(267.) The next class of simple machines, which present themselves to our attention, is that which we have called the cord. If a rope were perfectly flexible, and were capable of being bent over a sharp edge, and of moving upon it without friction, we should be enabled by its means to make a force in any one direction overcome resistance, or communicate motion in any other direction. Thus if P, fig. 112., be such an edge, a perfectly flexible rope passing over it would be capable of transmitting a force S F to a resistance Q R, so as to support or overcome R, or by a motion in the direction of S F to produce another motion in the direction R Q. But as no materials of which ropes can be constructed can give them perfect flexibility, and as in proportion to the strength by which they are enabled to transmit force their rigidity increases, it is necessary, in practice, to adopt means to remove or mitigate those effects which attend imperfect flexibility, and which would otherwise render cords practically inapplicable as machines.

When a cord is used to transmit a force from one direction to another, its stiffness renders some force necessary in bending it over the angle P, which the two directions form; and if the angle be sharp, the exertion of such a force may be attended with the rupture of the cord. If, instead of bending the rope at one point over a single angle, the change of direction were produced by successively deflecting it over several angles, each of which would be less sharp than a single one could be, the force requisite for the deflection, as well as the liability of rupturing the cord, would be considerably diminished. But this end will be still more perfectly attained if the deflection of the cord be produced by bending it over the surface of a curve.

If a rope were applied only to sustain, and not to move a weight, this would be sufficient to remove the inconveniences arising from its rigidity. But when motion is to be produced, the rope, in passing over the curved surface, would be subject to excessive friction, and consequently to rapid wear. This inconvenience is removed by causing the surface on which the rope runs to move with it, so that no more friction is produced than would arise from the curved surface rolling upon the rope.

(268.) All these ends are attained by the common pulley, which consists of a wheel called a sheave, fixed in a block and turning on a pivot. A groove is formed in the edge of the wheel in which the rope runs, the wheel revolving with it. Such an apparatus is represented in fig. 113.

We shall, for the present, omit the consideration of that part of the effects of the stiffness and friction of the machine which is not removed by the contrivance just explained, and shall consider the rope as perfectly flexible and moving without friction.

From the definition of a flexible cord, it follows, that its tension, or the force by which it is stretched throughout its entire length, must be uniform. From this principle, and this alone, all the mechanical properties of pulleys may be derived.

Although, as already explained, the whole mechanical efficacy of this machine depends on the qualities of the cord, and not on those of the block and sheave, which are only introduced to remove the accidental effects of stiffness and friction; yet it has been usual to give the name pulley to the block and sheave, and a combination of blocks, sheaves, and ropes is called a tackle.

(269.) When the rope passes over a single wheel, which is fixed in its position, as in fig. 113., the machine is called a fixed pulley. Since the tension of the cord is uniform throughout its length, it follows, that in this machine the power and weight are equal. For the weight stretches that part of the cord which is between the weight and pulley, and the power stretches that part between the power and the pulley. And since the tension throughout the whole length is the same, the weight must be equal to the power.

Hence it appears that no mechanical advantage is gained by this machine. Nevertheless, there is scarcely any engine, simple or complex, attended with more convenience. In the application of power, whether of men or animals, or arising from natural forces, there are always some directions in which it may be exerted to much greater convenience and advantage than others, and in many cases the exertion of these powers is limited to a single direction. A machine, therefore, which enables us to give the most advantageous direction to the moving power, whatever be the direction of the resistance opposed to it, contributes as much practical convenience as one which enables a small power to balance or overcome a great weight. In directing the power against the resistance, it is often necessary to use two fixed pulleys. Thus, in elevating a weight A, fig. 114., to the summit of a building, by the strength of a horse moving below, two fixed pulleys B and C may be used. The rope is carried from A over the pulley B; and, passing downwards, is brought under C, and finally drawn by the animal on the horizontal plane. In the same manner sails are spread, and flags hoisted on the yards and masts of a ship, by sailors pulling a rope on the deck.

By means of the fixed pulley a man may raise himself to a considerable height, or descend to any proposed depth. If he be placed in a chair or bucket attached to one end of a rope which is carried over a fixed pulley, by laying hold of this rope on the other side, as represented in fig. 115., he may, at will, descend to a depth equal to half of the entire length of the rope, by continually yielding rope on the one side, and depressing the bucket or chair by his weight on the other. Fire-escapes have been constructed on this principle, the fixed pulley being attached to some part of the building.

(270.) A single moveable pulley is represented in fig. 116. A cord is carried from a fixed point F, and passing through a block B, attached to a weight W, passes over a fixed pulley C, the power being applied at P. We shall first suppose the parts of the cord on each side the wheel B to be parallel; in this case, the whole weight W being sustained by the parts of the cords B C and B F, and these parts being equally stretched (268.), each must sustain half the weight, which is therefore the tension of the cord. This tension is resisted by the power at P, which must, therefore, be equal to half the weight. In this machine, therefore, the weight is twice the power.

(271.) If the parts of the cord B C and B F be not parallel, as in fig. 117., a greater power than half the weight is therefore necessary to sustain it. To determine the power necessary to support a given weight, in this case take the line B A in the vertical direction, consisting of as many inches as the weight consists of ounces; from A draw A D parallel to B C, and A E parallel to B F; the force of the weight represented by A B will be equivalent to two forces represented by B D and B E. (74.) The number of inches in these lines respectively will represent the number of ounces which are equivalent to the tensions of the parts B F and B C of the cord. But as these tensions are equal, B D and B E must be equal, and each will express the amount of the power P, which stretches the cord at P C.

It is evident that the four lines, A E, E B, B D, and D A, are equal. And as each of them represents the power, the weight which is represented by A B must be less than twice the power which is represented by A E and E B taken together. It follows, therefore, that as parts of the ropes which support the weight depart from parallelism the machine becomes less and less efficacious; and there are certain obliquities at which the equilibrating power would be much greater than the weight.

(272.) The mechanical power of pulleys admits of being almost indefinitely increased by combination. Systems of pulleys may be divided into two classes; those in which a single rope is used, and those which consist of several distinct ropes. Fig. 118. and 119. represent two systems of pulleys, each having a single rope. The weight is in each case attached to a moveable block, B, in which are fixed two or more wheels; A is a fixed block, and the rope is successively passed over the wheels above and below, and, after passing over the last wheel above, is attached to the power. The tension of that part of the cord to which the power is attached is produced by the power, and therefore equivalent to it, and the same tension must extend throughout its whole length. The weight is sustained by all those parts of the cord which pass from the lower block, and as the force which stretches them all is the same, viz. that of the power, the effect of the weight must be equally distributed among them, their directions being supposed to be parallel. It will be evident, from this reasoning, that the weight will be as many times greater than the power as the number of cords which support the lower block. Thus, if there be six cords, each cord will support a sixth part of the weight, that is, the weight will be six times the tension of the cord, or six times the power. In fig. 118. the cord is represented as being finally attached to a hook on the upper block. But it may be carried over an additional wheel fixed in that block, and finally attached to a hook in the lower block, as in fig. 119., by which one will be added to the power of the machine, the number of cords at the lower block being increased by one. In the system represented in fig. 118. the wheels are placed in the blocks one above the other; in fig. 119. they are placed side by side. In all systems of pulleys of this class, the weight of the lower block is to be considered as a part of the weight to be raised, and in estimating the power of the machine, this should always be attended to.

(273.) When the power of the machine, and therefore the number of wheels, is considerable, some difficulty arises in the arrangement of the wheels and cords. The celebrated Smeaton contrived a tackle, which takes its name from him, in which there are ten wheels in each block: five large wheels placed side by side, and five smaller ones similarly placed above them in the lower block, and below them in the upper. Fig. 120. represents Smeaton’s blocks without the rope. The wheels are marked with the numbers 1, 2, 3, &c., in the order in which the rope is to be passed over them. As in this pulley 20 distinct parts of the rope support the lower block, the weight, including the lower block, will be 20 times the equilibrating power.

(274.) In all these systems of pulleys, every wheel has a separate axle, and there is a distinct wheel for every turn of the rope at each block. Each wheel is attended with friction on its axle, and also with friction between the sheave and block. The machine is by this means robbed of a great part of its efficacy, since, to overcome the friction alone, a considerable power is in most cases necessary.

An ingenious contrivance has been suggested, by which all the advantage of a large number of wheels may be obtained without the multiplied friction of distinct sheaves and axles. To comprehend the excellence of this contrivance, it will be necessary to consider the rate at which the rope passes over the several wheels of such a system, as fig. 118. If one foot of the rope G F pass over the pulley F, two feet must pass over the pulley E, because the distance between F and E being shortened one foot, the total length of the rope G F E must be shortened two feet. These two feet of rope must pass in the direction E D, and the wheel D, rising one foot, three feet of rope must consequently pass over it. These three feet of rope passing in the direction D C, and the rope D C being also shortened one foot by the ascent of the lower block, four feet of rope must pass over the wheel C. In the same way it may be shown that five feet must pass over B, and six feet over A. Thus, whatever be the number of wheels in the upper and lower blocks, the parts of the rope which pass in the same time over the wheels in the lower block are in the proportion of the odd numbers 1, 3, 5, &c.; and those which pass over the wheels in the upper block in the same time, are as the even numbers 2, 4, 6, &c. If the wheels were all of equal size, as in fig. 119., they would revolve with velocities proportional to the rate at which the rope passes over them. So that, while the first wheel below revolves once, the first wheel above will revolve twice; the second wheel below three times; the second wheel above, four times, and so on. If, however, the wheels differed in size in proportion to the quantity of rope which must pass over them, they would evidently revolve in the same time. Thus, if the first wheel above were twice the size of the first wheel below, one revolution would throw off twice the quantity of rope. Again, if the second wheel below were thrice the size of the first wheel below, it would throw off in one revolution thrice the quantity of rope, and so on. Wheels thus proportioned, revolving in exactly the same time, might be all placed on one axle, and would partake of one common motion, or, what is to the same effect, several grooves might be cut upon the face of one solid wheel, with diameters in the proportion of the odd numbers 1, 3, and 5, &c., for the lower pulley, and corresponding grooves on the face of another solid wheel represented by the even numbers 2, 4, 6, &c., for the upper pulley. The rope being passed successively over the grooves of such wheels, would be thrown off exactly in the same manner as if every groove were upon a separate wheel, and every wheel revolved independently of the others. Such is White’s pulley, represented in fig. 121.

The advantage of this machine, when accurately constructed, is very considerable. The friction, even when great resistances are to be opposed, is very trifling; but, on the other hand, it has corresponding disadvantages which greatly circumscribe its practical utility. In the workmanship of the grooves great difficulty is found in giving them the exact proportions. In doing which, the thickness of the rope must be accurately allowed for; and consequently it follows, that the same pulley can never act except with a rope of a particular diameter. A very slight deviation from the true proportion of the grooves will cause the rope to be unequally stretched, and will throw on some parts of it an undue proportion of the weight, while other parts become nearly, and sometimes altogether slack. Besides these defects, the rope is so liable to derangement by being thrown out of the grooves, that the pulley can scarcely be considered portable.

For these and other reasons, this machine, ingenious as it unquestionably is, has never been extensively used.

(275.) In the several systems of pulleys just explained, the hook to which the fixed block is attached supports the entire of both the power and weight. When the machine is in equilibrium, the power only supports so much of the weight as is equal to the tension of the cord, all the remainder of the weight being thrown on the fixed point, according to what was observed in (225.)

If the power be moved so as to raise the weight, it will move with a velocity as many times greater than that of the weight as the weight itself is greater than the power. Thus in fig. 118. if the weight attached to the lower block ascend one foot, six feet of line will pass over the pulley A, according to what has been already proved. Thus, the power will descend through six feet, while the weight rises one foot. But, in this case, the weight is six times the power. All the observations in (226.) will therefore be applicable to the cases of great weights raised by small powers by means of the system of pulleys just described.

(276.) When two or more ropes are used, pulleys may be combined in various ways so as to produce any degree of mechanical effect. If to any of the systems already described a single moveable pulley be added, the power of the machine would be doubled. In this case, the second rope is attached to the hook of the lower block, as in fig. 122., and being carried through a moveable pulley attached to the weight, it is finally brought up to a fixed point. The tension of the second cord is equal to half the weight (270.); and therefore the power P, by means of the first cord, will have only half the tension which it would have if the weight were attached to the lower block. A moveable pulley thus applied is called a runner.

(277.) Two systems of pulleys, called Spanish bartons, having each two ropes, are represented in fig. 123. The tension of the rope P A B C in the first system is equal to the power; and therefore the parts B A and B C support a portion of the weight equal to twice the power. The rope E A supports the tensions of A P and A B; and therefore the tension of A E D is twice the power. Thus, the united tensions of the ropes which support the pulley B is four times the power, which is therefore the amount of the weight. In the second system, the rope P A D is stretched by the power. The rope A E B C acts against the united tensions A P and A D; and therefore the tension of A E or E B is twice the power. Thus, the weight acts against three tensions; two of which are equal to twice the power, and the remaining one is equal to the power. The weight is therefore equal to five times the power.

A single rope may be so arranged with one moveable pulley as to support a weight equal to three times the power. In fig. 124. this arrangement is represented, where the numbers sufficiently indicate the tension of the rope, and the proportion of the weight and power. In fig. 125. another method of producing the same effect with two ropes is represented.

(278.) If several single moveable pulleys be made successively to act upon each other, the effect is doubled by every additional pulley: such a system as this is represented in fig. 126. The tension of the first rope is equal to the power; the second rope acts against twice the tension of the first, and therefore it is stretched with a force equal to twice the power: the third rope acts against twice this tension, and therefore it is stretched with a force equal to four times the power, and so on. In the system represented in fig. 126. there are three ropes, and the weight is eight times the power. Another rope would render it sixteen times the power, and so on.

In this system, it is obvious that the ropes will require to have different degrees of strength, since the tension to which they are subject increases in a double proportion from the power to the weight.

(279.) If each of the ropes, instead of being attached to fixed points at the top, are carried over fixed pulleys, and attached to the several moveable pulleys respectively, as in fig. 127., the power of the machine will be greatly increased; for in that case the forces which stretch the successive ropes increase in a treble instead of a double proportion, as will be evident by attending to the numbers which express the tensions in the figure. One rope would render the weight three times the power, two ropes nine times, three ropes twenty-seven times, and so on. An arrangement of pulleys is represented in fig. 128., by which each rope, instead of being finally attached to a fixed point, as in fig. 126., is attached to the weight. The weight is in this case supported by three ropes; one stretched with a force equal to the power; another with a force equal to twice the power; and a third with a force equal to four times the power. The weight is therefore, in this case, seven times the power.

(280.) If the ropes, instead of being attached to the weight, pass through wheels, as in fig. 129., and are finally attached to the pulleys above, the power of the machine will be considerably increased. In the system here represented the weight is twenty-six times the power.

(281.) In considering these several combinations of pulleys, we have omitted to estimate the effects produced by the weights of the sheaves and blocks. Without entering into the details of this computation, it may be observed generally, that in the systems represented in figs. 126., 127. the weight of the wheel and blocks acts against the power; but that in figs. 128. and 129. they assist the powers in supporting the weight. In the systems represented in fig. 123. the weight of the pulleys, to a certain extent, neutralise each other.

(282.) It will in all cases be found, that that quantity by which the weight exceeds the power is supported by fixed points; and therefore, although it be commonly stated that a small power supports a great weight, yet in the pulley, as in all other machines, the power supports no more of the weight than is exactly equal to its own amount. It will not be necessary to establish this in each of the examples which have been given: having explained it in one instance, the student will find no difficulty in applying the same reasoning to others. In fig. 126., the fixed pulley sustains a force equal to twice the power, and by it the power giving tension to the first rope sustains a part of the weight equal to itself. The first hook sustains a portion of the weight equal to the tension of the first string, or to the power. The second hook sustains a force equal to twice the power; and the third hook sustains a force equal to four times the power. The three hooks therefore sustain a portion of the weight equal to seven times the power; and the weight itself being eight times the power, it is evident that the part of the weight which remains to be supported by the power is equal to the power itself.

(283.) When a weight is raised by any of the systems of pulleys which have been last described, the proportion between the velocity of the weight and the velocity of the power, so frequently noticed in other machines, will always be observed. In the system of pulleys represented in fig. 126. the weight being eight times the power, the velocity of the power will be eight times that of the weight. If the power be moved through eight feet, that part of the rope between the fixed pulley and the first moveable pulley will be shortened by eight feet. And since the two parts which lie above the first moveable pulley must be equally shortened, each will be diminished by four feet; therefore the first pulley will rise through four feet while the power moves through eight feet. In the same way it may be shown, that while the first pulley moves through four feet, the second moves through two; and while the second moves through two, the third, to which the weight is attached, is raised through one foot. While the power, therefore, is carried through eight feet, the weight is moved through one foot.

By reasoning similar to this, it may be shown that the space through which the power is moved in every case is as many times greater than the height through which the weight is raised, as the weight is greater than the power.

(284.) From its portable form, cheapness of construction, and the facility with which it may be applied in almost every situation, the pulley is one of the most useful of the simple machines. The mechanical advantage, however, which it appears in theory to possess is considerably diminished in practice, owing to the stiffness of the cordage, and the friction of the wheels and blocks. By this means it is computed that in most cases so great a proportion as two thirds of the power is lost. The pulley is much used in building, where weights are to be elevated to great heights. But its most extensive application is found in the rigging of ships, where almost every motion is accomplished by its means.

(285.) In all the examples of pulleys, we have supposed the parts of the rope sustaining the weight and each of the moveable pulleys to be parallel to each other. If they be subject to considerable obliquity, the relative tensions of the different ropes must be estimated according to the principle applied in (271.)