbs. at k will raise it; the mechanical efficiency of the machine is therefore about 4·1, which is almost exactly half the velocity ratio. We note that the load will only just run down when the power is removed; from this we might have inferred, by Art. 222, that nearly half the power is expended on friction, and that therefore the mechanical efficiency is about half the velocity ratio. The actual percentage of energy that is utilised with this particular load is 51. If we suspend 112 lbs. from the load hook, 26 lbs. is just enough to raise it; the mechanical efficiency that would be deduced from this result is 112 ÷ 26 = 4·3, which is slightly in excess [Pg 162]of the amount obtained by the former experiment. It is often found to be a property of the mechanical powers, that as the load increases the mechanical efficiency slightly improves.

329. In Table XX. will be found a record of experiments upon the relation between the power and the load with the wheel and pinion; the table will sufficiently explain itself, after the description of similar tables already given (Arts. 310, 324).

330. The large amount of friction present in this contrivance is the consequence of winding the rope directly upon the axle instead of upon a barrel, as already pointed out in Art. 319. We might place barrels upon these axles and demonstrate the truth of this statement; but we need not delay to do so, as we use the barrel in the machines which we shall next describe.

THE CRANE.

331. We have already explained (Art. 38) the construction of the lifting crane, so far as its framework is concerned. We now examine the mechanism by which the load is raised. We shall employ for this purpose [Pg 163]the model which is represented in Fig. 48. The jib is supported by a wooden bar as a tie, and the crane is steadied by means of the weights placed at h: some such counterpoise is necessary, for otherwise the machine would tumble over when a load is suspended from the hook.

332. The load is supported by a rope or chain which passes over the pulley e and thence to the barrel d, upon which it is to be wound. This barrel receives its motion from a large wheel a, which contains 200 teeth.

The wheel a is turned by the pinion b which contains 25 teeth. In the actual use of the crane, the axle which carries this pinion would be turned round by means of a handle; but for the purpose of experiments upon the relation of the power to the load, the handle would be inconvenient, and therefore we have placed upon the axle of the pinion a wheel c containing a groove in its circumference. Around this groove a string is wrapped, so that when a weight g is suspended from the string it will cause the wheel to revolve. This weight g will constitute the power by which the load may be raised.

333. Let us compute the velocity ratio of this machine before commencing experiments upon its mechanical efficiency. The effective circumference of the barrel d is found by trial to be 14"·9. Since there are 200 teeth on a and 25 on b, it follows that the pinion b must revolve eight times to produce one revolution of the barrel. Hence the wheel c at the circumference of which the power is applied must also revolve eight times for one revolution of the barrel. The effective circumference of c is 43"; the power must therefore have been applied through 8 × 43" = 344", in order to raise the load 15"·9. The velocity ratio is 344 ÷ 14·9 = 23 very nearly. We can easily verify this value of the velocity ratio by actually raising the load 1', when it appears that the number of revolutions of the wheel b is such that the power must have moved 23'.

Fig. 48.

[Pg 165]334. The mechanical efficiency is to be found as usual by trial. 56 lbs. placed at f is raised by 3·1 lbs. at G; hence the mechanical efficiency deduced from this experiment is 56 ÷ 3·1 = 18. The percentage of useful effect is easily shown to be 78 by the method of Art. 323. Here, then, we have a machine possessing very considerable efficiency, and being at the same time economical of energy.

335. A series of experiments made with this crane is recorded in Table XXI., and a comparison of the calculated and observed values will show that the formula P = 0·0556 R represents the experiments with considerable accuracy.

336. It may be noticed that in this formula the term independent of R, which we frequently meet with in the expression of the relation between the power and the load, is absent. The probable explanation is to be found in the fact that some minute irregularity in the form of the [Pg 166]barrel or of the wheel has been constantly acting like a small weight in favour of the power. In each experiment the motion is always started from the same position of the wheels, and hence any irregularity will be constantly acting in favour of the power or against it; here the former appears to have happened. In other cases doubtless the latter has occurred; the difference is, however, of extremely small amount. The friction of the machine itself when without a load is another source for the production of the constant term; it has happened in the present case that this friction has been almost exactly balanced by the accidental influence referred to.

337. In cranes it is usual to provide means of adding a second train of wheels, when the load is very heavy. In another model we applied the power to an axle with a pinion of 25 teeth, gearing into a wheel of 200 teeth; on the axle of the wheel with 200 teeth is a pinion of 30 teeth, which gears into a wheel of 180 teeth; the barrel is on the axle of the last wheel. A series of experiments with this crane is shown in Table XXII.

The velocity ratio is now 137, and the mechanical efficiency is 87; one man could therefore raise a ton with ease by applying a power of 26 lbs. to a crane of this kind.

CONCLUSION.

338. It will be useful to contrast the wheel and axle on which we have experimented (Art. 304) with the differential pulley (Art. 209). The velocity ratio of the former machine is nearly double that of the latter, and its mechanical efficiency is nearly four times as great. Less than half the applied power is wasted in the wheel and axle, while more than half is wasted in the differential pulley. This makes the wheel and axle both a more powerful machine, and a more economical machine than the differential pulley. On the other hand, the greater compactness of the latter, its facility of application, and the practical conveniences arising from the property of not allowing the load to run down, do often more than compensate for the superior mechanical advantage of the wheel and axle.

339. We may also contrast the wheel and axle with the screw (Art. 277). The screw is remarkable among the mechanical powers for its very high velocity ratio, and its excessive friction. Thus we have seen in Art. 291 how the velocity ratio of a screw-jack with an arm attached amounted to 414, while its mechanical efficiency was little more than one-fourth as great. No single wheel and axle could conveniently be made to give a mechanical efficiency of 116; but from Art. 337 we could easily design a combination of wheels and axles to yield an efficiency of quite this amount. The friction in the wheel and axle is very much less than in the screw, and consequently energy is saved by the use of the former machine.[Pg 168]

340. In practice, however, it generally happens that economy of energy does not weigh much in the selection of a mechanical power for any purpose, as there are always other considerations of greater consequence.

341. For example, let us take the case of a lifting crane employed in loading or unloading a vessel, and inquire why it is that a train of wheels is generally used for the purpose of producing the requisite power. The answer is simple, the train of wheels is convenient, for by their aid any length of chain can be wound upon the barrel; whereas if a screw were used, we should require a screw as long as the greatest height of lift. This screw would be inconvenient, and indeed impracticable, and the additional circumstance that a train of wheels is more economical of energy than a screw has no influence in the matter.

342. On the other hand, suppose that a very heavy load has to be overcome for a short distance, as for example in starting a ship launch, a screw-jack is evidently the proper machine to employ; it is easily applied, and has a high mechanical efficiency. The want of economy of energy is of no consequence in such an operation.

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