Chapter IX: Part 9

A third wheel revolves once, while that which carries the minute hand revolves twelve times; consequently this last wheel, which carries the _hour hand_, revolves at a rate twelve times less than that of the minute hand, and therefore seven hundred and twenty times less than the second hand. We shall now endeavour to explain the manner in which these motions are produced and regulated. Let A, B, C, D, E, _fig. 110._, represent a train of wheels, and _a_, _b_, _c_, _d_ represent their pinions, _e_ being a cylinder on the axis of the wheel E, round which a rope is coiled, sustaining a weight W. Let the effect of this weight transmitted through the train of wheels be opposed by a power P acting upon the wheel A, and let this power be supposed to be of such a nature as to cause the weight W to descend with a uniform velocity, and at any proposed rate. The wheel E carries on its circumference eighty-four teeth. The wheel D carries eighty teeth; the wheel C is also furnished with eighty teeth, and the wheel B with seventy-five. The pinions _d_ and _c_ are each furnished with twelve leaves, and the pinions _b_ and _a_ with ten.

If the power at P be so regulated as to allow the wheel A to revolve once in a minute, with a uniform velocity, a hand attached to the axis of this wheel will serve as the _second hand_. The pinion _a_ carrying ten teeth must revolve seven times and a half to produce one revolution of B, consequently fifteen revolutions of the wheel A will produce two revolutions of the wheel B; the wheel B, therefore, revolves twice in fifteen minutes. The pinion _b_ must revolve eight times to produce one revolution of the wheel C, and therefore the wheel C must revolve once in four quarters of an hour, or in one hour. If a hand be attached to the axis of this wheel, it will have the motion necessary for the minute hand. The pinion _c_ must revolve six times and two thirds to produce one revolution of the wheel D, and therefore this wheel must revolve once in six hours and two thirds. The pinion _d_ revolves seven times for one revolution of the wheel E, and therefore the wheel E will revolve once in forty-six hours and two thirds.

On the axis of the wheel C a second pinion may be placed, furnished with seven leaves, which may lead a wheel of eighty-four teeth, so that this wheel shall turn once during twelve turns of the wheel C. If a hand be fixed upon the axis, this hand will revolve once for twelve revolutions of the minute hand fixed upon the axis of the wheel C; that is, it will revolve once in twelve hours. If it play upon a dial divided into twelve equal parts, it will move over each part in an hour, and will serve the purpose of the hour hand of the chronometer.

We have here supposed that the second hand, the minute hand, and the hour hand move on separate dials. This, however, is not necessary. The axis of the hour hand is commonly a tube, inclosing within it that of the minute hand, so that the same dial serves for both. The second hand, however, is generally furnished with a separate dial.

(266.) We shall now explain the manner in which a power is applied to the wheel A, so as to regulate and equalise the effect of the weight W. Suppose the wheel A furnished with thirty teeth, as in _fig. 111._; if nothing check the motion, the weight W would descend with an accelerated velocity, and would communicate an accelerated motion to the wheel A. This effect, however, is interrupted by the following contrivance:--L M is a pendulum vibrating on the centre L, and so regulated that the time of its oscillation is one second. The pallets I and K are connected with the pendulum, so as to oscillate with it. In the position of the pendulum represented in the figure, the pallet I stops the motion of the wheel A, and entirely suspends the action of the weight W, _fig. 110._, so that for a moment the entire machine is motionless. The weight M, however, falls by its gravity towards the lowest position, and disengages the pallet I from the tooth of the wheel. The weight W begins then to take effect, and the wheel A turns from A towards B. Meanwhile the pendulum M oscillates to the other side, and the pallet K falls under a tooth of the wheel A, and checks for a moment its further motion. On the returning vibration the pallet K becomes again disengaged, and allows the tooth of the wheel to escape, and by the influence of the weight W another tooth passes before the motion of the wheel A is again checked by the interposition of the pallet I.

From this explanation it will appear that, in two vibrations of the pendulum, one tooth of the wheel A passes the pallet I, and therefore, if the wheel A be furnished with 30 teeth, it will be allowed to make one revolution during 60 vibrations of the pendulum. If, therefore, the pendulum be regulated so as to vibrate seconds, this wheel will revolve once in a minute. From the action of the pallets in checking the motion of the wheel A, and allowing its teeth alternately to _escape_, this has been called the _escapement_ wheel; and the wheel and pallets together are generally called the _escapement_, or _’scapement_.

We have already explained, that by reason of the friction on the points of support, and other causes, the swing of the pendulum would gradually diminish, and its vibration at length cease. This, however, is prevented by the action of the teeth of the scapement wheel upon the pallets, which is just sufficient to communicate that quantity of force to the pendulum which is necessary to counteract the retarding effects, and to maintain its motion. It thus appears, that although the effect of the gravity of the weight W in giving motion to the machine is at intervals suspended, yet this part of the force is not lost, being, during these intervals, employed in giving to the pendulum all that motion which it would lose by the resistances to which it is inevitably exposed.

In stationary clocks, and in other cases in which the bulk of the machine is not an objection, a descending weight is used as the moving power. But in watches and portable chronometers, this would be attended with evident inconvenience. In such cases, a spiral spring, called the _mainspring_, is the moving power. The manner in which this spring communicates rotation to an axis, and the ingenious method of equalising the effect of its variable elasticity by giving to it a leverage, which increases as the elastic force diminishes, have been already explained. (255.)

A similar objection lies against the use of a pendulum in portable chronometers. A spiral spring of a similar kind, but infinitely more delicate, called a _hair spring_, is substituted in its place. This spring is connected with a nicely-balanced wheel, called _the balance wheel_, which plays in pivots. When this wheel is turned to a certain extent in one direction, the hair spring is coiled up, and its elasticity causes the wheel to recoil, and return to a position in which the energy of the spring acts in the opposite direction. The balance wheel then returns, and continually vibrates in the same manner. The axis of this wheel is furnished with pallets similar to those of the pendulum, which are alternately engaged with the teeth of a crown wheel, which takes the place of the scapement wheel already described.

A general view of the work of a common watch is represented in _fig. 111._ _bis._ A is the balance wheel bearing pallets _p_ _p_ upon its axis; C is the crown wheel, whose teeth are suffered to escape alternately by those pallets in the manner already described in the scapement of a clock. On the axis of the crown wheel is placed a pinion _d_, which drives another crown wheel K. On the axis of this is placed the pinion _c_, which plays in the teeth of the third wheel L. The pinion _b_ on the axis of L is engaged with the wheel M, called the centre wheel. The axle of this wheel is carried up through the centre of the dial. A pinion _a_ is placed upon it, which works in the great wheel N. On this wheel the mainspring immediately acts. O P is the mainspring stripped of its barrel. The axis of the wheel M passing through the centre of the dial is squared at the end to receive the minute hand. A second pinion Q is placed upon this axle which drives a wheel T. On the axle of this wheel a pinion _g_ is placed, which drives the hour wheel V. This wheel is placed upon a tubular axis, which incloses within it the axis of the wheel M. This tubular axis passing through the centre of the dial, carries the hour hand. The wheels A, B, C, D, E, _fig. 110._, correspond to the wheels C, K, L, M, N, _fig. 112._; and the pinions _a_, _b_, _c_, _d_, _e_, _fig. 109._, correspond to the pinions _d_, _c_, _b_, _a_, _fig. 111_. From what has already been explained of these wheels, it will be obvious that the wheel M, _fig. 111._, revolves once in an hour, causing the minute hand to move round the dial once in that time. This wheel at the same time turns the pinion Q which leads the wheel T. This wheel again turns the pinion _g_ which leads the hour wheel V. The leaves and teeth of these pinions and wheels are proportioned, as already explained, so that the wheel V revolves once during twelve revolutions of the wheel M. The hour hand, therefore, which is carried by the tubular axle of the wheel V, moves once round the dial in twelve hours.

Our object here has not been to give a detailed account of watch and clock work, a subject for which we must refer the reader to the proper department of this work. Such a general account has only been attempted as may explain how tooth and pinion work may be applied to regulate motion.

_London, Pubd. by Longman & Co._]

CHAP. XV.

OF THE PULLEY.

(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_.

_London, Pubd. by Longman & Co._]

(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.)

CHAP. XVI.

ON THE INCLINED PLANE, WEDGE, AND SCREW.

(286.) The inclined plane is the most simple of all machines. It is a hard plane surface forming some angle with a horizontal plane, that angle not being a right angle. When a weight is placed on such a plane, a two-fold effect is produced. A part of the effect of the weight is resisted by the plane, and produces a pressure upon it; and the remainder urges the weight down the plane, and would produce a pressure against any surface resisting its motion placed in a direction perpendicular to the plane (131.)

Let A B, _fig. 130._, be such a plane, B C its horizontal base, A C its height, and A B C its angle of elevation. Let W be a weight placed upon it. This weight acts in the vertical direction W D, and is equivalent to two forces, W F perpendicular to the plane, and W E directed down the plane (74.) If a plane be placed at right angles to the inclined plane below W, it will resist the descent of the weight, and sustain a pressure expressed by W E. Thus, the weight W resting in the corner, instead of producing one pressure in the direction W D, will produce two pressures, one expressed by W F upon the inclined plane, and the other expressed by W E upon the resisting plane. These pressures respectively have the same proportion to the entire weight as W F and W E have to W D, or as D E and W E have to W D, because D E is equal to W F. Now the triangle W E D is in all respects similar to the triangle A B C, the one differing from the other only in the scale on which it is constructed. Therefore, the three lines A C, C B, and B A, are in the same proportion to each other as the lines W E, E D, and W D. Hence, A B has to A C the same proportion as the whole weight has to the pressure directed toward B, and A B has to B C the same proportion as the whole weight has to the pressure on the inclined plane.

We have here supposed the weight to be sustained upon the inclined plane by a hard plane fixed at right angles to it. But the power necessary to sustain the weight will be the same in whatever way it is applied, provided it act in the direction of the plane. Thus, a cord may be attached to the weight, and stretched towards A, or the hands of men may be applied to the weight below it, so as to resist its descent towards B. But in whatever way it be applied, the amount of the power will be determined in the same manner. Suppose the weight to consist of as many pounds as there are inches in A B, then the power requisite to sustain it upon the plane will consist of as many pounds as there are inches in A C, and the pressure on the plane will amount to as many pounds as there are inches in B C.

From what has been stated it may easily be inferred that the less the elevation of the plane is, the less will be the power requisite to sustain a given weight upon it, and the greater will be the pressure upon it. Suppose the inclined plane A B to turn upon a hinge at B, and to be depressed so that its angle of elevation shall be diminished, it is evident that as this angle decreases the height of the plane decreases, and its base increases. Thus, when it takes the position B A′, the height A′ C′ is less than the former height A C, while the base B C′ is greater than the former base B C. The power requisite to support the weight upon the plane in the position B A′ is represented by A′ C′, and is as much less than the power requisite to sustain it upon the plane A B, as the height A′ C′ is less than the height A C. On the other hand, the pressure upon the plane in the position B A′ is as much greater than the pressure upon the plane B A, as the base B C′ is greater than the base B C.

(287.) The power of an inclined plane, considered as a machine, is therefore estimated by the proportion which its length bears to its height. This power is always increased by diminishing the elevation of the plane.

Roads which are not level may be regarded as inclined planes, and loads drawn upon them in carriages, considered in reference to the powers which impel them, are subject to all the conditions which have been established for inclined planes. The inclination of the road is estimated by the height corresponding to some proposed length. Thus it is said to rise one foot in fifteen, one foot in twenty, &c., meaning that if fifteen or twenty feet of the road be taken as the length of an inclined plane, such as A B, the corresponding height will be one foot. Or the same may be expressed thus: that if fifteen or twenty feet be measured upon the road, the difference of the levels of the two extremities of the distance measured is one foot. According to this method of estimating the inclination of roads, the power requisite to sustain a load upon them (setting aside the effect of friction), is always proportional to that elevation. Thus, if a road rise one foot in twenty, a power of one ton will be sufficient to sustain twenty tons, and so on.

On a horizontal plane the only resistance which the power has to overcome is the friction of the load with the plane, and the consideration of this being for the present omitted, a weight once put in motion would continue moving for ever, without any further action of the power. But if the plane be inclined, the power will be expended in raising the weight through the perpendicular height of the plane. Thus, in a road which rises one foot in ten, the power is expended in raising the weight through one perpendicular foot for every ten feet of the road over which it is moved. As the expenditure of power depends upon the rate at which the weight is raised perpendicularly, it is evident that the greater the inclination of the road is, the slower the motion must be with the same force. If the energy of the power be such as to raise the weight at the rate of one foot per minute, the weight may be moved in each minute through that length of the road which corresponds to a rise of one foot. Thus, if two roads rise one at the rate of a foot in fifteen feet, and the other at the rate of one foot in twenty feet, the same expenditure of power will move the weight through fifteen feet of the one, and twenty feet of the other at the same rate.

From such considerations as these, it will readily appear that it may often be more expedient to carry a road through a circuitous route than to continue it in the most direct course; for though the measured length of road may be considerably greater than in the former case, yet more may be gained in speed with the same expenditure of power than is lost by the increase of distance. By attending to these circumstances, modern road-makers have greatly facilitated and expedited the intercourse between distant places.

(288.) If the power act obliquely to the plane, it will have a twofold effect; a part being expended in supporting or drawing the weight, and a part in diminishing or increasing the pressure upon the plane. Let W P, _fig. 130._, be the power. This will be equivalent to two forces, W F′, perpendicular to the plane, and W E′ in the direction of the plane. (74.) In order that the power should sustain the weight, it is necessary that that part W E′ of the power which acts in the direction of the plane should be equal to that part W E, _fig. 130._, of the weight which acts down the plane. The other part W F′ of the power acting perpendicular to the plane is immediately opposed to that part W F of the weight which produces pressure. The pressure upon the plane will therefore be diminished by the amount of W F′. The amount of the power which will equilibrate with the weight may, in this case, be found as follows. Take W E′ equal to W E, and draw E′ P perpendicular to the plane, and meeting the direction of the power. The proportion of the power to the weight will be that of W P to W D. And the proportion of the pressure to the weight will be that of the difference between W F and W F′ to W D. If the amount of the power have a less proportion to the weight than W P has to W D, it will not support the body on the plane, but will allow it to descend. And if it have a greater proportion, it will draw the weight up the plane towards A.

(289.) It sometimes happens that a weight upon one inclined plane is raised or supported by another weight upon another inclined plane. Thus, if A B and A B′, _fig. 131._, be two inclined planes forming an angle at A, and W W′ be two weights placed upon these planes, and connected by a cord passing over a pulley at A, the one weight will either sustain the other, or one will descend, drawing the other up. To determine the circumstances under which these effects will ensue, draw the lines W D and W′ D′ in the vertical direction, and take upon them as many inches as there are ounces in the weights respectively. W D and W′ D′ being the lengths thus taken, and therefore representing the weights, the lines W E and W′ E′ will represent the effects of these weights respectively down the planes. If W E and W′ E′ be equal, the weights will sustain each other without motion. But if W E be greater than W′ E′, the weight W will descend, drawing the weight W′ up. And if W′ E′ be greater than W E, the weight W′ will descend, drawing the weight W up. In every case the lines W F and W′ F′ will represent the pressures upon the planes respectively.

It is not necessary, for the effect just described, that the inclined planes should, as represented in the figure, form an angle with each other. They may be parallel, or in any other position, the rope being carried over a sufficient number of wheels placed so as to give it the necessary deflection. This method of moving loads is frequently applied in great public works where rail-roads are used. Loaded waggons descend one inclined plane, while other waggons, either empty or so loaded as to permit the descent of those with which they are connected, are drawn up the other.

(290.) In the application of the inclined plane which we have hitherto noticed, the machine itself is supposed to be fixed in its position, while the weight or load is moved upon it. But it frequently happens that resistances are to be overcome which do not admit of being thus moved. In such cases, instead of moving the load upon the planes, the plane is to be moved under or against the load. Let D E, _fig. 132._, be a heavy beam secured in a vertical position between guides F G and H I, so that it is free to move upwards and downwards, but not laterally. Let A B C be an inclined plane, the extremity of which is placed beneath the end of the beam. A force applied to the back of this plane A C, in the direction C B, will urge the plane under the beam so as to raise the beam to the position represented in _fig. 133._ Thus, while the inclined plane is moved through the distance C B, the beam is raised through the height C A.

(291.) When the inclined plane is applied in this manner, it is called a _wedge_. And if the power applied to the back were a continued pressure, its proportion to the weight would be that of A C to C B. It follows, therefore, that the more acute the angle B is, the more powerful will be the wedge.

In some cases, the wedge is formed of two inclined planes, placed base to base, as represented in _fig. 134._ The theoretical estimation of the power of this machine is not applicable in practice with any degree of accuracy. This is in part owing to the enormous proportion which the friction in most cases bears to the theoretical value of the power, but still more to the nature of the power generally used. The force of a blow is of a nature so wholly different from continued forces, such as the pressure of weights, or the resistance offered by the cohesion of bodies, that it admits of no numerical comparison with them. Hence we cannot properly state the proportion which the force of a blow bears to the amount of a weight or resistance. The wedge is almost invariably urged by percussion; while the resistances which it has to overcome are as constantly forces of the other kind. Although, however, no exact numerical comparison can be made, yet it may be stated in a general way that the wedge is more and more powerful as its angle is more acute.

_London, Pubd. by Longman & Co._]

In the arts and manufactures, wedges are used where enormous force is to be exerted through a very small space. Thus it is resorted to for splitting masses of timber or stone. Ships are raised in docks by wedges driven under their keels. The wedge is the principal agent in the oil-mill. The seeds from which the oil is to be extracted are introduced into hair bags, and placed between planes of hard wood. Wedges inserted between the bags are driven by allowing heavy beams to fall on them. The pressure thus excited is so intense, that the seeds in the bags are formed into a mass nearly as solid as wood. Instances have occurred in which the wedge has been used to restore a tottering edifice to its perpendicular position.

All cutting and piercing instruments, such as knives, razors, scissors, chisels, &c., nails, pins, needles, awls, &c. are wedges. The angle of the wedge, in these cases, is more or less acute, according to the purpose to which it is to be applied. In determining this, two things are to be considered--the mechanical power, which is increased by diminishing the angle of the wedge; and the strength of the tool, which is always diminished by the same cause. There is, therefore, a practical limit to the increase of the power, and that degree of sharpness only is to be given to the tool which is consistent with the strength requisite for the purpose to which it is to be applied. In tools intended for cutting wood, the angle is generally about 30°. For iron it is from 50° to 60°; and for brass, from 80° to 90°. Tools which act by pressure may be made more acute than those which are driven by a blow; and in general the softer and more yielding the substance to be divided is, and the less the power required to act upon it, the more acute the wedge may be constructed.