Chapter VII: Part 7

(204.) The effects upon the axis which have been now explained are those which arise from the motion of rotation, from whatever cause that motion may have arisen. The forces which produce that motion, however, are attended with effects on the axis which still remain to be noticed. When these forces, whether they be of the nature of instantaneous actions or continued forces, are entirely resisted by the axis, their directions must severally be in a plane passing through the axis, or they must, by the principles of the composition of force [(74.) et seq.], be mechanically equivalent to forces in that plane. In every other case the impressed forces _must_ produce motion, and, except in certain cases, must also produce effects upon the axis.

By the rules for the composition of force it is possible in all cases to resolve the impressed forces into others which are either in planes through the axis, or in planes perpendicular to it, or, finally, some in planes through it, and others in planes perpendicular to it. The effect of those which are in planes through the axis has been already explained; and we shall now confine our attention to those impelling forces which act at right angles to the axis, and which produce motion.

It will be sufficient to consider the effect of a single force at right angles to the axis; for whatever be the number of forces which act either simultaneously or successively, the effect of the whole will be decided by combining their separate effects. The effect which a single force produces depends on two circumstances, 1. The position of the axis with respect to the figure and mass of the body, and 2. The quantity and direction of the force itself.

In general the shock which the axis sustains from the impact may be represented by two impacts applied to it at different points, one parallel to the impressed force, and the other perpendicular to it, but both perpendicular to the axis. There are certain circumstances, however, under which this effect will be modified.

If the impulse which the body receives be in a direction perpendicular to a plane through the axis and the centre of gravity, and at a distance from the axis which bears to the radius of gyration (186.) the same proportion as that line bears to the distance of the centre of gravity from the axis, there are certain cases in which the impulse will produce no percussion. To characterise these cases generally would require analytical formulæ which cannot conveniently be translated into ordinary language. That point of the plane, however, where the direction of the impressed force meets it, when no percussion on the axis is produced, is called the _centre of percussion_.

If the axis of rotation be a principal axis, the centre of percussion must be in the right line drawn through the centre of gravity, intersecting the axis at right angles, and at the distance from the axis already explained.

If the axis of rotation be parallel to a principal axis through the centre of gravity, the centre of percussion will be determined in the same manner.

(205.) There are many positions which the axis may have in which there will be no centre of percussion; that is, there will be no direction in which an impulse could be applied without producing a shock upon the axis. One of these positions is when it is a principal axis through the centre of gravity. This is the only case of rotation round an axis in which no effect arises from the centrifugal force; and therefore it follows that the only case in which the axis sustains no effect from the motion produced, is one in which it must necessarily suffer an effect from that which produces the motion.

If the body be acted upon by continued forces, their effect is at each instant determined by the general principles for the composition of force.

CHAP. XI.

ON THE PENDULUM.

(206.) When a body is placed on a horizontal axis which does not pass through its centre of gravity, it will remain in permanent equilibrium only when the centre of gravity is immediately below the axis. If this point be placed in any other situation, the body will oscillate from side to side, until the atmospherical resistance and the friction of the axis destroy its motion. (159, 160.) Such a body is called a _pendulum_. The swinging motion which it receives is called _oscillation_ or _vibration_.

(207.) The use of the pendulum, not only for philosophical purposes, but in the ordinary economy of life, renders it a subject of considerable importance. It furnishes the most exact means of measuring time, and of determining with precision various natural phenomena. By its means the variation of the force of gravity in different latitudes is discovered, and the law of that variation experimentally exhibited. In the present chapter, we propose to explain the general principles which regulate the oscillation of pendulums. Minute details concerning their construction will be given in the twenty-first chapter of this volume.

(208.) A simple pendulum is composed of a heavy molecule attached to the end of a flexible thread, and suspended by a fixed point O, _fig. 73._ When the pendulum is placed in the position O C, the molecule being vertically below the point of suspension, it will remain in equilibrium; but if it be drawn into the position O A and there liberated, it will descend towards C, moving through the arc A C with accelerated motion. Having arrived at C and acquired a certain velocity, it will, by reason of its inertia, continue to move in the same direction. It will therefore commence to ascend the arc C A′ with the velocity so acquired. During its ascent, the weight of the molecule retards its motion in exactly the same manner as it had accelerated it in descending from A to C; and when the molecule has ascended through the arc C A′ equal to C A, its entire velocity will be destroyed, and it will cease to move in that direction. It will thus be placed at A′ in the same manner as in the first instance it had been placed at A, and consequently it will descend from A′ to C with accelerated motion, in the same manner as it first moved from A to C. It will then ascend from C to A, and so on, continually. In this case the thread, by which the molecule is suspended, is supposed to be perfectly flexible, inextensible, and of inconsiderable weight. The point of suspension is supposed to be without friction, and the atmosphere to offer no resistance to the motion.

It is evident from what has been stated, that the times of moving from A to A′ and from A′ to A are equal, and will continue to be equal so long as the pendulum continues to vibrate. If the number of vibrations performed by the pendulum were registered, and the time of each vibration known, this instrument would become a chronometer.

The rate at which the motion of the pendulum is accelerated in its descent towards its lowest position is not uniform, because the force which impels it is continually decreasing, and altogether disappears at the point C. The impelling force arises from the effect of gravity on the suspended molecule, and this effect is always produced in the vertical direction A V. The greater the angle O A V is, the less efficient the force of gravity will be in accelerating the molecule: this angle evidently increases as the molecule approaches C, which will appear by inspecting _fig. 73._ At C, the force of gravity acting in the direction C B is totally expended in giving tension to the thread, and is inefficient in moving the molecule. It follows, therefore, that the impelling force is greatest at A, and continually diminishes from A to C, where it altogether vanishes. The same observations will be applicable to the retarding force from C to A′, and to the accelerating force from A′ to C, and so on.

When the length of the thread and the intensity of the force of gravity are given, the time of vibration depends on the length of the arc A C, or on the magnitude of the angle A O C. If, however, this angle do not exceed a certain limit of magnitude, the time of vibration will be subject to no sensible variation, however that angle may vary. Thus the time of oscillation will be the same, whether the angle A O C be 2°, or 1° 30′, or 1°, or any lesser magnitude. This property of a pendulum is expressed by the word _isochronism_. The strict demonstration of this property depends on mathematical principles, the details of which would not be suitable to the present treatise. It is not difficult, however, to explain generally how it happens that the same pendulum will swing through greater and smaller arcs of vibration in the same time. If it swing from A, the force of gravity at the commencement of its motion impels it with an effect depending on the obliquity of the lines O A and A V. If it commence its motion from _a_, the impelling effect from the force of gravity will be considerably less than at A; consequently, the pendulum begins to move at a slower rate, when it swings from _a_ than when it moves from A: the greater magnitude of the swing is therefore compensated by the increased velocity, so that the greater and the smaller arcs of vibration are moved through in the same time.

(209.) To establish this property experimentally, it is only necessary to suspend a small ball of metal, or other heavy substance, by a flexible thread, and to put it in a state of vibration, the entire arc of vibration not exceeding 4° or 5°, the friction on the point of suspension and other causes will gradually diminish the arc of vibration, so that after the lapse of some hours it will be so small, that the motion will scarcely be discerned without microscopic aid. If the vibration of this pendulum be observed in reference to a correct timekeeper, at the commencement, at the middle, and towards the end of its motion, the rate will be found to suffer no sensible change.

This remarkable law of isochronism was one of the earliest discoveries of Galileo. It is said, that when very young, he observed a chandelier suspended from the roof of a church in Pisa swinging with a pendulous motion, and was struck with the uniformity of the rate even when the extent of the swing was subject to evident variation.

(210.) It has been stated in (117.) that the attraction of gravity affects all bodies equally, and moves them with the same velocity, whatever be the nature or quantity of the materials of which they are composed. Since it is the force of gravity which moves the pendulum, we should therefore expect that the circumstances of that motion should not be affected either by the quantity or quality of the pendulous body. And we find this, in fact, to be the case; for if small pieces of different heavy substances such as lead, brass, ivory, &c., be suspended by fine threads of equal length, they will vibrate in the same time, provided their weights bear a considerable proportion to the atmospherical resistance, or that they be suspended _in vacuo_.

(211.) Since the time of vibration of a pendulum, which oscillates in small arcs, depends neither on the magnitude of the arc of vibration nor on the quality or weight of the pendulous body, it will be necessary to explain the circumstances on which the variation of this time depends.

The first and most striking of these circumstances is the length of the suspending thread. The rudest experiments will demonstrate the fact, that every increase in the length of this thread will produce a corresponding increase in the time of vibration; but according to what law does this increase proceed? If the length of the thread be doubled or trebled, will the time of vibration also be increased in a double or treble proportion? This problem is capable of exact mathematical solution, and the result shows that the time of vibration increases not in the proportion of the increased length of the thread, but as the square root of that length; that is to say, if the length of the thread be increased in a four-fold proportion, the time of vibration will be augmented in a two-fold proportion. If the thread be increased to nine times its length, the time of vibration will be trebled, and so on. This relation is exactly the same as that which was proved to subsist between the spaces through which a body falls freely, and the times of fall. In the table, page 89, if the figures representing the height be understood to express the length of different pendulums, the figures immediately above them will express the corresponding times of vibration.

This law of the proportion of the lengths of pendulums to the squares of the time of vibration may be experimentally established in the following manner:--

Let A, B, C, _fig. 74._, be three small pieces of metal each attached by threads to two points of suspension, and let them be placed in the same vertical line under the point O; suppose them so adjusted that the distances O A, O B, and O C shall be in the proportion of the numbers 1, 4, and 9. Let them be removed from the vertical in a direction at right angles to the plane of the paper, so that the threads shall be in the same plane, and therefore the three pendulums will have the same angle of vibration. Being now liberated, the pendulum A will immediately gain upon B, and B upon C, so that A will have completed one vibration before B or C. At the end of the second vibration of A, the pendulum B will have arrived at the end of its first vibration, so that the suspending threads of A and B will then be separated by the whole angle of vibration; at the end of the fourth vibration of A the suspending threads of A and B will return to their first position, B having completed two vibrations; thus the proportion of the times of vibration of B and A will be 2 to 1, the proportion of their lengths being 4 to 1. At the end of the third vibration of A, C will have completed one vibration, and the suspending strings will coincide in the position distant by the whole angle of vibration from their first position. So that three vibrations of A are performed in the same time as one of C: the proportion of the time of vibration of C and A are, therefore, 3 to 1, the proportion of their lengths being 9 to 1, conformably to the law already explained.

(212.) In all the preceding observations we have assumed that the material of the pendulous body is of inconsiderable magnitude, its whole weight being conceived to be collected in a physical point. This is generally called a simple pendulum; but since the conditions of a suspending thread without weight, and a heavy molecule without magnitude, cannot have practical existence, the simple pendulum must be considered as imaginary, and merely used to establish hypothetical theorems, which, though inapplicable in practice, are nevertheless the means of investigating the laws which govern the real phenomena of pendulous bodies.

A pendulous body being of determinate magnitude, its several parts will be situated at different distances from the axis of suspension. If each component part of such a body were separately connected with the axis of suspension by a fine thread, it would, being unconnected with the other particles, be an independent simple pendulum, and would oscillate according to the laws already explained. It therefore follows that those particles of the body which are nearest to the axis of suspension would, if liberated from their connection with the others, vibrate more rapidly than those which are more remote. The connection, however, which the particles of the body have, by reason of their solidity, compels them all to vibrate in the same time. Consequently, those particles which are nearer the axis are retarded by the slower motion of those which are more remote; while the more remote particles, on the other hand, are urged forward by the greater tendency of the nearer particles to rapid vibration. This will be more readily comprehended, if we conceive two particles of matter A and B, _fig. 75._, to be connected with the same axis O by an inflexible wire O C, the weight of which may be neglected. If B were removed, A would vibrate in a certain time depending upon the distance O A. If A were removed, and B placed upon the wire at a distance B O equal to four times A O, B would vibrate in twice the former time. Now if both be placed on the wire at the distances just mentioned, the tendency of A to vibrate more rapidly will be transmitted to B by means of the wire, and will urge B forward more quickly than if A were not present: on the other hand, the tendency of B to vibrate more slowly will be transmitted by the wire to A, and will cause it to move more slowly than if B were not present. The inflexible quality of the connecting wire will in this case compel A and B to vibrate simultaneously, the time of vibration being greater than that of A, and less than that of B, if each vibrated unconnected with the other.

If, instead of supposing two particles of matter placed on the wire, a greater number were supposed to be placed at various distances from O, it is evident the same reasoning would be applicable. They would mutually affect each other’s motion; those placed nearest to point O accelerating the motion of those more remote, and being themselves retarded by the latter. Among these particles one would be found in which all these effects would be mutually neutralised, all the particles nearer O being retarded in reference to that motion which they would have if unconnected with the rest, and those more remote being in the same respect accelerated. The point at which such a particle is placed is called _the centre of oscillation_.

What has been here observed of the effects of particles of matter placed upon rigid wire will be equally applicable to the particles of a solid body. Those which are nearer to the axis are urged forward by those which are more remote, and are in their turn retarded by them; and as with the particles placed upon the wire, there is a certain particle of the body at which the effects are mutually neutralised, and which vibrates in the same time as it would if it were unconnected with the other parts of the body, and simply connected by a fine thread to the axis. By this centre of oscillation the calculations respecting the vibration of a solid body are rendered as simple as those of a molecule of inconsiderable magnitude. All the properties which have been explained as belonging to a simple pendulum may thus be transferred to a vibrating body of any magnitude and figure, by considering it as equivalent to a single particle of matter vibrating at its centre of oscillation.

(213.) It follows from this reasoning, that the virtual length of a pendulum is to be estimated by the distance of its centre of oscillation from the axis of suspension, and therefore that the times of vibration of different pendulums are in the same proportion as the square roots of the distances of their centres of oscillation from their axes.

The investigation of the position of the centre of oscillation is, in most cases, a subject of intricate mathematical calculation. It depends on the magnitude and figure of the pendulous body, the manner in which the mass is distributed through its volume, or the density of its several parts, and the position of the axis on which it swings.

The place of the centre of oscillation may be determined when the position of the centre of gravity and the centre of gyration are known; for the distance of the centre of oscillation from the axis will always be obtained by dividing the square of the radius of gyration (186.) by the distance of the centre of gravity from the axis. Thus if 6 be the radius of gyration, and 9 the distance of gravity from the axis, 36 divided by 9, which is 4, will be the distance of the centre of oscillation from the axis. Hence it may be inferred generally, that the greater the proportion which the radius of gyration bears to the distance of the centre of gravity from the axis, the greater will be the distance of the centre of oscillation.

It follows from this reasoning, that the length of a pendulum is not limited by the dimensions of its volume. If the axis be so placed that the centre of gravity is near it, and the centre of gyration comparatively removed from it, the centre of oscillation may be placed far beyond the limits of the pendulous body. Suppose the centre of gravity is at a distance of one inch from the axis, and the centre of gyration 12 inches, the centre of oscillation will then be at the distance of 144 inches, or 12 feet. Such a pendulum may not in its greatest dimensions exceed one foot, and yet its time of vibration would be equal to that of a simple pendulum whose length is 12 feet.

By these means pendulums of small dimensions may be made to vibrate as slowly as may be desired. The instruments called _metronomes_, used for marking the time of musical performances, are constructed on this principle.

(214.) The centre of oscillation is distinguished by a very remarkable property in relation to the axis of suspension. If A, _fig. 76._, be the point of suspension, and O the corresponding centre of oscillation, the time of vibration of the pendulum will not be changed if it be raised from its support, inverted, and suspended from the point O. It follows, therefore, that if O be taken as the point of suspension, A will be the corresponding centre of oscillation. These two points are, therefore, convertible. This property may be verified experimentally in the following manner. A pendulum being put into a state of vibration, let a small heavy body be suspended by a fine thread, the length of which is so adjusted that it vibrates simultaneously with the pendulum. Let the distance from the point of suspension to the centre of the vibrating body be measured, and take this distance on the pendulum from the axis of suspension downwards; the place of the centre of oscillation will thus be obtained, since the distance so measured from the axis is the length of the equivalent simple pendulum. If the pendulum be now raised from its support, inverted, and suspended from the centre of oscillation thus obtained, it will be found to vibrate simultaneously with the body suspended by the thread.

(215.) This property of the interchangeable nature of the centres of oscillation and suspension has been, at a late period, adopted by Captain Kater, as an accurate means of determining the length of a pendulum. Having ascertained with great accuracy two points of suspension at which the same body will vibrate in the same time, the distance between these points being accurately measured, is the length of the equivalent simple pendulum. See Chapter XXI.

(216.) The manner in which the time of vibration of a pendulum depends on its length being explained, we are next to consider how this time is affected by the attraction of gravity. It is obvious that, since the pendulum is moved by this attraction, the rapidity of its motion will be increased, if the impelling force receive any augmentation; but it still is to be decided, in what exact proportion the time of oscillation will be diminished by any proposed increase in the intensity of the earth’s attraction. It can be demonstrated mathematically, that the time of one vibration of a pendulum has the same proportion to the time of falling freely in the perpendicular direction, through a height equal to half the length of the pendulum, as the circumference of a circle has to its diameter. Since, therefore, the times of vibration of pendulums are in a fixed proportion to the times of falling freely through spaces equal to the halves of their lengths, it follows that these times have the same relation to the force of attraction as the times of falling freely through their lengths have to that force. If the intensity of the force of gravity were increased in a four-fold proportion, the time of falling through a given height would be diminished in a two-fold proportion; if the intensity were increased to a nine-fold proportion, the time of falling through a given space would be diminished in a three-fold proportion, and so on; the rate of diminution of the time being always as the square root of the increased force. By what has been just stated this law will also be applicable to the vibration of pendulums. Any increase in the intensity of the force of gravity would cause a given pendulum to vibrate more rapidly, and the increased rapidity of the vibration would be in the same proportion as the square root of the increased intensity of the force of gravity.

(217.) The laws which regulate the times of vibration of pendulums in relation to one another being well understood, the whole theory of these instruments will be completed, when the method of ascertaining the actual time of vibration of any pendulum, in reference to its length, has been explained. In such an investigation, the two elements to be determined are, 1. the exact time of a single vibration, and, 2. the exact distance of the centre of oscillation from the point of suspension.

The former is ascertained by putting a pendulum in motion in the presence of a good chronometer, and observing precisely the number of oscillations which are made in any proposed number of hours. The entire time during which the pendulum swings, being divided by the number of oscillations made during that time, the exact time of one oscillation will be obtained.

The distance of the centre of oscillation from the point of suspension may be rendered a matter of easy calculation, by giving a certain uniform figure and material to the pendulous body.

(218.) The time of vibration of one pendulum of known length being thus obtained, we shall be enabled immediately to solve either of the following problems.

“To find the length of a pendulum which shall vibrate in a given time.”

“To find the time of vibration of a pendulum of a given length.”

The former is solved as follows: the time of vibration of the known pendulum is to the time of vibration of the required pendulum, as the square root of the length of the known pendulum is to the square root of the length of the required pendulum. This length is therefore found by the ordinary rules of arithmetic.

The latter may be solved as follows: the length of the known pendulum is to the length of the proposed pendulum, as the square of the time of vibration of the known pendulum is to the square of the time of vibration of the proposed pendulum. The latter time may therefore be found by arithmetic.

(219.) Since the rate of a pendulum has a known relation to the intensity of the earth’s attraction, we are enabled, by this instrument, not only to detect certain variations in that attraction in various parts of the earth, but also to discover the actual amount of the attraction at any given place.

The actual amount of the earth’s attraction at any given place is estimated by the height through which a body would fall freely at that place in any given time, as in one second. To determine this, let the length of a pendulum which would vibrate in one second at that place be found. As the circumference of a circle is to its diameter[2] (a known proportion), so will one second be to the time of falling through a height equal to half the length of this pendulum. This time is therefore a matter of arithmetical calculation. It has been proved in (120.), that the heights, through which a body falls freely, are in the same proportion as the squares of the times; from whence it follows, that the square of the time of falling through a height equal to half the length of the pendulum is to one second as half the length of that pendulum is to the height through which a body would fall in one second. This height, therefore, may be immediately computed, and thus the actual amount of the force of gravity at any given place may be ascertained.

[2] This ratio is that of 31,416 to 10,000 very nearly.

(220.) To compare the force of gravity in different parts of the earth, it is only necessary to swing the same pendulum in the places under consideration, and to observe the rapidity of its vibrations. The proportion of the force of gravity in the several places will be that of the squares of the velocity of the vibration. Observations to this effect have been made at several places, by Biot, Kater, Sabine, and others.

The earth being a mass of matter of a form nearly spherical, revolving with considerable velocity on an axis, its component parts are affected by a centrifugal force; in virtue of which, they have a tendency to fly off in a direction perpendicular to the axis. This tendency increases in the same proportion as the distance of any part from the axis increases, and consequently those parts of the earth which are near the equator, are more strongly affected by this influence than those near the pole. It has been already explained (145.) that the figure of the earth is affected by this cause, and that it has acquired a spheroidal form. The centrifugal force, acting in opposition to the earth’s attraction, diminishes its effects; and consequently, where this force is more efficient, a pendulum will vibrate more slowly. By these means the rate of vibration of a pendulum becomes an indication of the amount of the centrifugal force. But this latter varies in proportion to the distance of the place from the earth’s axis; and thus the rate of a pendulum indicates the relation of the distances of different parts of the earth’s surface from its axis. The figure of the earth may be thus ascertained, and that which theory assigns to it, it may be practically proved to have.

This, however, is not the only method by which the figure of the earth may be determined. The meridians being sections of the earth through its axis, if their figure were exactly determined, that of the earth would be known. Measurements of arcs of meridians on a large scale have been executed, and are still being made in various parts of the earth, with a view to determine the curvature of a meridian at different latitudes. This method is independent of every hypothesis concerning the density and internal structure of the earth, and is considered by some to be susceptible of more accuracy than that which depends on the observations of pendulums.

(221.) It has been stated that, when the arc of vibration of a pendulum is not very small, a variation in its length will produce a sensible effect on the time of vibration. To construct a pendulum such that the time of vibration may be independent of the extent of the swing, was a favourite speculation of geometers. This problem was solved by Huygens, who showed that the curve called a _cycloid_, previously discovered and described by Galileo, possessed the isochronal property; that is, that a body moving in it by the force of gravity, would vibrate in the same time, whatever be the length of the arc described.

Let O A, _fig. 77._, be a horizontal line, and let O B be a circle placed below this line, and in contact with it. If this circle be rolled upon the line from O towards A, a point upon its circumference, which at the beginning of the motion is placed at O, will during the motion trace the curve O C A. This curve is called a _cycloid_. If the circle be supposed to roll in the opposite direction towards A′, the same point will trace another cycloid O C′ A′. The points C and C′ being the lowest points of the curves, if the perpendiculars C D and C′ D′ be drawn, they will respectively be equal to the diameter of the circle. By a known property of this curve, the arcs O C and O C′ are equal to twice the diameter of the circle. From the point O suppose a flexible thread to be suspended, whose length is twice the diameter of the circle, and which sustains a pendulous body P at its extremity. If the curves O C and O C′, from the plane of the paper, be raised so as to form surfaces to which the thread may be applied, the extremity P will extend to the points C and C′, when the entire thread has been applied to either of the curves. As the thread is deflected on either side of its vertical position, it is applied to a greater or lesser portion of either curve, according to the quantity of its deflection from the vertical. If it be deflected on each side until the point P reaches the points C and C′, the extremity would trace a cycloid C P C′ precisely equal and similar to those already mentioned. Availing himself of this property of the curve, Huygens constructed his cycloidal pendulum. The time of vibration was subject to no variation, however the arc of vibration might change, provided only that the length of the string O P continued the same. If small arcs of the cycloid be taken on either side of the point P, they will not sensibly differ from arcs of a circle described with the centre O and the radius O P; for, in slight deflections from the vertical position, the effect of the curves O C and O C′ on the thread O P is altogether inconsiderable. It is for this reason that when the arcs of vibration of a circular pendulum are small, they partake of the property of isochronism peculiar to those of a cycloid. But when the deflection of P from the vertical is great, the effect of the curves O C and O C′ on the thread produces a considerable deviation of the point P from the arc of the circle whose centre is O and whose radius is O P, and consequently the property of isochronism will no longer be observed in the circular pendulum.

CHAP. XII.

OF SIMPLE MACHINES.

(222.) A MACHINE is an instrument by which force or motion may be transmitted and modified as to its quantity and direction. There are two ways in which a machine may be applied, and which give rise to a division of mechanical science into parts denominated STATICS and DYNAMICS; the one including the theory of equilibrium, and the other the theory of motion. When a machine is considered statically, it is viewed as an instrument by which forces of determinate quantities and direction are made to balance other forces of other quantities and other directions. If it be viewed dynamically, it is considered as a means by which certain motions of determinate quantity and direction may be made to produce other motions in other directions and quantities. It will not be convenient, however, in the present treatise, to follow this division of the subject. We shall, on the other hand, as hitherto, consider the phenomena of equilibrium and motion together.

The effects of machinery are too frequently described in such a manner as to invest them with the appearance of paradox, and to excite astonishment at what appears to contradict the results of the most common experience. It will be our object here to take a different course, and to attempt to show that those effects which have been held up as matters of astonishment are the necessary, natural, and obvious results of causes adapted to produce them in a manner analogous to the objects of most familiar experience.

(223.) In the application of a machine there are three things to be considered. 1. The force or resistance which is required to be sustained, opposed, or overcome. 2. The force which is used to sustain, support, or overcome that resistance. 3. The machine itself by which the effect of this latter force is transmitted to the former. Of whatever nature be the force or the resistance which is to be sustained or overcome, it is technically called the _weight_, since, whatever it be, a weight of equivalent effect may always be found. The force which is employed to sustain or overcome it is technically called the _power_.

(224.) In expressing the effect of machinery it is usual to say that the power sustains the weight; but this, in fact, is not the case, and hence arises that appearance of paradox which has already been alluded to. If, for example, it is said that a power of one ounce sustains the weight of one ton, astonishment is not unnaturally excited, because the fact, as thus stated, if the terms be literally interpreted, is physically impossible. No power less than a ton can, in the ordinary acceptation of the word, support the weight of a ton. It will, however, be asked how it happens that a machine _appears_ to do this? how it happens that by holding a silken thread, which an ounce weight would snap, many hundred weight may be sustained? To explain this it will only be necessary to consider the effect of a machine, when the power and weight are in equilibrium.

(225.) In every machine there are some fixed points or props; and the arrangement of the parts is always such, that the pressure, excited by the power or weight, or both, is distributed among these props. If the weight amount to twenty hundred, it is possible so to distribute it, that any proportion, however great, of it may be thrown on the fixed points or props of the machine; the remaining part only can properly be said to be supported by the power, and this part can never be greater than the power. Considering the effect in this way, it appears that the power supports just so much of the weight and no more as is equal to its own force, and that all the remaining part of the weight is sustained by the machine. The force of these observations will be more apparent when the nature and properties of the mechanic powers and other machines have been explained.

(226.) When a machine is considered dynamically, its effects are explained on different principles. It is true that, in this case, a very small power may elevate a very great weight; but nevertheless, in so doing, whatever be the machine used, the total expenditure of power, in raising the weight through any height, is never less than that which would be expended if the power were immediately applied to the weight without the intervention of any machine. This circumstance arises from an universal property of machines by which the velocity of the weight is always less than that of the power, in exactly the same proportion as the power itself is less than the weight; so that when a certain power is applied to elevate a weight, the rate at which the elevation is effected is always slow in the same proportion as the weight is great. From a due consideration of this remarkable law, it will easily be understood, that a machine can never diminish the total expenditure of power necessary to raise any weight or to overcome any resistance. In such cases, all that a machine ever does or ever can do, is to enable the power to be expended at a slow rate, and in a more advantageous direction than if it were immediately applied to the weight or the resistance.

Let us suppose that P is a power amounting to an ounce, and that W is a weight amounting to 50 ounces, and that P elevates W by means of a machine. In virtue of the property already stated, it follows, that while P moves through 50 feet, W will be moved through 1 foot; but in moving P through 50 feet, 50 distinct efforts are made, by each of which 1 ounce is moved through 1 foot, and by which collectively 50 distinct ounces might be successively raised through 1 foot. But the weight W is 50 ounces, and has been raised through 1 foot; from whence it appears, that the expenditure of power is equal to that which would be necessary to raise the weight without the intervention of any machine.

This important principle may be presented under another aspect, which will perhaps render it more apparent. Suppose the weight W were actually divided into 50 equal parts, or suppose it were a vessel of liquid weighing 50 ounces, and containing 50 equal measures; if these 50 measures were successively lifted through a height of 1 foot; the efforts necessary to accomplish this would be the same as those used to move the power P through 50 feet, and it is obvious, that the total expenditure of force would be the same as that which would be necessary to lift the entire contents of the vessel through 1 foot.

When the nature and properties of the mechanic powers and other machines have been explained, the force of these observations will be more distinctly perceived. The effects of props and fixed points in sustaining a part of the weight, and sometimes the whole, both of the weight and power, will then be manifest, and every machine will furnish a verification of the remarkable proportion between the velocities of the weight and power, which has enabled us to explain what might otherwise be paradoxical and difficult of comprehension.

(227.) The most simple species of machines are those which are commonly denominated the MECHANIC POWERS. These have been differently enumerated by different writers. If, however, the object be to arrange in distinct classes, and in the smallest possible number of them, those machines which are alike in principle, the mechanic powers may be reduced to three.

1. The lever.
2. The cord.
3. The inclined plane.

To one or other of these classes all simple machines whatever may be reduced, and all complex machines may be resolved into simple elements which come under them.

(228.) The first class includes every machine which is composed of a solid body revolving on a fixed axis, although the name lever has been commonly confined to cases where the machine affects certain particular forms. This is by far the most useful class of machines, and will require in subsequent chapters very detailed development. The general principle, upon which equilibrium is established between the power and weight in machines of this class has been already explained in (183.) The power and weight are always supposed to be applied in directions at right angles to the axis. If lines be drawn from the axis perpendicular to the directions of power and weight, equilibrium will subsist, provided the power multiplied by the perpendicular distance of its direction from the axis, be equal to the weight multiplied by the perpendicular distance of its direction from the axis. This is a principle to which we shall have occasion to refer in explaining the various machines of this class.

(229.) If the moment of the power (184.) be greater than that of the weight, the effect of the power will prevail over that of the weight, and elevate it; but if, on the other hand, the moment of the power be less than that of the weight, the power will be insufficient to support the weight, and will allow it to fall.

(230.) The second class of simple machines includes all those cases in which force is transmitted by means of flexible threads, ropes, or chains. The principle, by which the effects of these machines are estimated, is, that the tension throughout the whole length of the same cord, provided it be perfectly flexible, and free from the effects of friction, must be the same. Thus, if a force acting at one end be balanced by a force acting at the other end, however the cord may be bent, or whatever course it may be compelled to take, by any causes which may affect it between its ends, these forces must be equal, provided the cord be free to move over any obstacles which may deflect it.

Within this class of machines are included all the various forms of _pulleys_.

(231.) The third class of simple machines includes all those cases in which the weight or resistance is supported or moved on a hard surface inclined to the vertical direction.

The effects of such machines are estimated by resolving the whole weight of the body into two elements by the parallelogram of forces. One of these elements is perpendicular to the surface, and supported by its resistance; the other is parallel to the surface, and supported by the power. The proportion, therefore, of the power to the weight will always depend on the obliquity of the surface to the direction of the weight. This will be easily understood by referring to what has been already explained in Chapter VIII.

Under this class of machines come the inclined plane, commonly so called, the wedge, the screw, and various others.

(232.) In order to simplify the development of the elementary theory of machines, it is expedient to omit the consideration of many circumstances, of which, however, a strict account must be taken before any practically useful application of that theory can be attempted. A machine, as we must for the present contemplate it, is a thing which can have no real or practical existence. Its various parts are considered to be free from friction: all surfaces which move in contact are supposed to be infinitely smooth and polished. The solid parts are conceived to be absolutely inflexible. The weight and inertia of the machine itself are wholly neglected, and we reason upon it as if it were divested of these qualities. Cords and ropes are supposed to have no stiffness, to be infinitely flexible. The machine, when it moves, is supposed to suffer no resistance from the atmosphere, and to be in all respects circumstanced as if it were _in vacuo_.

It is scarcely necessary to state, that, all these suppositions being false, none of the consequences deduced from them can be true. Nevertheless, as it is the business of art to bring machines as near to this state of ideal perfection as possible, the conclusions which are thus obtained, though false in a strict sense, yet deviate from the truth in but a small degree. Like the first outline of a picture, they resemble in their general features that truth to which, after many subsequent corrections, they must finally approximate.

After a first approximation has been made on the several false suppositions which have been mentioned, various effects, which have been previously neglected, are successively taken into account. Roughness, rigidity, imperfect flexibility, the resistance of air and other fluids, the effects of the weight and inertia of the machine, are severally examined, and their laws and properties detected. The modifications and corrections, thus suggested as necessary to be introduced into our former conclusions, are applied, and a second approximation, but still _only_ an approximation, to truth is made. For, in investigating the laws which regulate the several effects just mentioned, we are compelled to proceed upon a new group of false suppositions. To determine the laws which regulate the friction of surfaces, it is necessary to assume that every part of the surfaces of contact are uniformly rough; that the solid parts which are imperfectly rigid, and the cords which are imperfectly flexible, are constituted throughout their entire dimensions of a uniform material; so that the imperfection does not prevail more in one part than another. Thus, all irregularity is left out of account, and a general average of the effects taken. It is obvious, therefore, that by these means we have still failed in obtaining a result exactly conformable to the real state of things; but it is equally obvious, that we have obtained one much more conformable to that state than had been previously accomplished, and sufficiently near it for most practical purposes.

This apparent imperfection in our instruments and powers of investigation is not peculiar to mechanics: it pervades all departments of natural science. In astronomy, the motions of the celestial bodies, and their various changes and appearances as developed by theory, assisted by observation and experience, are only approximations to the real motions and appearances which take place in nature. It is true that these approximations are susceptible of almost unlimited accuracy; but still they are, and ever will continue to be, only approximations. Optics and all other branches of natural science are liable to the same observations.

CHAP. XIII.

OF THE LEVER.

(233.) An inflexible, straight bar, turning on an axis, is commonly called a _lever_. The _arms_ of the lever are those parts of the bar which extend on each side of the axis.

The axis is called the _fulcrum_ or _prop_.

(234.) Levers are commonly divided into three kinds, according to the relative positions of the power, the weight, and the fulcrum.

In a lever of the first kind, as in _fig. 78._, the fulcrum is between the power and weight.

In a lever of the second kind, as in _fig. 79._, the weight is between the fulcrum and power.

In a lever of the third kind, as in _fig. 80._, the power is between the fulcrum and weight.

(235.) In all these cases, the power will sustain the weight in equilibrium, provided its moment be equal to that of the weight. (184.) But the moment of the power is, in this case, equal to the product obtained by multiplying the power by its distance from the fulcrum; and the moment of the weight by multiplying the weight by its distance from the fulcrum. Thus, if the number of ounces in P, being multiplied by the number of inches in P F, be equal to the number of ounces in W, multiplied by the number of inches in W F, equilibrium will be established. It is evident from this, that as the distance of the power from the fulcrum increases in comparison to the distance of the weight from the fulcrum, in the same degree exactly will the proportion of the power to the weight diminish. In other words, 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.

In cases where a small power is required to sustain or elevate a great weight, it will therefore be necessary either to remove the power to a great distance from the fulcrum, or to bring the weight very near it.

(236.) Numerous examples of levers of the first kind may be given. A crow-bar, applied to elevate a stone or other weight, is an instance. The fulcrum is another stone placed near that which is to be raised, and the power is the hand placed at the other end of the bar.

A handspike is a similar example.

A poker applied to raise fuel is a lever of the first kind, the fulcrum being the bar of the grate.

Scissors, shears, nippers, pincers, and other similar instruments are composed of two levers of the first kind; the fulcrum being the joint or pivot, and the weight the resistance of the substance to be cut or seized; the power being the fingers applied at the other end of the levers.

The brake of a pump is a lever of the first kind; the pump-rods and piston being the weight to be raised.

(237.) Examples of levers of the second kind, though not so frequent as those just mentioned, are not uncommon.

An oar is a lever of the second kind. The reaction of the water against the blade is the fulcrum. The boat is the weight, and the hand of the boatman the power.

The rudder of a ship or boat is an example of this kind of lever, and explained in a similar way.

The chipping knife is a lever of the second kind. The end attached to the bench is the fulcrum, and the weight the resistance of the substance to be cut, placed beneath it.

A door moved upon its hinges is another example.