part C G F D of the paper will exactly surround the
ruler once: the part C G will form one convolution of the thread, and may be considered as the length of one inclined plane surrounding the cylinder, C H being the corresponding height, and G H the base. The power of the screw does not, as in the ordinary cases of the inclined plane, act parallel to the plane or thread, but at right angles to the length of the cylinder A B, or, what is to the same effect, parallel to the base H G; therefore the proportion of the power to the weight will be, according to principles already explained, the same as that of C H to the space through which the power moves parallel to H G in one revolution of the screw. H C is evidently the distance between the successive positions of the thread as it winds round the cylinder; and it appears from what has been just stated, that the less this distance is, or, in other words, the finer the thread is, the more powerful the machine will be.
(293.) In the application of the screw the weight or resistance is not, as in the inclined plane and wedge, placed upon the surface of the plane or thread. The power is usually transmitted by causing the screw to move in a concave cylinder, on the interior surface of which a spiral cavity is cut, corresponding exactly to the thread of the screw, and in which the thread will move by turning round the screw continually in the same direction. This hollow cylinder is usually called the _nut_ or _concave screw_. The screw surrounded by its spiral thread is represented in _fig. 137._; and a section of the same playing in the nut is represented in _fig. 138._
There are several ways in which the effect of the power may be conveyed to the resistance by this apparatus.
First, let us suppose that the nut A B is fixed. If the screw be continually turned on its axis, by a lever E F inserted in one end of it, it will be moved in the direction C D, advancing every revolution through a space equal to the distance between two contiguous threads. By turning the lever in an opposite direction, the screw will be moved in the direction D C.
If the screw be fixed, so as to be incapable either of moving longitudinally or revolving on its axis, the nut A B may be turned upon the screw by a lever, and will move on the screw towards C or towards D, according to the direction in which the lever is turned.
In the former case we have supposed the nut to be absolutely immoveable, and in the latter case the screw to be absolutely immoveable. It may happen, however, that the nut, though capable of revolving, is incapable of moving longitudinally; and that the screw, though incapable of revolving, is capable of moving longitudinally. In that case, by turning the nut A B upon the screw by the lever, the screw will be urged in the direction C D or D C, according to the way in which the nut is turned.
The apparatus may, on the contrary, be so arranged, that the nut, though incapable of revolving, is capable of moving longitudinally; and the screw, though capable of revolving, is incapable of moving longitudinally. In this case, by turning the screw in the one direction or in the other, the nut A B will be urged in the direction C D or D C.
All these various arrangements may be observed in different applications to the machine.
(294.) A screw may be cut upon a cylinder by placing the cylinder in a turning lathe, and giving it a rotatory motion upon its axis. The cutting point is then presented to the cylinder, and moved in the direction of its length, at such a rate as to be carried through the distance between the intended thread, while the cylinder revolves once. The relative motions of the cutting point and the cylinder being preserved with perfect uniformity, the thread will be cut from one end to the other. The shape of the threads may be either square, as in _fig. 137._, or triangular, as in _fig. 139._
(295.) The screw is generally used in cases where severe pressure is to be excited through small spaces; it is therefore the agent in most presses. In _fig. 140._, the nut is fixed, and by turning the lever, which passes through the head of the screw, a pressure is excited upon any substance placed upon the plate immediately under the end of the screw. In _fig. 141._, the screw is incapable of revolving, but is capable of advancing in the direction of its length. On the other hand, the nut is capable of revolving, but does not advance in the direction of the screw. When the nut is turned by means of the screw inserted in it, the screw advances in the direction of its length, and urges the board which is attached to it upwards, so as to press any substance placed between it and the fixed board above.
In cases where liquids or juices are to be expressed from solid bodies, the screw is the agent generally employed. It is also used in coining, where the impression of a die is to be made upon a piece of metal, and in the same way in producing the impression of a seal upon wax or other substance adapted to receive it. When soft and light materials, such as cotton, are to be reduced to a convenient bulk for transportation, the screw is used to compress them, and they are thus reduced into hard dense masses. In printing, the paper is urged by a severe and sudden pressure upon the types, by means of a screw.
(296.) As the mechanical power of the screw depends upon the relative magnitude of the circumference through which the power revolves, and the distance between the threads, it is evident, that, to increase the efficacy of the machine, we must either increase the length of the lever by which the power acts, or diminish the magnitude of the thread. Although there is no limit in theory to the increase of the mechanical efficacy by these means, yet practical inconvenience arises which effectually prevents that increase being carried beyond a certain extent. If the lever by which the power acts be increased, the same difficulty arises as was already explained in the wheel and axle (254.); the space through which the power should act would be so unwieldy, that its application would become impracticable. If, on the other hand, the power of the machine be increased by diminishing the size of the thread, the strength of the thread will be so diminished, that a slight resistance will tear it from the cylinder. The cases in which it is necessary to increase the power of the machine, being those in which the greatest resistances are to be overcome, the object will evidently be defeated, if the means chosen to increase that power deprive the machine of the strength which is necessary to sustain the force to which it is to be submitted.
(297.) These inconveniences are removed by a contrivance of Mr. Hunter, which, while it gives to the machine all the requisite strength and compactness, allows it to have an almost unlimited degree of mechanical efficacy.
This contrivance consists in the use of two screws, the threads of which may have any strength and magnitude, but which have a very small difference of breadth. While the working point is urged forward by that which has the greater thread, it is drawn back by that which has the less; so that during each revolution of the screw, instead of being advanced through a space equal to the magnitude of either of the threads, it moves through a space equal to their difference. The mechanical power of such a machine will be the same as that of a single screw having a thread, whose magnitude is equal to the difference of the magnitudes of the two threads just mentioned.
Thus, without inconveniently increasing the sweep of the power, on the one hand, or, on the other, diminishing the thread until the necessary strength is lost, the machine will acquire an efficacy limited by nothing but the smallness of the difference between the two threads.
This principle was first applied in the manner represented in _fig. 142._ A is the greater thread, playing in the fixed nut; B is the lesser thread, cut upon a smaller cylinder, and playing in a concave screw, cut within the greater cylinder. During every revolution of the screw, the cylinder A descends through a space equal to the distance between its threads. At the same time the smaller cylinder B ascends through a space equal to the distance between the threads cut upon it: the effect is, that the board D descends through a space equal to the difference between the threads upon A and the threads upon B, and the machine has a power proportionate to the smallness of this difference.
Thus, suppose the screw A has twenty threads in an inch, while the screw B has twenty-one; during one revolution, the screw A will descend through a space equal to the 20th part of an inch. If, during this motion, the screw B did not turn within A, the board D would be advanced through the 20th of an inch; but because the hollow screw within A turns upon B, the screw B will, relatively to A, be raised in one revolution through a space equal to the 21st part of an inch. Thus, while the board D is depressed through the 20th of an inch by the screw A, it is raised through the 21st of an inch by the screw B. It is, therefore, on the whole, depressed through a space equal to the excess of the 20th of an inch above the 21st of an inch, that is, through the 420th of an inch.
The power of this machine will, therefore, be expressed by the number of times the 420th of an inch is contained in the circumference through which the power moves.
(298.) In the practical application of this principle at present the arrangement is somewhat different. The two threads are usually cut on different parts of the same cylinder. If nuts be supposed to be placed upon these, which are capable of moving in the direction of the length, but not of revolving, it is evident that by turning the screw once round, each nut will be advanced through a space equal to the breadth of the respective threads. By this means the two nuts will either approach each other, or mutually recede, according to the direction in which the screw is turned, through a space equal to the difference of the breadth of the threads, and they will exert a force either in compressing or extending any substance placed between them, proportionate to the smallness of that difference.
(299.) A toothed wheel is sometimes used instead of a nut, so that the same quality by which the revolution of the screw urges the nut forward is applied to make the wheel revolve. The screw is in this case called an endless screw, because its action upon the wheel may be continued without limit. This application of the screw is represented in _fig. 143._ P is the winch to which the power is applied; and its effect at the circumference of the wheel is estimated in the same manner as the effect of the screw upon the nut. This effect is to be considered as a power acting upon the circumference of the wheel; and its proportion to the weight or resistance is to be calculated in the same manner as the proportion of the power to the weight in the wheel and axle.
(300.) We have hitherto considered the screw as an engine used to overcome great resistances. It is also eminently useful in several departments of experimental science, for the measurement of very minute motions and spaces, the magnitude of which could scarcely be ascertained by any other means. The very slow motion which may be imparted to the end of a screw, by a very considerable motion in the power, renders it peculiarly well adapted for this purpose. To explain the manner in which it is applied--suppose a screw to be so cut as to have fifty threads in an inch, each revolution of the screw will advance its point through the fiftieth part of an inch. Now, suppose the head of the screw to be a circle, whose diameter is an inch, the circumference of the head will be something more than three inches: this may be easily divided into a hundred equal parts distinctly visible. If a fixed index be presented to this graduated circumference, the hundredth part of a revolution of the screw may be observed, by noting the passage of one division of the head under the index. Since one entire revolution of the head moves the point through the fiftieth of an inch, one division will correspond to the five thousandth of an inch. In order to observe the motion of the point of the screw in this case, a fine wire is attached to it, which is carried across the field of view of a powerful microscope, by which the motion is so magnified as to be distinctly perceptible.
A screw used for such purposes is called a _micrometer screw_. Such an apparatus is usually attached to the limbs of graduated instruments, for the purposes of astronomical and other observation. Without the aid of this apparatus, no observation could be taken with greater accuracy than the amount of the smallest division upon the limb. Thus, if an instrument for measuring angles were divided into small arcs of one minute, and an angle were observed which brought the index of the instrument to some point between two divisions, we could only conclude that the observed angle must consist of a certain number of degrees and minutes, together with an additional number of seconds, which would be unknown, inasmuch as there would be no means of ascertaining the fraction of a minute between the index and the adjacent division of the instrument. But if a screw be provided, the point of which moves through a space equal to one division of the instrument, with sixty revolutions of the head, and that the head itself be divided into one hundred equal parts, each complete revolution of the screw will correspond to the sixtieth part of a minute, or to one second, and each division on the head of the screw will correspond to the hundredth part of a second. The index being attached to this screw, let the head be turned until the index be moved from its observed position to the adjacent division of the limb. The number of complete revolutions of the screw necessary to accomplish this will be the number of seconds; and the number of parts of a revolution over the complete number of revolutions will be the hundredth parts of a second necessary to be added to the degrees and minutes primarily observed.
It is not, however, only to such instruments that the micrometer screw is applicable; any spaces whatever may be measured by it. An instance of its mechanical application may be mentioned in a steel-yard, an instrument for ascertaining the amount of weights by a given weight, sliding on a long graduated arm of a lever. The distance from the fulcrum, at which this weight counterpoises the weight to be ascertained, serves as a measure to the amount of that weight. When the sliding weight happens to be placed between two divisions of the arm, a micrometer screw is used to ascertain the fraction of the division.
Hunter’s screw, already described, seems to be well adapted to micrometrical purposes; since the motion of the point may be rendered indefinitely slow, without requiring an exquisitely fine thread, such as in the single screw would be necessary.
CHAP. XVII.
ON THE REGULATION AND ACCUMULATION OF FORCE.
(301.) It is frequently indispensable, and always desirable, that the operation of a machine should be regular and uniform. Sudden changes in its velocity, and desultory variations in the effective energy of its power, are often injurious or destructive to the apparatus itself, and when applied to manufactures never fail to produce unevenness in the work. To invent methods for insuring the regular motion of machinery, by removing those causes of inequality which may be avoided, and by compensating others, has therefore been a problem to which much attention and ingenuity have been directed. This is chiefly accomplished by controlling, and, as it were, measuring out the power according to the exigencies of the machine, and causing its effective energy to be always commensurate with the resistance which it has to overcome.
Irregularity in the motion of machinery may proceed from one or more of the following causes:--1. irregularity in the prime mover; 2. occasional variation in the amount of the load or resistance; and, 3. because, in the various positions which the parts of the machine assume during its motion, the power may not be transmitted with equal effect to the working point.
The energy of the prime mover is seldom if ever regular. The force of water varies with the copiousness of the stream. The power which impels the windmill is proverbially capricious. The pressure of steam varies with the intensity of the furnace. Animal power, the result of will, temper, and health is difficult of control. Human labour is most of all unmanageable; hence no machine works so irregularly as one which is manipulated. In some cases the moving force is subject, by the very conditions of its existence, to constant variation, as in the example of a spring, which gradually loses its energy as it recoils. (255.) In many instances the prime mover is liable to regular intermission, and is actually suspended for certain intervals of time. This is the case in the single acting steam-engine, where the pressure of the steam urges the descent of the piston, but is suspended during its ascent.
The load or resistance to which the machine is applied is not less fluctuating. In mills there are a multiplicity of parts which are severally liable to be occasionally disengaged, and to have their operation suspended. In large factories for spinning, weaving, printing, &c. a great number of separate spinning machines, looms, presses, or other engines, are usually worked by one common mover, such as a water-wheel or steam-engine. In these cases the number of machines employed from time to time necessarily varies with the fluctuating demand for the articles produced, and from other causes. Under such circumstances the velocity with which every part of the machinery is moved would suffer corresponding changes, increasing its rapidity with every augmentation of the moving power or diminution of the resistance, or being retarded in its speed by the contrary circumstances.
But even when the prime mover and the resistance are both regular, or rendered so by proper contrivances, still it will rarely happen that the machine by which the energy of the one is transmitted to the other conveys this with unimpaired effect in all the phases of its operation. To give a general notion of this cause of inequality to those who have not been familiar with machinery would not be easy, without having recourse to an example. For the present we shall merely state, that the several moving parts of every machine assume in succession a variety of positions; that at regular periods they return to their first position, and again undergo the same succession of changes. In the different positions through which they are carried in every period of motion, the efficacy of the machine to transmit the power to the resistance is different, and thus the effective energy of the machine in acting upon the resistance would be subject to continual fluctuation. This will be more clearly understood when we come to explain the methods of counteracting the defect or equalising the action of the power upon the resistance.
Such are the chief causes of the inequalities incidental to the motion of machinery, and we now propose to describe a few of the many ingenious contrivances which the skill of engineers has produced to remove the consequent inconveniences.
(302.) Setting aside, for the present, the last cause of inequality, and considering the machinery, whatever it be, to transmit the power to the resistance without irregular interruption, it is evident that every contrivance, having for its object to render the velocity uniform, can only accomplish this by causing the variations of the power and resistance to be proportionate to each other. This may be done either by increasing or diminishing the power as the resistance increases or diminishes; or by increasing or diminishing the resistance as the power increases or diminishes.
According to the facilities or convenience presented by the peculiar circumstances of the case either of these methods is adopted.
The contrivances for effecting this are called _regulators_. Most regulators act upon that part of the machine which commands the supply of the power by means of levers, or some other mechanical contrivance, so as to check the quantity of the moving principle conveyed to the machine when the velocity has a tendency to increase; and, on the other hand, to increase that supply upon any undue abatement of its speed. In a water-mill this is done by acting upon the shuttle; in a wind-mill, by an adjustment of the sail-cloth; and in a steam-engine, by opening or closing, in a greater or less degree, the valve by which the cylinder is supplied with steam.
(303.) Of all the contrivances for regulating machinery, that which is best known and most commonly used is the _governor_. This regulator, which had been long in use in mill-work and other machinery, has of late years attracted more general notice by its beautiful adaptation in the steam-engines of Watt. It consists of heavy balls B B, _fig. 144._, attached to the extremities of rods B F. These rods play upon a joint at E, passing through a mortise in the vertical stem D D′. At F they are united by joints to the short rods F H, which are again connected by joints at H to a ring which slides upon the vertical shaft D D′. From this description it will be apparent that when the balls B are drawn from the axis, their upper arms E F are caused to increase their divergence in the same manner as the blades of a scissors are opened by separating the handles. These, acting upon the ring by means of the short links F H, draw it down the vertical axis from D towards E. A contrary effect is produced when the balls B are brought closer to the axis, and the divergence of the rods B E diminished. A horizontal wheel W is attached to the vertical axis D D′, having a groove to receive a rope or strap upon its rim. This strap passes round the wheel or axis by which motion is transmitted to the machinery to be regulated, so that the spindle or shaft D D′ will always be made to revolve with a speed proportionate to that of the machinery.
As the shaft D D′ revolves, the balls B are carried round it with a circular motion, and consequently acquire a centrifugal force which causes them to recede from the axle, and therefore to depress the ring H. On the edge or rim of this ring is formed a groove, which is embraced by the prongs of a fork I, at the extremity of one arm of a lever whose fulcrum is at G. The extremity K of the other arm is connected by some means with the part of the machine which supplies the power. In the present instance we shall suppose it a steam-engine, in which case the rod K I communicates with a flat circular valve V, placed in the principal steam-pipe, and so arranged that, when K is elevated as far as by their divergence the balls B have power over it, the passage of the pipe will be closed by the valve V, and the passage of steam entirely stopped; and, on the other hand, when the balls subside to their lowest position, the valve will be presented with its edge in the direction of the tube, so as to intercept no part of the steam.
The property which renders this instrument so admirably adapted to the purpose to which it is applied is, that when the divergence of the balls is not very considerable, they must always revolve with the same velocity, whether they move at a greater or lesser distance from the vertical axis. If any circumstance increases that velocity, the balls instantly recede from the axis, and closing the valve V, check the supply of steam, and thereby diminishing the speed of the motion, restore the machine to its former rate. If, on the contrary, that fixed velocity be diminished, the centrifugal force being no longer sufficient to support the balls, they descend towards the axle, open the valve V, and, increasing the supply of steam, restore the proper velocity of the machine.
When the governor is applied to a water-wheel it is made to act upon the shuttle through which the water flows, and controls its quantity as effectually, and upon the same principle, as has just been explained in reference to the steam-engine. When applied to a windmill it regulates the sail-cloth so as to diminish the efficacy of the power upon the arms as the force of the wind increases, or _vice versâ_.
In cases where the resistance admits of easy and convenient change, the governor may act so as to accommodate it to the varying energy of the power. This is often done in corn-mills, where it acts upon the shuttle which metes out the corn to the millstones. When the power which drives the mill increases, a proportionally increased feed of corn is given to the stones, so that the resistance being varied in the ratio of the power, the same velocity will be maintained.
(304.) In some cases the centrifugal force of the revolving balls is not sufficiently great to control the power or the resistance, and regulators of a different kind must be resorted to. The following contrivance is called the _water-regulator_:--
A common pump is worked by the machine, whose motion is to be regulated, and water is thus raised and discharged into a cistern. It is allowed to flow from this cistern through a pipe of a given magnitude. When the water is pumped up with the same velocity as it is discharged by this pipe, it is evident that the level of the water in the cistern will be stationary, since it receives from the pump the exact quantity which it discharges from the pipe. But if the pump throw in more water in a given time than is discharged by the pipe, the cistern will begin to be filled, and the level of the water will rise. If, on the other hand, the supply from the pump be less than the discharge from the pipe, the level of the water in the cistern will subside. Since the rate at which water is supplied from the pump will always be proportional to the velocity of the machine, it follows that every fluctuation in this velocity will be indicated by the rising or subsiding of the level of the water in the cistern, and that level never can remain stationary, except at that exact velocity which supplies the quantity of water discharged by the pipe. This pipe may be constructed so as by an adjustment to discharge the water at any required rate; and thus the cistern may be adapted to indicate a constant velocity of any proposed amount.
If the cistern were constantly watched by an attendant, the velocity of the machine might be abated by regulating the power when the level of the water is observed to rise, or increased when it falls; but this is much more effectually and regularly performed by causing the surface of the water itself to perform the duty. A float or large hollow metal ball is placed upon the surface of the water in the cistern. This ball is connected with a lever acting upon some part of the machinery, which controls the power or regulates the amount of resistance, as already explained in the case of the governor. When the level of the water rises, the buoyancy of the ball causes it to rise also with a force equal to the difference between its own weight and the weight of as much water as it displaces. By enlarging the floating ball, a force may be obtained sufficiently great to move those parts of the machinery which act upon the power or resistance, and thus either to diminish the supply of the moving principle or to increase the amount of the resistance, and thereby retard the motion and reduce the velocity to its proper limit. When the level of the water in the cistern falls, the floating ball being no longer supported on the liquid surface, descends with the force of its own weight, and producing an effect upon the power or resistance contrary to the former, increases the effective energy of the one, or diminishes that of the other, until the velocity proper to the machine be restored.
The sensibility of these regulators is increased by making the surface of water in the cistern as small as possible; for then a small change in the rate at which the water is supplied by the pump will produce a considerable change in the level of the water in the cistern.
Instead of using a float, the cistern itself may be suspended from the lever which controls the supply of the power, and in this case a sliding weight may be placed on the other arm, so that it will balance the cistern when it contains that quantity of water which corresponds to the fixed level already explained. If the quantity of water in the cistern be increased by an undue velocity of the machine, the weight of the cistern will preponderate, draw down the arm of the lever, and check the supply of the power. If, on the other hand, the supply of water be too small, the cistern will no longer balance the counterpoise, the arm by which it is suspended will be raised, and the energy of the power will be increased.
(305.) In the steam-engine the self-regulating principle is carried to an astonishing pitch of perfection. The machine itself raises in due quantity the cold water necessary to condense the steam. It pumps off the hot water produced by the steam, which has been cooled, and lodges it in a reservoir for the supply of the boiler. It carries from this reservoir exactly that quantity of water which is necessary to supply the wants of the boiler, and lodges it therein according as it is required. It breathes the boiler of redundant steam, and preserves that which remains fit, both in quantity and quality, for the use of the engine. It blows its own fire, maintaining its intensity, and increasing or diminishing it according to the quantity of steam which it is necessary to raise; so that when much work is expected from the engine, the fire is proportionally brisk and vivid. It breaks and prepares its own fuel, and scatters it upon the bars at proper times and in due quantity. It opens and closes its several valves at the proper moments, works its own pumps, turns its own wheels, and is only not alive. Among so many beautiful examples of the self-regulating principle, it is difficult to select. We shall, however, mention one or two, and for others refer the reader to our treatise on this subject.[3]
[3] Lardner on the Steam-Engine, Steam-Navigation, Roads, and Railways. 8th edition. 1851.
It is necessary in this machine that the water in the boiler be maintained constantly at the same level, and, therefore, that as much be supplied, from time to time, as is consumed by evaporation. A pump which is wrought by the engine itself supplies a cistern C, _fig. 145._, with hot water. At the bottom of this cistern is a valve V opening into a tube which descends into the boiler. This valve is connected by a wire with the arm of a lever on the fulcrum D, the other arm E of which is also connected by a wire with a stone float F, which is partially immersed in the water of the boiler, and is balanced by a sliding weight A. The weight A only counterpoises the stone float F by the aid of its buoyance in the water; for if the water be removed, the stone F will preponderate, and raise the weight A. When the water in the boiler is at its proper level, the length of the wire connecting the valve V with the lever is so adjusted that this valve shall be closed, the wire at the same time being fully extended. When, by evaporation, the water in the boiler begins to be diminished, the level falls, and the stone weight F, being no longer supported, overcomes the counterpoise A, raises the arm of the lever, and, pulling the wire, opens the valve V. The water in the cistern C then flows through the tube into the boiler, and continues to flow until the level be so raised that the stone weight F is again elevated, the valve V closed, and the further supply of water from the cistern C suspended.
In order to render the operation of this apparatus easily intelligible, we have here supposed an imperfection which does not exist. According to what has just been stated, the level of the water in the boiler descends from its proper height, and subsequently returns to it. But, in fact, this does not happen. The float F and valve V adjust themselves, so that a constant supply of water passes through the valve, which proceeds exactly at the same rate as that at which the water in the boiler is consumed.
(306.) In the same machine there occurs a singularly happy example of self-adjustment, in the method by which the strength of the fire is regulated. The governor regulates the supply of steam to the engine, and proportions it to the work to be done. With this work, therefore, the demands upon the boiler increase or diminish, and with these demands the production of steam in the boiler ought to vary. In fact, the rate at which steam is generated in the boiler, ought to be equal to that at which it is consumed in the engine, otherwise one of two effects must ensue: either the boiler will fail to supply the engine with steam, or steam will accumulate in the boiler, being produced in undue quantity, and, escaping at the safety valve, will thus be wasted. It is, therefore, necessary to control the agent which generates the steam, namely, the fire, and to vary its intensity from time to time, proportioning it to the demands of the engine. To accomplish this, the following contrivance has been adopted:--Let T, _fig. 146._, be a tube inserted in the top of the boiler, and descending nearly to the bottom. The pressure of the steam confined in the boiler, acting upon the surface of the water, forces it to a certain height in the tube T. A weight F, half immersed in the water in the tube, is suspended by a chain, which passes over the wheels P P′, and is balanced by a metal plate D, in the same manner as the stone float, _fig. 145._, is balanced by the weight A. The plate D passes through the mouth of the flue E as it issues finally from the boiler; so that when the plate D falls it stops the flue, suspending thereby the draught of air through the furnace, mitigating the intensity of the fire, and checking the production of steam. If, on the contrary, the plate D be drawn up, the draught is increased, the fire is rendered more active, and the production of steam in the boiler is stimulated. Now, suppose that the boiler produces steam faster than the engine consumes it, either because the load on the engine has been diminished, and, therefore, its consumption of steam reduced, or because the fire has become too intense; the consequence is, that the steam, beginning to accumulate in the boiler, will press upon the surface of the water with increased force, and the water will be raised in the tube T. The weight F will, therefore, be lifted, and the plate D will descend, diminish, or stop the draught, mitigate the fire, and retard the production of steam, and will continue to do so until the rate at which steam is produced shall be commensurate to the wants of the engine. If, on the other hand, the production of steam be inadequate to the exigency of the machine, either because of an increased load, or of the insufficient force of the fire, the steam in the boiler will lose its elasticity, and the surface of the water not sustaining its wonted pressure, the water in the tube T will fall; consequently the weight F will descend, and the plate D will be raised. The flue being thus opened, the draught will be increased, and the fire rendered more intense. Thus the production of steam becomes more rapid, and is rendered sufficiently abundant for the purposes of the engine. This apparatus is called the _self-acting damper_.
(307.) When a perfectly uniform rate of motion has not been attained, it is often necessary to indicate small variations of velocity. The following contrivance, called a _tachometer_[4], has been invented to accomplish this. A cup, _fig. 147._, is filled to the level C D with quicksilver, and is attached to a spindle, which is whirled by the machine in the same manner as the governor already described. It is well known that the centrifugal force produced by this whirling motion will cause the mercury to recede from the centre and rise upon the sides of the cup, so that its surface will assume the concave appearance represented in _fig. 148._ In this case the centre of the surface will obviously have fallen below its original level, _fig. 147._, and the edges will have risen above that level. As this effect is produced by the velocity of the machine, so it is proportionate to that velocity, and subject to corresponding variations. Any method of rendering visible small changes in the central level of the surface of the quicksilver will indicate minute variations in the velocity of the machine.
[4] From the Greek words _tachos_ speed, and _metron_ measure.
A glass tube A, open at both ends, and expanding at one extremity into a bell B, is immersed with its wider end in the mercury, the surface of which will stand at the same level in the bell B, and in the cup C D. The tube is so suspended as to be unconnected with the cup. This tube is then filled to a certain height A, with spirits tinged with some colouring matter, to render it easily observable. When the cup is whirled by the machine to which it is attached, the level of the quicksilver in the bell falls, leaving more space for the spirits, which, therefore, descends in the tube. As the motion is continued, every change of velocity causes a corresponding change in the level of the mercury, and, therefore, also in the level A of the spirits. It will be observed, that, in consequence of the capacity of the bell B being much greater than that of the tube A, a very small change in the level of the quicksilver in the bell will produce a considerable change in the height of the spirits in the tube. Thus this ingenious instrument becomes a very delicate indicator of variations in the motion of machinery.
(308.) The governor, and other methods of regulating the motion of machinery which have been just described, are adapted principally to cases in which the proportion of the resistance to the load is subject to certain fluctuations or gradual changes, or at least to cases in which the resistance is not at any time entirely withdrawn, nor the energy of the power actually suspended. Circumstances, however, frequently occur in which, while the power remains in full activity, the resistance is at intervals suddenly removed and as suddenly again returns. On the other hand, cases also present themselves, in which, while the resistance is continued, the impelling power is subject to intermission at regular periods. In the former case, the machine would be driven with a ruinous rapidity during those periods at which it is relieved from its load, and on the return of the load every part would suffer a violent strain, from its endeavour to retain the velocity which it had acquired, and the speedy destruction of the engine could not fail to ensue. In the latter case, the motion would be greatly retarded or entirely suspended during those periods at which the moving power is deprived of its activity, and, consequently, the motion which it would communicate would be so irregular as to be useless for the purposes of manufactures.
It is also frequently desirable, by means of a weak but continued power, to produce a severe but instantaneous effect. Thus a blow may be required to be given by the muscular action of a man’s arm with a force to which, unaided by mechanical contrivance, its strength would be entirely inadequate.
In all these cases, it is evident that the object to be attained is, an effectual method of accumulating the energy of the power so as to make it available after the action by which it has been produced has ceased. Thus, in the case in which the load is at periodical intervals withdrawn from the machine, if the force of the power could be imparted to something by which it would be preserved, so as to be brought against the load when it again returned, the inconvenience would be removed. In like manner, in the case where the power itself is subject to intermission, if a part of the force which it exerts in its intervals of action could be accumulated and preserved, it might be brought to bear upon the machine during its periods of suspension. By the same means of accumulating force, the strength of an infant, by repeated efforts, might produce effects which would be vainly attempted by the single and momentary action of the strongest man.
(309.) The property of inertia, explained and illustrated in the third and fourth chapters of this volume furnishes an easy and effectual method of accomplishing this. A mass of matter retains, by virtue of its inertia, the whole of any force which may have been given to it, except that part of which friction and the atmospheric resistance deprives it. By contrivances which are well known and present no difficulty, the part of the moving force thus lost may be rendered comparatively small, and the moving mass may be regarded as retaining nearly the whole of the force impressed upon it. To render this method of accumulating force fully intelligible, let us first imagine a polished level plane on which a heavy globe of metal, also polished, is placed. It is evident that the globe will remain at rest on any part of the plane without a tendency to move in any direction. As the friction is nearly removed by the polish of the surfaces, the globe will be easily moved by the least force applied to it. Suppose a slight impulse given to it, which will cause it to move at the rate of one foot in a second. Setting aside the effects of friction, it will continue to move at this rate for any length of time. The same impulse repeated will increase its speed to two feet per second. A third impulse to three feet, and so on. Thus 10,000 repetitions of the impulse will cause it to move at the rate of 10,000 feet per second. If the body to which these impulses were communicated were a cannon ball, it might, by a constant repetition of the impelling force, be at length made to move with as much force as if it were projected from the most powerful piece of ordnance. The force with which the ball in such a case would strike a building might be sufficient to reduce it to ruins, and yet such force would be nothing more than the accumulation of a number of weak efforts not beyond the power of a child to exert, which are stored up, and preserved, as it were, by the moving mass, and thereby brought to bear, at the same moment, upon the point to which the force is directed. It is the sum of a number of actions exerted successively, and, during a long interval, brought into operation at one and the same moment.
But the case which is here supposed cannot actually occur; because we have not usually any practical means of moving a body for any considerable time in the same direction without much friction, and without encountering numerous obstacles which would impede its progress. It is not, however, essential to the effect which is to be produced, that the motion should be in a straight line. If a leaden weight be attached to the end of a light rod or cord, and be whirled by the force of the arm in a circle, it will gradually acquire increased speed and force, and at length may receive an impetus which would cause it to penetrate a piece of board as effectually as if it were discharged from a musket.
The force of a hammer or sledge depends partly on its weight, but much more on the principle just explained. Were it allowed merely to fall by the force of its weight upon the head of a nail, or upon a bar of heated iron which is to be flattened, an inconsiderable effect would be produced. But when it is wielded by the arm of a man, it receives at every moment of its motion increased force, which is finally expended in a single instant on the head of the nail, or on the bar of iron.
The effects of flails in threshing, of clubs, whips, canes, and instruments for striking, axes, hatchets, cleavers, and all instruments which cut by a blow, depend on the same principle, and are similarly explained.
The bow-string which impels the arrow does not produce its effect at once. It continues to act upon the shaft until it resumes its straight position, and then the arrow takes flight with the force accumulated during the continuance of the action of the string, from the moment it was disengaged from the finger of the bow-man.
Fire-arms themselves act upon a similar principle, as also the air-gun and steam-gun. In these instruments the ball is placed in a tube, and suddenly exposed to the pressure of a highly elastic fluid, either produced by explosion as in fire-arms, by previous condensation as in the air-gun, or by the evaporation of highly heated liquids as in the steam-gun. But in every case this pressure continues to act upon it until it leaves the mouth of the tube, and then it departs with the whole force communicated to it during its passage along the tube.
(310.) From all these considerations it will easily be perceived that a mass of inert matter may be regarded as a magazine in which force may be deposited and accumulated, to be used in any way which may be necessary. For many reasons, which will be sufficiently obvious, the form commonly given to the mass of matter used for this purpose in machinery is that of a wheel, in the rim of which it is principally collected. Conceive a massive ring of metal, _fig. 149._, connected with a central box or nave by light spokes, and turning on an axis with little friction. Such an apparatus is called a fly-wheel. If any force be applied to it, with that force (making some slight deduction for friction) it will move, and will continue to move until some obstacle be opposed to its motion, which will receive from it a part of the force it has acquired. The uses of this apparatus will be easily understood by examples of its application.
Suppose that a heavy stamper or hammer is to be raised to a certain height, and thence to be allowed to fall, and that the power used for this purpose is a water-wheel. While the stamper ascends, the power of the wheel is nearly balanced by its weight, and the motion of the machine is slow. But the moment the stamper is disengaged and allowed to fall, the power of the wheel, having no resistance, nor any object on which to expend itself, suddenly accelerates the machine, which moves with a speed proportioned to the amount of the power, until it again engages the stamper, when its velocity is as suddenly checked. Every part suffers a strain, and the machine moves again slowly until it discharges its load, when it is again accelerated, and so on. In this case, besides the certainty of injury and wear, and the probability of fracture from the sudden and frequent changes of velocity, nearly the whole force exerted by the power in the intervals between the commencement of each descent of the stamper and the next ascent is lost. These defects are removed by a fly-wheel. When the stamper is discharged, the energy of the power is expended in moving the wheel, which, by reason of its great mass, will not receive an undue velocity. In the interval between the descent and ascent of the stamper, the force of the power is lodged in the heavy rim of the fly-wheel. When the stamper is again taken up by the machine, this force is brought to bear upon it, combined with the immediate power of the water-wheel, and the stamper is elevated with nearly the same velocity as that with which the machine moved in the interval of its descent.
(311.) In many cases, when the moving power is not subject to variation, the efficacy of the machine to transmit it to the working point is subject to continual change. The several parts of every machine have certain periods of motion, in which they pass through a variety of positions, to which they continually return after stated intervals. In these different positions the effect of the power transmitted to the working point is different; and cases even occur in which this effect is altogether annihilated, and the machine is brought into a predicament in which the power loses all influence over the weight. In such cases the aid of a fly-wheel is effectual and indispensable. In those phases of the machine, which are most favourable to the transmission of force, the fly-wheel shares the effect of the power with the load, and retaining the force thus received directs it upon the load at the moments when the transmission of power by the machine is either feeble or altogether suspended. These general observations will, perhaps, be more clearly apprehended by an example of an application of the fly-wheel, in a case such as those now alluded to.
Let A B C D E F, _fig. 150._, be a _crank_, which is a double winch ((252.) and _fig. 89._), by which an axle, A B E F, is to be turned. Attached to the middle of C D by a joint is a rod, which is connected with a beam, worked with an alternate motion on a centre, like the brake of a pump, and driven by any constant power, such as a steam-engine. The bar C D is to be carried with a circular motion round the axis A E. Let the machine, viewed in the direction A B E F of the axis, be conceived to be represented in _fig. 151._, where A represents the centre round which the motion is to be produced, and G the point where the connecting rod G H is attached to the arm of the crank. The circle through which G is to be urged by the rod is represented by the dotted line. In the position represented in _fig. 151._, the rod acting in the direction H G has its full power to turn the crank G A round the centre A. As the crank comes into the position represented in _fig. 152._, this power is diminished, and when the point G comes immediately below A, as in _fig. 153._, the force in the direction H G has no effect in turning the crank round A, but, on the contrary, is entirely expended in pulling the crank in the direction A G, and, therefore, only acts upon the pivots or gudgeons which support the axle. At this crisis of the motion, therefore, the whole effective energy of the power is annihilated.
After the crank has passed to the position represented in _fig. 154._, the direction of the force which acts upon the connecting rod is changed, and now the crank is drawn upward in the direction G H. In this position the moving force has some efficacy to produce rotation round A, which efficacy continually increases until the crank attains the position shown in _fig. 155._, when its power is greatest. Passing from this position its efficacy is continually diminished, until the point G comes immediately above the axis A, _fig. 156._ Here again the power loses all its efficacy to turn the axle. The force in the direction G H or H G can obviously produce no other effect than a strain upon the pivots or gudgeons.
In the critical situations represented in _fig. 153._, and _fig. 156._, the machine would be incapable of moving, were the immediate force of the power the only impelling principle. But having been previously in motion by virtue of the inertia of its various parts, it has a tendency to continue in motion; and if the resistance of the load and the effects of friction be not too great, this disposition to preserve its state of motion will extricate the machine from the dilemma in which it is involved in the cases just mentioned, by the peculiar arrangement of its parts. In many cases, however, the force thus acquired during the phases of the machine, in which the power is active, is insufficient to carry it through the dead points (_fig. 153._ and _fig. 156._); and in all cases the motion would be very unequal, being continually retarded as it approached these points, and continually accelerated after it passed them. A fly-wheel attached to the axis A, or to some other part of the machinery, will effectually remove this defect. When the crank assumes the positions in _fig. 151._ and _fig. 155._, the power is in full play upon it, and a share of the effect is imparted to the massive rim of the fly-wheel. When the crank gets into the predicament exhibited in _fig. 153._ and _fig. 156._, the momentum which the fly-wheel received when the crank acted with most advantage, immediately extricates the machine, and, carrying the crank beyond the dead point, brings the power again to bear upon it.
The astonishing effects of a fly-wheel, as an accumulator of force, have led some into the error of supposing that such an apparatus increases the actual power of a machine. It is hoped, however, that after what has been explained respecting the inertia of matter and the true effects of machines, the reader will not be liable to a similar mistake. On the contrary, as a fly cannot act without friction, and as the amount of the friction, like that of inertia, is in proportion to the weight, a portion of the actual moving force must unavoidably be lost by the use of a fly. In cases, however, where a fly is properly applied this loss of power is inconsiderable, compared with the advantageous distribution of what remains.
As an accumulator of force, a fly can never have more force than has been applied to put it in motion. In this respect it is analogous to an elastic spring, or the force of condensed air, or any other power which derives its existence from causes purely mechanical. In bending a spring a gradual expenditure of power is necessary. On the recoil this power is exerted in a much shorter time than that consumed in its production, but its total amount is not altered. Air is condensed by a succession of manual efforts, one of which alone would be incapable of projecting a leaden ball with any considerable force, and all of which could not be immediately applied to the ball at the same instant. But the reservoir of condensed air is a magazine in which a great number of such efforts are stored up, so as to be brought at once into action. If a ball be exposed to their effect, it may be projected with a destructive force.
In mills for rolling metal the fly-wheel is used in this way. The water-wheel or other moving power is allowed for some time to act upon the fly-wheel alone, no load being placed upon the machine. A force is thus gained which is sufficient to roll a large piece of metal, to which without such means the mill would be quite inadequate. In the same manner a force may be gained by the arm of a man acting on a fly for a few seconds, sufficient to impress an image on a piece of metal by an instantaneous stroke. The fly is, therefore, the principal agent in coining presses.
(312.) The power of a fly is often transmitted to the working point by means of a screw. At the extremities of the cross arm A B, _fig. 157._, which works the screw, two heavy balls of metal are placed. When the arm A B is whirled round, those masses of metal acquire a momentum, by which the screw, being driven downward, urges the die with an immense force against the substance destined to receive the impression.
Some engines used in coining have flies with arms four feet long, bearing one hundred weight at each of their extremities. By turning such an arm at the rate of one entire circumference in a second, the die will be driven against the metal with the same force as that with which 7500 pounds weight would fall from the height of 16 feet; an enormous power, if the simplicity and compactness of the machine be considered.
The place to be assigned to a fly-wheel relatively to the other parts of the machinery is determined by the purpose for which it is used. If it be intended to equalise the action, it should be near the working point. Thus, in a steam-engine, it is placed on the crank which turns the axle by which the power of the engine is transmitted to the object it is finally designed to affect. On the contrary, in handmills, such as those commonly used for grinding coffee, &c., it is placed upon the axis of the winch by which the machine is worked.
The open work of fenders, fire-grates, and similar ornamental articles constructed in metal, is produced by the action of a fly, in the manner already described. The cutting tool, shaped according to the pattern to be executed, is attached to the end of the screw; and the metal being held in a proper position beneath it, the fly is made to urge the tool downwards with such force as to stamp out pieces of the required figure. When the pattern is complicated, and it is necessary to preserve with exactness the relative situation of its different parts, a number of punches are impelled together, so as to strike the entire piece of metal at the same instant, and in this manner the most elaborate open work is executed by a single stroke.
CHAP. XVIII.
MECHANICAL CONTRIVANCES FOR MODIFYING MOTION.
(313.) The classes of simple machines denominated mechanic powers, have relation chiefly to the peculiar principle which determines the action of the power on the weight or resistance. In explaining this arrangement various other reflections have been incidentally mixed up with our investigations; yet still much remains to be unfolded before the student can form a just notion of those means by which the complex machinery used in the arts and manufactures so effectually attains the ends, to the accomplishment of which it is directed.
By a power of a given energy to oppose a resistance of a different energy, or by a moving principle having a given velocity to generate another velocity of a different amount, is only one of the many objects to be effected by a machine. In the arts and manufactures the _kind_ of motion produced is generally of greater importance than its _rate_. The latter may affect the quantity of work done in a given time, but the former is essential to the performance of the work in any quantity whatever. In the practical application of machines, the object to be attained is generally to communicate to the working point some peculiar sort of motion suitable to the uses for which the machine is intended; but it rarely happens that the moving power has this sort of motion. Hence, the machine must be so contrived that, while that part on which this power acts is capable of moving in obedience to it, its connection with the other parts shall be such that the working point may receive that motion which is necessary for the purposes to which the machine is applied.
To give a perfect solution of this problem it would be necessary to explain, first, all the varieties of moving powers which are at our disposal; secondly, all the variety of motions which it may be necessary to produce; and, thirdly, to show all the methods by which each variety of prime mover may be made to produce the several species of motion in the working point. It is obvious that such an enumeration would be impracticable, and even an approximation to it would be unsuitable to the present treatise. Nevertheless, so much ingenuity has been displayed in many of the contrivances for modifying motion, and an acquaintance with some of them is so essential to a clear comprehension of the nature and operation of complex machines, that it would be improper to omit some account of those at least which most frequently occur in machinery, or which are most conspicuous for elegance and simplicity.
The varieties of motion which most commonly present themselves in the practical application of mechanics may be divided into _rectilinear_ and _rotatory_. In rectilinear motion the several parts of the moving body proceed in parallel straight lines with the same speed. In rotatory motion the several points revolve round an axis, each performing a complete circle, or similar parts of a circle, in the same time.
Each of these may again be resolved into continued and reciprocating. In a continued motion, whether rectilinear or rotatory, the parts move constantly in the same direction, whether that be in parallel straight lines, or in rotation on an axis. In reciprocating motion the several parts move alternately in opposite directions, tracing the same spaces from end to end continually. Thus, there are four principal species of motion which more frequently than any others act upon, or are required to be transmitted by, machines:--
1. _Continued rectilinear motion._
2. _Reciprocating rectilinear motion._
3. _Continued circular motion._
4. _Reciprocating circular motion._
These will be more clearly understood by examples of each kind.
Continued rectilinear motion is observed in the flowing of a river, in a fall of water, in the blowing of the wind, in the motion of an animal upon a straight road, in the perpendicular fall of a heavy body, in the motion of a body down an inclined plane.
Reciprocating rectilinear motion is seen in the piston of a common syringe, in the rod of a common pump, in the hammer of a pavier, the piston of a steam-engine, the stampers of a fulling mill.
Continued circular motion is exhibited in all kinds of wheel-work, and is so common, that to particularise it is needless.
Reciprocating circular motion is seen in the pendulum of a clock, and in the balance-wheel of a watch.
We shall now explain some of the contrivances by which a power having one of these motions may be made to communicate either the same species of motion changed in its velocity or direction, or any of the other three kinds of motion.
(314.) By a continued rectilinear motion another continued rectilinear motion in a different direction may be produced, by one or more fixed pulleys. A cord passed over these, one end of it being moved by the power, will transmit the same motion unchanged to the other end. If the directions of the two motions cross each other, one fixed pulley will be sufficient; see _fig. 113._, where the hand takes the direction of the one motion, and the weight that of the other. In this case the pulley must be placed in the angle at which the directions of the two motions cross each other. If this angle be distant from the places at which the objects in motion are situate, an inconvenient length of rope may be necessary. In this case the same may be effected by the use of two pulleys, as in _fig. 158._
If the directions of the two motions be parallel, two pulleys must be used as in _fig. 158._, where P′ A′ is one motion, and B W the other. In these cases the axles of the two wheels are parallel.
It may so happen that the directions of the two motions neither cross each other nor are parallel. This would happen, for example, if the direction of one were upon the paper in the line P A, while the other were perpendicular to the paper from the point O. In this case two pulleys should be used, the axle of one O′ being perpendicular to the paper, while the axle of the other O should be on the paper. This will be evident by a little reflection.
In general, the axle of each pulley must be perpendicular to the two directions in which the rope passes from its groove; and by due attention to this condition it will be perceived, that a continued rectilinear motion may be transferred from any one direction to any other direction, by means of a cord and two pulleys, without changing its velocity.
If it be necessary to change the velocity, any of the systems of pulleys described in chap. XV. may be used in addition to the fixed pulleys.
By the wheel and axle any one continued rectilinear motion may be made to produce another in any other direction, and with any other velocity. It has been already explained (250.) that the proportion of the velocity of the power to that of the weight is as the diameter of the wheel to the diameter of the axle. The thickness of the axle being therefore regulated in relation to the size of the wheel, so that their diameters shall have that proportion which subsists between the proposed velocities, one condition of the problem will be fulfilled. The rope coiled upon the axle may be carried, by means of one or more fixed pulleys, into the direction of one of the proposed motions, while that which surrounds the wheel is carried into the direction of the other by similar means.
(315.) By the wheel and axle a continued rectilinear motion may be made to produce a continued rotatory motion, or _vice versâ_. If the power be applied by a rope coiled upon the wheel, the continued motion of the power in a straight line will cause the machine to have a rotatory motion. Again, if the weight be applied by a rope coiled upon the axle, a power having a rotatory motion applied to the wheel will cause the continued ascent of the weight in a straight line.
Continued rectilinear and rotatory motions may be made to produce each other, by causing a toothed wheel to work in a straight bar, called a _rack_, carrying teeth upon its edge. Such an apparatus is represented in _fig. 159._
In some cases the teeth of the wheel work in the links of a chain. The wheel is then called a _rag-wheel_, _fig. 160._
Straps, bands, or ropes, may communicate rotation to a wheel, by their friction in a groove upon its edge.
A continued rectilinear motion is produced by a continued circular motion in the case of a screw. The lever which turns the screw has a continued circular motion, while the screw itself advances with a continued rectilinear motion.
The continued rectilinear motion of a stream of water acting upon a wheel produces continued circular motion in the wheel, _fig. 93_, _94_, _95_. In like manner the continued rectilinear motion of the wind produces a continued circular motion in the arms of a windmill.
Cranes for raising and lowering heavy weights convert a circular motion of the power into a continued rectilinear motion of the weight.
(316.) Continued circular motion may produce reciprocating rectilinear motion, by a great variety of ingenious contrivances.
Reciprocating rectilinear motion is used when heavy stampers are to be raised to a certain height, and allowed to fall upon some object placed beneath them. This may be accomplished by a wheel bearing on its edge curved teeth, called _wipers_. The stamper is furnished with a projecting arm or peg, beneath which the wipers are successively brought by the revolution of the wheel. As the wheel revolves the wiper raises the stamper, until its extremity passes the extremity of the projecting arm of the stamper, when the latter immediately falls by its own weight. It is then taken up by the next wiper, and so the process is continued.
A similar effect is produced if the wheel be partially furnished with teeth, and the stamper carry a rack in which these teeth work. Such an apparatus is represented in _fig. 161._
It is sometimes necessary that the reciprocating rectilinear motion shall be performed at a certain varying rate in both directions. This may be accomplished by the machine represented in _fig. 162._ A wheel upon the axle C turns uniformly in the direction A B D E. A rod _mn_ moves in guides, which only permit it to ascend and descend perpendicularly. Its extremity _m_ rests upon a path or groove raised from the face of the wheel, and shaped into such a curve that as the wheel revolves the rod _mn_ shall be moved alternately in opposite directions through the guides, with the required velocity. The manner in which the velocity varies will depend on the form given to the groove or channel raised upon the face of the wheel, and this may be shaped so as to give any variation to the motion of the rod _mn_ which may be required for the purpose to which it is to be applied.
The _rose-engine_ in the turning-lathe is constructed on this principle. It is also used in spinning machinery.
It is often necessary that the rod to which reciprocating motion is communicated shall be urged by the same force in both directions. A wheel partially furnished with teeth, acting on two racks placed on different sides of it, and both connected with the bar or rod to which the reciprocating motion is to be communicated, will accomplish this. Such an apparatus is represented in _fig. 163._, and needs no further explanation.
Another contrivance for the same purpose is shown in _fig. 164._, where A is a wheel turned by a winch H, and connected with a rod or beam moving in guides by the joint _ab_. As the wheel A is turned by the winch H the beam is moved between the guides alternately in opposite directions, the extent of its range being governed by the length of the diameter of the wheel. Such an apparatus is used for grinding and polishing plane surfaces, and also occurs in silk machinery.
An apparatus applied by M. Zureda in a machine for pricking holes in leather is represented in _fig. 165._ The wheel A B has its circumference formed into teeth, the shape of which may be varied according to the circumstances under which it is to be applied. One extremity of the rod _ab_ rests upon the teeth of the wheel upon which it is pressed by a spring at the other extremity. When the wheel revolves, it communicates to this rod a reciprocating rectilinear motion.
Leupold has applied this mechanism to move the pistons of pumps.[5] Upon the vertical axis of a horizontal hydraulic wheel is fixed another horizontal wheel, which is furnished with seven teeth in the manner of a crown wheel (263.). These teeth are shaped like inclined planes, the intervals between them being equal to the length of the planes. Projecting arms attached to the piston rods rest upon the crown of this wheel; and, as it revolves, the inclined surfaces of the teeth, being forced under the arm, raise the rod upon the principle of the wedge. To diminish the obstruction arising from friction, the projecting arms of the piston rods are provided with rollers, which run upon the teeth of the wheel. In one revolution of the wheel each piston makes as many ascents and descents as there are teeth.
[5] Theatrum Machinarum, tom. ii. pl. 36. fig. 3.
(317.) Wheel-work furnishes numerous examples of continued circular motion round one axis, producing continued circular motion round another. If the axles be in parallel directions, and not too distant, rotation may be transmitted from one to the other by two spur-wheels (263.); and the relative velocities may be determined by giving a corresponding proportion to the diameter of the wheels.
If a rotary motion is to be communicated from one axis to another parallel to it, and at any considerable distance, it cannot in practice be accomplished by wheels alone, for their diameters would be too large. In this case a strap or chain is carried round the circumferences of both wheels. If they are intended to turn in the same direction, the strap is arranged as in _fig. 100._; but if in contrary directions it is crossed, as in _fig. 101._ In this case, as with toothed wheels, the relative velocities are determined by the proportion of the diameters of the wheels.
If the axles be distant and not parallel, the cord, by which the motion is transmitted, must be passed over grooved wheels, or fixed pulleys, properly placed between the two axles.
It may happen that the strain upon the wheel, to which the motion is to be transmitted, is too great to allow of a strap or cord being used. In this case, a shaft extending from the one axis to another, and carrying two bevelled wheels (263.), will accomplish the object. One of these bevelled wheels is placed upon the shaft near to, and in connection with, the wheel from which the motion is to be taken, and the other at a part of it near to, and in connection with, that wheel to which the motion is to be conveyed, _fig. 166._
The methods of transmitting rotation from one axis to another perpendicular to it, by crown and by bevelled wheels, have been explained in (263.).
The endless screw (299.) is a machine by which a rotatory motion round one axis may communicate a rotatory motion round another perpendicular to it. The power revolves round an axis coinciding with the length of the screw, and the axis of the wheel driven by the screw is at right angles to this.
The axis to which rotation is to be given, or from which it is to be taken, is sometimes variable in its position. In such cases, an ingenious contrivance, called a _universal joint_, invented by the celebrated Dr. Hooke, may be used. The two shafts or axles A B, _fig. 167._, between which the motion is to be communicated, terminate in semicircles, the diameters of which, C D and E F, are fixed in the form of a cross, their extremities moving freely in bushes placed in the extremities of the semicircles. Thus, while the central cross remains unmoved, the shaft A and its semicircular end may revolve round C D as an axis; and the shaft B and its semicircular end may revolve round E F as an axis. If the shaft A be made to revolve without changing its direction, the points C D will move in a circle whose centre is at the middle of the cross. The motion thus given to the cross will cause the points E F to move in another circle round the same centre, and hence the shaft B will be made to revolve.
This instrument will not transmit the motion if the angle under the directions of the shafts be less than 140°. In this case a double joint, as represented in _fig. 168._, will answer the purpose. This consists of four semicircles united by two crosses, and its principle and operation is the same as in the last case.
Universal joints are of great use in adjusting the position of large telescopes, where, while the observer continues to look through the tube, it is necessary to turn endless screws or wheels whose axes are not in an accessible position.
The cross is not indispensably necessary in the universal joint. A hoop, with four pins projecting from it at four points equally distant from each other, or dividing the circle of the hoop into four equal arcs, will answer the purpose. These pins play in the bushes of the semicircles in the same manner as those of the cross.
The universal joint is much used in cotton-mills, where shafts are carried to a considerable distance from the prime mover, and great advantage is gained by dividing them into convenient lengths, connected by a joint of this kind.
(318.) In the practical application of machinery, it is often necessary to connect a part having a continued circular motion with another which has a reciprocating or alternate motion, so that either may move the other. There are many contrivances by which this may be effected.
One of the most remarkable examples of it is presented in the scapements of watches and clocks. In this case, however, it can scarcely be said with strict propriety that it is the rotation of the scapement-wheel (266.) which _communicates_ the vibration to the balance-wheel or pendulum. That vibration is produced in the one case by the peculiar nature of the spiral spring fixed upon the axis of the balance-wheel, and in the other case by the gravity of the pendulum. The force of the scapement-wheel only _maintains_ the vibration, and prevents its decay by friction and atmospheric resistance. Nevertheless, between the two parts thus moving there exists a mechanical connection, which is generally brought within the class of contrivances now under consideration.
A beam vibrating on an axis, and driven by the piston of a steam-engine, or any other power, may communicate rotary motion to an axis by a connector and a crank. This apparatus has been already described in (311.). Every steam-engine which works by a beam affords an example of this. The working beam is generally placed over the engine, the piston rod being attached to one end of it, while the connecting rod unites the other end with the crank. In boat-engines, however, this position would be inconvenient, requiring more room than could easily be spared. The piston rod, in these cases, is, therefore, connected with the end of the beam by long rods, and the beam is placed beside and below the engine. The use of a fly-wheel here would also be objectionable. The effect of the dead points explained in (311.) is avoided without the aid of a fly, by placing two cranks upon the revolving axle, and working them by two pistons. The cranks are so placed that when either is at its dead point, the other is in its most favourable position.
A wheel A, _fig. 169._, armed with wipers, acting upon a sledge-hammer B, fixed upon a centre or axle C, will, by a continued rotatory motion, give the hammer the reciprocating motion necessary for the purposes to which it is applied. The manner in which this acts must be evident on inspecting the figure.
The treddle of the lathe furnishes an obvious example of a vibrating circular motion producing a continued circular one. The treddle acts upon a crank, which gives motion to the principal wheel, in the same manner as already described in reference to the working beam and crank in the steam-engine.
By the following ingenious mechanism an alternate or vibrating force may be made to communicate a circular motion continually in the same direction. Let A B, _fig. 170._, be an axis receiving an alternate motion from some force applied to it, such as a swinging weight. Two ratchet wheels (253.) _m_ and _n_ are fixed on this axle, their teeth being inclined in opposite directions. Two toothed wheels C and D are likewise placed upon it, but so arranged that they turn upon the axle with a little friction. These wheels carry two catches _p_, _q_, which fall into the teeth of the ratchet wheels _m_, _n_, but fall on opposite sides conformably to the inclination of the teeth already mentioned. The effect of these catches is, that if the axis be made to revolve in one direction, one of the two toothed wheels is always compelled (by the catch _against_ which the motion is directed) to revolve with it, while the other is permitted to remain stationary in obedience to any force sufficiently great to overcome its friction with the axle on which it is placed. The wheels C and D are both engaged by bevelled teeth (263.) with the wheel E.
According to this arrangement, in whichever direction the axis A B is made to revolve, the wheel E will continually turn in the same direction, and, therefore, if the axle A B be made to turn alternately in the one direction and the other, the wheel E will not change the direction of its motion. Let us suppose that the axle A B is turned against the catch _p_. The wheel C will then be made to turn with the axle. This will drive the wheel E in the same direction. The teeth on the opposite side of the wheel E being engaged with those of the wheel D, the latter will be turned upon the axle, the friction, which alone resists its motion in that direction, being overcome. Let the motion of the axle A B be now reversed. Since the teeth of the ratchet wheel _n_ are moved against the catch _q_, the wheel D will be compelled to revolve with the axle. The wheel E will be driven in the same direction as before, and the wheel C will be moved on the axle A B, and in a contrary direction to the motion of the axle, the friction being overcome by the force of the wheel E. Thus, while the axle A B is turned alternately in the one direction and the other, the wheel E is constantly moved in the same direction.
It is evident that the direction in which the wheel E moves may be reversed by changing the position of the ratchet wheels and catches.
(319.) It is often necessary to communicate an alternate circular motion, like that of a pendulum, by means of an alternate motion in a straight line. A remarkable instance of this occurs in the steam engine. The moving force in this machine is the pressure of steam, which impels a piston from end to end alternately in a cylinder. The force of this piston is communicated to the working beam by a strong rod, which passes through a collar in one end of the piston. Since it is necessary that the steam included in the cylinder should not escape between the piston rod and the collar through which it moves, and yet, that it should move as freely and be subject to as little resistance as possible, the rod is turned so as to be truly cylindrical, and is well polished. It is evident that, under these circumstances, it must not be subject to any lateral or cross strain, which would bend it towards one side or the other of the cylinder. But the end of the beam to which it communicates motion, if connected immediately with the rod by a joint, would draw it alternately to the one side and the other, since it moves in the arc of a circle, the centre of which is at the centre of the beam. It is necessary, therefore, to contrive some method of connecting the rod and the end of the beam, so that while the one shall ascend and descend in a straight line, the other may move in the circular arc.
The method which first suggests itself to accomplish this is, to construct an arch-head upon the end of the beam, as in _fig. 171._ Let C be the centre on which the beam works, and let B D be an arch attached to the end of the beam, being a part of a circle having C for its centre. To the highest point B of the arch a chain is attached, which is carried upon the face of the arch B A, and the other end of which is attached to the piston rod. Under these circumstances it is evident, that when the force of the steam impels the piston downwards, the chain P A B will draw the end of the beam down, and will, therefore, elevate the other end.
When the steam-engine is used for certain purposes, such as pumping, this arrangement is sufficient. The piston in that case is not forced upwards by the pressure of steam. During its ascent it is not subject to the action of any force of steam, and the other end of the beam falls by the weight of the pump-rods drawing the piston, at the opposite end A, to the top of the cylinder. Thus the machine is in fact passive during the ascent of the piston, and exerts its power only during the descent.
If the machine, however, be applied to purposes in which a constant action of the moving force is necessary, as is always the case in manufactures, the force of the piston must drive the beam in its ascent as well as in its descent. The arrangement just described cannot effect this; for although a chain is capable of transmitting any force, by which its extremities are drawn in opposite directions, yet it is, from its flexibility, incapable of communicating a force which drives one extremity of it towards the other. In the one case the piston first _pulls_ down the beam, and then the beam _pulls_ up the piston. The chain, because it is inextensible, is perfectly capable of both these actions; and being flexible, it applies itself to the arch-head of the beam, so as to maintain the direction of its force upon the piston continually in the same straight line. But when the piston acts upon the beam in both ways, in pulling it down and pushing it up, the chain becomes inefficient, being from its flexibility incapable of the latter action.
The problem might be solved by extending the length of the piston rod, so that its extremity shall be above the beam, and using two chains; one connecting the highest point of the rod with the lowest point of the arch-head, and the other connecting the highest point of the arch-head with a point on the rod below the point which meets the arch-head when the piston is at the top of the cylinder, _fig. 172._
The connection required may also be made by arming the arch-head with teeth, _fig. 173._, and causing the piston rod to terminate in a rack. In cases where, as in the steam-engine, smoothness of motion is essential, this method is objectionable; and under any circumstances such an apparatus is liable to rapid wear.
The method contrived by Watt, for connecting the motion of the piston with that of the beam, is one of the most ingenious and elegant solutions ever proposed for a mechanical problem. He conceived the motion of two straight rods A B, C D, _fig. 174._, moving on centres or pivots A and C, so that the extremities B and D would move in the arcs of circles having their centres at A and C. The extremities B and D of these rods he conceived to be connected with a third rod B D united with them by pivots on which it could turn freely. To the system of rods thus connected let an alternate motion on the centres A and C be communicated: the points B and D will move upwards and downwards in the arcs expressed by the dotted lines, but the middle point P of the connecting rod B D will move upwards and downwards without any sensible deviation from a straight line.
To prove this demonstratively would require some abstruse mathematical investigation. It may, however, be rendered in some degree apparent by reasoning of a looser and more popular nature. As the point B is raised to E, it is also drawn aside towards the right. At the same time the other extremity D of the rod B D is raised to E′, and is drawn aside towards the left. The ends of the rod B D being thus at the same time drawn equally towards opposite sides, its middle point P will suffer no lateral derangement, and will move directly upwards. On the other hand, if B be moved downwards to F, it will be drawn laterally to the right, while D being moved to F′ will be drawn to the left. Hence, as before, the middle point P sustains no lateral derangement, but merely descends. Thus, as the extremities B and D move upwards and downwards in circles, the middle point P moves upwards and downwards in a straight line.[6]
[6] In a strictly mathematical sense, the path of the point P is a curve, and not a straight line; but in the play given to it in its application to the steam-engine, it moves through a part only of its entire locus, and this part extending equally on each side of a point of inflection, the radius of curvature is infinite, so that in practice the deviation from a straight line, when proper proportions are observed in the rods, is imperceptible.
The application of this geometrical principle in the steam-engine evinces much ingenuity. The same arm of the beam usually works two pistons, that of the cylinder and that of the _air-pump_. The apparatus is represented on the arm of the beam in _fig. 175._ The beam moves alternately upwards and downwards on its axis A. Every point of it, therefore, describes a part of a circle of which A is the centre. Let B be the point which divides the arm A G into two equal parts A B and B G; and let C D be a straight rod, equal in length to G B, and fixed on a centre or pivot C on which it is at liberty to play. The end D of this rod is connected by a straight bar with the point B, by pivots on which the rod B D turns freely. If the beam be now supposed to rise and fall alternately, the points B and D will move upwards and downwards in circular arcs, and, as already explained with respect to the points B D, _fig. 174._, the middle point P of the connecting rod B D will move upwards and downwards without lateral deflection. To this point one of the piston rods which are to be worked is attached.
To comprehend the method of working the other piston, conceive a rod G P′, equal in length to B D, to be attached to the end G of the beam by a pivot on which it moves freely; and let its extremity P′ be connected with D by another rod P′ D, equal in length to G B, and playing on points at P′ and D. The piston rod of the cylinder is attached to the point P′, and this point has a motion precisely similar to that of P, without any lateral derangement, but with a range in the perpendicular direction twice as great. This will be apparent by conceiving a straight line drawn from the centre A of the beam to P′, which will also pass through P. Since G P′ is always parallel to B P, it is evident that the triangle P′ G A is always similar to P B A, and has its sides and angles similarly placed, but those sides are each twice the magnitude of the corresponding sides of the other triangle. Hence the point P′ must be subject to the same changes of position as the point P, with this difference only, that in the same time it moves over a space of twice the magnitude. In fact, the line traced by P′ is the same as that traced by P, but on a scale twice as large. This contrivance is usually called the _parallel motion_, but the same name is generally applied to all contrivances by which a circular motion is made to produce a rectilinear one.
CHAP. XIX.
OF FRICTION AND THE RIGIDITY OF CORDAGE.
(320.) With a view to the simplification of the elementary theory of machines, the consideration of several mechanical effects of great practical importance has been postponed, and the attention of the student has been directed exclusively to the way in which the moving power is modified in being transmitted to the resistance independently of such effects. A machine has been regarded as an instrument by which a moving principle, inapplicable in its existing state to the purpose for which it is required, may be changed either in its velocity or direction, or in some other character, so as to be adapted to that purpose. But in accomplishing this, the several parts of the machine have been considered as possessing in a perfect degree qualities which they enjoy only in an imperfect degree; and accordingly the conclusions to which by such reasoning we are conducted are infected with errors, the amount of which will depend on the degree in which the machinery falls short of perfection in those qualities which theoretically are imputed to it.
Of the several parts of a machine, some are designed to move, while others are fixed; and of those which move, some have motions differing in quantity and direction from those of others. The several parts, whether fixed or movable, are subject to various strains and pressures, which they are intended to resist. These forces not only vary according to the load which the machine has to overcome, but also according to the peculiar form and structure of the machine itself. During the operation the surfaces of the movable parts move in immediate contact with the surfaces either of fixed parts or of parts having other motions. If these surfaces were endued with perfect smoothness or polish, and the several parts subject to strains possessed perfect inflexibility and infinite strength, then the effects of machinery might be practically investigated by the principles already explained. But the materials of which every machine is formed are endued with limited strength, and therefore the load which is placed upon it must be restricted accordingly, else it will be liable to be distorted by the flexure, or even to be destroyed by the fracture of those parts which are submitted to an undue strain. The surfaces of the movable parts, and those surfaces with which they move in contact, cannot in practice be rendered so smooth but that such roughness and inequality will remain as sensibly to impede the motion. To overcome such an impediment requires no inconsiderable part of the moving power. This part is, therefore, intercepted before its arrival at the working point, and the resistance to be finally overcome is deprived of it. The property thus depending on the imperfect smoothness of surfaces, and impeding the motion of bodies whose surfaces are in immediate contact, is called _friction_. Before we can form a just estimate of the effects of machinery, it is necessary to determine the force lost by this impediment, and the laws which under different circumstances regulate that loss.
When cordage is engaged in the formation of any part of a machine, it has hitherto been considered as possessing perfect flexibility. This is not the case in practice; and the want of perfect flexibility, which is called _rigidity_, renders a certain quantity of force necessary to bend a cord or rope over the surface of an axle or the groove of a wheel. During the motion of the rope a different part of it must thus be continually bent, and the force which is expended in producing the necessary flexure must be derived from the moving power, and is thus intercepted on its way to the working point. In calculating the effects of cordage, due regard must be had to this waste of power; and therefore it is necessary to enquire into the laws which govern the flexure of imperfectly flexible ropes, and the way in which these affect the machines in which ropes are commonly used.
To complete, therefore, the elementary theory of machinery, we propose in the present and following chapter to explain the principal laws which determine the effects of friction, the rigidity of cordage, and the strength of materials.
(321.) If a horizontal plane surface were perfectly smooth, and free from the smallest inequalities, and a body having a flat surface also perfectly smooth were placed upon it, any force applied to the latter would put it in motion, and that motion would continue undiminished as long as the body would remain upon the smooth horizontal surface. But if this surface, instead of being every where perfectly even, had in particular places small projecting eminences, a certain quantity of force would be necessary to carry the moving body over these, and a proportional diminution in its rate of motion would ensue. Thus, if such eminences were of frequent occurrence, each would deprive the body of a part of its speed, so that between that and the next it would move with a less velocity than it had between the same and the preceding one. This decrease being continued by a sufficient number of such eminences encountering the body in succession, the velocity would at last be so much diminished that the body would not have sufficient force to carry it over the next eminence, and its motion would thus altogether cease.
Now, instead of the eminences being at a considerable distance asunder, suppose them to be contiguous, and to be spread in every direction over the horizontal plane, and also suppose corresponding eminences to be upon the surface of the moving body; these projections incessantly encountering one another will continually obstruct the motion of the body, and will gradually diminish its velocity, until it be reduced to a state of rest.
Such is the cause of friction. The amount of this resisting force increases with the magnitude of these asperities, or with the roughness of the surfaces; but it does not solely depend on this. The surfaces remaining the same, a little reflection on the method of illustration just adopted, will show that the amount of friction ought also to depend upon the force with which the surfaces moving one upon the other are pressed together. It is evident, that as the weight of the body supposed to move upon the horizontal plane is increased, a proportionally greater force will be necessary to carry it over the obstacles which it encounters, and therefore it will the more speedily be deprived of its velocity and reduced to a state of rest.
(322.) Thus we might predict with probability, that which accurate experimental enquiry proves to be true, that the resistance from friction depends conjointly on the roughness of the surfaces and the force of the pressure. When the surfaces are the same, a double pressure will produce a double amount of friction, a treble pressure a treble amount of friction, and so on.
Experiment also, however, gives a result which, at least at first view, might not have been anticipated from the mode of illustration we have adopted. It is found that the resistance arising from friction does not at all depend on the magnitude of the surface of contact; but provided the nature of the surfaces and the amount of pressure remain the same, this resistance will be equal, whether the surfaces which move one upon the other be great or small. Thus, if the moving body be a flat block of wood, the face of which is equal to a square foot in magnitude, and the edge of which does not exceed a square inch, it will be subject to the same amount of friction, whether it move upon its broad face or upon its narrow edge. If we consider the effect of the pressure in each case, we shall be able to perceive why this must be the case. Let us suppose the weight of the block to be 144 ounces. When it rests upon its face, a pressure to this amount acts upon a surface of 144 square inches, so that a pressure of one ounce acts upon each square inch. The total resistance arising from friction will, therefore, be 144 times that resistance which would be produced by a surface of one square inch under a pressure of one ounce. Now, suppose the block placed upon its edge, there is then a pressure of 144 ounces upon a surface equal to one square inch. But it has been already shown, that when the surface is the same, the friction must increase in proportion to the pressure. Hence we infer that the friction produced in the present case is 144 times the friction which would be produced by a pressure of one ounce acting on one square inch of surface, which is the same resistance as that which the body was proved to be subject to when resting on its face.
These two laws, that friction is independent of the magnitude of the surface, and is proportional to the pressure when the quality of the surfaces is the same, are useful in practice, and _generally_ true. In very extreme cases they are, however, in error. When the pressure is very intense, in proportion to the surface, the friction is somewhat _less_ than it would be by these laws; and when it is very small in proportion to the surface, it is somewhat _greater_.
(323.) There are two methods of establishing by experiment the laws of friction, which have been just explained.
First. The surfaces between which the friction is to be determined being rendered perfectly flat, let one be fixed in the horizontal position on a table T T′, _fig. 176._; and let the other be attached to the bottom of a box B C, adapted to receive weights, so as to vary the pressure. Let a silken cord S P, attached to the box, be carried parallel to the table over a wheel at P, and let a dish D be suspended from it. If no friction existed between the surfaces, the smallest weight appended to the cord would draw the box towards P with a continually increasing speed. But the friction which always exists interrupts this effect, and a small weight may act upon the string without moving the box at all. Let weights be put in the dish D, until a sufficient force is obtained to overcome the friction without giving the box an accelerated motion. Such a weight is equivalent to the amount of the friction.
The amount of the weight of the box being previously ascertained, let this weight be now doubled by placing additional weights in the box. The pressure will thus be doubled, and it will be found that the weight of the dish D and its load, which before was able to overcome the friction, is now altogether inadequate to it. Let additional weights be placed in the dish until the friction be counteracted as before, and it will be observed, that the whole weight necessary to produce this effect is exactly twice the weight which produced it in the former case. Thus it appears that a double amount of pressure produces a double amount of friction; and in a similar way it may be proved, that any proposed increase or decrease of the pressure will be attended with a proportionate variation in the amount of the friction.
Second. Let one of the surfaces be attached to a flat plane A B, _fig. 177._, which can be placed at any inclination with an horizontal plane B C, the other surface being, as before, attached to the box adapted to receive weights. The box being placed upon the plane, let the latter be slightly elevated. The tendency of the box to descend upon A B, will bear the same proportion to its entire weight as the perpendicular A E bears to the length of the plane A B (286.). Thus if the length A B be 36 inches, and the height A E be three inches, that is a twelfth part of the length, then the tendency of the weight to move down the plane is equal to a twelfth part of its whole amount. If the weight were twelve ounces, and the surfaces perfectly smooth, a force of one ounce acting up the plane would be necessary to prevent the descent of the weight.
In this case also the pressure on the plane will be represented by the length of the base B E (286.), that is, it will bear the same proportion to the whole weight as B E bears to B A. The relative amounts of the weight, the tendency to descend, and the pressure, will always be exhibited by the relative lengths of A B, A E, and B E.
This being premised, let the elevation of the plane A B be gradually increased until the tendency of the weight to descend just overcomes the friction, but not so much as to allow the box to descend with accelerated speed. The proportion of the whole weight, which then acts down the plane, will be found by measuring the height A E, and the pressure will be determined by measuring the base B E. Now let the weight in the box be increased, and it will be found that the same elevation is necessary to overcome the friction; nor will this elevation suffer any change, however the pressure or the magnitude of the surfaces which move in contact may be varied.
Since, therefore, in all these cases, the height A E and the base B E remain the same, it follows that the proportion between the friction and pressure is undisturbed.
(324.) The law that friction is proportional to the pressure, has been questioned by the late professor Vince of Cambridge, who deduced from a series of experiments, that although the friction increases with the pressure, yet that it increases in a somewhat less ratio; and from this it would follow, that the variation of the surface of contact must produce some effect upon the amount of friction. The law, as we have explained it, however, is sufficiently near the truth for most practical purposes.
(325.) There are several circumstances regarding the quality of the surfaces which produce important effects on the quantity of friction, and which ought to be noticed here.
This resistance is different in the surfaces of different substances. When the surfaces are those of wood newly planed, it amounts to about half the pressure, but is different in different kinds of wood. The friction of metallic surfaces is about one fourth of the pressure.
In general the friction between the surfaces of bodies of different kinds is less than between those of the same kind. Thus, between wood and metal the friction is about one fifth of the pressure.
It is evident that the smoother the surfaces are the less will be the friction. On this account, the friction of surfaces, when first brought into contact, is often greater than after their attrition has been continued for a certain time, because that process has a tendency to remove and rub off those minute asperities and projections on which the friction depends. But this has a limit, and after a certain quantity of attrition the friction ceases to decrease. Newly planed surfaces of wood have at first a degree of friction which is equal to half the entire pressure, but after they are worn by attrition it is reduced to a third.
If the surfaces in contact be placed with their grains in the same direction, the friction will be greater than if the grains cross each other.
Smearing the surfaces with unctuous matter diminishes the friction, probably by filling the cavities between the minute projections which produce the friction.
When the surfaces are first placed in contact, the friction is less than when they are suffered to rest so for some time; this is proved by observing the force which in each case is necessary to move the one upon the other, that force being less if applied at the first moment of contact than when the contact has continued. This, however, has a limit. There is a certain time, different in different substances, within which this resistance attains its greatest amount. In surfaces of wood this takes place in about two minutes; in metals the time is imperceptibly short; and when a surface of wood is placed upon a surface of metal, it continues to increase for several days. The limit is larger when the surfaces are great, and belong to substances of different kinds.
The velocity with which the surfaces move upon one another produces but little effect upon the friction.
(326.) There are several ways in which bodies may move one upon the other, in which friction will produce different effects. The principal of these are, first, the case where one body _slides_ over another; the second, where a body having a round form _rolls_ upon another; and, _thirdly_, where an axis revolves within a hollow cylinder, or the hollow cylinder revolves upon the axis.
With the same amount of pressure and a like quality of surface, the quantity of friction is greatest in the first case and least in the second. The friction in the second case also depends on the diameter of the body which rolls, and is small in proportion as that diameter is great. Thus a carriage with large wheels is less impeded by the friction of the road than one with small wheels.
In the third case, the leverage of the wheel aids the power in overcoming the friction. Let _fig. 178._ represent a section of the wheel and axle; let C be the centre of the axle, and let B E be the hollow cylinder in the nave of the wheel in which the axle is inserted. If B be the part on which the axle presses, and the wheel turn in the direction N D M, the friction will act at B in the direction B F, and with the leverage B C. The power acts against this at D in the direction D A, and with the leverage D C. It is therefore evident, that as D C is greater than B C, in the same proportion does the power act with mechanical advantage on the friction.
(327.) Contrivances for diminishing the effects of friction depend on the properties just explained, the motion of rolling being as much as possible substituted for that of sliding; and where the motion of rolling cannot be applied, that of a wheel upon its axle is used. In some cases both these motions are combined.
If a heavy load be drawn upon a plane in the manner of a sledge, the motion will be that of sliding, the species which is attended with the greatest quantity of friction; but if the load be placed upon cylindrical rollers, the nature of the motion is changed, and becomes that in which there is the least quantity of friction. Thus large blocks of stone, or heavy beams of timber, which would require an enormous power to move them on a level road, are easily advanced when rollers are put under them.
When very heavy weights are to be moved through small spaces, this method is used with advantage; but when loads ore to be transported to considerable distances, the process is inconvenient and slow, owing to the necessity of continually replacing the rollers in front of the load as they are left behind by its progressive advancement.
The wheels of carriages may be regarded as rollers which are continually carried forward with the load. In addition to the friction of the rolling motion on the road, they have, it is true, the friction of the axle in the nave; but, on the other hand, they are free from the friction of the rollers with the under surface of the load, or the carriage in which the load is transported. The advantages of wheel carriages in diminishing the effects of friction is sometimes attributed to the slowness with which that axle moves within the box, compared with the rate at which the wheel moves over the road; but this is erroneous. The quantity of friction does not in any case vary considerably with the velocity of the motion, but least of all does it in that particular kind of motion here considered.
In certain cases, where it is of great importance to remove the effects of friction, a contrivance called _friction-wheels_, or friction-rollers, is used. The axle of a friction-wheel, instead of revolving within a hollow cylinder, which is fixed, rests upon the edges of wheels which revolve with it; the species of motion thus becomes that in which the friction is of least amount.
Let A B and D C, _fig. 179._, be two wheels revolving on pivots P Q with as little friction as possible, and so placed that the axle O of a third wheel E F may rest between their edges. As the wheel E F revolves, the axle O, instead of grinding its surface on the surface on which it presses, carries that surface with it, causing the wheels A B, C D, to revolve.
In wheel carriages, the roughness of the road is more easily overcome by large wheels than by small ones. The cause of this arises partly from the large wheels not being so liable to sink into holes as small ones, but more because, in surmounting obstacles, the load is elevated less abruptly. This will be easily understood by observing the curves in _fig. 180._, which represent the elevation of the axle in each case.
(328.) If a carriage were capable of moving on a road without friction, the most advantageous direction in which a force could be applied to draw it would be parallel to the road. When the motion is impeded by friction, it is better, however, that the line of draught should be inclined to the road, so that the drawing force may be expended partly in lessening the pressure on the road, and partly in advancing the load.
Let W, _fig. 181._, be a load which is to be moved upon the plane surface A B. If the drawing force be applied in the direction C D, parallel to the plane A B, it will have to overcome the friction produced by the pressure of the whole weight of the load upon the plane; but if it be inclined upwards in the direction C E, it will be equivalent to two forces expressed (74.) by C G and C F. The part C G has the effect of lightening the pressure of the carriage upon the road, and therefore of diminishing the friction in the same proportion. The part C F draws the load along the plane. Since C F is less than C E or C D the whole moving force, it is evident that a part of the force of draught is lost by this obliquity; but, on the other hand, a part of the opposing resistance is also removed. If the latter exceed the former, an advantage will be gained by the obliquity; but if the former exceed the latter, force will be lost.
By mathematical reasoning, founded on these considerations, it is proved that the best angle of draught is exactly that obliquity which should be given to the road in order to enable the carriage to move of itself. This obliquity is sometimes called the _angle of repose_, and is that angle which determines the proportion of the friction to the pressure in the second method, explained in (323.). The more rough the road is, the greater will this angle be; and therefore it follows, that on bad roads the obliquity of the traces to the road should be greater than on good ones. On a smooth Macadamised way, a very slight declivity would cause a carriage to roll by its own weight: hence, in this case, the traces should be nearly parallel to the road.
In rail roads, for like reasons, the line of draught should be parallel to the road, or nearly so.
(329.) When ropes or cords form a part of machinery, the effects of their imperfect flexibility are in a certain degree counteracted by bending them over the grooves of wheels. But although this so far diminishes these effects as to render ropes practically useful, yet still, in calculating the powers of machinery, it is necessary to take into account some consequences of the rigidity of cordage which even by these means are not removed.
To explain the way in which the stiffness of a rope modifies the operation of a machine, we shall suppose it bent over a wheel and stretched by weights A B, _fig. 182._, at its extremities. The weights A and B being equal, and acting at C and D in opposite ways, balance the wheel. If the weight A receive an addition, it will overcome the resistance of B, and turn the wheel in the direction D E C. Now, for the present, let us suppose that the rope is perfectly inflexible; the wheel and weights will be turned into the position represented in _fig. 183._ The leverage by which A acts will be diminished, and will become O F, having been before O C; and the leverage by which B acts will be increased to O G, having been before O D.
But the rope not being inflexible will yield partially to the effects of the weights A and B, and the parts A C and B D will be bent into the forms represented in _fig. 184._ The form of the curvature which the rope on each side of the wheel receives is still such that the descending weight A works with a diminished leverage F O, while the ascending weight resists it with an increased leverage G O. Thus so much of the moving power is lost, by the stiffness of the rope, as is necessary to compensate this disadvantageous change in the power of the machine.
CHAP. XX.
ON THE STRENGTH OF MATERIALS.
(330.) Experimental enquiries into the laws which regulate the strength of solid bodies, or their power to resist forces variously applied to tear or break them, are obstructed by practical difficulties, the nature and extent of which are so discouraging that few have ventured to encounter them at all, and still fewer have had the steadiness to persevere until any result showing a general law has been obtained. These difficulties arise, partly from the great forces which must be applied, but more from the peculiar nature of the objects of those experiments. The end to which such an enquiry must be directed is the development of a _general law_; that is, such a rule as would be rigidly observed if the materials, the strength of which is the object of enquiry, were perfectly uniform in their texture, and subject to no casual inequalities. In proportion as these inequalities are frequent, experiments must be multiplied, that a long average may embrace cases varying in both extremes, so as to eliminate each other’s effects in the final result.
The materials of which structures and works of art are composed are liable to so many and so considerable inequalities of texture, that any rule which can be deduced, even by the most extensive series of experiments, must be regarded as a mean result, from which individual examples will be found to vary in so great a degree, that more than usual caution must be observed in its practical application. The details of this subject belong to engineering, more properly than to the elements of mechanics. Nevertheless, a general view of the most important principles which have been established respecting the strength of materials will not be misplaced in this treatise.
A piece of solid matter may be submitted to the action of a force tending to separate its parts in several ways; the principal of which are,--
1. To a _direct pull_,--as when a rope or wire is stretched by a weight. When a tie-beam resists the separation of the sides of a structure, &c.
2. To a direct pressure or thrust,--as when a weight rests upon a pillar.
3. To a transverse strain,--as when weights on the ends of a lever press it on the fulcrum.
(331.) If a solid be submitted to a force which draws it in the direction of its length, having a tendency to pull its ends in opposite directions, its strength or power to resist such a force is proportional to the magnitude of its transverse section. Thus, suppose a square rod of metal A B, _fig. 185._, of the breadth and thickness of one inch, be pulled by a force in the direction A B, and that a certain force is found sufficient to tear it; a rod of the same metal of twice the breadth and the same thickness will require double the force to break it; one of treble the breadth and the same thickness will require treble the force to break it, and so on.
The reason of this is evident. A rod of double or treble the thickness, in this case, is equivalent to two or three equal and similar rods which equally and separately resist the drawing force, and therefore possess a degree of strength proportionate to their number.
It will easily be perceived, that whatever be the section, the same reasoning will be applicable, and the power of resistance will, in general, be proportional to its magnitude or area.
If the material were perfectly uniform throughout its dimensions, the resistance to a direct pull would not be affected by the length of the rod. In practice, however, the increase of length is found to lessen the strength. This is to be attributed to the increased chance of inequality.
(332.) No satisfactory results have been obtained either by theory or experiment respecting the laws by which solids resist compression. The power of a perpendicular pillar to support a weight placed upon it evidently depends on its thickness, or the magnitude of its base, and on its height. It is certain that when the height is the same, the strength increases with every increase of the base, but it seems doubtful whether the strength be exactly proportional to the base. That is, if two columns of the same material have equal heights, and the base of one be double the base of the other, the strength of one will be greater, but it is not certain whether it will exactly double that of the other. According to the theory of Euler, which is in a certain degree verified by the experiments of Musschenbrock, the strength will be increased in a greater proportion than the base, so that, if the base be doubled, the strength will be more than doubled.
When the base is the same, the strength is diminished by increasing the height, and this decrease of strength is proportionally greater than the increase of height. According to Euler’s theory, the decrease of strength is proportional to the square of the height; that is, when the height is increased in a two-fold proportion, the strength is diminished in a four-fold proportion.
(333.) The strain to which solids forming the parts of structures of every kind are most commonly exposed is the lateral or transverse strain, or that which acts at right angles to their lengths. If any strain act obliquely to the direction of their length it may be resolved into two forces (76.), one in the direction of the length, and the other at right angles to the length. That part which acts in the direction of the length will produce either compression or a direct pull, and its effect must be investigated accordingly.
Although the results of theory, as well as those of experimental investigations, present great discordances respecting the transverse strength of solids, yet there are some particulars, in which they, for the most part, agree; to this it is our object here to confine our observations, declining all details relating to disputed points.
Let A B C D, _fig. 186._, be a beam, supported at its ends A and B. Its strength to support a weight at E pressing downwards at right angles to its length is evidently proportional to its breadth, the other things being the same. For a beam of double or treble breadth, and of the same thickness, is equivalent to two or three equal and similar beams placed side by side. Since each of these would possess the same strength, the whole taken together would possess double or treble the strength of any one of them.
When the breadth and length are the same the strength obviously increases with the depth, but not in the same proportion. The increase of strength is found to be much greater in proportion than the increase of depth. By the theory of Galileo, a double or treble thickness ought to increase the strength in a four-fold or nine-fold proportion, and experiments in most cases do not materially vary from this rule.
If while the breadth and depth remain the same, the length of the beam, or rather, the distance between the points of support, vary, the strength will vary accordingly, decreasing in the same proportion as the length increases.
From these observations it appears, that the transverse strength of a beam depends more on its thickness than its breadth. Hence we find that a broad thin board is much stronger when its edge is presented upwards. On this principle the joists or rafters of floors and roofs are constructed.
If two beams be in all respects similar, their strengths will be in the proportion of the squares of their lengths. Let the length, breadth, and depth of the one be respectively double the length, breadth, and depth of the other. By the double breadth the beam doubles its strength, but by doubling the length half this strength is lost. Thus the increase of length and breadth counteract each other’s effects, and as far as they are concerned the strength of the beam is not changed. But by doubling the thickness the strength is increased in a four-fold proportion, that is, as the square of the length. In the same manner it may be shown, that when all the dimensions are trebled, the strength is increased in a nine-fold proportion, and so on.
(334.) In all structures the materials have to support their own weight, and therefore their available strength is to be estimated by the excess of their absolute strength above that degree of strength which is just sufficient to support their own weight. This consideration leads to some conclusions, of which numerous and striking illustrations are presented in the works of nature and art.
We have seen that the absolute strength with which a lateral strain is resisted is in the proportion of the square of the linear dimensions of similar parts of a structure, and therefore the amount of this strength increases rapidly with every increase of the dimensions of a body. But at the same time the weight of the body increases in a still more rapid proportion. Thus, if the several dimensions be doubled, the strength will be increased in a four-fold but the weight in an eight-fold proportion. If the dimensions be trebled, the strength will be multiplied nine times, but the weight twenty-seven times. Again, if the dimensions be multiplied four times, the strength will be multiplied sixteen times, and the weight sixty-four times, and so on.
Hence it is obvious, that although the strength of a body of small dimensions may greatly exceed its weight, and, therefore, it may be able to support a load many times its own weight; yet by a great increase in the dimensions the weight increasing in a much greater degree the available strength may be much diminished, and such a magnitude may be assigned, that the weight of the body must exceed its strength, and it not only would be unable to support any load, but would actually fall to pieces by its own weight.
The strength of a structure of any kind is not, therefore, to be determined by that of its model, which will always be much stronger in proportion to its size. All works natural and artificial have limits of magnitude which, while their materials remain the same, they cannot surpass.
In conformity with what has just been explained, it has been observed, that small animals are stronger in proportion than large ones; that the young plant has more available strength in proportion than the large forest tree; that children are less liable to injury from accident than men, &c. But although to a certain extent these observations are just, yet it ought not to be forgotten, that the mechanical conclusions which they are brought to illustrate are founded on the supposition, that the smaller and greater bodies which are compared are composed of precisely similar materials. This is not the case in any of the examples here adduced.
CHAP. XXI.
ON BALANCES AND PENDULUMS.
(335.) The preceding chapters have been confined almost wholly to the consideration of the laws of mechanics, without entering into a particular description of the machinery and instruments dependant upon those laws. Such descriptions would have interfered too much with the regular progress of the subject, and it therefore appeared preferable to devote a chapter exclusively to this portion of the work.
Perhaps there are no ideas which man receives through the medium of sense which may not be referred ultimately to matter and motion. In proportion, therefore, as he becomes acquainted with the properties of the one and the laws of the other, his knowledge is extended, his comforts are multiplied; he is enabled to bend the powers of nature to his will, and to construct machinery which effects with ease that which the united labour of thousands would in vain be exerted to accomplish.
Of the properties of matter, one of the most important is its weight, and the element which mingles inseparably with the laws of motion is time.
In the present chapter it is our intention to describe such instruments as are usually employed for determining the weight of bodies. To attempt a description of the various machines which are used for the measurement of time, would lead us into too wide a field for the present occasion, and we shall, therefore, confine ourselves to an account of the methods which have been practised to perfect, to perfect that instrument which affords the most correct means of measuring time, the pendulum.
The instrument by which we are enabled to determine, with greater accuracy than by any other means, the relative weight of a body, compared with the weight of another body assumed as a standard, is the balance.
_Of the Balance._
The balance may be described as consisting of an inflexible rod or lever, called the beam, furnished with three axes; one, the fulcrum or centre of motion situated in the middle, upon which the beam turns, and the other two near the extremities, and at equal distances from the middle. These last are called the points of support, and serve to sustain the pans or scales.
The points of support and the fulcrum are in the same right line, and the centre of gravity of the whole should be a little below the fulcrum when the position of the beam is horizontal.
The arms of the lever being equal, it follows that if equal weights be put into the scales no effect will be produced on the position of the balance, and the beam will remain horizontal.
If a small addition be made to the weight in one of the scales, the horizontality of the beam will be disturbed; and after oscillating for some time, it will, on attaining a state of rest, form an angle with the horizon, the extent of which is a measure of the delicacy or sensibility of the balance.
As the sensibility of a balance is of the utmost importance in nice scientific enquiries, we shall enter somewhat at large into a consideration of the circumstances by which this property is influenced.
In _fig. 187._ let A B represent the beam drawn from the horizontal position by a very small weight placed in the scale suspended from the point of support B; then the force tending to draw the beam from the horizontal position may be expressed by P B, multiplied by such very small weight acting upon the point B.
Let the centre of gravity of the whole be at G; then the force acting against the former will be G P multiplied into the weight of the beam and scales, and when these forces are equal, the beam will rest in an inclined position. Hence we may perceive that as the centre of gravity is nearer to or further from the fulcrum S, (every thing else remaining the same) the sensibility of the balance will be increased or diminished.
For, suppose the centre of gravity were removed to _g_, then to produce an opposing force equal to that acting upon the extremity of the beam, the distance _g p_ from the perpendicular line must be increased until it becomes nearly equal to G P; but for this purpose the end of the beam B must descend, which will increase the angle H S B.
As all weights placed in the scales are referred to the line joining the points of support, and as this line is above the centre of gravity of the beam when not loaded, such weights will raise the centre of gravity; but it will be seen that the sensibility of the balance, as far as it depends upon this cause, will remain unaltered.
For, calling the distance S G unity, the distance of the centre of gravity from the point S (to which the weight which has been added is referred) will be expressed by the reciprocal of the weight of the beam so increased; that is, if the weight of the beam be doubled by weights placed in the scales, S _g_ will be one half of S G; and if the weight of the beam be in like manner trebled, S _g_ will be one third of S G, and so on. And as G P varies as S G, _g p_ will be inversely proportionate to the increased weight of the beam, and consequently, the product obtained by multiplying _g p_ by the weight of the beam and its load will be a constant quantity, and the sensibility of the balance, as before stated, will suffer no alteration.
We will now suppose that the fulcrum S, _fig. 188._, is situated below the line joining the points of support, and that the centre of gravity of the beam when not loaded is at G. Also that when a very small weight is placed in the scale suspended from the point B, the beam is drawn from its horizontal position, the deviation being a measure of the sensibility of the balance. Then, as before stated, G P multiplied by the weight of the beam will be equal to P′ B multiplied by the very small additional weight acting on the point B.
Now if we place equal weights in both scales, such additional weights will be referred to the point W, and the resulting distance of the centre of gravity from the point W, calling W G unity, will be expressed as before by the reciprocal of the increased weight of the loaded beam. But G P will decrease in a greater proportion than W G: thus, supposing the weight of the beam to be doubled, W _g_ would be one half of W G; but _g p_, as will be evident on an inspection of the figure, will be less than half of G P; and the same small weight which was before applied to the point B, if now added, would depress the point B, until the distance _g p_ became such as that, when multiplied by the weight of the whole, the product would be as before equal to P′ B, multiplied by the before mentioned very small added weight. The sensibility of the balance, therefore, in this case would be increased.
If the beam be sufficiently loaded, the centre of gravity will at length be raised to the fulcrum S, and the beam will rest indifferently in any position. If more weight be then added, the centre of gravity will be raised above the fulcrum, and the beam will turn over.
Lastly, if the fulcrum S, _fig. 189._, is above the line joining the two points of support, as any additional weights placed in the scales will be referred to the point W, in the line joining A and B, if the weight of the beam be doubled by such added weights, and the centre of gravity be consequently raised to _g_, W _g_ will become equal to half of W G. But _g p_, being greater than one half of G P, the end of the beam B will rise until _g p_ becomes such as to be equal, when multiplied by the whole increased weight of the beam, to P B, multiplied by the small weight, which we suppose to have been placed as in the preceding examples, in the scale.
From what has been said it will be seen that there are three positions of the fulcrum which influence the sensibility of the balance: first, when the fulcrum and the points of support are in a right line, when the sensibility of the balance will remain the same, though the weight with which the beam is loaded should be varied: secondly, when the fulcrum is below the line joining the two points of support, in which case the sensibility of the balance will be increased by additional weights, until at length the centre of gravity is raised above the fulcrum, when the beam will turn over; and, thirdly, when the fulcrum is above the line joining the two points of support, in which case the sensibility of the balance will be diminished as the weight with which the beam is loaded is increased.
The sensibility of a balance, as here defined, is the angular deviation of the beam occasioned by placing an additional constant small weight in one of the scales; but it is frequently expressed by the proportion which such small additional weight bears to the weight of the beam and its load, and sometimes to the weight the value of which is to be determined.
This proportion, however, will evidently vary with different weights, except in the case where the centre of gravity of the beam is in the line joining the points supporting the scales, the fulcrum being above this line, and it is therefore necessary, in every other case, when speaking of the sensibility of the balance, to designate the weight with which it is loaded: thus, if a balance has a troy pound in each scale, and the horizontality of the beam varies a certain small quantity, just perceptible on the addition of one hundredth of a grain, we say that the balance is sensible to 1/1152000 part of its load with a pound in each scale, or that it will determine the weight of a troy pound within 1/576000 part of the whole.
The nearer the centre of gravity of a balance is to its fulcrum the slower will be the oscillations of the beam. The number of oscillations, therefore, made by the beam in a given time (a minute for example), affords the most accurate method of judging of the sensibility of the balance, which will be the greater as the oscillations are fewer.
Balances of the most perfect kind, and of such only it is our present object to treat, are usually furnished with adjustments, by means of which the length of the arms, or the distances of the fulcrum from the points of support, may be equalised, and the fulcrum and the two points of support be placed in a right line; but these adjustments, as will hereafter be seen, are not absolutely necessary.
The beam is variously constructed, according to the purposes to which the balance is to be applied. Sometimes it is made of a rod of solid steel; sometimes of two hollow cones joined at their bases; and, in some balances, the beam is a frame in the form of a rhombus: the principal object in all, however, is to combine strength and inflexibility with lightness.
A balance of the best kind, made by Troughton, is so contrived as to be contained, when not in use, in a drawer below the case; and when in use, it is protected from any disturbance from currents of air, by being enclosed in the case above the drawer, the back and front of which are of plate glass. There are doors in the sides, through which the scale-pans are loaded, and there is a door at the top through which the beam may be taken out.
A strong brass pillar, in the centre of the box, supports a square piece, on the front and back of which rise two arches, nearly semicircular, on which are fixed two horizontal planes of agate, intended to support the fulcrum. Within the pillar is a cylindrical tube, which slides up and down by means of a handle on the outside of the case. To the top of this interior tube is fixed an arch, the terminations of which pass beneath and outside of the two arches before described. These terminations are formed into Y _s_, destined to receive the ends of the fulcrum, which are made cylindrical for this purpose, when the interior tube is elevated in order to relieve the axis when the balance is not in use. On depressing the interior tube, the Y _s_ quit the axis, and leave it in its proper position on the agate planes. The beam is about eighteen inches long, and is formed of two hollow cones of brass, joined at their bases. The thickness of the brass does not exceed 0·02 of an inch, but by means of circular rings driven into the cones at intervals they are rendered almost inflexible. Across the middle of the beam passes a cylinder of steel, the lower side of which is formed into an edge, having an angle of about thirty degrees, which, being hardened and well polished, constitutes the fulcrum, and rests upon the agate planes for the length of about 0·05 of an inch.
Each point of suspension is formed of an axis having two sharp concave edges, upon which rest at right angles two other sharp concave edges formed in the spur-shaped piece to which the strings carrying the scale-pan are attached. The two points are adjustable, the one horizontally, for the purpose of equalising the arms of the beam, and the other vertically, for bringing the points of suspension and the fulcrum into a right line.
Such is the form of Troughton’s balance: we shall now give the description of a balance as constructed by Mr. Robinson of Devonshire Street, Portland Place:--
The beam of this balance is only ten inches long. It is a frame of bell-metal in the form of a rhombus. The fulcrum is an equilateral triangular prism of steel one inch in length; but the edge on which the beam vibrates is formed to an angle of 120°, in order to prevent any injury from the weight with which it may be loaded. The chief peculiarity in this balance consists in the knife-edge which forms the fulcrum bearing upon an agate plane throughout its whole length, whereas we have seen in the balance before described that the whole weight is supported by portions only of the knife-edge, amounting together to one tenth of an inch. The supports for the scales are knife-edges each six tenths of an inch long. These are each furnished with two pressing screws, by means of which they may be made parallel to the central knife-edge.
Each end of the beam is sprung obliquely upwards and towards the middle, so as to form a spring through which a pushing screw passes, which serves to vary the distance of the point of support from the fulcrum, and, at the same time, by its oblique action to raise or depress it, so as to furnish a means of bringing the points of support and the fulcrum into a right line.
A piece of wire, four inches long, on which a screw is cut, proceeds from the middle of the beam downwards. This is pointed to serve as an index, and a small brass ball moves on the screw, by changing the situation of which the place of the centre of gravity may be varied at pleasure.
The fulcrum, as before remarked, rests upon an agate plane throughout its whole length, and the scale-pans are attached to planes of agate which rest upon the knife-edges forming the points of support. This method of supporting the scale-pans, we have reason to believe, is due to Mr. Cavendish. Upon the lower half of the pillar to which the agate plane is fixed, a tube slides up and down by means of a lever which passes to the outside of the case. From the top of this tube arms proceed obliquely towards the ends of the balance, serving to support a horizontal piece, carrying at each extremity two sets of Y _s_, one a little above the other. The upper Y _s_ are destined to receive the agate planes to which the scale-pans are attached, and thus to relieve the knife-edges from their pressure; the lower to receive the knife-edges which, form the points of support, consequently these latter Y _s_, when in action, sustain the whole beam.
When the lever is freed from a notch in which it is lodged, a spring is allowed to act upon the tube we have mentioned, and to elevate it. The upper Y _s_ first meet the agate planes carrying the scale-pans and free them from the knife-edges. The lower Y _s_ then come into action and raise the whole beam, elevating the central knife-edge above the agate plane. This is the usual state of the balance when not in use: when it is to be brought into action, the reverse of what we have described takes place. On pressing down the lever, the central knife-edge first meets the agate plane, and afterwards the two agate planes carrying the scale-pans are deposited upon their supporting knife-edges.
A balance of this construction was employed by the writer of this article in adjusting the national standard pound. With a pound troy in each scale, the addition of one hundredth of a grain caused the index to vary one division, equal to one tenth of an inch, and Mr. Robinson adjusts these balances so that with one thousand grains in each scale, the index varies perceptibly on the addition of one thousandth of a grain, or of one-millionth part of the weight to be determined.
It may not be uninteresting to subjoin, from the Philosophical Transactions for 1826, the description of a balance perhaps the most sensible that has yet been made, constructed for verifying the national standard bushel. The author says,--
“The weight of the bushel measure, together with the 80 lbs. of water it should contain, was about 250 lbs.; and as I could find no balance capable of determining so large a weight with sufficient accuracy, I was under the necessity of constructing one for this express purpose.
“I first tried cast iron; but though the beam was made as light as was consistent with the requisite degree of strength, the inertia of such a mass appeared to be so considerable, that much time must have been lost before the balance would have answered to the small differences I wished to ascertain. Lightness was a property essentially necessary, and bulk was very desirable, in order to preclude such errors as might arise from the beam being partially affected by sudden alterations of temperature. I therefore determined to employ wood, a material in which the requisites I sought were combined. The beam was made of a plank of mahogany, about 7O inches long, 22 inches wide, and 2-1/4 thick, tapering from the middle to the extremities. An opening was cut in the centre, and strong blocks screwed to each side of the plank, to form a bearing for the back of a knife-edge which passed through the centre. Blocks were also screwed to each side at the extremities of the beam on which rested the backs of the knife-edges for supporting the pans. The opening in the centre was made sufficiently large to admit the support hereafter to be described, upon which the knife-edge rested.
“In all beams which I have seen, with the exception of those made by Mr. Robinson, the whole weight is sustained by short portions at the extremities of the knife-edge; and the weight being thus thrown upon a few points, the knife-edge becomes more liable to change its figure and to suffer injury.
“To remedy this defect, the central knife-edge of the beam I am describing was made 6 inches, and the two others 5 inches long. They were triangular prisms with equal sides of three fourths of an inch, very carefully finished, and the edges ultimately formed to an angle of 120°.
“Each knife-edge was screwed to a thick plate of brass, the surfaces in contact having been previously ground together; and these plates were screwed to the beam, the knife-edges being placed in the same plane, and as nearly equidistant and parallel to each other as could be done by construction.
“The support upon which the central knife-edge rested throughout its whole length was formed of a plate of polished hard steel, screwed to a block of cast iron. This block was passed through the opening before mentioned in the centre of the beam, and properly attached to a frame of cast iron.
“The stirrups to which the scales were hooked rested upon plates of polished steel to which they were attached, and the under surfaces of which were formed by careful grinding into cylindrical segments. These were in contact with the knife-edges their whole length, and were known to be in their proper position by the correspondence of their extremities with those of the knife-edges. A well imagined contrivance was applied by Mr. Bate for raising the beam when loaded, in order to prevent unnecessary wear of the knife-edge, and for the purpose of adjusting the place of the centre of gravity, when the beam was loaded with the weight required to be determined, a screw carrying a movable ball projected vertically from the middle of die beam.
“The performance of this balance fully equalled my expectations. With two hundred and fifty pounds in each scale, the addition of a single grain occasioned an immediate variation in the index of one twentieth of an inch, the radius being fifty inches.”
From the preceding account it appears that this balance is sensible to 1/1750000 part of the weight which was to be determined.
We shall now describe the method to be pursued in adjusting a balance.
1. To bring the points of suspension and the fulcrum into a right line.
Make the vibrations of the balance very slow by moving the weight which influences the centre of gravity, and bring the beam into a horizontal position, by means of small bits of paper thrown into the scales. Then load the scales with nearly the greatest weight the beam is fitted to carry. If the vibrations are performed in the same time as before, no further adjustment is necessary; but if the beam vibrates quicker, or if it oversets, cause it to vibrate in the same time as at first, by moving the adjusting weight, and note the distance through which the weight has passed. Move the weight then in the contrary direction through double this distance, and then produce the former slow motion by means of the screw acting vertically on the point of support. Repeat this operation until the adjustment is perfect.
2. To make the arms of the beam of an equal length.
Put weights in the scales as before; bring the beam as nearly as possible to a horizontal position, and note the division at which the index stands; unhook the scales, and transfer them with their weights to the other ends of the beam, when, if the index points to the same division, the arms are of an equal length; but if not, bring the index to the division which had been noted, by placing small weights in one or the other scale. Take away half these weights, and bring the index again to the observed division by the adjusting screw, which acts horizontally on the point of support. If the scale-pans are known to be of the same weight, it will not be necessary to change the scales, but merely to transfer the weights from one scale-pan to the other.
_Of the Use of the Balance._
Though we have described the method of adjusting the balance, these adjustments, as we have before remarked, may be dispensed with. Indeed, in all delicate scientific operations, it is advisable never to rely upon adjustments, which, after every care has been employed in effecting them, can only be considered as approximations to the truth. We shall, therefore, now describe the best method of ascertaining the weight of a body, and which does not depend on the accuracy of these adjustments.
Having levelled the case which contains the balance, and thrown the beam out of action, place a weight in each scale-pan nearly equal to the weight which is to be determined. Lower the beam very gently till it is in action, and by means of the adjustment for raising or lowering the centre of gravity, cause the beam to vibrate very slowly. Remove these weights, and place the substance, the weight of which is to be determined in one of the scale-pans; carefully counterpoise it by means of any convenient substances put into the other scale-pan, and observe the division at which the index stands; remove the body, the weight of which is to be ascertained, and substitute standard weights for it so as to bring the index to the same division as before. These weights will be equal to the weight of the body.
If it be required to compare two weights together which are intended to be equal, and to ascertain their difference, if any, the method of proceeding will be nearly the same. The standard weight is to be carefully counterpoised, and the division at which the index stands, noted. And now it will be convenient to add in either of the scales some small weight, such as one or two hundredths of a grain, and mark the number of divisions passed over in consequence by the index, by which the value of one division of the scale will be known. This should be repeated a few times, and the mean taken for greater certainty.
Having noted the division at which the index rests, the standard weight is to be removed, and the weight which is to be compared with it substituted for it. The index is then again to be noted, and the difference between this and the former indication will give the difference between the weights in parts of a grain.
If the balance is adjusted so as to be very sensible, it will be long before it comes to a state of rest. It may, therefore, sometimes be advisable to take the mean of the extent of the vibrations of the index as the point where it would rest, and this may be repeated several times for greater accuracy. It must, however, be remembered, that it is not safe to do this when the extent of the vibrations is beyond one or two divisions of the scale; but with this limitation it is, perhaps, as good a method as can be pursued.
Many precautions are necessary to ensure a satisfactory result. The weights should never be touched by the hand; for not only would this oxydate the weight, but by raising its temperature it would appear lighter, when placed in the scale-pan, than it should do, in consequence of the ascent of the heated air. For the larger weights a wooden fork or tongs, according to the form of the weight, should be employed; and for the smaller, a pair of forceps made of copper will be found the most convenient. This metal possessing sufficient elasticity to open the forceps on their being released from pressure, and yet not opposing a resistance sufficient to interfere with that delicacy of touch which is desirable in such operations.
_Of Weights._
It must be obvious, that the excellence of the balance would be of little use, unless the weights employed were equally to be depended upon. The weights may either be accurately adjusted, or the difference between each weight and the standard may be determined, and, consequently, its true value ascertained. It has been already shown how the latter may be effected, in the instructions which have been given for comparing two weights together; and we shall now show the readiest mode of adjusting weights to an exact equality with a given standard.
The material of the weight may be either brass or platina, and its form may be cylindrical: the diameter being nearly twice the height. A small spherical knob is screwed into the centre, a space being left under the screw to receive the portions of fine wire used in the adjustment. It will be convenient to form a cavity in the bottom of each weight to receive the knob of the weight upon which it may be placed.
Each weight is now to be compared with the standard, and should it be too heavy, it is to be reduced till it becomes in a very small degree too light, when the amount of the deficiency is to be carefully determined.
Some very fine silver wire is now to be taken, and the weight of three or four feet of it ascertained. From this it will be known what length of the wire is equal to the error of the weight to be adjusted; and this length being cut off is to be enclosed under the screw. To guard against any possible error, it will be advisable before the screw is firmly fixed in its place, again to compare the weight with the standard.
The most approved method of making weights expressing the decimal parts of a grain, is to determine, as before, with great care, the weight of a certain length of fine wire, and then to cut off such portions as are equal to the weights required.
Before we conclude this article we shall give a description, from the Annals of Philosophy for 1825, of “a very sensible balance,” used by the late Dr. Black:--
“A thin piece of fir wood, not thicker than a shilling, and a foot long, three tenths of an inch broad in the middle, and one tenth and a half at each end, is divided by transverse lines into twenty parts; that is, ten parts on each side of the middle. These are the principal divisions, and each of them is subdivided into halves and quarters. Across the middle is fixed one of the smallest needles I could procure, to serve as an axis, and it is fixed in its place by means of a little sealing wax. The numeration of the divisions is from the middle to each end of the beam. The fulcrum is a bit of plate brass, the middle of which lies flat on my table when I use the balance, and the two ends are bent up to a right angle so as to stand upright. These two ends are ground at the same time on a flat hone, that the extreme surfaces of them may be in the same plane; and their distance is such that the needle, when laid across them, rests on them at a small distance from the sides of the beam. They rise above the surface of the table only one tenth and a half or two tenths of an inch, so that the beam is very limited in its play. See _fig. 190._
“The weights I use are one globule of gold, which weighs one grain, and two or three others which weigh one tenth of a grain each; and also a number of small rings of fine brass wire, made in the manner first mentioned by Mr. Lewis, by appending a weight to the wire, and coiling it with the tension of that weight round a thicker brass wire in a close spiral, after which, the extremity of the spiral being tied hard with waxed thread, I put the covered wire into a vice, and applying a sharp knife, which is struck with a hammer, I cut through a great number of the coils at one stroke, and find them as exactly equal to one another as can be desired. Those I use happen to be the 1/30 part of a grain each, or 300 of them weigh ten grains; but I have others much lighter.
“You will perceive that by means of these weights placed on different parts of the beam, I can learn the weight of any little mass from one grain, or a little more, to the 1/1200 of a grain. For if the thing to be weighed weighs one grain, it will, when placed on one extremity of the beam, counterpoise the large gold weight at the other extremity. If it weighs half a grain it will counterpoise the heavy gold weight placed at 5. If it weigh 6/10 of a grain, you must place the heavy gold weight at 5, and one of the lighter ones at the extremity to counterpoise it, and if it weighs only one or two, or three or four hundredths of a grain, it will be counterpoised by one of the small gold weights placed at the first or second, or third or fourth division. If, on the contrary, it weighs one grain and a fraction, it will be counterpoised by the heavy gold weight at the extremity, and one or more of the lighter ones placed in some other part of the beam.
“This beam has served me hitherto for every purpose; but had I occasion for a more delicate one, I could make it easily by taking a much thinner and lighter slip of wood, and grinding the needle to give it an edge. It would also be easy to make it carry small scales of paper for particular purposes.”
The writer of this article has used a balance of this kind, and finds that it is sensible to 1/1000 of a grain when loaded with ten grains. It is necessary, however, where accuracy is required, to employ a scale-pan. This may be made of thin card paper, shaped as in _fig. 191._
A thread is to be passed through the two ends, by tightening which they may be brought near each other.
The most convenient weights for this beam appear to be two of one grain each, and one of one tenth of a grain. They should be made of straight wire; and if the beam be notched at the divisions, they may be lodged in these notches very conveniently. Ten divisions on each side of the middle will be sufficient. The weight of the scale-pan must first be carefully ascertained, in order that it may be deducted from the weight, afterwards determined, of the scale-pan and the substance it may contain.
If the scale-pan be placed at the tenth division of the beam, it is evident that by means of the two grain weights, a greater weight cannot be determined than one grain and nine tenths; but if the scale-pan be placed at any other division of the beam, the resulting apparent weight must be increased by multiplying it by ten, and dividing by the number of the division at which the scale-pan is placed; and in this manner it is evident that if the scale-pan be placed at the division numbered 1, a weight amounting to nineteen grains may be determined.
We have been tempted to describe this little apparatus, because it is extremely simple in its construction, may be easily made, and may be very usefully employed on many occasions where extreme accuracy is not necessary.
_Description of the Steelyard._
The steelyard is a lever, having unequal arms; and in its most simple form it is so arranged, that one weight alone serves to determine a great variety of others, by sliding it along the longer arm of the lever, and thus varying its distance from the fulcrum.
It has been demonstrated, chapter xiii., that in the lever the proportion of the power to the weight will be always the same as that of their distances from the fulcrum, taken in a reverse order; consequently, when a constant weight is used, and an equilibrium established by sliding this weight on the longer arm of the lever, the relative weight of the substance weighed, to the constant weight, will be in the same proportion as the distance of the constant weight from the fulcrum is to the length of the shorter arm.
Thus, suppose the length of the shorter arm, or the distance of the fulcrum from the point from which the weight to be determined is suspended, to be one inch; let the longer arm of the lever be divided into parts of one inch each, beginning at the fulcrum. Now let the constant weight be equal to one pound, and let the steelyard be so constructed that the shorter arm shall be sufficiently heavy to counterpoise the longer when the bar is unloaded. Then suppose a substance, the weight of which is five pounds, to be suspended from the shorter arm. It will be found that when the constant weight is placed at the distance of five inches from the fulcrum, the weights will be in equilibrium, and the bar consequently horizontal. In this steelyard, therefore, the distance of each inch from the fulcrum indicates a weight of one pound. An instrument of this form was used by the Romans, and it is usually described as the Roman statera or steelyard. A representation of it is given at _fig. 192._
The steelyard is in very general use for the coarser purposes of commerce, but constructed differently from that which we have described. The beam with the scales or hooks is seldom in equilibrium upon the point F, when the weight P is removed; but the longer arm usually preponderates, and the commencement of the graduations, therefore, is not at F, but at some point between B and F. The common steelyard, which we have represented at _fig. 193._, is usually furnished with two points, from either of which the substance, the weight of which is to be determined, may be suspended. The value of the divisions is in this case increased in proportion as the length of the shorter arm is decreased. Thus, in the steelyard which we have described, if there be a second point of suspension at the distance of half an inch from the fulcrum, each division of the longer arm will indicate two pounds instead of one, and these divisions are usually marked upon the opposite edge of the steelyard, which is made to turn over.
This instrument is very convenient, because it requires but one weight; and the pressure on the fulcrum is less than in the balance, when the substance to be weighed is heavier than the constant weight. But, on the contrary, when the constant weight exceeds the substance to be weighed, the pressure on the fulcrum is greater in the steelyard than in the balance, and the balance is, therefore, preferable in determining small weights. There is also an advantage in the balance, because the subdivision of weights can be effected with a greater degree of precision than the subdivision of the arm of the steelyard.
_C. Paul’s Steelyard._
A steelyard has been constructed by Mr. C. Paul, inspector of weights and measures at Geneva, which is much to be preferred to that in common use. Mr. C. Paul states, that steelyards have two advantages over balances: 1. That their axis of suspension is not loaded with any other weight than that of the merchandise, the constant weight of the apparatus itself excepted; while the axis of the balance, besides the weight of the instrument, sustains a weight double to that of the merchandise. 2. The use of the balance requires a considerable assortment of weights, which causes a proportional increase in the price of the apparatus, independently of the chances of error which it multiplies, and of the time employed in producing an equilibrium.
1. In C. Paul’s steelyard the centres of the movement of suspension, or the two constant centres, are placed on the exact line of the divisions of the beam; an elevation almost imperceptible in the axis of the beam, destined to compensate for the very slight flexion of the bar, alone excepted.
2. The apparatus, by the construction of the beam, is balanced below its centre of motion, so that when no weight is suspended the beam naturally remains horizontal, and resumes that position when removed from it, as also when the steelyard is loaded, and the weight is at the division which ought to show how much the merchandise weighs. The horizontal situation in this steelyard, as well as in the others, is known by means of the tongue which rises vertically above the axis of suspension.
3. It may be discovered, that the steelyard is deranged if, when not loaded, the beam does not remain horizontal.
4. The advantage of a great and a small side (which in the other augments the extent of their power of weighing) is supplied by a very simple process, which accomplishes the same end with some additional advantages. This process is to employ on the same division different weights. The numbers of the divisions on the bar, point out the degree of heaviness expressed by the corresponding weights. For example, when the large weight of the large steelyard weighs 16 lbs., each division it passes over on the bar is equivalent to a pound; the small weight, weighing sixteen times less than the large one, will represent on each of these divisions the sixteenth part of a pound, or one ounce; and the opposite face of the bar is marked by pounds at each sixteenth division. In this construction, therefore, we have the advantage of being able, by employing both weights at once, to ascertain, for example, almost within an ounce, the weight of 500 pounds of merchandise. It will be sufficient to add what is indicated by the small weight in ounces, to that of the large one in pounds, after an equilibrium has been obtained by the position of the two weights, viz. the large one placed at the next pound below its real weight, and the small one at the division which determines the number of ounces to be added.
5. As the beam is graduated only on one edge, it may have the form of a thin bar, which renders it much less susceptible of being bent by the action of the weight, and affords room for making the figures more visible on both the faces.
6. In these steelyards the disposition of the axes is not only such that the beam represents a mathematical lever without weight, but in the principle of its division, the interval between every two divisions is a determined and aliquot part of the distance between the two fixed points of suspension; and each of the two weights employed has for its absolute weight the unity of the weight it represents, multiplied by the number of the divisions contained in the interval between the two centres of motion.
Thus, supposing the arms of the steelyard divided in such a manner that ten divisions are exactly contained in the distance between the two constant centres of motion, a weight to express the pounds on each division of the beam must really weigh ten pounds; that to point out the ounces on the same divisions must weigh ten ounces, &c. So that the same steelyard may be adapted to any system of measures whatever, and in particular to the decimal system, by varying the absolute heaviness of the weights, and their relation with each other.
But to trace out, in a few words, the advantages of the steelyards constructed by C. Paul for commercial purposes, we shall only observe,--
1. That the buyer and seller are certain of the correctness of the instrument, if the beam remains horizontal when it is unloaded and in its usual position. 2. That these steelyards have one suspension less than the old ones, and are so much more simple. 3. That by these means we obtain, with the greatest facility, by employing two weights, the exact weight of merchandise, with all the approximation that can be desired, and even with a greater precision than that given by common balances. There are few of these which, when loaded with 500 pounds at each end, give decided indication of an ounce variation; and the steelyards of C. Paul possess that advantage, and cost one half less than balances of equal dominion. 4. In the last place, we may verify at pleasure the justness of the weights, by the transposition which their ratio to each other will permit; for example, by observing whether, when the weight of one pound is brought back one division, and the weight of one ounce carried forward sixteen divisions, the equilibrium still remains.
It is on this simple and advantageous principle that C. Paul has constructed his universal steelyard. It serves for weighing in the usual manner, and according to any system of weights, all ponderable bodies to the precision of half a grain in the weight of a hundred ounces; that is to say, of a ten-thousandth part. It is employed, besides, for ascertaining the specific gravity of solids, of liquids, and of air, by processes extremely simple, and which do not require many subdivisions in the weights.
We think the description above given will be sufficiently intelligible without a representation of this instrument. An account of its application to the determination of specific gravities will be found in vol. iii. of the Philosophical Magazine.
_The Chinese Steelyard._
This instrument is used in China and the East Indies for weighing gems, precious metals, &c. The beam is a small rod of ivory, about a foot in length. Upon this are three lines of divisions, marked by fine silver studs, all beginning from the end of the beam, whence the first is extended 8 inches, the second 6-1/2, and the third 8-1/2. The first is European weight, and the other two Chinese. At the other end of the beam hangs a round scale, and at three several distances from this end are holes, through which pass so many fine strings, serving as different points of suspension. The first distance makes 1-3/5 inches, the second 3-1/5, or double the former, and the third 4-4/5, or triple the same. The instrument, when used, is held by one of the strings, and a sealed weight of about 1-1/4 oz. troy, is slid upon the beam until an equilibrium is produced; the weight of the body is then indicated by the graduated scale above mentioned.
_The Danish Balance._
The Danish balance is a straight bar or lever, having a heavy weight fixed to one end, and a hook or scale-pan to receive the substance, the weight of which is to be determined, suspended from the other end. The fulcrum is moveable, and is made to slide upon the bar, till the beam rests in a horizontal position, when the place of the fulcrum indicates the weight required. In order to construct a balance of this kind, let the distance of the centre of gravity from that point to which the substance to be weighed is suspended be found by experiment, when the beam is unloaded. Multiply this distance by the weight of the whole apparatus, and divide the product by the weight of the apparatus increased by the weight of the body. This will give the distance from the point of suspension, at which the fulcrum being placed, the whole will be in equilibrio: for example, supposing the distance of the centre of gravity from the point of suspension to be 10 inches, and the weight of the whole apparatus to be ten pounds; suppose, also, it were required to mark the divisions which should indicate weights of one, two, or three pounds, &c. First, for the place of the division indicating one pound we have (10 × 10)/(10 + 1) = 100/(10 + 1) = 9-1/11 inches, the place of the division marking one pound. For two pounds we have 100/(10 + 2) = 8-1/3 inches, the place of the division indicating two pounds; and for three pounds 100/(10 + 3) = 7-9/13 inches for the place of the division indicating three pounds, and so on.
This balance is subject to the inconvenience of the divisions becoming much shorter as the weight increases. The distance between the divisions indicating one and two pounds being, in the example we have given, about seven tenths of an inch, whilst that between 20 and 21 pounds is only one tenth of an inch; consequently a very small error in the place of the divisions indicating the larger weights would occasion very inaccurate results. The Danish balance is represented at _fig. 194._
_The Bent Lever Balance._
This instrument is represented at _fig. 195._ The weight at C, is fixed at the end of the bent lever A B C, which is supported by its axis B on the pillar I H. A scale-pan E, is suspended from the other end of the lever at A. Through the centre of motion B draw the horizontal line K B G, upon which, from A and C let fall the perpendiculars A K and C D. Then if B K and B D are reciprocally proportional to the weights at A and C, they will be in equilibrio, but if not, the weight C will move upwards or downwards along the arc F G till that ratio is obtained. If the lever be so bent that when A coincides with the line G K, C coincides with the vertical B H, then as C moves from F to G, its momentum will increase while that of the weight in the scale-pan E will decrease. Hence the weight in E, corresponding to different positions of the balance, may be expressed on the graduated arc F G.
_Brady’s Balance, or Weighing Apparatus._
This partakes of the properties both of the bent lever balance and of the steelyard. It is represented, at _fig. 196._ A B C is a frame of cast iron having a great part of its weight towards A. F is a fulcrum, and E a moveable suspender, having a scale and hook at its lower extremity. E K G are three distinct places, to which the suspender E may be applied, and to which belong respectively the three graduated scales of division expressing weights, _f_ C, _c d_, and _a b_. When the scale and suspender are applied at G, the apparatus is in equilibrio, with the edge A B horizontal, and the suspender cuts the zero on the scale _a b_. Now, any substance, the weight of which is to be ascertained, being put into the scale, the whole apparatus turns about F, and the part towards B descends till the equilibrium is again established, when the weight of the body is read off from the scale _a b_, which registers to ounces and extends to two pounds. If the weight of the body exceed two pounds, and be less than eleven pounds, the suspender is placed at K; and when the scale is empty, the number 2 is found to the right of the index of the suspender. If now weights exceeding two pounds be placed in the scale, the whole again turns about F, and the weight of the body is shown on the graduated arc _c d_, which extends to eleven pounds, and registers to every two ounces.
If the weight of the body exceed eleven pounds, the suspender is hung on at E, and the weights are ascertained in the same manner on the scale _f_ C to thirty pounds, the subdivisions being on this scale quarters of pounds. The same principles would obviously apply to weights greater or less than the above. To prevent mistake, the three points of support G, K, E, are numbered 1, 2, 3; and the corresponding arcs are respectively numbered in the same manner. When the hook is used instead of the scale, the latter is turned upwards, there being a joint at _m_ for that purpose.
_The Weighing Machine for Turnpike Roads._
This machine is for the purpose of ascertaining the weight of heavy bodies, such as wheel carriages. It consists of a wooden platform placed over a pit made in the line of the road, and which contains the machinery. The pit is walled withinside, and the platform is fitted to the walls of the pit, but without touching them, and it is therefore at liberty to move freely up and down. The platform is supported by levers placed beneath it, and is exactly level with the surface of the road, so that a carriage is easily drawn on it, the wheels being upon the platform whilst the horses are upon the solid ground beyond it. The construction of this machine will be readily understood by reference to _fig. 197._, in which the platform is supposed to be transparent so as to allow of the levers being seen below it.
A, B, C, D, represent four levers tending towards the centre of the platform, and each moveable on its fulcrum at A, B, C, D; the fulcrum of each rests upon a piece securely fixed in the corner of the pit. The platform is supported upon the cross pins _a_, _b_, _c_, _d_, by means of pieces of iron which project from it near its corners, and which are represented in the plate by the short dark lines crossing the pins _a_, _b_, _c_, _d_. The four levers are connected under the centre of the platform, but not so as to prevent their free motion, and are supported by a long lever at the point F, the fulcrum of which rests upon a piece of masonry at E: the end of this last lever passes below the surface of the road into the turnpike house, and is there attached to one arm of a balance, or, as in Salmon’s patent weighing machine, to a strap passing round a cylinder which winds up a small weight round a spiral, and indicates, by means of an index, the weight placed upon the platform.
Suppose the distance from A to F to be ten times as great as that from A to _a_, then a force of one pound applied beneath F would balance ten pounds applied at _a_, or upon the platform. Again: let the distance from E to G be also ten times greater than the distance from the fulcrum E to F; then a force of one pound applied to raise up the end of the lever G would counterpoise a weight of ten pounds placed upon F. Now, as we gain ten times the power by the first levers, and ten times more by the lever E G, it follows, that a force of one pound tending to elevate G, would balance 100 lbs. placed on the platform; so that if the end of the lever G be attached to one arm of a balance, a weight of 10 lbs. placed in a scale suspended from the other arm, will express the value of 1000 lbs. placed upon the platform. The levers are counterpoised, when the platform is not loaded, by a weight H applied to the end of the last lever, continued beyond the fulcrum for that purpose.
_Of Instruments for weighing by means of a Spring._
The spring is well adapted to the construction of a weighing machine, from the property it possesses of yielding in proportion to the force impressed, and consequently giving a scale of equal parts for equal additions of weight. It is liable, however, to suffer injury, unless the steel of which it is composed be very well tempered, from a want of perfect elasticity, and, consequently, from not returning to its original place after it has been forcibly compressed. This, however, must be considered to arise, in a great measure, from imperfection of workmanship, or of the material employed, or from its having been subjected to too great a force.
_The Spring Steelyard._
The little instrument known by this name is in very general use, and is particularly convenient where great accuracy is not necessary, as a spring which will ascertain weights from one pound to fifty, is contained in a cylinder only 4 inches long and 3/4 inch diameter.
This instrument is represented at _fig. 198._ It consists of a tube of iron, of the dimensions just stated, closed at the bottom, to which is attached an iron hook for supporting the substance to be weighed; a rod of iron _a b_, four tenths of an inch wide and one tenth thick, is firmly fixed in the circular plate _c d_, which slides smoothly in the iron tube.
A strong steel spring is also fastened to this plate, and passed round the rod _a b_ without touching it, and without coming in contact with the interior of the cylindrical tube. The tube is closed at the top by a circular piece of iron through which the piece _a b_ passes.
Upon the face of _a b_ the weight is expressed by divisions, each of which indicates one pound, and five of such divisions in the instrument now before us occupy two tenths of an inch. The divisions, notwithstanding, are of sufficient size to enable them to be subdivided by the eye.
To use this instrument, the substance to be weighed is suspended by the hook, the instrument being held by a ring passing through the rod at the other end. The spring then suffers a compression proportionate to the weight, and the number of pounds is indicated by the division on the rod which is cut by the top of the cylindrical tube.
_Salter’s improved Spring Balance._
A very neat form of the instrument last described has been recently brought before the public by Mr. Salter, under the name of the Improved Spring Balance. It is represented at _fig. 199._ The spring is contained in the upper half of a cylinder behind the brass plate forming the face of the instrument; and the rod is fixed to the lower extremity of the spring, which is consequently extended, instead of being compressed, by the application of the weight. The divisions, each indicating half a pound, are engraved upon the face of the brass plate, and are pointed out by an index attached to the rod.
_Marriott’s Patent Dial Weighing Machine._
The exterior of this instrument is represented at _fig. 200._, and the interior at _fig. 201._ A B C is a shallow brass box, having a solid piece as represented at A, to which the spring D E F is firmly fixed by a nut at D. The other end of the spring at F is pinned to the brass piece G H, to the part of which at G is also fixed the iron racked plate I. A screw L serves as a stop to keep this rack in its place. The teeth of the rack fit into those of the pinion M, the axis of which passes through the centre of the dial-plate, and carries an index which points out the weight. The brass piece G H is merely a plate where it passes over the spring, and the tail piece H, to which the weight is suspended, passes through an opening in the side of the box.
_Of the Dynamometer._
This is an important instrument in mechanics, calculated to measure the muscular strength exerted by men and animals. It consists essentially of a spring steelyard, such as that we first described. This is sometimes employed alone, and sometimes in combination with various levers, which allow of the spring being made more delicate, and consequently increase the extent of the divisions indicating the weight.
The first instrument of this kind appears to have been invented by Mr. Graham, but it was too bulky and inconvenient for use. M. le Roy made one of a more simple construction. It consisted of a metal tube, about a foot long, placed vertically upon a stand, and containing in the inside a spiral spring, having above it a graduated rod terminating in a globe. This rod entered the tube more or less in proportion to the force applied to the globe, and the divisions indicated the quantity of this force. Therefore, when a man pressed upon the globe with all his strength, the divisions upon the rod showed the number of pounds weight to which it was equal.
An instrument of this kind for determining the force of a blow struck by a man with his fist was lately exhibited at the National Repository. It was fixed to a wall, from which it projected horizontally. In place of the globe there was a cushion to receive the blow, and as the suddenness with which the spring returned rendered it impossible to read the division upon the rod, another rod similarly divided was forced in by the plate forming the basis of the cushion, and remained stationary when the spring returned. The common spring steelyard, however, which we first described, is in principle the same as M. le Roy’s dynamometer, and is much more conveniently constructed for the purpose we are considering. The ring at one end may be fixed to an immovable object, and the hook at the other attached to a man, or to an animal, and the extent to which the graduated rod is drawn out of the cylinder shows at once the force which is applied. Though this is perhaps the best, and certainly the most simple dynamometer, others have been contrived, which are, however, but modifications of the spring steelyard. One of these is represented at _fig. 202._ The spiral spring acts in the manner before described, but its divisions are increased in size, and therefore rendered more perceptible by means of a rack fixed to the plate, acting against the spiral spring, the teeth of which move a pinion upon which the arm I is fixed, pointing to the graduated arc K.
Another dynamometer has been invented by Mr. Salmon; it is represented at _fig. 203._ and is a combination of levers with the spring. By means of these levers a much more delicate spring, and which is therefore more sensible, may be employed than in the dynamometer last described.
The manner in which these levers and spring act will be readily understood by an inspection of the figure. Like the weighing machine for carriages, the fulcrum of each lever is at one end, and the force is diminished in passing to the spring, in the ratio of the length of its arms. The spring moves a pinion by means of a rack, upon which pinion a hand is placed, indicating by divisions upon a circular dial-plate, the amount of the force employed.
The spring used in this machine is calculated to weigh only about 50 lbs. instead of about 5 cwt., as in the last described; but by means of the levers which intervene between it and the force applied, it will serve to estimate a force equal to 6 cwt., and might obviously be made to go to a much greater extent, by varying the ratio of the length of the arms of the levers.
ON COMPENSATION PENDULUMS.
(336.) It is said of Galileo that, when very young, he observed a lamp suspended from the roof of a church at Pisa, swinging backwards and forwards with a pendulous motion. This, if it had been remarked at all by an uneducated mind, would, most probably, have been passed by as a common occurrence, unworthy of the slightest notice; but to the mind imbued with science no incident is insignificant; and a circumstance apparently the most trivial, when subjected to the giant force of expanded intellect, may become of immense importance to the improvement and to the well-being of man. The fall of an apple, it is said, suggested to Newton the theory of gravitation, and his powerful mind speedily extended to all creation that great law which brings an apple to the ground. The swinging of a lamp in a church at Pisa, viewed by the piercing intellect of Galileo, gave rise to an instrument which affords the most perfect measure of time, which serves to determine the figure of the earth, and which is inseparably connected with all the refinements of modern astronomy.
The properties of the pendulum, and the manner in which it serves to measure time, have been fully explained in chapter xi.; and if a substance could be found not susceptible of any change in its dimensions from a change of temperature, nothing more would be necessary, as the centre of oscillation would always remain at the same distance from the point of suspension. As every known substance, however, expands with heat, and contracts with cold, the length of the pendulum will vary with every alteration of temperature, and thus the time of its vibration will suffer a corresponding change. The effect of a difference of temperature of 25°, or that which usually occurs between winter and summer, would occasion a clock furnished with a pendulum having an iron rod to gain or lose six seconds in twenty-four hours.
It became, then, highly important to discover some means of counteracting this variation to which the length of the pendulum was liable, or, in other words, to devise a method by which the centre of oscillation should, under every change of temperature, remain at the same distance from the point of suspension: happily, the difference in the rate of expansion of different metals presented a ready means of effecting this.
Graham, in the year 1715, made several experiments to ascertain the relative expansions of various metals, with a view of availing himself of the difference of the expansions of two or more of them when opposed to each other, to construct a compensating pendulum. But the difference he found was so small, that he gave up all hope of being able to accomplish his object in that way. Knowing, however, that mercury was much more affected by a given change of temperature than any other substance, he saw that if the mercury could be made to ascend while the rod of the pendulum became longer, and _vice versâ_, the centre of oscillation might always be kept at the same distance from the point of suspension. This idea happily gave birth to the mercurial pendulum, which is now in very general use.
In the mean time, Graham’s suggestion excited the ingenuity of Harrison, originally a carpenter at Barton in Lincolnshire, who, in 1726, produced a pendulum formed of parallel brass and steel rods, known by the name of the gridiron pendulum.
In the mercurial pendulum, the bob or weight is the material affording the compensation; but in the gridiron pendulum the object is attained by the greater expansion of the brass rods, which raise the bob upwards towards the point of suspension as much as the steel rods elongate downwards.
In the present article, we shall describe such compensation pendulums as appear to us likely to answer best in practice; and we trust we shall be able to simplify the subject so as to render a knowledge of mathematics in the construction of this important instrument unnecessary.
The following table contains the linear expansion of various substances in parts of their length, occasioned by a change of temperature amounting to one degree. We have taken the liberty of extracting it from a very valuable paper by F. Bailey, Esq., on the mercurial compensation pendulum, published in the Memoirs of the Astronomical Society of London for 1824.
TABLE I.
_Linear Expansion of various Substances for One Degree of Fahrenheit’s Thermometer._
+----------------------+-------------+-----------------------+ | Substances. | Expansions. | Authors. | +----------------------+-------------+-----------------------+ |White Deal, { | ·0000022685 | Captain Kater. | | { | ·0000028444 | Dr. Struve. | |English Flint Glass, | ·0000047887 | Dulong and Petit. | |Iron (cast), { | ·0000061700 | General Roy. | | { | ·0000065668 | Dulong and Petit. | |Iron (wire), | ·0000068613 | Lavoisier and L. | |Iron (bar), | ·0000069844 | Hasslar. | |Steel (rod), | ·0000063596 | General Roy. | | | | {Commissioners of | |Brass, | ·0000104400 | {Weights and Measures | | | | {--mean of several | | | | {experiments. | |Lead, | ·0000159259 | Smeaton. | |Zinc, | ·0000163426 | Ditto. | |Zinc (hammered), | ·0000172685 | Ditto. | |Mercury _in bulk_, | ·00010010 | Dulong and Petit. | +----------------------+-------------+-----------------------+
From this table it is easy to determine the length of a rod of any substance the expansion of which shall be equal to that of a rod of given length of any other substance.
The lengths of such rods will be inversely proportionate to their expansions. If, therefore, we divide the lesser expansion by the greater (supposing the rod the length of which is given to be made of the lesser expansible material), and multiply the given length by this quotient, we shall have the required length of a rod, the expansion of which will be equal to that of the rod given. For example:--The expansion of a rod of steel being, from the above table, ·0000063596, and that of brass, ·0000104400; if it were required to determine the length of a rod of brass which should expand as much as a rod of steel of 39 inches in length, we have ·0000063596/·0000104400 = ·6091, which, multiplied by 39, gives 23·75 inches for the length of brass required.
We shall here, in order to facilitate calculation, give the ratio of the lengths of such substances as may be employed in the construction of compensation pendulums.
TABLE II.
+---------------------------------------------------+ | Steel rod and brass compensation, as 1: ·6091 | | Iron wire rod and lead compensation, ·4308 | | Steel rod and lead compensation, ·3993 | | Iron wire rod and zinc compensation, ·3973 | | Steel rod and zinc compensation, ·3682 | | Glass rod and lead compensation, ·3007 | | Glass rod and zinc compensation, ·2773 | | Deal rod and lead compensation, ·1427 | | Deal rod and zinc compensation, ·1313 | | Steel rod and mercury in a steel cylinder, ·0728 | | Steel rod and mercury in a glass cylinder, ·0703 | | Glass rod and mercury in a glass cylinder, ·0529 | +---------------------------------------------------+
It is evident that in this table the decimals express the length of a rod of the compensating material, the expansion of which is equal to that of a pendulum rod whose length is unity.
As we are not aware of the existence of any work which contains instructions that might enable an artist or an amateur to make a compensation pendulum, we shall endeavour to give such detailed information as may free the subject from every difficulty.
The pendulum of a clock is generally suspended by a spring, fixed to its upper extremity, and passing through a slit made in a piece which is called the cock of the pendulum. The point of suspension is, therefore, that part of the spring which meets the lower surface of the cock. Now the distance of the centre of oscillation of the pendulum from this point may be varied in two ways; the one by drawing up the spring through this slit, and the other by raising the bob of the pendulum. Either of these methods may be practised in the compensation pendulum, but the former is subject to objections from which the latter is exempt.
Suppose it were required to compensate a pendulum of 39 inches in length, of steel, by means of the expansion of a brass rod. Here, referring to _fig. 204._, we have S C 39 inches (which is to remain constant) of steel; the pendulum spring, passing through the cock at S, is attached to another rod of steel, which is fixed to the cross piece R A at A. The other end of the cross piece at R is fastened to a brass rod, the lower extremity of which is fixed to the cock of the pendulum at B. Now the brass rod B R must expand upwards, as much as the steel rod A C expands downwards; and the length of the brass must be such as to effect this, leaving 39 inches of the steel rod below the cock of the pendulum.
Let us first try 80 inches of steel. Multiplying this by ·6091, we have 48·73 inches for the length of brass, which compensates 80 inches of steel. But as 48·73 inches of the steel, equal in length to the brass, would in this case be above the cock of the pendulum, it would leave only 31·27 inches below it, instead of 39 inches.
Let us now try 100 inches of steel. This, multiplied as before by ·6091, gives 60·91 inches, according to the expansions which we have used, for the length of the brass rod, and leaves 39·09 inches below the cock of the pendulum, which is sufficiently near for our present purpose.
From what has been said we may perceive that the total length of the material of which the pendulum rod is composed must be always equal to the length of the pendulum added to the length of the compensation.
In this instance we have effected our object, by drawing the pendulum-spring through the slit; but we will now show how the same thing may be done by moving the bob of the pendulum. At _fig. 205._, let S C, as before, be equal to 39 inches. Let the steel rod S D turn off at right angles at D, and let a rod of brass B R, of 61 inches in length, ascend perpendicularly from this cross piece to R. To the upper part of the brass rod fix another cross piece R A, and from the extremity A let a steel rod descend to E, bending it as in the figure till it reaches C. Now the total length of the pieces of steel expanding downwards is equal to S D, D F, and F C (amounting together to 39 inches), to which must be added a length of steel equal to that of the brass rod B R, (61 inches), making together 100 inches of steel as before, the expansion of which downwards is compensated by that of the brass rod, of 61 inches in length, expanding upwards.
This form, however, is evidently inconvenient, from the great length of brass and steel which is carried above the cock of the pendulum; but it is the same thing whether the brass and steel be each in one piece, or divided into several, provided the pieces of steel be all so arranged as to expand downwards, and those of brass upwards. Thus, at _fig. 206._, the portions of steel expanding downwards are together equal, as before, to 100 inches, and the two brass pieces expanding upwards are together equal to 61 inches. So that, in fact, the two last forms of compensation which we have described differ in no respect from each other in principle, but only in the arrangement of the materials. The last is the half of the gridiron pendulum, the remaining bars being merely duplicates of those we have described, and serving no other purpose but to form a secure frame-work.
_Harrison’s Gridiron Pendulum._
After what has been said, little more is necessary than to give a representation of this pendulum. This is done at _fig. 207._, in which the darker lines represent the steel rods, and the lighter those of brass. The central rod is fixed at its lower extremity to the middle of the third cross piece from the bottom, and passes freely through holes in the cross pieces which are above, whilst the other rods are secured near their extremities to the cross pieces by pins passing through them. In order to render the whole more secure, the bars pass freely through holes made in two other cross pieces, the extremities of which are fixed to the exterior steel wires. As different kinds of the same metal vary in their rate of expansion, the pendulum when finished may be found upon trial to be not duly compensated. In this case one or more of the cross pieces is shifted higher or lower upon the bars, and secured by pins passed through fresh holes.
_Troughton’s Tubular Pendulum._
This is an admirable modification of Harrison’s gridiron pendulum. It is represented at _fig. 208._, where it may be seen that it has the appearance of a simple pendulum, as the whole compensation is concealed within a tube six tenths of an inch in diameter.
A steel wire, about one tenth of an inch in diameter, is fixed in the usual manner to the spring by which the pendulum is suspended. This wire passes to the bottom of an interior brass tube, in the centre of which it is firmly screwed. The top of this tube is closed, the steel rod passing freely through a hole in the centre. Into the top of this interior tube two steel wires, of one tenth of an inch in diameter, are screwed into holes made in that diameter, which is at right angles to the motion of the pendulum. These wires pass down the tube without touching either it or the central rod, through holes made in the piece which closes the bottom of the interior tube. The lower extremities of these wires, which project a little beyond the inner tube, are securely fixed in a piece which closes the bottom of an exterior brass tube, which is of such a diameter as just to allow the interior tube to pass freely through it, and of a sufficient length to extend a little above it. The top of the exterior tube is closed like that of the interior, having also a hole in its centre, to allow the first steel rod to pass freely through it. Into the top of the exterior tube, in that diameter which coincides with the motion of the pendulum, a second pair of steel wires of the same diameter as the former are screwed, their distance from the central rod being equal to the distance of each from the first pair. They consequently pass down within the interior tube, and through holes made in the pieces closing the lower ends of both the interior and exterior tubes. The lower ends of these wires are fastened to a short cylindrical piece of brass of the same diameter as the exterior tube, to which the bob is suspended by its centre.
_Fig. 209._ is a full sized section of the rod; the three concentric circles represent the two tubes, and the rectangular position of the two pair of wires round the middle one is shown by the five small circles.
_Fig. 210._ is the part which closes the upper end of the interior tube. The two small circles are the two wires which proceed from it, and the three large circles show the holes through which the middle wire and the other pair of wires pass.
_Fig. 211._ is the bottom of the interior tube. The small circle in the centre is where the central rod is fastened to it, the others the holes for the other four wires to pass through.
_Fig. 212._ is the part which closes the top of the external tube. In the large circle in the centre a small brass tube is fixed, which serves as a covering for the upper part of the middle wire, and the two small circles are to receive the wires of the last expansion.
_Fig. 213._ represents the bottom of the exterior tube, in which the small circles show the places where the wires of the second expansion are fastened, and the larger ones the holes for the other pair of wires to pass through.
_Fig. 214._ is a cylindrical piece of brass, showing the manner in which the lower ends of the wires of the last expansion are fastened to it, and the hole in the middle is that by which it is pinned to the centre of the bob. The upper ends of the two pair of wires are, as we have observed, fastened by screwing them into the pieces which stop up the ends of the tubes, but at the lower ends they are all fixed as represented in _fig. 214._ The pieces represented by _figs. 213._ and _214._ have each a jointed motion, by means of which the fellow wires of each pair would be equally stretched, although they were not exactly of the same length.
The action of this pendulum is evidently the same as that of the gridiron pendulum, as we have three lengths of steel expanding downwards, and two of brass expanding upwards. The weight of the pendulum has a tendency to straighten the steel rods, and the tubular form of the brass compensation effectually precludes the fear of its bending; an advantage not possessed by the gridiron pendulum, in which brass rods are employed.
Mr. Troughton, to the account he has given of this pendulum in Nicholson’s Journal, for December, 1804, has added the lengths of the different parts of which it was composed, and the expansions of brass and steel from which these lengths were computed. The length of the interior tube was 31·9 inches, and that of the exterior one 32·8 inches, to which must be added 0·4, the quantity by which in this pendulum the centre of oscillation is higher than the centre of the bob. These are all of brass. The parts which are of steel are,--the middle wire, which, including 0·6, the length of the suspension spring, is 39·3 inches. The first pair of wires 32·5 inches; and the second pair, 33·2 inches. The expansions used were, for brass ·00001666, and for steel ·00000661, in parts of their length for one degree of temperature.
_Benzenberg’s Pendulum._
This pendulum is mentioned in Nicholson’s Journal for April, 1804, and is taken from Voigt’s Magazin für den Neuesten Zustande der Naturkunde, vol. iv. p. 787. The compensation appears to have been effected by a single rod of lead in the centre, of about half an inch thick; the descending rods were made of the best thick iron wire.
As this pendulum deserves attention from the ease with which it may be made, and as others which have since been produced resemble it in principle, we have given a representation of it at _fig. 215._, where A B C D are two rods of iron wire riveted into the cross pieces A C B D. E F is a rod of lead pinned to the middle of the piece B D, and also at its upper extremity to the cross piece G H, into which the second pair of iron wires are fixed, which pass downwards freely through holes made in the cross piece B D. The lower extremities of these last iron wires are fastened into the piece K L, which carries the bob of the pendulum.
To determine the length of lead necessary for the compensation, we must recollect, as before, that the distance from the point of suspension to the centre of the bob (speaking always of a pendulum intended to vibrate seconds) must be 39 inches. Let us suppose the total length of the iron wire to be 60 inches; then, from the table which we have given, we have ·4308 for the length of a rod of lead, the expansion of which is equivalent to that of an iron rod whose length is unity. Multiplying 60 inches by ·4308, we have 25·84 inches of lead, which would compensate 60 inches of iron; but this, taken from 60 inches, leaves only 34·16 instead of 39 inches. Trying again, in like manner, 68·5 inches of iron, we find 29·5 inches of lead for the length, affording an equivalent compensation, and which, taken from 68·5 inches, leaves 39 inches.
The length of the rod of lead then required as a compensation in this pendulum is about 29-1/2 inches.
The writer of this article would suggest another form for this pendulum, which has the advantage of greater simplicity of construction.
S A, _fig. 216._, is a rod of iron wire, to which the pendulum spring is attached. Upon this passes a cylindrical tube of lead, 29-1/2 inches long, which is either pinned at its lower extremity to the end of the iron rod S A, or rests upon a nut firmly screwed upon the extremity of this rod.
A tube of sheet iron passes over the tube of lead, and is furnished at top with a flanche, by which it is supported upon the leaden tube; or it may be fastened to the top of this tube in any manner that may be thought convenient.
The bob of the pendulum may be either passed upon the iron tube (continued to a sufficient length) and secured by a pin passing through the centre of the bob, or the iron tube may be terminated by an iron wire serving the same purpose.
Here we have evidently the same expansions upwards and downwards as in the gridiron form, given to this pendulum by Mr. Benzenberg, joined to the compactness of Troughton’s tubular pendulum.
_Ward’s Compensation Pendulum._
In the year 1806, Mr. Henry Ward, of Blandford in Dorsetshire, received the silver medal of the Society of Arts for the compensation pendulum which we are about to describe.
_Fig. 217._ is a side view of the pendulum rod when together. H H and I I are two flat rods of iron about an eighth of an inch thick. K K is a bar of zinc placed between them, and is nearly a quarter of an inch thick. The corners of the iron bars are bevelled off, which gives them a much lighter appearance. These bars are kept together by means of three screws, O O O, which pass through oblong holes in the bars H H and K K, and screw into the rod I I. The bar H H is fastened to the bar of zinc K K, by the screw _m_, which is called the adjusting screw. This screw is tapped into H H, and passes just through K K; but that part of the screw which passes K K has its threads turned off. The iron bar I I has a shoulder at its upper end, and rests on the top of the zinc bar K K and is wholly supported by it. There are several holes for the screw _m_, in order to adjust the compensation.
The action of this pendulum is similar to that last described, the zinc expanding upwards as much as the iron rods expand downwards, and consequently the instance from the point of suspension to the centre of oscillation remains the same.
Mr. Ward states that the expansion of the zinc he used (hammered zinc) was greater than that given in the tables. He found that the true length of the zinc bar should be about 23 inches; our computation would make it nearly 26.
_The Compensation Tube of Julien le Roy._
We mention this merely to state that it is similar in principal to the apparatus represented at _fig. 204._, with merely this difference, that, instead of the steel rod being fixed to a cross piece proceeding from the brass bar B R, it is attached to a cap fixed upon a brass tube (through which it passes) of the same length as that of the brass rod B R. Cassini spoke well of this pendulum, and it was used in the observatory of Cluny about the year 1748.
_Deparcieux’s Compensation._
This was contrived in the same year as that invented by Julien le Roy. It is represented at _fig. 218._, where A B D F is a steel bar, the ends of which are to be fixed to the lower sides of pieces forming a part of the cock of the pendulum. G E I H is of brass, and stands with its extremities resting on the horizontal part B D of the steel frame. The upper part E I of the brass frame passes above the cock of the pendulum, and admits the tapped wire K, to which the pendulum spring is fixed through a squared hole in the middle. A nut upon this tapped wire gives the adjustment for time. The spring passes through the slit in the cock in the usual manner.
It may be easily perceived that this pendulum is in principle the same as that of Le Roy; the expansion of the total length of steel A B S C downwards being compensated by the equivalent expansion of the brass bar G E upwards. It is, however, preferable to Le Roy’s, because the compensation is contained in the clock case.
Deparcieux had previously published, in the year 1739, an improvement of an imperfectly compensating pendulum, proposed in the year 1733 by Regnauld, a clockmaker of Chalons. In this pendulum Deparcieux employed a lever with unequal arms to increase the effect of the expansion of the brass rod, which was too short.
We may here remark, that all fixed compensations are liable to the same objection, namely, that of not moving with the pendulum, and therefore not taking precisely the same temperature.
_Captain Kater’s Compensation Pendulum._
In Nicholson’s Journal, for July, 1808, is the description of a compensation pendulum by the writer of this article. In this pendulum the rod is of white deal, three quarters of an inch wide, and a quarter of an inch thick. It was placed in an oven, and suffered to remain there for a long time until it became a little charred. The ends were then soaked in melted sealing-wax; and the rod, being cleaned, was coated several times with copal varnish. To the lower extremity of the rod a cap of brass was firmly fixed, from which a strong steel screw proceeded for the purpose of regulating the pendulum for time in the usual manner.
A square tube of zinc was cast, seven inches long and three quarters of an inch square; the internal dimensions being four tenths of an inch. The lower part of the pendulum rod was cut away on the two sides, so as to slide with perfect freedom within the tube of zinc. To the bottom of this zinc tube a piece of brass a quarter of an inch thick was soldered, in which a circular hole was made nearly four tenths of an inch in diameter, having a screw on the inside. A cylinder of zinc, furnished with a corresponding screw on its surface, fitted into this aperture, and a thin plate of brass screwed upon the cylinder, served as a clamp to prevent any shake after the length of zinc necessary for compensation should have been determined. A hole was made through the axis of the cylinder, through which passed the steel screw terminating the pendulum rod.
An opening was made through the bob of the pendulum, extending to its centre, to admit the square tube of zinc which was fixed at its upper extremity to the centre of the bob. The pendulum rod passed through the bob in the usual manner, and the whole was supported by a nut on the steel screw at the extremity.
In this form the compensation acts immediately upon the centre of the bob, elevating it along the rod as much as the rod elongates downwards: the method of calculating the length of the required compensation is precisely the same as that we have before given.
Assuming the length of the deal rod to be 43 inches, and multiplying this by ·1313 from Table II., we have 5·64 inches for the length of the zinc necessary to counteract the expansion of the deal. The length of the steel screw between the termination of the pendulum rod and the nut was two inches, and that of the suspension spring one inch. Now, 3 inches of steel multiplied by ·3682 would give 1·10 inches for the length of zinc which would compensate the steel, and, adding this to 5·64 inches, we have 6·74 inches for the whole length of zinc required.
In this pendulum, the length of the compensating part may be varied by means of the zinc cylinder furnished with a screw for that purpose. The bob of this pendulum and its compensation are represented at _fig. 219._
It has been objected to the use of wooden pendulum rods, that it is difficult, if not impossible, to secure them from the action of moisture, which would at once be fatal to their correct performance. The pendulum now before us has, however, been going with but little intermission since it was first constructed: it is attached to a sidereal clock, not of a superior description, and exposed to very considerable variations of moisture and dryness; yet the change in its rate has been so very trifling as to authorize the belief that moisture has little or no effect upon a wooden rod prepared in the manner we have described. Its rate, under different temperatures, shows that it is over-compensated; the length of the zinc remaining, as stated in Nicholson’s Journal 7·42 inches, instead of which it appears, by our present compensation, that it should be 6·78 inches.
_Reid’s Compensation Pendulum._
Mr. Adam Reid of Woolwich presented to the Society of Arts, in 1809, a compensation pendulum, for which he was rewarded with fifteen guineas. This pendulum is the same in principle with that last described; the rod, however, is of steel instead of wood, and the compensation possesses no means of adjustment. This pendulum is represented at _fig. 220._, where S B is the steel rod, a little thicker where it enters the bob C, and of a lozenge shape to prevent the bob turning, but above and below it is cylindrical.
A tube of zinc D passes to the centre of the bob from below, and the bob is supported upon it by a piece which crosses its centre, and which meets the upper end of the tube.
The rod being passed through the bob and zinc tube, a nut is applied upon a screw at the lower extremity of the rod in the usual manner. If the compensation should be too much, the zinc tube is to be shortened until it is correct.
The length of the zinc tube will be the same in this pendulum as in that of Mr. Ward--about 23 inches, if his experiments are to be relied upon.
The objection to this pendulum appears to be its great length, which amounts to 62 inches. We conceive it would be preferable to place the zinc above the bob, as in the modification which we have suggested of Benzenberg’s pendulum.
_Ellicott’s Pendulum._
It appears that the idea of combining the expansions of different metals with a lever, so as to form a compensation pendulum, originated with Mr. Graham; for Mr. Short, in the Philosophical Transactions for 1752, states that he was informed by Mr. Shelton, that Mr. Graham, in the year 1737, made a pendulum, consisting of three bars, one of steel between two of brass; and that the steel bar acted upon a lever so as to raise the pendulum when lengthened by heat, and to let it down when shortened by cold.
This pendulum, however, was found upon trial to move by jerks, and was therefore laid aside by the inventor to make way for the mercurial pendulum.
Mr. Short also says that Mr. Fotheringham, a quaker of Lincolnshire, caused a pendulum to be made, in the year 1738 or 1739, consisting of two bars, one of brass and the other of steel, fastened together by screws with levers to raise or let down the bob, and that these levers were placed above the bob.
Mr. John Ellicott of London had made very, accurate experiments on the relative expansions of seven different metals, which, however, will be found to differ more or less from the results of the experiments of others. It is not, however, from this to be concluded that Ellicott’s determinations were erroneous; for the expansion of a metal will suffer considerable change even by the processes to which it is necessarily subjected in the construction of a pendulum. It is therefore desirable, whenever a compensation pendulum is to be made, that the expansions of the materials employed should be determined after the processes of drilling, filing, and hammering have been gone through.
It had been objected to Harrison’s gridiron pendulum, that the adjustments of the rods was inconvenient, and that the expansion of the bob supported at its lower edge would, unless taken into the account, vitiate the compensation. These considerations, it is supposed, gave rise to Ellicott’s pendulum, which is nearly similar to those we have just mentioned.
Ellicott’s pendulum is thus constructed:--A bar of brass and a bar of iron are firmly fixed together at their upper ends, the bar of brass lying upon the bar of iron, which is the rod of the pendulum. These bars are held near each other by screws passing through oblong holes in the brass, and tapped into the iron, and thus the brass is allowed to expand or contract freely upon the iron with any change of temperature. The brass bar passes to the centre of the bob of the pendulum, a little above and below which the iron is left broader for the purpose of attaching the levers to it, and the iron is made of a sufficient length to pass quite through the bob of the pendulum.
The pivots of two strong steel levers turn in two holes drilled in the broad part of the iron bar. The short arms of these levers are in contact with the lower extremity of the brass bar, and their longer arms support the bob of the pendulum by meeting the heads of two screws which pass horizontally from each side of the bob towards its centre. By advancing these screws towards the centre of the bob, the longer arms of the lever are shortened, and thus the compensation may be readily adjusted. At the lower end of the iron rod, under the bob, a strong double spring is fixed, to support the greater part of the weight of the bob by its pressure upwards against two points at equal distances from the pendulum rod. Mr. Ellicott gave a description of this pendulum to the Royal Society in 1752, but he says the thought was executed in 1738. As this pendulum is very seldom met with, we think it unnecessary to give a representation of it.
_Compensation by means of a Compound Bar of Steel and Brass._
Several compensations for pendulums have been proposed, by means of a compound bar formed of steel and brass soldered together. In a bar of this description, the brass expanding more than the steel, the bar becomes curved by a change of temperature, the brass side becoming convex and the steel concave with heat. Now, if a bar of this description have its ends resting on supports on each side the cock of the pendulum, the bar passing above the cock with the brass uppermost, if the pendulum spring be attached to the middle of the bar, and it pass in the usual manner through the slit of the cock, it is evident that, by an increase of temperature, the bar will become curved upwards, and the pendulum spring be drawn upwards through the slit, and thus the elongation of the pendulum downwards will be compensated. The compensation may be adjusted by varying the distance of the points of support from the middle of the bar.
Such was one of the modes of compensation proposed by Nicholson. Others of the same description (that is, with compound bars) have been brought before the public by Mr. Thomas Doughty and Mr. David Ritchie; but as they are supposed to be liable to many practical objections, we do not think it requisite to describe them more particularly.
There is, however, a mode of compensation by means of a compound bar, described by M. Biot in the first volume of his Traité de Physique, which appears to possess considerable merit, of which he mentions having first witnessed the successful employment by the inventor, a clockmaker named Martin. At _fig. 221._, S C, is the rod of the pendulum, made, in the usual manner, of iron or steel; this rod passes through the middle of a compound bar of brass and steel (the brass being undermost), which should be furnished with a short tube and screws, by means of which, or by passing a pin through the tube and rod, it may be securely fixed at any part of the pendulum rod.
Two small equal weights W W slide along the compound bar, and, when their proper position has been determined, may be securely clamped.
The manner in which this compensation acts is thus:--Suppose the temperature to increase, the brass expanding more than the steel, the bar becomes curved, and its extremities carrying the weights W and W are elevated, and thus the place of the centre of oscillation is made to approach the point of suspension as much, when the compensation is properly adjusted, as it had receded from it by the elongation of the pendulum rod.
There are three methods of adjusting this compensation: the first, by increasing or diminishing the weights W and W; the second, by varying the distance of the weights W and W from the middle of the bar; and the third, by varying the distance of the bar from the bob of the pendulum, taking care not to pass the middle of the rod. The effect of the compensation is greater as the weights W and W are greater or more distant from the centre of the bar, and also as the bar is nearer to the bob of the pendulum.
M. Biot says that he and M. Matthieu employed a pendulum of this kind for a long time in making astronomical observations in which they were desirous of attaining an extreme degree of precision, and that they found its rate to be always perfectly regular.
In all the pendulums which we have described, the bob is supposed to be fixed to the rod by a pin passing through its centre, and the adjustment for time is to be made by means of a small weight sliding upon the rod.
_Of the Mercurial Pendulum._
We have been guided, in our arrangement of the pendulums which we have described, by the similarity in the mode of compensation employed; and we have now to treat of that method of compensation which is effected by the expansion of the material of which the bob itself of the pendulum is composed.
On this subject, as we have before observed, an admirable paper, from the pen of Mr. Francis Baily, may be found in the Memoirs of the Astronomical Society of London, which leaves nothing to be desired by the mathematical reader. But as our object is to simplify, and to render our subjects as popular as may be, we must endeavour to substitute for the perfect accuracy which Mr. Baily’s paper presents, such rules as may be found not only readily intelligible, but practically applicable, within the limits of those inevitable errors which arise from a want of knowledge of the exact expansion of the materials employed.
At _fig. 222._, let S B represent the rod of a pendulum, and F C B a metallic tube or cylinder, supported by a nut at the extremity of the pendulum rod, in the usual manner, and having a greater expansibility than that of the rod. Now C, the centre of gravity, supposing the rod to be without weight, will be in the middle of the cylinder; and if C B, or half the cylinder, be of such a length as to expand upwards as much as the pendulum rod S B expands downwards, it is evident that the centre of gravity C will remain, under any change of temperature, at the same distance from the point of suspension S. M. Biot imagined that, in effecting this, a compensation sufficiently accurate would be obtained; but Mr. Baily has shown that this is by no means the fact.
Let us suppose the place of the centre of oscillation to be at O, about three or four tenths of an inch, in a pendulum of the usual construction, below the centre of gravity. Now, the object of the compensation is to preserve the distance from S to O invariable, and not the distance from S to C.
The distance of the centre of oscillation varies with the length of the cylinder F B, and hence suffers an alteration in its distance from the point of suspension by the elongation of the cylinder, although the distance of the centre of gravity C from the point of suspension remains unaltered.
We shall endeavour to render this perfectly familiar. Suppose a metallic cylinder, 6 inches long, to be suspended by a thread 36 inches long, thus forming a pendulum in which the distance of the centre of gravity from the point of suspension is 39 inches: the centre of oscillation in such a pendulum will be nearly one tenth of an inch below the centre of gravity. Now let us imagine cylindrical portions of equal lengths to be added to each end of the cylinder, until it reaches the point of suspension; we shall then have a cylinder of 78 inches in length, the centre of gravity of which will still be at the distance of 39 inches from the point of suspension. But it is well known that the centre of oscillation of such a cylinder is at the distance of about two thirds of its length from the point of suspension. The centre of oscillation, therefore, has been removed, by the elongation of the cylinder, about 13 inches below the centre of gravity, whilst the centre of gravity has remained stationary.
Now the same thing as that which we have just described takes place, though in a very minor degree, with our former cylinder, employed as a compensating bob to a pendulum. The rod expands downwards, the centre of gravity remains at the same distance from the point of suspension, and the cylinder elongates both above and below this point; the consequence of which is, that though the centre of gravity has remained stationary, the distance of the centre of oscillation from the point of suspension has increased. It is, therefore, evident that the length of the compensation must be such as to carry the centre of gravity a little nearer to the point of suspension than it was before the expansion took place; by which means the centre of oscillation will be restored to its former distance from the point of suspension.
Let us suppose the expansions to have taken place, and that the centre of gravity, remaining at the same distance from the point of suspension, the centre of oscillation is removed to a greater distance, as we have before explained. It is well known that the product obtained by multiplying the distance from the point of suspension to the centre of gravity, by the distance from the centre of gravity to the centre of oscillation, is a constant quantity; if, therefore, the distance from the centre of gravity to the point of suspension be lessened, the distance from the centre of gravity to the centre of oscillation will be proportionally, though not equally, increased, and the centre of oscillation will, therefore, be elevated. We see, then, if we elevate the centre of gravity precisely the requisite quantity, by employing a sufficient length of the compensating material, that although the distance from the centre of gravity to the point of suspension is lessened, yet the distance from the point of suspension to the centre of oscillation will suffer no change.
The following rule for finding the length of the compensating material in a pendulum of the kind we have been considering will be found sufficiently accurate for all practical purposes:--
_Find in the manner before directed the length of the compensating material, the expansion of which will be equal to that of the rod of the pendulum. Double this length, and increase the product by its one-tenth part, which will give the total length required._ We shall give examples of this as we proceed.
_Graham’s Mercurial Pendulum._
It was in the year 1721 that Graham first put up a pendulum of this description, and subjected it to the test of experiment; but it appears to have been afterwards set aside to make way for Harrison’s gridiron pendulum, or for others of a similar description. For some years past, however, its merits have been more generally known, and it is not surprising that it should be considered as preferable to others, both from the simplicity of its construction, and the perfect ease with which the compensation may be adjusted.
We have already alluded to Mr. Baily’s very able paper on this pendulum, and we shall take the liberty of extracting from it the following description:--
At _fig. 223._ is a drawing of the mercurial pendulum, as constructed in the manner proposed by Mr. Baily.
“The rod S F is made of steel, and perfectly straight; its form may be either cylindrical, of about a quarter of an inch in diameter, or a flat bar, three eighths of an inch wide, and one eighth of an inch thick: its length from S to F, that is, from the bottom of the spring to the bottom of the rod at F, should be 34 inches. The lower part of this rod, which passes through the top of the stirrup, and about half an inch above and below the same, must be formed into a _coarse_ and _deep_ screw, about two tenths of an inch in diameter, and having about thirty turns in an inch. A steel nut with a milled head must be placed at the end of the rod, in order to support the stirrup; and a similar nut must also be placed on the rod _above_ the head of the stirrup, in order to screw firmly down on the same, and thus secure it in its position, after it has been adjusted _nearly_ to the required rate. These nuts are represented at B and C. A small slit is cut in the rod, where it passes through the head of the stirrup, through which a steel pin E is screwed, in order to keep the stirrup from turning round on the rod. The stirrup itself is also made of steel, and the side pieces should be of the same form as the rod, in order that they may readily acquire the same temperature. The top of the stirrup consists of a flat piece of steel, shaped as in the drawing, somewhat more than three eighths of an inch thick. Through the middle of the top (which at this part is about one inch deep) a hole must be drilled sufficiently large to enable the screw of the rod to pass _freely_, but without _shaking_. The inside height of the stirrup from A to D may be 8-1/2 inches, and the inside width between the bars about three inches. The bottom piece should be about three eighths of an inch thick, and hollowed out nearly a quarter of an inch deep, so as to admit the glass cylinder freely. This glass cylinder should have a brass or iron cover G, which should fit the mouth of it freely, with a shoulder projecting on each side, by means of which it should be screwed to the side bars of the stirrup, and thus be secured always in the same position. This cap should not _press_ on the glass cylinder, so as to prevent its expansion. The measures above given may require a slight modification, according to the weight of the mercury employed, and the magnitude of the cylinder: the final adjustment, however, may be safely left to the artist. Some persons have recommended that a circular piece of thick plate glass should float on the mercury, in order to preserve its surface uniformly level.[7] The part at the bottom marked H is a piece of brass fastened with screws to the front of the bottom of the stirrup, through a small hole, in which a steel wire or common needle is passed, in order to indicate (on a scale affixed to the case of the clock) the arc of vibration. This wire should merely rest in the hole, whereby it may be easily removed when it is required to detach the pendulum from the clock, in order that the stirrup might then stand securely on its base. One of the screw holes should be rather larger than the body of the screw, in order to admit of a small adjustment, in case the steel wire should not stand exactly perpendicular to the axis of motion. The scale should be divided into _degrees_, and not _inches_, observing that with a radius of 44 inches (the estimated distance from the bend of the spring to the end of the steel wire) the length of each degree on the scale must be 0·768 inch.”
[7] The variation produced in the height of the column of mercury (supposed to be 6-1/2 inches high) by an alteration of ± 16° in the temperature will be only ± 1/100 of an inch, or in other words, 1/100 of an inch will be the total variation from its _mean_ state, by an alteration of 32° in the temperature. It is therefore probable that, in most cases of moderate alteration in the temperature, the _centre_ only of the column of mercury is subject to elevation and depression, whilst the exterior parts remain attached to the sides of the glass vessel. It was with a view to obviate this inconvenience that Henry Browne, Esq. of Portland Place (I believe) first suggested the piece of floating glass.
In order to determine the length of the mercurial column necessary to form the compensation for this pendulum, we must proceed in the following manner:--
Let us suppose the length of the steel rod and stirrup together to be 42 inches. The absolute expansion of the mercury is ·00010010; but it is not the absolute expansion, but the vertical expansion in a glass cylinder, which is required, and this will evidently be influenced by the expansion of the base of this cylinder. It is easily demonstrable that, if we multiply the linear expansion of any substance (always supposed to be a very small part of its length) by 3, we may in all cases take the result for the cubical or absolute expansion of such substance. In like manner, if we multiply the linear expansion by 2, we shall have the superficial expansion.
If we want the apparent expansion of mercury, the absolute or cubical expansion of the glass vessel must be deducted from the absolute expansion of the mercury, which will leave its excess or apparent expansion. In like manner, deducting the superficial expansion of glass from the absolute expansion of mercury, we shall have its relative vertical expansion. Now, taking the rate of expansion of glass to be ·00000479, and multiplying it by 2, the relative vertical expansion of the mercury in the glass cylinder will be ·00010010 - ·00000958 = ·00009052.
The expansion of a steel rod, according to our table, is ·0000063596; which, divided by ·00009052, gives ·0703 for the length of a column of mercury, the expansion of which is equal to that of a steel rod whose length is unity.
We have now to multiply 42 inches by ·0703, which gives 2·95 inches; and this, deducted from 42, leaves 39·1 inches; so that the length of rod we have chosen is sufficiently near the truth. Now, double 2·95 inches, and add one tenth of its product, and we shall have 6·49 inches for the length of the mercurial column forming the requisite compensation. Mr. Baily’s more accurate calculation gives 6·31 inches.
A mercurial compensation pendulum may be formed, having a cylinder of steel or iron, with its top constructed in the same manner as the top of the stirrup, so as to receive the screw of the rod. To find the length of the mercurial column necessary in a pendulum of this description (that is, with a cylinder made of steel), we must double the linear expansion of steel, and take it from the absolute expansion of mercury to obtain the relative vertical expansion of the mercury. This will be ·00010010 - ·00001272 = ·00008738; and, proceeding as before, we have ·0000063596/·00008738 = ·07279.
Let the length of the steel rod be, as before, 42 inches. Multiplying this by ·07279, we have 3·057, which being doubled, and one tenth of the product added, we obtain 6·72 inches for the length of the compensating mercurial column; which Mr. Baily states to be 6·59.
A mercurial compensation pendulum having a rod of glass has been employed by the writer of this article, who has had reason to think well of its performance. Its cheapness and simplicity much recommend it. It is merely a cylinder of glass of about 7 inches in depth, and 2-1/2 inches diameter, terminated by a long neck, which forms the rod of the pendulum, the whole blown in one piece. A cap of brass is clamped by means of screws to the top of the rod, and to this the pendulum spring is pinned.
We have unquestionable authority for saying, that the mercurial pendulum of the usual construction, that is, with a steel rod and glass cylinder, is not affected by a change of temperature simultaneously in all its parts. Now, the pendulum of which we are treating being formed throughout of the same material in a single piece, and in every part of the same thickness, it is presumed it cannot expand in a linear direction, until the temperature has penetrated to the whole interior surface of the glass, when it is rapidly diffused through the mass of mercury. M. Biot mentions that a pendulum of this kind was formerly used in France, and expresses his surprise that it was no longer employed, as he had heard it very highly spoken of. The writer of this article has also used a pendulum with a glass rod, which differs from that we have just mentioned, in having the lower end of the rod firmly fixed in a socket attached to the centre of a circular iron plate, on the circumference of which a screw is cut, which fits into a collar of iron, supporting the cylinder (to which it is cemented) by means of a circular lip.
This arrangement, though perhaps less perfect than that we have just described, the pendulum not being in one piece, has the advantage of allowing a circular plate of glass to be placed upon the surface of the mercury, as practised by Mr. Browne. To determine the length of a column of mercury for a glass pendulum, let us suppose the glass, including the cylinder, to be 41 inches in length. Multiplying this by ·0529, the number taken from Table II. for a glass rod and mercury in a glass cylinder, we have 2·17 inches for the uncorrected length of mercury, which compensates 41 inches of glass. Suppose the steel spring to be one inch and a half long: multiplying this by ·0703, the appropriate decimal taken from Table II., we have 0·1, the length of mercury due to the steel, making with the former 2·27 inches, which, being doubled, and the product increased by its one-tenth part, we obtain five inches for the length of the required column of mercury.
_Compensation Pendulum of Wood and Lead, on the Principle of the Mercurial Pendulum._
If by any contrivance wood could be rendered impervious to moisture, it would afford one of the most convenient substances known for a compensation pendulum. It does not appear that sufficient experiments have been made upon this subject to decide the question. Mr. Browne of Portland Place, who has devoted much of his time and attention to the most delicate enquiries of this kind, has, we believe, found that if a teak rod is well gilded, it will not afterwards be affected by moisture. At all events, it makes a far superior pendulum, when thus prepared, to what it does when such preparation is omitted.
Mr. Baily, in the paper we have before alluded to, proposes an economical pendulum to be constructed by means of a leaden cylinder and a deal rod. He prefers lead to zinc, on account of its inferior price, and the ease with which it may be formed into the required shape; and as there is no considerable difference in their rates of expansion, it is equally applicable to the purpose.
Let the length of the deal rod be taken at 46 inches. Then, to find the length of the cylinder of lead to compensate this, we have, in Table II., ·1427 for such a pendulum; which, being multiplied by 46, the product doubled, and one tenth of the result added to it, gives 14·44 inches for the length of the leaden cylinder. Mr. Baily’s compensation gives 14·3 inches.
The rod is recommended to be made of about three eighths of an inch in diameter: the leaden cylinder is to be cast with a hole through its centre, which will admit with perfect freedom the cylindrical end of the rod. The cylinder is supported upon a nut, which screws on the end of the rod in the usual manner. This pendulum is represented at _fig. 224._
Mr. Baily proposes that the pendulum should be adjusted nearly to the given rate by means of the screw at the bottom, and that the final adjustment be made by means of a slider moving along the rod. Indeed, this is a means of adjustment which we would recommend to be employed in every pendulum.
_Smeaton’s Pendulum._
We shall conclude our account of compensation pendulums with a description of that invented by Mr. Smeaton. The compensation for temperature in this pendulum is effected by combining the two modes, which have been so fully described in the preceding part of this article.
The pendulum rod is of solid glass, and is furnished with a steel screw and nut at the bottom in the usual manner. Upon the glass rod a hollow cylinder of zinc, about the eighth of an inch thick, and about 12 inches long, passes freely, and rests upon the nut at the bottom of the pendulum rod.
Over the zinc cylinder passes a tube made of sheet-iron. The edge of this tube at the top is turned inwards, and is notched so as to allow of this being effected. A flanche is thus formed, by which the iron tube is supported, upon the zinc cylinder. The lower edge of the iron tube is turned outwards, so as to form a base destined to support a leaden cylinder, which we are about to describe.
A cylinder of lead, rather more than 12 inches long, is cast with a hole through its axis, of such a diameter as to allow of its sliding freely, but without shake, upon the iron tube over which it passes, and by the lower extremity of which it is supported.
Now the zinc, resting upon the nut and expanding upwards, will raise the whole of the remaining part of the compensation. This expansion upwards will be slightly counteracted by the lesser expansion downwards of the iron tube, which carries with it the leaden cylinder. The cylinder of lead now acts upon the principle of the mercurial pendulum, and, expanding upwards, contributes that which was wanting to restore the centre of oscillation to its proper distance from the point of suspension.
This pendulum, we have been informed, does well in practice, and we are not aware that any description of it has been before published.
The method of calculating the length of the tubes required to form the compensation is very simple; nothing more is necessary than to find the length of zinc, the expansion of which is equal to that of the pendulum rod.
Let the pendulum rod be composed of 43 inches of glass, the spring being an inch and a half long, and the screw between the end of the glass rod and the nut half an inch, making in the whole two inches of steel and 43 inches of glass.
Now to find the length of zinc that will compensate the glass, we have, from Table II., for glass and zinc ·2773, which, multiplied by 43, gives 11·92 inches. In like manner we obtain as a compensation for two inches of steel 0·74 of zinc, which, added to 11·92, gives 12·66 inches for the total length of the zinc cylinder.
Now if the iron tube and the lead cylinder be each made of the same length as the zinc, and arranged as we have described, the compensation will be perfect.
To prove this, find, by means of the expansions given in Table I., the actual expansion of each of the substances employed in the pendulum, and we shall have the following results:--
The expansion of 12·66 inches of zinc expanding upwards is ·0002186
Deduct that of 12·66 inches of iron expanding downwards ·0000869 -------- Remaining effect of expansion upwards, referred to the lower extremity of the iron tube ·0001317
Now, for the lead.--On the principle of the mercurial compensation, subtract one tenth part of the length of the cylinder, and take half the remainder, and we shall have six inches of lead, the expansion of which upwards is ·0000955 -------- Total expansion of the compensation upwards ·0002272 -------- To find the expansion of the rod, we have the expansion of 43 inches of glass ·0002059
Of two inches of steel ·0000127 -------- Total expansion of the pendulum rod ·0002186
Agreeing near enough with that of the compensation before found.
As we conceive we have been sufficiently explicit in our description of this pendulum, in the construction of which no difficulty presents itself, we think an engraved representation of it would be superfluous.
We have hitherto treated only of compensations for temperature; but there is another kind of error, which has been sometimes insisted upon, arising from a variation in the density of the atmosphere. If the density of the atmosphere be increased, the pendulum will experience a greater resistance, the arc of vibration will in consequence be diminished, and the pendulum will vibrate faster. This, however, is in some measure counteracted by the increased buoyancy of the atmosphere, which, acting in opposition to gravity, occasions the pendulum to vibrate slower. If the one effect exactly equalled the other, it is evident no error would arise; and in a paper by Mr. Davies Gilbert, President of the Royal Society of London, published in the Quarterly Journal for 1826, he has proved that, by a happy chance, the arc in which pendulums of clocks are usually made to vibrate is the arc at which this compensation of error takes place. This arc, for a pendulum having a brass bob, is 1° 56′ 30″ on each side of the perpendicular; and for a mercurial pendulum, 1° 31′ 44″, or about one degree and a half.
It is well known that, if a pendulum vibrates in a circular arc, the times of vibration will vary nearly as the squares of the arcs; but if the pendulum could be made to vibrate in a cycloid, the time of its vibration in arcs of different extent would then remain the same. Huygens and others, therefore, endeavoured to effect this by placing the spring of the pendulum between cheeks of a cycloidal form.
When escapements are employed which do not insure an unvarying impulse to the pendulum, the force may be unequally transmitted through the train of the clock in consequence of unavoidable imperfections of workmanship, and the arc of vibration may suffer some increase or diminution from this cause. To discover a remedy for this is certainly desirable.
The writer of this article some years ago imagined a mode, which he believes has also been suggested by others, by which he conceived a pendulum might be made to describe an arc approaching in form to that of a cycloid. The pendulum spring was of a triangular form, and the point or vertex was pinned into the top of the pendulum rod, the base of the triangle forming the axis of suspension. Now it is evident that when the pendulum is in motion, the spring will resist bending at the axis of suspension, with a force in some sort proportionate to the base of the triangle.
Suppose the pendulum to have arrived at the extent of its vibrations; the spring will present a curved appearance; and if the distance from the point of suspension to the centre of oscillation be then measured, it will evidently, in consequence of the curvature of the spring, be shorter than the distance from the point of suspension to the centre of oscillation, measured when the pendulum is in a perpendicular position, and consequently when the spring is perfectly straight.
The base of the triangle may be diminished, or the spring be made thinner; either of which will lessen its effect. We cannot say how this plan might answer upon further trial, as sufficient experiments were not made at the time to authorize a decisive conclusion.
We have thus completed our account of compensation pendulums; but before we conclude, it may not be unacceptable if we offer a few remarks on some points which may be found of practical utility.
The cock of the pendulum should be firmly fixed either to the wall or to the case of the clock, and not to the clock itself, as is sometimes done, and which has occasioned much irregularity in its rate, from the motion communicated to the point of suspension. We prefer a bracket or shelf of cast iron or brass, upon which the clock may be fixed, and the cock carrying the pendulum attached to its perpendicular back. This bracket may either be screwed to the back of the clock-case, or, which is the better mode, securely fixed to the wall; and if the latter be adopted, the whole may be defended from the atmosphere, or from dust, by the clock-case, which thus has no connection either with the clock or with the pendulum.
The point of suspension should be distinctly defined and immovable. This may be readily effected, after the pendulum shall have taken the direction of gravity, by means of a strong screw entering the cock (which should be very stout) on one side, and pressing a flat piece of brass into firm contact with the spring.
The impulse should be given in that plane of the rod which coincides with the plane of vibration passing through the axis of the rod. If the impulse be given at any point either before or behind this plane, the probable result will be a tremulous unsteady motion of the pendulum.
A few rough trials, and moving the weight, will bring the pendulum near its intended time of vibration, which should be left a little too slow; when the bob should be firmly fixed to the rod, if the form of the pendulum will admit of it, by a pin or screw passing through its centre.
The more delicate adjustment may be completed by shifting the place of the slider with which the pendulum is supposed to be furnished on the rod.
Mr. Browne (of whom we have before spoken) practises the following very delicate mode of adjustment for rate, which will be found extremely convenient, as it is not necessary to stop the pendulum in order to make the required alteration. Having ascertained, by experiment, the effect produced on the rate of the clock, by placing a weight upon the bob equal to a given number of grains, he prepares certain smaller weights of sheet-lead, which are turned up at the corners, that they may be conveniently laid hold of by a pair of forceps, and the effect of these small weights on the rate of the clock will be, of course, known by proportion. The rate being supposed to be in defect, the weights necessary to correct this may be deposited, without difficulty, upon the bob of the pendulum, or upon some convenient plane surface, placed in order to receive them: and should it be necessary to remove any one of the weights, this may readily be done by employing a delicate pair of forceps, without producing the slightest disturbance in the motion of the pendulum.
INDEX.
A.
Action and reaction, 34.
Aeriform fluids, 26.
Animalcules, 12.
Atmosphere, impenetrability of, 22. Compressibility and elasticity of, 23.
Atoms, 6. Coherence of, 7.
Attraction, magnetic, of gravitation, 8, 50, 64. Molecular or atomic, 69. Cohesion, 70.
Attwood, machine of, 92.
Axes, principal, 138.
Axis, mechanical properties of, 128.
B.
Balance, 279. Of Bates, 288. Use of, 289. Danish, 299. Bent-lever of Brady, 301.
Bodies, 2. Lines, surfaces, edges, area, length of, 4. Figure, volume, shape of, 5. Porosity of, 17. Compressibility of, 18. Elasticity, dilatibility of, 19. Inertia of, 27. Rule for determining velocity of; motion of two bodies after impact, 38.
C.
Capillary attraction, 73.
Capstan, 179.
Cause and effect, 7.
Circle of curvature, 99.
Cog, hunting, 191.
Components, 51.
Cord, 163.
Cordage, friction and rigidity of, 260.
Crank, 241.
Crystallisation, 14.
Cycloid, 158.
D.
Damper, self-acting, 234.
Deparcieux’s compensation pendulum, 319.
Diagonal, 51.
Dynamics, 160.
Dynamometer, 305.
E.
Electricity, 76.
Electro-magnetism, 76.
Equilibrium, neutral, instable, and stable, 118.
F.
Figure, 5.
Fly-wheel, 239.
Force, 6. Composition and resolution of, 49. Centrifugal, 98. Moment of; leverage of, 135. Regulation and accumulation of, 224.
Friction, effects of, 96. Laws of, 264.
G.
Governor, 227.
Gravitation, attraction of, 77. Terrestrial, 84.
Gravity, centre of, 107.
Gyration, radius of, centre of, 137.
H.
Hooke’s universal joint, 252.
Hydrophane, porosity of, 18.
I.
Impact, 39.
Impulse, 65.
Inclined plane, 163–209.
Inclined roads, 211.
Inertia, 27. Laws of, 32. Moment of, 137.
J.
Julien le Roy, compensation tube of, 319.
L.
Lever, 163. Fulcrum of; three kinds of, 167. Equivalent, 176.
Line of direction, 110.
Liquids, compressibility of, 24.
Loadstone, 68.
M.
Machines, simple, 160. Power of, 175. Regulation of, 225.
Magnet, 68.
Magnetic attraction, 8.
Magnetism, 76.
Magnitude, 4.
Marriott’s patent weighing machine, 305.
Materials, strength of, 272.
Matter, properties of, 2. Impenetrability of, 4. Atoms of; molecules of, 6. Divisibility of, 9. Examples of the subtilty of, 12. Limit to the divisibility of, 13. Porosity of; density of, 17. Compressibility of, 18. Elasticity and dilatability of, 19. Impenetrability of, 22. Inertia of, 27.
Mechanical science, foundation of, 16.
Metronomes, principles of, 153.
Molecules, 6.
Motion, laws of, 46. Uniformly accelerated, 87. Table illustrative of, 90. Retarded; of bodies on inclined planes and curves, 94. Rotary and progressive, 127. Mechanical contrivances for the modification of, 245. Continued rectilinear; reciprocatory rectilinear; continued circular; reciprocating circular, 246.
N.
Newton, method of, for determining the thickness of transparent substances, 10. Laws of motion of, 46.
O.
Oscillation, 129. Of the pendulum, 145. Centre of, 152.
P.
Parallelogram, 51.
Particle, 6.
Pendulum, oscillation or vibration of, 145. Isochronism of, 147. Centre of oscillation of, 152. Of Troughton, 284. Compensation, 307. Of Harrison, 313. Tubular, of Troughton, 314. Of Benzenberg, 316. Ward’s compensation, 318. Captain Kater’s compensation, 320. Reid’s; Ellicott’s compensation, 322. Steel and brass compensation, 324. Mercurial, 326. Graham’s mercurial, 329. Wood and lead, 334. Smeaton’s, 335.
Percussion, 130. Centre of, 144.
Planes of cleavage, 15.
Porosity, 17.
Power, 161.
Properties, 2.
Projectiles, curvilinear path of, 82.
Pulley, 164. Tackle; fixed, 198. Single moveable, 200. Called a runner; Spanish bartons, 205.
R.
Rail-roads, 213.
Regulating damper, 233.
Regulators, 227.
Repulsion, 8. Molecular, 74.
Resultant, 51.
Rose-engine, 250.
S.
Salters, spring balance of, 305.
Screw, 209. Concave, 217. Micrometer, 223.
Shape, 5.
Siphon, capillary, 73.
Spring, 304.
Statics, 160.
Steelyard, 294. C. Paul’s, 296. Chinese, 299.
T.
Table, whirling, 99.
Tachometer, 234.
Tread-mill, 179.
V.
Velocity, angular, 99.
Vibration, 129. Of the pendulum, 145. Centre of, 152.
Volume, 5–17.
W.
Watch, mainspring of; balance wheel of, 195.
Water regulator, 229.
Wedge, 209. Use of, 215.
Weight, 161–291.
Weighing machines, 278. For turnpike roads, 302. By means of a spring, 303.
Wheels, spur, crown, bevelled, 189. Escapement, 194.
Wheel and axle, 177.
Wheel-work, 176.
Winch, 179.
Windlass, 178.
Wollaston’s wire, 10.
Z.
Zureda, apparatus of; Leupold’s application of, 251.
END OF MECHANICS.
LONDON: SPOTTISWOODES and SHAW New-street-Square.