Chapter IV. The impelling force being equally distributed among all the
parts, the velocity will be found by dividing the numerical value of that force by the number expressing the mass.
If any number of impacts be given simultaneously to different points of a body, a certain complex motion will generally ensue. The mass will have a relative motion round the centre of gravity as if it were fixed, while that point will move forward uniformly in a straight line, carrying the body with it. The relative motion of the mass round the centre of gravity may be found by considering the centre of gravity as a fixed point, round which the mass is free to move, and then determining the motion which the applied forces would produce. This motion being supposed to continue uninterrupted, let all the forces be imagined to be applied in their proper directions and quantities to the centre of gravity. By the principles for the composition of force they will be mechanically equivalent to a single force through that point. In the direction of this single force the centre of gravity will move and have the same velocity as if the whole mass were there concentrated and received the impelling forces.
(179.) These general properties, which are entirely independent of gravity, render the “centre of gravity” an inadequate title for this important point. Some physical writers have, consequently, called it the “centre of inertia.” The “centre of gravity,” however, is the name by which it is still generally designated.
CHAP. X.
THE MECHANICAL PROPERTIES OF AN AXIS.
(180.) When a body has a motion of rotation, the line round which it revolves is called an _axis_. Every point of the body must in this case move in a circle, whose centre lies in the axis, and whose radius is the distance of the point from the axis. Sometimes while the body revolves, the axis itself is moveable, and not unfrequently in a state of actual motion. The motions of the earth and planets, or that of a common spinning-top, are examples of this. The cases, however, which will be considered in the present chapter, are chiefly those in which the axis is immovable, or at least where its motion has no relation to the phenomena under investigation. Instances of this are so frequent and obvious, that it seems scarcely necessary to particularise them. Wheel-work of every description, the moving parts of watches and clocks, turning lathes, mill-work, doors and lids on hinges, are all obvious examples. In tools or other instruments which work on joints or pivots, such as scissors, shears, pincers, although the joint or pivot be not absolutely fixed, it is to be considered so in reference to the mechanical effect.
In some cases, as in most of the wheels of watches and clocks, fly-wheels and chucks of the turning lathe, and the arms of wind-mills, the body turns continually in the same direction, and each of its points traverses a complete circle during every revolution of the body round its axis. In other instances the motion is alternate or reciprocating, its direction being at intervals reversed. Such is the case in pendulums of clocks, balance-wheels of chronometers, the treddle of the lathe, doors and lids on hinges, scissors, shears, pincers, &c. When the alternation is constant and regular, it is called _oscillation_ or _vibration_, as in pendulums and balance-wheels.
(181.) To explain the properties of an axis of rotation it will be necessary to consider the different kinds of forces to the action of which a body moveable on such an axis may be submitted, to show how this action depends on their several quantities and directions, to distinguish the cases in which the forces neutralise each other and mutually equilibrate from those in which motion ensues, to determine the effect which the axis suffers, and, in the cases where motion is produced, to estimate the effects of those centrifugal forces (137.) which are created by the mass of the body whirling round the axis.
Forces in general have been distinguished by the duration of their action into instantaneous and continued forces. The effect of an instantaneous force is produced in an infinitely short time. If the body which sustains such an action be previously quiescent and free, it will move with a uniform velocity in the direction of the impressed force. (93.) If, on the other hand, the body be not free, but so restrained that the impulse cannot put it in motion, then the fixed points or lines which resist the motion sustain a corresponding shock at the moment of the impulse. This effect, which is called _percussion_, is, like the force which causes it, instantaneous.
A continued force produces a continued effect. If the body be free and previously quiescent, this effect is a continual increase of velocity. If the body be so restrained that the applied force cannot put it in motion, the effect is a continued pressure on the points or lines which sustain it. (94.)
It may happen, however, that although the body be not absolutely free to move in obedience to the force applied to it, yet still it may not be altogether so restrained as to resist the effect of that force and remain at rest. If the point at which a force is applied be free to move in a certain direction not coinciding with that of the applied force, that force will be resolved into two elements; one of which is in the direction in which the point is free to move, and the other at right angles to that direction. The point will move in obedience to the former element, and the latter will produce percussion or pressure on the points or lines which restrain the body. In fact, in such cases the resistance offered by the circumstances which confine the motion of the body modifies the motion which it receives, and as every change of motion must be the consequence of a force applied (44.), the fixed points or lines which offer the resistance must suffer a corresponding effect.
It may happen that the forces impressed on the body, whether they be continued or instantaneous, are such as, were it free, would communicate to it a motion which the circumstances which restrain it do not forbid it to receive. In such a case the fixed points or lines which restrain the body sustain no force, and the phenomena will be the same in all respects as if these points or lines were not fixed.
It will be easy to apply these general reflections to the case in which a solid body is moveable on a fixed axis. Such a body is susceptible of no motion except one of rotation on that axis. If it be submitted to the action of instantaneous forces, one or other of the following effects must ensue. 1. The axis may resist the forces, and prevent any motion. 2. The axis may modify the effect of the forces sustaining a corresponding percussion, and the body receiving a motion of rotation. 3. The forces applied may be such as would cause the body to spin round the axis even were it not fixed, in which case the body will receive a motion of rotation, but the axis will suffer no percussion.
What has been just observed of the effect of instantaneous forces is likewise applicable to continued ones. 1. The axis may entirely resist the effect of such forces, in which case it will suffer a pressure which may be estimated by the rules for the composition of force. 2. It may modify the effect of the applied forces, in which case it must also sustain a pressure, and the body must receive a motion of rotation which is subject to constant variation, owing to the incessant action of the forces. 3. The forces may be such as would communicate to the body the same rotatory motion if the axis were not fixed. In this case the forces will produce no pressure on the axis.
The impressed forces are not the only causes which affect the axis of a body during the phenomenon of rotation. This species of motion calls into action other forces depending on the inertia of the mass, which produce effects upon the axis, and which play a prominent part in the theory of rotation. While the body revolves on its axis, the component particles of its mass move in circles, the centres of which are placed in the axis. The radius of the circle in which each particle moves is the line drawn from that particle perpendicular to the axis. It has been already proved that a particle of matter, moving round a centre, is attended with a centrifugal force proportionate to the radius of the circle in which it moves and to the square of its angular velocity. When a solid body revolves on its axis, all its parts are whirled round together, each performing a complete revolution in the same time. The angular velocity is consequently the same for all, and the difference of the centrifugal forces of different particles must entirely depend upon their distances from the axis. The tendency of each particle to fly from the axis, arising from the centrifugal force, is resisted by the cohesion of the parts of the mass, and in general this tendency is expended in exciting a pressure or strain upon the axis. It ought to be recollected, however, that this pressure or strain is altogether different from that already mentioned, and produced by the forces which give motion to the body. The latter depends entirely upon the quantity and directions of the applied forces in relation to the axis: the former depends on the figure and density of the body, and the velocity of its motion.
These very complex effects render a simple and elementary exposition of the mechanical properties of a fixed axis a matter of considerable difficulty. Indeed, the complete mathematical development of this theory long eluded the skill of the most acute geometers, and it was only at a comparatively late period that it yielded to the searching analysis of modern science.
(182.) To commence with the most simple case, we shall consider the body as submitted to the action of a single force. The effect of this force will vary according to the relation of its direction to that of the axis. There are two ways in which a body may be conceived to be moveable around an axis. 1. By having pivots at two points which rest in sockets, so that when the body is moved it must revolve round the right line joining the pivots as an axis. 2. A thin cylindrical rod may pass through the body, on which it may turn in the same manner as a wheel upon its axle.
If the force be applied to the body in the direction of the axis, it is evident that no motion can ensue, and the effect produced will be a pressure on that pivot towards which the force is directed. If in this case the body revolved on a cylindrical rod, the tendency of the force would be to make it slide along the rod without revolving round it.
Let us next suppose the force to be applied not in the direction of the axis itself, but parallel to it. Let A B, _fig. 70._, be the axis, and let C D be the direction of the force applied. The pivots being supposed to be at A and B, draw A G and B F perpendicular to A B. The force C D will be equivalent to three forces, one acting from B towards A, equal in quantity to the force C D. This force will evidently produce a corresponding pressure on the pivot A. The other two forces will act in the directions A G and B F, and will have respectively to the force C D the same proportion as A E has to A B. Such will be the mechanical effect of a force C D parallel to the axis. And as these effects are all directed on the pivots, no motion can ensue.
If the body revolve on a cylindrical rod, the forces A G and B F would produce a strain upon the axis, while the third force in the direction B A would have a tendency to make the body slide along it.
(183.) If the force applied to the body be directed upon the axis, and at right angles to it, no motion can be produced. In this case, if the body be supported by pivots at A and B, the force K L, perpendicular to the line A B, will be distributed between the pivots, producing a pressure on each proportional to its distance from the other. The pressure on A having to the pressure on B the same proportion as L B has to L A.
If the force K H be directed obliquely to the axis, it will be equivalent to two forces (76.), one K L perpendicular to the axis, and the other K M parallel to it. The effect of each of these may be investigated as in the preceding cases.
In all these observations the body has been supposed to be submitted to the action of one force only. If several forces act upon it, the direction of each of them crossing the axis either perpendicularly or obliquely, or taking the direction of the axis or any parallel direction, their effects may be similarly investigated. In the same manner we may determine the effects of any number of forces whose combined results are mechanically equivalent to forces which either intersect the axis or are parallel to it.
(184.) If any force be applied whose direction lies in a plane oblique to the axis, it can always be resolved into two elements (76.), one of which is parallel to the axis, and the other in a plane perpendicular to it. The effect of the former has been already determined, and therefore we shall at present confine our attention to the latter.
Suppose the axis to be perpendicular to the paper, and to pass through the point G, _fig. 71._ and let A B C be a section of the body. It will be convenient to consider the section vertical and the axis horizontal, omitting, however, any notice of the effect of the weight of the body.
Let a weight W be suspended by a cord Q W from any point Q. This weight will evidently have a tendency to turn the body round in the direction A B C. Let another cord be attached to any other point P, and, being carried over a wheel R, let a dish S be attached to it, and let fine sand be poured into this dish until the tendency of S to turn the body round the axis in the direction of C B A balances the opposite tendency of W. Let the weights of W and S be then exactly ascertained, and also let the distances G I and G H of the cords from the axis be exactly measured. It will be found that, if the number of ounces in the weight S be multiplied by the number of inches in G H, and also the number of ounces in W by the number of inches in G I, equal products will be obtained. This experiment may be varied by varying the position of the wheel R, and thereby changing the direction of the string P R, in which cases it will be always found necessary to vary the weight of S in such a manner, that when the number of ounces in it is multiplied by the number of inches in the distance of the string from the axis, the product obtained shall be equal to that of the weight W by the distance G I. We have here used ounces and inches as the measures of weight and distance; but it is obvious that any other measures would be equally applicable.
From what has been just stated it follows, that the energy of the weight of S to move the body on its axis, does not depend alone upon the actual amount of that weight, but also upon the distance of the string from the axis. If, while the position of the string remains unaltered, the weight of S be increased or diminished, the resisting weight W must be increased or diminished in the same proportion. But if, while the weight of S remains unaltered, the distance of the string P R from the axis G be increased or diminished, it will be found necessary to increase or diminish the resisting weight W in exactly the same proportion. It therefore appears that the increase or diminution of the distance of the direction of a force from the axis has the same effect upon its power to give rotation as a similar increase or diminution of the force itself. The power of a force to produce rotation is, therefore, accurately estimated, not by the force alone, but by the product found by multiplying the force by the distance of its direction from the axis. It is frequently necessary in mechanical science to refer to this power of a force, and, accordingly, the product just mentioned has received a particular denomination. It is called the _moment_ of the force round the axis.
(185.) The distance of the direction of a force from the axis is sometimes called the _leverage_ of the force. The _moment_ of a force is therefore found by multiplying the force by its leverage, and the energy of a given force to turn a body round an axis is proportional to the leverage of that force.
From all that has been observed it may easily be inferred that, if several forces affect a body moveable on an axis, having tendencies to turn it in different directions, they will mutually neutralise each other and produce equilibrium, if the sum of the moments of those forces which tend to turn the body in one direction be equal to the sum of the moments of those which tend to turn it in the opposite direction. Thus, if the forces A, B, C, ... tend to turn the body from right to left, and the distances of their directions from the axis be _a_, _b_, _c_, ... and the forces A′, B′, C′, ... tend to move it from left to right, and the distances of their directions from the axis be _a′_, _b′_, _c′_, ...; then these forces will produce equilibrium, if the products found by multiplying the ounces in A, B, C, ... respectively by the inches in _a_, _b_, _c_, ... when added together be equal to the products found by multiplying the ounces in A′, B′, C′, ... by the inches in _a′_, _b′_, _c′_, ... respectively when added together. But if either of these sets of products when added together exceed the other, the corresponding set of forces will prevail, and the body will revolve on its axis.
(186.) When a body receives an impulse in a direction perpendicular to the axis, but not crossing it, a uniform rotatory motion is produced. The velocity of this motion depends on the force of the impulse, the distance of the direction of the impulse from the axis, and the manner in which the mass of the body is distributed round the axis. It is to be considered that the whole force of the impulse is shared amongst the various parts of the mass, and is transmitted to them from the point where the impulse is applied by reason of the cohesion and tenacity of the parts, and the impossibility of one part yielding to a force without carrying all the other parts with it. The force applied acts upon those particles nearer to the axis than its own direction under advantageous circumstances; for, according to what has been already explained, their power to resist the effect of the applied force is small in the same proportion with their distance. On the other hand, the applied force acts upon particles of the mass, at a greater distance than its own direction, under circumstances proportionably disadvantageous; for their resistance to the applied force is great in proportion to their distances from the axis.
Let C D, _fig. 72._, be a section of the body made by a plane passing through the axis A B. Suppose the impulse to be applied at P, perpendicular to this plane, and at the distance P O from the axis. The effect of the impulse being distributed through the mass will cause the body to revolve on A B, with a uniform velocity. There is a certain point G, at which, if the whole mass were concentrated, it would receive from the impulse the same velocity round the axis. The distance O G is called the _radius of gyration_ of the axis A B, and the point G is called the _centre of gyration_ relatively to that axis. The effect of the impulse upon the mass concentrated at G is great in exactly the same proportion as O G is small. This easily follows from the property of moments which has been already explained; from whence it may be inferred, that the greater the radius of gyration is, the less will be the velocity which the body will receive from a given impulse.
(187.) Since the radius of gyration depends on the manner in which the mass is arranged round the axis, it follows that for different axes in the same body there will be different radii of gyration. Of all axes taken in the same body parallel to each other, that which passes through the centre of gravity has the least radius of gyration. If the radius of gyration of any axis passing through the centre of gravity be given, that of any parallel axis can be found; for the square of the radius of gyration of any axis is equal to the square of the distance of that axis from the centre of gravity added to the square of the radius of gyration of the parallel axis through the centre of gravity.
(188.) The product of the numerical expressions for the mass of the body and the square of the radius of gyration is a quantity much used in mechanical science, and has been called the _moment of inertia_. The moments of inertia, therefore, for different axes in the same body are proportional to the squares of the corresponding radii of gyration; and consequently increase as the distances of the axes from the centre of gravity increase. (187.)
(189.) From what has been explained in (187.), it follows, that the moment of inertia of any axis may be computed by common arithmetic, if the moment of inertia of a parallel axis through the centre of gravity be previously known. To determine this last, however, would require analytical processes altogether unsuitable to the nature and objects of the present treatise.
The velocity of rotation which a body receives from a given impulse is great in exactly the same proportion as the moment of inertia is small. Thus the moment of inertia may be considered in rotatory motion analogous to the mass of the body in rectilinear motion.
From what has been explained in (187.) it follows that a given impulse at a given distance from the axis will communicate the greatest angular velocity when the axis passes through the centre of gravity, and that the velocity which it will communicate round other axes will be diminished in the same proportion as the squares of their distances from the centre of gravity added to the square of the radius of gyration for a parallel axis through the centre of gravity are augmented.
(190.) If any point whatever be assumed in a body, and right lines be conceived to diverge in all directions from that point, there are generally two of these lines, which being taken as axes of rotation, one has a greater and the other a less moment of inertia than any of the others. It is a remarkable circumstance, that, whatever be the nature of the body, whatever be its shape, and whatever be the position of the point assumed, these two axes of greatest and least moment will always be at right angles to each other.
These axes and a third through the same point, and at right angles to both of them, are called the _principal axes_ of that point from which they diverge. To form a distinct notion of their relative position, let the axis of greatest moment be imagined to lie horizontally from north to south, and the axis of least moment from east to west; then the third principal axis will be presented perpendicularly upwards and downwards. The first two being called the principal axes of greatest and least moment, the third may be called the _intermediate principal axis_.
(191.) Although the moments of the three principal axes be in general unequal, yet bodies may be found having certain axes for which these moments may be equal. In some cases the moment of the intermediate axis is equal to that of the principal axis of greatest moment: in others it is equal to that of the principal axis of least moment, and in others the moments of all the three principal axes are equal to each other.
If the moments of any two of three principal axes be equal, the moments of all axes through the same point and in their plane will also be equal; and if the moments of the three principal axes through a point be equal, the moments of all axes whatever, through the same point, will be equal.
(192.) If the moments of the principal axes through the centre of gravity be known, the moments for all other axes through that point may be easily computed. To effect this it is only necessary to multiply the moments of the principal axes by the squares of the co-sines of the angles formed by them respectively with the axis whose moment is sought. The products being added together will give the required moment.
(193.) By combining this result with that of (189.), it will be evident that the moment of all axes whatever may be determined, if those of the principal axes through the centre of gravity be known.
(194.) It is obvious that the principal axis of least moment through the centre of gravity has a less moment of inertia than any other axis whatever. For it has, by its definition (190.) a less moment of inertia than any other axis through the centre of gravity, and every other axis through the centre of gravity has a less moment of inertia than a parallel axis through any other point (187.) and (189.)
(195.) If two of the principal axes through the centre of gravity have equal moments of inertia, all axes in any plane parallel to the plane of these axes, and passing through the point where a perpendicular from the centre of gravity meets that plane, must have equal moments of inertia. For by (191.) all axes in the plane of those two have equal moments, and by (189.) the axes in the parallel plane have moments which exceed these by the same quantity, being equally distant from them. (187.)
Hence it is obvious that if the three principal axes through the centre of gravity have equal moments, all axes situated in any given plane, and passing through the point where the perpendicular from the centre of gravity meets that plane, will have equal moments, being equally distant from parallel axes through the centre of gravity.
(196.) If the three principal axes through the centre of gravity have unequal moments, there is no point whatever for which all axes will have equal moments; but if the principal axis of least moment and the intermediate principal axis through the centre of gravity have equal moments, then there will be two points on the principal axis of greatest moment, equally distant at opposite sides of the centre of gravity, at which all axes will have equal moments. If the three principal axes through the centre of gravity have equal moments, no other point of the body can have principal axes of equal moment.
(197.) When a body revolves on a fixed axis, the parts of its mass are whirled in circles round the axis; and since they move with a common angular velocity, they will have centrifugal forces proportional to their distances from the axis. If the component parts of the mass were not united together by cohesive forces of energies greater than these centrifugal forces, they would be separated, and would fly off from the axis; but their cohesion prevents this, and causes the effects of the different centrifugal forces, which affect the different parts of the mass, to be transmitted so as to modify each other, and finally to produce one or more forces mechanically equivalent to the whole, and which are exerted upon the axis and resisted by it. We propose now to explain these effects, as far as it is possible to render them intelligible without the aid of mathematical language.
It is obvious that any number of equal parts of the mass, which are uniformly arranged in a circle round the axis, have equal centrifugal forces acting from the centre of the circle in every direction. These mutually neutralise each other, and therefore exert no force on the axis. The same may be said of all parts of the mass which are regularly and equally distributed on every side of the axis.
Also if equal masses be placed at equal distances on opposite sides of the axis, their centrifugal forces will destroy each other. Hence it appears that the pressure which the axis of rotation sustains from the centrifugal forces of the revolving mass, arises from the unequal distribution of the matter around it.
From this reasoning it will be easily perceived that in the following examples the axis of rotation will sustain no pressure.
A globe revolving on any of its diameters, the density being the same at equal distances from the centre.
A spheroid or a cylinder revolving on its axis, the density being equal at equal distances from the axis.
A cube revolving on an axis which passes through the centre of two opposite bases, being of uniform density.
A circular plate of uniform thickness and density revolving on one of its diameters as an axis.
(198.) In all these examples it will be observed that the axis of rotation passes through the centre of gravity. The general theorem, of which they are only particular instances, is, “if a body revolve on a principal axis, passing through the centre of gravity, the axis will sustain no pressure from the centrifugal force of the revolving mass.” This is a property in which the principal axes through the centre of gravity are unique. There is no other axis on which a body could revolve without pressure.
If two of the principal axes through the centre of gravity have equal moments, every axis in their plane has the same moment, and is to be considered equally as a principal axis. In this case the body would revolve on any of these axes without pressure.
A homogeneous spheroid furnishes an example of this. If any of the diameters of the earth’s equator were a fixed axis, the earth would revolve on it without producing pressure.
If the three principal axes through the centre of gravity have equal moments, all axes through the centre of gravity are to be considered as principal axes. In this case the body would revolve without pressure on any axis through the centre of gravity.
A globe, in which the density of the mass at equal distances from the centre is the same, is an example of this. Such a body would revolve without pressure on any axis through its centre.
(199.) Since no pressure is excited on the axis in these cases, the state of the body will not be changed, if during its rotation the axis cease to be fixed. The body will notwithstanding continue to revolve round the axis, and the axis will maintain its position.
Thus a spinning-top of homogeneous material and symmetrical form will revolve steadily in the same position, until the friction of its point with the surface on which it rests deprives it of motion. This is a phenomenon which can only be exhibited when the axis of rotation is a principal axis through the centre of gravity.
(200.) If the body revolve round any axis through the centre of gravity, which is not a principal axis, the centrifugal pressure is represented by two forces, which are equal and parallel, but which act in opposite directions on different points of the axis. The effect of these forces is to produce a strain upon the axis, and give the body a tendency to move round another axis at right angles to the former.
(201.) If the fixed axis on which a body revolves be a principal axis through any point different from the centre of gravity, then a pressure will be produced by the centrifugal force of the revolving mass, and this pressure will act at right angles to the axis on the point to which it is a principal axis, and in the plane through that axis and the centre of gravity. The amount of the pressure will be proportional to the mass of the body, the distance of the centre of gravity from the axis, and the square of the velocity of rotation.
(202.) Since the whole pressure is in this case excited on a single point, the stability of the axis will not be disturbed, provided that point alone be fixed. So that even though the axis should be free to turn on that point, no motion will ensue as long as no external forces act upon the body.
(203.) If the axis of rotation be not a principal axis, the centrifugal forces will produce an effect which cannot be represented by a single force. The effect may be understood by conceiving two forces to act on _different points_ of the axis at right angles to it and to each other. The quantities of these pressures and their directions depend on the figure and density of the mass and the position of the axis, in a manner which cannot be explained without the aid of mathematical language and principles.
(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.
Nut-crackers are two levers of the second kind; the hinge which unites them being the fulcrum, the resistance of the shell placed between them being the weight, and the hand applied to the extremity being the power.
A wheelbarrow is a lever of the second kind; the fulcrum being the point at which the wheel presses on the ground, and the weight being that of the barrow and its load, collected at their centre of gravity.
The same observation may be applied to all two-wheeled carriages, which are partly sustained by the animal which draws them.
(238.) In a lever of the third kind, the weight, being more distant from the fulcrum than the power, must be proportionably less than it. In this instrument, therefore, the power acts upon the weight to a mechanical disadvantage, inasmuch as a greater power is necessary to support or move the weight than would be required if the power were immediately applied to the weight, without the intervention of a machine. We shall, however, hereafter show that the advantage which is lost in force is gained in despatch, and that in proportion as the weight is less than the power which moves it, so will the speed of its motion be greater than that of the power.
Hence a lever of the third kind is only used in cases where the exertion of great power is a consideration subordinate to those of rapidity and despatch.
The most striking example of levers of the third kind is found in the animal economy. The limbs of animals are generally levers of this description. The socket of the bone is the fulcrum; a strong muscle attached to the bone near the socket is the power; and the weight of the limb, together with whatever resistance is opposed to its motion, is the weight. A slight contraction of the muscle in this case gives a considerable motion to the limb: this effect is particularly conspicuous in the motion of the arms and legs in the human body; a very inconsiderable contraction of the muscles at the shoulders and hips giving the sweep to the limbs from which the body derives so much activity.
The treddle of the turning lathe is a lever of the third kind. The hinge which attaches it to the floor is the fulcrum, the foot applied to it near the hinge is the power, and the crank upon the axis of the fly-wheel, with which its extremity is connected, is the weight.
Tongs are levers of this kind, as also the shears used in shearing sheep. In these cases the power is the hand placed immediately below the fulcrum or point where the two levers are connected.
(239.) When the power is said to support the weight by means of a lever or any other machine, it is only meant that the power keeps the machine in equilibrium, and thereby enables it to sustain the weight. It is necessary to attend to this distinction, to remove the difficulty which may arise from the paradox of a small power sustaining a great weight.
In a lever of the first kind, the fulcrum F, _fig. 78._, or axis, sustains the united forces of the power and weight.
In a lever of the second kind, if the power be supposed to act over a wheel R, _fig. 79._, the fulcrum F sustains a pressure equal to the difference between the power and weight, and the axis of the wheel R sustains a pressure equal to twice the power; so that the total pressures on F and R are equivalent to the united forces of the power and weight.
In a lever of the third kind similar observations are applicable. The wheel R, _fig. 80._, sustains a pressure equal to twice the power, and the fulcrum F sustains a pressure equal to the difference between the power and weight.
These facts may be experimentally established by attaching a string to the lever immediately over the fulcrum, and suspending the lever by that string from the arm of a balance. The counterpoising weight, when the fulcrum is removed, will, in the first case, be equal to the sum of the weight and power, and in the last two cases equal to their difference.
(240.) We have hitherto omitted the consideration of the effect of the weight of the lever itself. If the centre of gravity of the lever be in the vertical line through the axis, the weight of the instrument will have no other effect than to increase the pressure on the axis by its own amount. But if the centre of gravity be on the same side of the axis with the weight, as at G, it will oppose the effect of the power, a certain part of which must therefore be allowed to support it. To ascertain what part of the power is thus expended, it is to be considered that the moment of the weight of the lever collected at G, is found by multiplying that weight by the distance G F. The moment of that part of the power which supports this must be equal to it; therefore, it is only necessary to find how much of the power multiplied by P F will be equal to the weight of the lever multiplied by G F. This is a question in common arithmetic.
If the centre of gravity of the lever be at a different side of the axis from the weight, as at G′, the weight of the instrument will co-operate with the power in sustaining the weight W. To determine what portion of the weight W is thus sustained by the weight of the lever, it is only necessary to find how much of W, multiplied by the distance W F, is equal to the weight of the lever multiplied by G′ F.
In these cases the pressure on the fulcrum, as already estimated, will always be increased by the weight of the lever.
(241.) The sense in which a small power is said to sustain a great weight, and the manner of accomplishing this, being explained, we shall now consider how the power is applied in moving the weight. Let P W, _fig. 81._, be the places of the power and weight, and F that of the fulcrum, and let the power be depressed to P′ while the weight is raised to W′. The space P P′ evidently bears the same proportion to W W′, as the arm P F to W F. Thus if P F be ten times W F, P P′ will be ten times W W′. A power of one pound at P being moved from P to P′, will carry a weight of ten pounds from W to W′. But in this case it ought not to be said, that a lesser weight moves a greater, for it is not difficult to show, that the total expenditure of force in the motion of one pound from P to P′ is exactly the same as in the motion of ten pounds from W to W′. If the space P P′ be ten inches, the space W W′ will be one inch. A weight of one pound is therefore moved through ten successive inches, and in each inch the force expended is that which would be sufficient to move one pound through one inch. The total expenditure of force from P to P′ is ten times the force necessary to move one pound through one inch, or what is the same, it is that which would be necessary to move ten pounds through one inch. But this is exactly what is accomplished by the opposite end W of the lever; for the weight W is ten pounds, and the space W W′ is one inch.
If the weight W of ten pounds could be conveniently divided into ten equal parts of one pound each, each part might be separately raised through one inch, without the intervention of the lever or any other machine. In this case, the same quantity of power would be expended, and expended in the same manner as in the case just mentioned.
It is evident, therefore, that when a machine is applied to raise a weight or to overcome resistance, as much force must be really used as if the power were immediately applied to the weight or resistance. All that is accomplished by the machine is to enable the power to do that by a succession of distinct efforts which should be otherwise performed by a single effort. These observations will be found to be applicable to all machines whatever.
(242.) Weighing machines of almost every kind, whether used for commercial or philosophical purposes, are varieties of the lever. The common balance, which, of all weighing machines, is the most perfect and best adapted for ordinary use, whether in commerce or experimental philosophy, is a lever with equal arms. In the steel-yard one weight serves as a counterpoise and measure of others of different amount, by receiving a leverage variable according to the varying amount of the weight against which it acts. A detailed account of such instruments will be found in Chapter XXI.
(243.) We have hitherto considered the power and weight as acting on the lever, in directions perpendicular to its length and parallel to each other. This does not always happen. Let A B, _fig. 83._, be a lever whose fulcrum is F, and let A R be the direction of the power, and B S the direction of the weight. If the lines R A and S B be continued, and perpendiculars F C and F D drawn from the fulcrum to those lines, the moment of the power will be found by multiplying the power by the line F C, and the moment of the weight by multiplying the weight by F D. If these moments be equal, the power will sustain the weight in equilibrium. (185).
It is evident, that the same reasoning will be applicable when the arms of the lever are not in the same direction. These arms may be of any figure or shape, and may be placed relatively to each other in any position.
(244.) In the rectangular lever the arms are perpendicular to each other, and the fulcrum F, _fig. 84._, is at the right angle. The moment of the power, in this case, is P multiplied by A F, and that of the weight W multiplied by B F. When the instrument is in equilibrium these moments must be equal.
When the hammer is used for drawing a nail, it is a lever of this kind: the claw of the hammer is the shorter arm; the resistance of the nail is the weight; and the hand applied to the handle the power.
(245.) When a beam rests on two props A B, _fig. 85._, and supports, at some intermediate place C, a weight W, this weight is distributed between the props in a manner which may be determined by the principles already explained. If the pressure on the prop B be considered as a power sustaining the weight W, by means of the lever of the second kind B A, then this power multiplied by B A must be equal to the weight multiplied by C A. Hence the pressure on B will be the same fraction of the weight as the part A C is of A B. In the same manner it may be proved, that the pressure on A is the same fraction of the weight as B C is of B A. Thus, if A C be one third, and therefore B C two thirds of B A, the pressure on B will be one third of the weight, and the pressure on A two thirds of the weight.
It follows from this reasoning, that if the weight be in the middle, equally distant from B and A, each prop will sustain half the weight. The effect of the weight of the beam itself may be determined by considering it to be collected at its centre of gravity. If this point, therefore, be equally distant from the props, the weight of the beam will be equally distributed between them.
According to these principles, the manner in which a load borne on poles between two bearers is distributed between them may be ascertained. As the efforts of the bearers and the direction of the weight are always parallel; the position of the poles relatively to the horizon makes no difference in the distribution of the weights between the bearers. Whether they ascend or descend, or move on a level plane, the weight will be similarly shared between them.
If the beam extend beyond the prop, as in _fig. 86._, and the weight be suspended at a point not placed between them, the props must be applied at different sides of the beam. The pressures which they sustain may be calculated in the same manner as in the former case. The pressure of the prop B may be considered as a power sustaining the weight W by means of the lever B C. Hence, the pressure of B, multiplied by B A, must be equal to the weight W multiplied by A C. Therefore, the pressure on B bears the same proportion to the weight as A C does to A B. In the same manner, considering B as a fulcrum, and the pressure of the prop A as the power, it may be proved that the pressure of A bears the same proportion to the weight as the line B C does to A B. It therefore appears, that the pressure on the prop A is greater than the weight.
(246.) When great power is required, and it is inconvenient to construct a long lever, a combination of levers may be used. In _fig. 87._ such a system of levers is represented, consisting of three levers of the first kind. The manner in which the effect of the power is transmitted to the weight may be investigated by considering the effect of each lever successively. The power at P produces an upward force at P′, which bears to P the same proportion as P′ F to P F. Therefore, the effect at P′ is as many times the power as the line P F is of P′ F. Thus, if P F be ten times P′ F, the upward force at P′ is ten times the power. The arm P′ F′ of the second lever is pressed upwards by a force equal to ten times the power at P. In the same manner this may be shown to produce an effect at P″ as many times greater than P′ as P′ F′ is greater than P″ F′. Thus, if P′ F′ be twelve times P″ F′, the effect at P″ will be twelve times that of P′. But this last was ten times the power, and therefore the P″ will be one hundred and twenty times the power. In the same manner it may be shown that the weight is as many times greater than the effect at P″ as P″ F″ is greater than W F″. If P″ F″ be five times W F″, the weight will be five times the effect at P″. But this effect is one hundred and twenty times the power, and therefore the weight would be six hundred times the power.
In the same manner the effect of any compound system of levers may be ascertained by taking the proportion of the weight to the power in each lever separately, and multiplying these numbers together. In the example given, these proportions are 10, 12, and 5, which multiplied together give 600. In _fig. 87._ the levers composing the system are of the first kind; but the principles of the calculation will not be altered if they be of the second or third kind, or some of one kind and some of another.
(247.) That number which expresses the proportion of the weight to the equilibrating power in any machine, we shall call the _power of the machine_. Thus, if, in a lever, a power of one pound support a weight of ten pounds, the power of the machine is _ten_. If a power of 2lbs. support a weight of 11lbs., the power of the machine is 5-1/2, 2 being contained in 11 5-1/2 times.
(248.) As the distances of the power and weight from the fulcrum of a lever may be varied at pleasure, and any assigned proportion given to them, a lever may always be conceived having a power equal to that of any given machine. Such a lever may be called, in relation to that machine, the _equivalent lever_.
As every complex machine consists of a number of simple machines acting one upon another, and as each simple machine may be represented by an equivalent lever, the complex machine will be represented by a compound system of equivalent levers. From what has been proved in (246.), it therefore follows that the power of a complex machine may be calculated by multiplying together the powers of the several simple machines of which it is composed.
CHAP. XIV.
OF WHEEL-WORK.
(249.) When a lever is applied to raise a weight, or overcome a resistance, the space through which it acts at any one time is small, and the work must be accomplished by a succession of short and intermitting efforts. In _fig. 81._, after the weight has been raised from W to W′, the lever must again return to its first position, to repeat the action. During this return the motion of the weight is suspended, and it will fall downwards unless some provision be made to sustain it. The common lever is, therefore, only used in cases where weights are required to be raised through small spaces, and under these circumstances its great simplicity strongly recommends it. But where a continuous motion is to be produced, as in raising ore from the mine, or in weighing the anchor of a vessel, some contrivance must be adopted to remove the intermitting action of the lever, and render it continual. The various forms given to the lever, with a view to accomplish this, are generally denominated the _wheel and axle_.
In _fig. 88._, A B is a horizontal axle, which rests in pivots at its extremities, or is supported in gudgeons, and capable of revolving. Round this axis a rope is coiled, which sustains the weight W. On the same axis a wheel C is fixed, round which a rope is coiled in a contrary direction, to which is appended the power P. The moment of the power is found by multiplying it by the radius of a wheel, and the moment of the weight, by multiplying it by the radius of its axle. If these moments be equal (185.), the machine will be in equilibrium. Whence it appears that the power of the machine (247.) is expressed by the proportion which the radius of the wheel bears to the radius of the axle; or, what is the same, of the diameter of the wheel to the diameter of the axle. This principle is applicable to the wheel and axle in every variety of form under which it can be presented.
(250.) It is evident that as the power descends continually, and the rope is uncoiled from the wheel, the weight will be raised continually, the rope by which it is suspended being at the same time coiled upon the axle.
When the machine is in equilibrium, the forces of both the weight and power are sustained by the axle, and distributed between its props, in the manner explained in (245.)
When the machine is applied to raise a weight, the velocity with which the power moves is as many times greater than that with which the weight rises, as the weight itself is greater than the power. This is a principle which has already been noticed, and which is common to all machines whatsoever. It may hence be proved, that in the elevation of the weight a quantity of power is expended equal to that which would be necessary to elevate the weight if the power were immediately applied to it, without the intervention of any machine. This has been explained in the case of the lever in (241.), and may be explained in the present instance in nearly the same words.
In one revolution of the machine the length of rope uncoiled from the wheel is equal to the circumference of the wheel, and through this space the power must therefore move. At the same time the length of rope coiled upon the axle is equal to the circumference of the axle, and through this space the weight must be raised. The spaces, therefore, through which the power and weight move in the same time, are in the proportion of the circumferences of the wheel and axle; but these circumferences are in the same proportion as their diameters. Therefore the velocity of the power will bear to the velocity of the weight the same proportion as the diameter of the wheel bears to the diameter of the axle, or, what is the same, as the weight bears to the power (249).
(251.) We have here omitted the consideration of the thickness of the rope. When this is considered, the force must be conceived as acting in the direction of the centre of the rope, and therefore the thickness of the rope which supports the power ought to be added to the diameter of the wheel, and the thickness of the rope which supports the weight to the diameter of the axle. It is the more necessary to attend to this circumstance, as the strength of the rope necessary to support the weight causes its thickness to bear a considerable proportion to the diameter of the axle; while the rope which sustains the power not requiring the same strength, and being applied to a larger circle, bears a very inconsiderable proportion to its diameter.
(252.) In numerous forms of the wheel and axle, the weight or resistance is applied by a rope coiled upon the axle; but the manner in which the power is applied is very various, and not often by means of a rope. The circumference of a wheel sometimes carries projecting pins, as represented in _fig. 88._, to which the hand is applied to turn the machine. An instance of this occurs in the wheel used in the steerage of a vessel.
In the common _windlass_, the power is applied by means of a _winch_, which is a rectangular lever, as represented in _fig. 89._ The arm B C of the winch represents the radius of the wheel, and the power is applied to C D at right angles to B C.
In some cases no wheel is attached to the axle; but it is pierced with holes directed towards its centre, in which long levers are incessantly inserted, and a continuous action produced by several men working at the same time; so that while some are transferring the levers from hole to hole, others are working the windlass.
The axle is sometimes placed in a vertical position, the wheel or levers being moved horizontally. The _capstan_ is an example of this: a vertical axis is fixed in the deck of the ship; the circumference is pierced with holes presented towards its centre. These holes receive long levers, as represented in _fig. 90._ The men who work the capstan walk continually round the axle, pressing forward the levers near their extremities.
In some cases the wheel is turned by the weight of animals placed at its circumference, who move forward as fast as the wheel descends, so as to maintain their position continually at the extremity of the horizontal diameter. The _treadmill_, _fig. 91._, and certain _cranes_, such as _fig. 92._, are examples of this.
In water-wheels, the power is the weight of water contained in buckets at the circumference, as in _fig. 93._, which is called an over-shot wheel: and sometimes by the impulse of water against float-boards at the circumference, as in the under-shot wheel, _fig. 94._ Both these principles act in the breast-wheel, _fig. 95._
In the paddle-wheel of a steam-boat, the power is the resistance which the water offers to the motion of the paddle-boards.
In windmills, the power is the force of the wind acting on various parts of the arms, and may be considered as different powers simultaneously acting on different wheels having the same axle.
(253.) In most cases in which the wheel and axle is used, the action of the power is liable to occasional suspension or intermission, in which case some contrivance is necessary to prevent the recoil of the weight. A ratchet wheel R, _fig. 88._, is provided for this purpose, which is a contrivance which permits the wheel to turn in one direction; but a catch which falls between the teeth of a fixed wheel prevents its motion in the other direction. The effect of the power or weight is sometimes transmitted to the wheel or axle by means of a straight bar, on the edge of which teeth are raised, which engage themselves in corresponding teeth on the wheel or axle. Such a bar is called a rack; and an instance of its use may be observed in the manner of working the pistons of an air-pump.
(254.) The power of the wheel and axle being expressed by the number of times the diameter of the axle is contained in that of the wheel, there are obviously only two ways by which this power may be increased; viz. either by increasing the diameter of the wheel, or diminishing that of the axle. In cases where great power is required, each of these methods is attended with practical inconvenience and difficulty. If the diameter of the wheel be considerably enlarged, the machine will become unwieldy, and the power will work through an unmanageable space. If, on the other hand, the power of the machine be increased by reducing the thickness of the axle, the strength of the axle will become insufficient for the support of that weight, the magnitude of which had rendered the increase of the power of the machine necessary. To combine the requisite strength with moderate dimensions and great mechanical power is, therefore, impracticable in the ordinary form of the wheel and axle. This has, however, been accomplished by giving different thicknesses to different parts of the axle, and carrying a rope, which is coiled on the thinner part, through a wheel attached to the weight, and coiling it in the opposite direction on the thicker part, as in _fig. 96._ To investigate the proportion of the power to the weight in this case, let _fig. 97._ represent a section of the apparatus at right angles to the axis. The weight is equally suspended by the two parts of the rope, S and S′, and therefore each part is stretched by a force equal to half the weight. The moment of the force, which stretches the rope S, is half the weight multiplied by the radius of the thinner part of the axle. This force being at the same side of the centre with the power, co-operates with it in supporting the force which stretches S′, and which acts at the other side of the centre. By the principle established in (185.), the moments of P and S must be equal to that of S′; and therefore if P be multiplied by the radius of the wheel, and added to half the weight multiplied by the radius of the thinner part of the axle, we must obtain a sum equal to half the weight multiplied by the radius of the thicker part of the axle. Hence it is easy to perceive, that the power multiplied by the radius of the wheel is equal to half the weight multiplied by the difference of the radii of the thicker and thinner parts of the axle; or, what is the same, the power multiplied by the diameter of the wheel, is equal to the weight multiplied by half the difference of the diameters of the thinner and thicker parts of the axle.
A wheel and axle constructed in this manner is equivalent to an ordinary one, in which the wheel has the same diameter, and whose axle has a diameter equal to half the difference of the diameters of the thicker and thinner parts. The power of the machine is expressed by the proportion which the diameter of the wheel bears to half the difference of these diameters; and therefore this power, when the diameter of the wheel is given, does not, as in the ordinary wheel and axle, depend on the smallness of the axle, but on the smallness of the difference of the thinner and thicker parts of it. The axle may, therefore, be constructed of such a thickness as to give it all the requisite strength, and yet the difference of the diameters of its different parts may be so small as to give it all the requisite power.
(255.) It often happens that a varying weight is to be raised, or resistance overcome by a uniform power. If, in such a case, the weight be raised by a rope coiled upon a uniform axle, the action of the power would not be uniform, but would vary with the weight. It is, however, in most cases desirable or necessary that the weight or resistance, even though it vary, shall be moved uniformly. This will be accomplished if by any means the leverage of the weight is made to increase in the same proportion as the weight diminishes, and to diminish in the same proportion as the weight increases: for in that case the moment of the weight will never vary, whatever it gains by the increase of weight being lost by the diminished leverage, and whatever it loses by the diminished weight being gained by the increased leverage. An axle, the surface of which is curved in such a manner, that the thickness on which the rope is coiled continually increased or diminishes in the same proportion as the weight or resistance diminishes or increases, will produce this effect.
It is obvious that all that has been said respecting a variable weight or resistance, is also applicable to a variable power, which, therefore, may, by the same means, be made to produce a uniform effect. An instance of this occurs in a watch, which is moved by a spiral spring. When the watch has been wound up, this spring acts with its greatest intensity, and as the watch goes down, the elastic force of the spring gradually loses its energy. This spring is connected by a chain with an axle of varying thickness, called a _fusee_. When the spring is at its greatest intensity, the chain acts upon the thinnest part of the fusee, and as it is uncoiled it acts upon a part of the fusee which is continually increasing in thickness, the spring at the same time losing its elastic power in exactly the same proportion. A representation of the fusee, and the cylindrical box which contains the spring, is given in _fig. 98._, and of the spring itself in _fig. 99._
(256.) When great power is required, wheels and axles may be combined in a manner analogous to a compound system of levers, explained in (246.) In this case the power acts on the circumference of the first wheel, and its effect is transmitted to the circumference of the first axle. That circumference is placed in connection with the circumference of the second wheel, and the effect is thereby transmitted to the circumference of the second axle, and so on. It is obvious from what was proved in (248.), that the power of such a combination of wheels and axles will be found by multiplying together the powers of the several wheels of which it is composed. It is sometimes convenient to compute this power by numbers expressing the proportions of the circumferences or diameters of the several wheels, to the circumferences or diameters of the several axles respectively. This computation is made by first multiplying the numbers together which express the circumferences or diameters of the wheels, and then multiplying together the numbers which express the circumferences or diameters of the several axles. The proportion of the two products will express the power of the machine. Thus, if the circumferences or diameters be as the numbers 10, 14, and 15, their product will be 2100; and if the circumferences or diameters of the axles be expressed by the numbers 3, 4, and 5, their product will be 60, and the power of the machine will be expressed by the proportion of 2100 and 60, or 35 to 1.
(257.) The manner in which the circumferences of the axles act upon the circumferences of the wheels in compound wheel-work is various. Sometimes a strap or cord is applied to a groove in the circumference of the axle, and carried round a similar groove in the circumference of the succeeding wheel. The friction of this cord or strap with the groove is sufficient to prevent its sliding and to communicate the force from the axle to the wheel, or _vice versa_. This method of connecting wheel-work is represented in _fig. 100._
Numerous examples of wheels and axles driven by straps or cords occur in machinery applied to almost every department of the arts and manufactures. In the turning lathe, the wheel worked by the treddle is connected with the mandrel by a catgut cord passing through grooves in the wheel and axle. In all great factories, revolving shafts are carried along the apartments, on which, at certain intervals, straps are attached passing round their circumferences and carried round the wheels which give motion to the several machines. If the wheels, connected by straps or cords, are required to revolve in the same direction, these cords are arranged as in _fig. 100._; but if they are required to revolve in contrary directions, they are applied as in _fig. 101._
One of the chief advantages of the method of transmitting motion between wheels and axles by straps or cords, is that the wheel and axle may be placed at any distance from each other which may be found convenient, and may be made to turn either in the same or contrary directions.
(258.) When the circumference of the wheel acts immediately on the circumference of the succeeding axle, some means must necessarily be adopted to prevent the wheel from moving in contact with the axle without compelling the latter to turn. If the surfaces of both were perfectly smooth, so that all friction were removed, it is obvious that either would slide over the surface of the other, without communicating motion to it. But, on the other hand, if there were any asperities, however small, upon these surfaces, they would become mutually inserted among each other, and neither the wheel nor axle could move without causing the asperities with which its edge is studded to encounter those asperities which project from the surface of the other; and thus, until these projections should be broken off, both wheel and axle must be moved at the same time. It is on this account that if the surfaces of the wheels and axles are by any means rendered rough, and pressed together with sufficient force, the motion of either will turn the other, provided the load or resistance be not greater than the force necessary to break off these small projections which produce the friction.
In cases where great power is not required, motion is communicated in this way through a train of wheel-work, by rendering the surface of the wheel and axle rough, either by facing them with buff leather, or with wood cut across the grain. This method is sometimes used in spinning machinery, where one large buffed wheel, placed in a horizontal position, revolves in contact with several small buffed rollers, each roller communicating motion to a spindle. The position of the wheel W, and the rollers R R, &c., are represented in _fig. 102._ Each roller can be thrown out of contact with the wheel, and restored to it at pleasure.
The communication of motion between wheels and axles by friction has the advantage of great smoothness and evenness, and of proceeding with little noise; but this method can only be used in cases where the resistance is not very considerable, and therefore is seldom adopted in works on a large scale. Dr. Gregory mentions an instance of a saw mill at Southampton, where the wheels act upon each other by the contact of the end grain of wood. The machinery makes very little noise, and wears very well, having been used not less than 20 years.
(259.) The most usual method of transmitting motion through a train of wheel-work is by the formation of teeth upon their circumferences, so that these indentures of each wheel fall between the corresponding ones of that in which it works, and ensure the action so long as the strain is not so great as to fracture the tooth.
In the formation of teeth very minute attention must be given to their figure, in order that the motion may be communicated from wheel to wheel with smoothness and uniformity. This can only be accomplished by shaping the teeth according to curves of a peculiar kind, which mathematicians have invented, and assigned rules for drawing. The ill consequences of neglecting this will be very apparent, by considering the nature of the action which would be produced if the teeth were formed of square projecting pins, as in _fig. 103._ When the tooth A comes into contact with B, it acts obliquely upon it, and, as it moves, the corner of B slides upon the plane surface of A in such a manner as to produce much friction, and to grind away the side of A and the end of B. As they approach the position C D, they sustain a jolt the moment their surfaces come into full contact; and after passing the position of C D, the same scraping and grinding effect is produced in the opposite direction, until by the revolution of the wheels the teeth become disengaged. These effects are avoided by giving to the teeth the curved forms represented in _fig. 104._ By such means the surfaces of the teeth roll upon each other with very inconsiderable friction, and the direction in which the pressure is excited is always that of a line M N, touching the two wheels, and at right angles to the radii. Thus the pressure being always the same, and acting with the same leverage, produces a uniform effect.
(260.) When wheels work together, their teeth must necessarily be of the same size, and therefore the proportion of their circumferences may always be estimated by the number of teeth which they carry. Hence it follows, that in computing the power of compound wheel-work, the number of teeth may always be used to express the circumferences respectively, or the diameters which are proportional to these circumferences. When teeth are raised upon an axle, it is generally called a _pinion_, and in that case the teeth are called _leaves_. The rule for computing the train of wheel-work given in (256.) will be expressed as follows: when the wheel and axle carry teeth, multiply together the number of teeth in each of the wheels, and next the number of leaves in each of the pinions; the proportion of the two products will express the power of the machine. If some of the wheels and axles carry teeth, and others not, this computation may be made by using for those circumferences which do not bear teeth the number of teeth which would fill them. _Fig. 105._ represents a train of three wheels and pinions. The wheel F which bears the power, and the axle which bears the weight, have no teeth; but it is easy to find the number of teeth which they would carry.
(261.) It is evident that each pinion revolves much more frequently in a given time than the wheel which it drives. Thus, if the pinion C be furnished with ten teeth, and the wheel E, which it drives, have sixty teeth, the pinion C must turn six times, in order to turn the wheel E once round. The velocities of revolution of every wheel and pinion which work in one another will therefore have the same proportion as their number of teeth taken in a reverse order, and by this means the relative velocity of wheels and pinions may be determined according to any proposed rate.
Wheel-work, like all other machinery, is used to transmit and modify force in every department of the arts and manufactures; but it is also used in cases where motion alone, and not force, is the object to be attained. The most remarkable example of this occurs in watch and clock-work, where the object is merely to produce uniform motions of rotation, having certain proportions, and without any regard to the elevation of weights, or the overcoming of resistances.
(262.) A _crane_ is an example of combination of wheel-work used for the purpose of raising or lowering great weights. _Fig. 106._ represents a machine of this kind. A B is a strong vertical beam, resting on a pivot, and secured in its position by beams in the floor. It is capable, however, of turning on its axis, being confined between rollers attached to the beams and fixed in the floor. C D is a projecting arm called a _gib_, formed of beams which are mortised into A B. The wheel-work is mounted in two cast-iron crosses, bolted on each side of the beams, one of which appears at E F G H. The winch at which the power is applied is at I. This carries a pinion immediately behind H. This pinion works in a wheel K, which carries another pinion upon its axle. This last pinion works in a larger wheel L, which carries upon its axis a barrel M, on which a chain or rope is coiled. The chain passes over a pulley D at the top of the gib. At the end of the chain a hook O is attached, to support the weight W. During the elevation of the weight it is convenient that its recoil should be hindered in case of any occasional suspension of the power. This is accomplished by a ratchet wheel attached to the barrel M, as explained in (253.); but when the weight W is to be lowered, the catch must be removed from this ratchet wheel. In this case the too rapid descent of the weight is in some cases checked by pressure excited on some part of the wheel-work, so as to produce sufficient friction to retard the descent in any required degree, or even to suspend it, if necessary. The vertical beam at B resting on a pivot, and being fixed between rollers, allows the gib to be turned round in any direction; so that a weight raised from one side of the crane may be carried round, and deposited on another side, at any distance within the range of the gib. Thus, if a crane be placed upon a wharf near a vessel, weights may be raised, and when elevated, the gib may be turned round so as to let them descend into the hold.
The power of this machine may be computed upon the principles already explained. The magnitude of the circle, in which the power at I moves, may be determined by the radius of the winch, and therefore the number of teeth which a wheel of that size would carry may be found. In like manner we may determine the number of leaves in a pinion whose magnitude would be equal to the barrel M. Let the first number be multiplied by the number of teeth in the wheel K, and that product by the number of teeth in the wheel L. Next let the number of leaves in the pinion H be multiplied by the number of leaves in the pinion attached to the axle of the wheel K, and let that product be multiplied by the number of leaves in a pinion, whose diameter is equal to that of the barrel M. These two products will express the power of the machine.
(263.) Toothed wheels are of three kinds, distinguished by the position which the teeth bear with respect to the axis of the wheel. When they are raised upon the edge of the wheel as in _fig. 105._, they are called _spur wheels_, or _spur gear_. When they are raised parallel to the axis, as in _fig. 107._, it is called a _crown wheel_. When the teeth are raised on a surface inclined to the plane of the wheel, as in _fig. 108._, they are called _bevelled wheels_.
If a motion round one axis is to be communicated to another axis parallel to it, spur gear is generally used. Thus, in _fig. 105._, the three axes are parallel to each other. If a motion round one axis is to be communicated to another at right angles to it, a crown wheel, working in a spur pinion, as in _fig. 107._, will serve. Or the same object may be obtained by two bevelled wheels, as in _fig. 108._
If a motion round one axis is required to be communicated to another inclined to it at any proposed angle, two bevelled wheels can always be used. In _fig. 109._ let A B and A C be the two axles; two bevelled wheels, such as D E and E F, on these axles will transmit the motion or rotation from one to the other, and the relative velocity may, as usual, be regulated by the proportional magnitude of the wheels.
(264.) In order to equalise the wear of the teeth of a wheel and pinion, which work in one another, it is necessary that every leaf of the pinion should work in succession through every tooth of the wheel, and not continually act upon the same set of teeth. If the teeth could be accurately shaped according to mathematical principles, and the materials of which they are formed be perfectly uniform, this precaution would be less necessary; but as slight inequalities, both of material and form, must necessarily exist, the effects of these should be as far as possible equalised, by distributing them through every part of the wheel. For this purpose it is usual, especially in mill-work, where considerable force is used, so to regulate the proportion of the number of teeth in the wheel and pinion, that the same leaf of the pinion shall not be engaged twice with any one tooth of the wheel, until after the action of a number of teeth, expressed by the product of the number of teeth in the wheel and pinion. Let us suppose that the pinion contains ten leaves, which we shall denominate by the numbers 1, 2, 3, &c., and that the wheel contains 60 teeth similarly denominated. At the commencement of the motion suppose the leaf 1 of the pinion engages the tooth 1 of the wheel; then after one revolution the leaf 1 of the pinion will engage the tooth 11 of the wheel, and after two revolutions the leaf 1 of the pinion will engage the tooth 21 of the wheel; and in like manner, after 3, 4, and 5 revolutions of the pinion, the leaf 1 will engage successively the teeth 31, 41, and 51 of the wheel. After the sixth revolution, the leaf 1 of the pinion will again engage the tooth 1 of the wheel. Thus it is evident, that in the case here supposed the leaf 1 of the pinion will continually be engaged with the teeth 1, 11, 21, 31, 41, and 51 of the wheel, and no others. The like may be said of every leaf of the pinion. Thus the leaf 2 of the pinion will be successively engaged with the teeth 2, 12, 22, 32, 42, and 52 of the wheel, and no others. Any accidental inequalities of these teeth will therefore continually act upon each other, until the circumference of the wheel be divided into parts of ten teeth each, unequally worn. This effect would be avoided by giving either the wheel or pinion one tooth more or one tooth less. Thus, suppose the wheel, instead of having sixty teeth, had sixty-one, then after six revolutions of the pinion the leaf 1 of the pinion would be engaged with the tooth 61 of the wheel; and after one revolution of the wheel, the leaf 2 of the pinion would be engaged with the tooth 1 of the wheel. Thus, during the first revolution of the wheel the leaf 1 of the pinion would be successively engaged with the teeth 1, 11, 21, 31, 41, 51, and 61 of the wheel: at the commencement of the second revolution of the wheel the leaf 2 of the pinion would be engaged with the tooth 1 of the wheel; and during the second revolution of the wheel the leaf 1 of the pinion would be successively engaged with the teeth 10, 20, 30, 40, 50, and 60 of the wheel. In the same manner it may be shown, that in the third revolution of the wheel the leaf 1 of the pinion would be successively engaged with the teeth 9, 19, 29, 39, 49, and 59 of the wheel: during the fourth revolution of the wheel the leaf 1 of the pinion would be successively engaged with the teeth 8, 18, 28, 38, 48, and 58 of the wheel. By continuing this reasoning it will appear, that during the tenth revolution of the wheel the leaf 1 of the pinion will be engaged successively with the teeth 2, 12, 22, 32, 42, and 52 of the wheel. At the commencement of the eleventh revolution of the wheel the leaf 1 of the pinion will be engaged with the tooth 1 of the wheel, as at the beginning of the motion. It is evident, therefore, that during the first ten revolutions of the wheel each leaf of the pinion has been successively engaged with every tooth of the wheel, and that during these ten revolutions the pinion has revolved sixty-one times. Thus the leaves of the pinion have acted six hundred and ten times upon the teeth of the wheel, before two teeth can have acted twice upon each other.
The odd tooth which produces this effect is called by millwrights the _hunting cog_.
(265.) The most familiar case in which wheel-work is used to produce and regulate motion merely, without any reference to weights to be raised or resistances to be overcome, is that of chronometers. In watch and clock work the object is to cause a wheel to revolve with a uniform velocity, and at a certain rate. The motion of this wheel is indicated by an index or hand placed upon its axis, and carried round with it. In proportion to the length of the hand the circle over which its extremity plays is enlarged, and its motion becomes more perceptible. This circle is divided, so that very small fractions of a revolution of the hand may be accurately observed. In most chronometers it is required to give motion to two hands, and sometimes to three. These motions proceed at different rates, according to the subdivisions of time generally adopted. One wheel revolves in a minute, bearing a hand which plays round a circle divided into sixty equal parts; the motion of the hand over each part indicating one second, and a complete revolution of the hand being performed in one minute. Another wheel revolves once, while the former revolves sixty times; consequently the hand carried by this wheel revolves once in sixty minutes, or one hour. The circle on which it plays is, like the former, divided into sixty equal parts, and the motion of the hand over each division is performed in one minute. This is generally called the _minute hand_, and the former the _second hand_.
A third wheel revolves once, while that which carries the minute hand revolves twelve times; consequently this last wheel, which carries the _hour hand_, revolves at a rate twelve times less than that of the minute hand, and therefore seven hundred and twenty times less than the second hand. We shall now endeavour to explain the manner in which these motions are produced and regulated. Let A, B, C, D, E, _fig. 110._, represent a train of wheels, and _a_, _b_, _c_, _d_ represent their pinions, _e_ being a cylinder on the axis of the wheel E, round which a rope is coiled, sustaining a weight W. Let the effect of this weight transmitted through the train of wheels be opposed by a power P acting upon the wheel A, and let this power be supposed to be of such a nature as to cause the weight W to descend with a uniform velocity, and at any proposed rate. The wheel E carries on its circumference eighty-four teeth. The wheel D carries eighty teeth; the wheel C is also furnished with eighty teeth, and the wheel B with seventy-five. The pinions _d_ and _c_ are each furnished with twelve leaves, and the pinions _b_ and _a_ with ten.
If the power at P be so regulated as to allow the wheel A to revolve once in a minute, with a uniform velocity, a hand attached to the axis of this wheel will serve as the _second hand_. The pinion _a_ carrying ten teeth must revolve seven times and a half to produce one revolution of B, consequently fifteen revolutions of the wheel A will produce two revolutions of the wheel B; the wheel B, therefore, revolves twice in fifteen minutes. The pinion _b_ must revolve eight times to produce one revolution of the wheel C, and therefore the wheel C must revolve once in four quarters of an hour, or in one hour. If a hand be attached to the axis of this wheel, it will have the motion necessary for the minute hand. The pinion _c_ must revolve six times and two thirds to produce one revolution of the wheel D, and therefore this wheel must revolve once in six hours and two thirds. The pinion _d_ revolves seven times for one revolution of the wheel E, and therefore the wheel E will revolve once in forty-six hours and two thirds.
On the axis of the wheel C a second pinion may be placed, furnished with seven leaves, which may lead a wheel of eighty-four teeth, so that this wheel shall turn once during twelve turns of the wheel C. If a hand be fixed upon the axis, this hand will revolve once for twelve revolutions of the minute hand fixed upon the axis of the wheel C; that is, it will revolve once in twelve hours. If it play upon a dial divided into twelve equal parts, it will move over each part in an hour, and will serve the purpose of the hour hand of the chronometer.
We have here supposed that the second hand, the minute hand, and the hour hand move on separate dials. This, however, is not necessary. The axis of the hour hand is commonly a tube, inclosing within it that of the minute hand, so that the same dial serves for both. The second hand, however, is generally furnished with a separate dial.
(266.) We shall now explain the manner in which a power is applied to the wheel A, so as to regulate and equalise the effect of the weight W. Suppose the wheel A furnished with thirty teeth, as in _fig. 111._; if nothing check the motion, the weight W would descend with an accelerated velocity, and would communicate an accelerated motion to the wheel A. This effect, however, is interrupted by the following contrivance:--L M is a pendulum vibrating on the centre L, and so regulated that the time of its oscillation is one second. The pallets I and K are connected with the pendulum, so as to oscillate with it. In the position of the pendulum represented in the figure, the pallet I stops the motion of the wheel A, and entirely suspends the action of the weight W, _fig. 110._, so that for a moment the entire machine is motionless. The weight M, however, falls by its gravity towards the lowest position, and disengages the pallet I from the tooth of the wheel. The weight W begins then to take effect, and the wheel A turns from A towards B. Meanwhile the pendulum M oscillates to the other side, and the pallet K falls under a tooth of the wheel A, and checks for a moment its further motion. On the returning vibration the pallet K becomes again disengaged, and allows the tooth of the wheel to escape, and by the influence of the weight W another tooth passes before the motion of the wheel A is again checked by the interposition of the pallet I.
From this explanation it will appear that, in two vibrations of the pendulum, one tooth of the wheel A passes the pallet I, and therefore, if the wheel A be furnished with 30 teeth, it will be allowed to make one revolution during 60 vibrations of the pendulum. If, therefore, the pendulum be regulated so as to vibrate seconds, this wheel will revolve once in a minute. From the action of the pallets in checking the motion of the wheel A, and allowing its teeth alternately to _escape_, this has been called the _escapement_ wheel; and the wheel and pallets together are generally called the _escapement_, or _’scapement_.
We have already explained, that by reason of the friction on the points of support, and other causes, the swing of the pendulum would gradually diminish, and its vibration at length cease. This, however, is prevented by the action of the teeth of the scapement wheel upon the pallets, which is just sufficient to communicate that quantity of force to the pendulum which is necessary to counteract the retarding effects, and to maintain its motion. It thus appears, that although the effect of the gravity of the weight W in giving motion to the machine is at intervals suspended, yet this part of the force is not lost, being, during these intervals, employed in giving to the pendulum all that motion which it would lose by the resistances to which it is inevitably exposed.
In stationary clocks, and in other cases in which the bulk of the machine is not an objection, a descending weight is used as the moving power. But in watches and portable chronometers, this would be attended with evident inconvenience. In such cases, a spiral spring, called the _mainspring_, is the moving power. The manner in which this spring communicates rotation to an axis, and the ingenious method of equalising the effect of its variable elasticity by giving to it a leverage, which increases as the elastic force diminishes, have been already explained. (255.)
A similar objection lies against the use of a pendulum in portable chronometers. A spiral spring of a similar kind, but infinitely more delicate, called a _hair spring_, is substituted in its place. This spring is connected with a nicely-balanced wheel, called _the balance wheel_, which plays in pivots. When this wheel is turned to a certain extent in one direction, the hair spring is coiled up, and its elasticity causes the wheel to recoil, and return to a position in which the energy of the spring acts in the opposite direction. The balance wheel then returns, and continually vibrates in the same manner. The axis of this wheel is furnished with pallets similar to those of the pendulum, which are alternately engaged with the teeth of a crown wheel, which takes the place of the scapement wheel already described.
A general view of the work of a common watch is represented in _fig. 111._ _bis._ A is the balance wheel bearing pallets _p_ _p_ upon its axis; C is the crown wheel, whose teeth are suffered to escape alternately by those pallets in the manner already described in the scapement of a clock. On the axis of the crown wheel is placed a pinion _d_, which drives another crown wheel K. On the axis of this is placed the pinion _c_, which plays in the teeth of the third wheel L. The pinion _b_ on the axis of L is engaged with the wheel M, called the centre wheel. The axle of this wheel is carried up through the centre of the dial. A pinion _a_ is placed upon it, which works in the great wheel N. On this wheel the mainspring immediately acts. O P is the mainspring stripped of its barrel. The axis of the wheel M passing through the centre of the dial is squared at the end to receive the minute hand. A second pinion Q is placed upon this axle which drives a wheel T. On the axle of this wheel a pinion _g_ is placed, which drives the hour wheel V. This wheel is placed upon a tubular axis, which incloses within it the axis of the wheel M. This tubular axis passing through the centre of the dial, carries the hour hand. The wheels A, B, C, D, E, _fig. 110._, correspond to the wheels C, K, L, M, N, _fig. 112._; and the pinions _a_, _b_, _c_, _d_, _e_, _fig. 109._, correspond to the pinions _d_, _c_, _b_, _a_, _fig. 111_. From what has already been explained of these wheels, it will be obvious that the wheel M, _fig. 111._, revolves once in an hour, causing the minute hand to move round the dial once in that time. This wheel at the same time turns the pinion Q which leads the wheel T. This wheel again turns the pinion _g_ which leads the hour wheel V. The leaves and teeth of these pinions and wheels are proportioned, as already explained, so that the wheel V revolves once during twelve revolutions of the wheel M. The hour hand, therefore, which is carried by the tubular axle of the wheel V, moves once round the dial in twelve hours.
Our object here has not been to give a detailed account of watch and clock work, a subject for which we must refer the reader to the proper department of this work. Such a general account has only been attempted as may explain how tooth and pinion work may be applied to regulate motion.
CHAP. XV.
OF THE PULLEY.
(267.) The next class of simple machines, which present themselves to our attention, is that which we have called the _cord_. If a rope were perfectly flexible, and were capable of being bent over a sharp edge, and of moving upon it without friction, we should be enabled by its means to make a force in any one direction overcome resistance, or communicate motion in any other direction. Thus if P, _fig. 112._, be such an edge, a perfectly flexible rope passing over it would be capable of transmitting a force S F to a resistance Q R, so as to support or overcome R, or by a motion in the direction of S F to produce another motion in the direction R Q. But as no materials of which ropes can be constructed can give them perfect flexibility, and as in proportion to the strength by which they are enabled to transmit force their rigidity increases, it is necessary, in practice, to adopt means to remove or mitigate those effects which attend imperfect flexibility, and which would otherwise render cords practically inapplicable as machines.
When a cord is used to transmit a force from one direction to another, its stiffness renders some force necessary in bending it over the angle P, which the two directions form; and if the angle be sharp, the exertion of such a force may be attended with the rupture of the cord. If, instead of bending the rope at one point over a single angle, the change of direction were produced by successively deflecting it over several angles, each of which would be less sharp than a single one could be, the force requisite for the deflection, as well as the liability of rupturing the cord, would be considerably diminished. But this end will be still more perfectly attained if the deflection of the cord be produced by bending it over the surface of a curve.
If a rope were applied only to sustain, and not to move a weight, this would be sufficient to remove the inconveniences arising from its rigidity. But when motion is to be produced, the rope, in passing over the curved surface, would be subject to excessive friction, and consequently to rapid wear. This inconvenience is removed by causing the surface on which the rope runs to move with it, so that no more friction is produced than would arise from the curved surface rolling upon the rope.
(268.) All these ends are attained by the common pulley, which consists of a wheel called a _sheave_, fixed in a block and turning on a pivot. A groove is formed in the edge of the wheel in which the rope runs, the wheel revolving with it. Such an apparatus is represented in _fig. 113._
We shall, for the present, omit the consideration of that part of the effects of the stiffness and friction of the machine which is not removed by the contrivance just explained, and shall consider the rope as perfectly flexible and moving without friction.
From the definition of a flexible cord, it follows, that its tension, or the force by which it is stretched throughout its entire length, must be uniform. From this principle, and this alone, all the mechanical properties of pulleys may be derived.
Although, as already explained, the whole mechanical efficacy of this machine depends on the qualities of the cord, and not on those of the block and sheave, which are only introduced to remove the accidental effects of stiffness and friction; yet it has been usual to give the name pulley to the block and sheave, and a combination of blocks, sheaves, and ropes is called a _tackle_.
(269.) When the rope passes over a single wheel, which is fixed in its position, as in _fig. 113._, the machine is called a _fixed pulley_. Since the tension of the cord is uniform throughout its length, it follows, that in this machine the power and weight are equal. For the weight stretches that part of the cord which is between the weight and pulley, and the power stretches that part between the power and the pulley. And since the tension throughout the whole length is the same, the weight must be equal to the power.
Hence it appears that no mechanical advantage is gained by this machine. Nevertheless, there is scarcely any engine, simple or complex, attended with more convenience. In the application of power, whether of men or animals, or arising from natural forces, there are always some directions in which it may be exerted to much greater convenience and advantage than others, and in many cases the exertion of these powers is limited to a single direction. A machine, therefore, which enables us to give the most advantageous direction to the moving power, whatever be the direction of the resistance opposed to it, contributes as much practical convenience as one which enables a small power to balance or overcome a great weight. In directing the power against the resistance, it is often necessary to use two fixed pulleys. Thus, in elevating a weight A, _fig. 114._, to the summit of a building, by the strength of a horse moving below, two fixed pulleys B and C may be used. The rope is carried from A over the pulley B; and, passing downwards, is brought under C, and finally drawn by the animal on the horizontal plane. In the same manner sails are spread, and flags hoisted on the yards and masts of a ship, by sailors pulling a rope on the deck.
By means of the fixed pulley a man may raise himself to a considerable height, or descend to any proposed depth. If he be placed in a chair or bucket attached to one end of a rope which is carried over a fixed pulley, by laying hold of this rope on the other side, as represented in _fig. 115._, he may, at will, descend to a depth equal to half of the entire length of the rope, by continually yielding rope on the one side, and depressing the bucket or chair by his weight on the other. Fire-escapes have been constructed on this principle, the fixed pulley being attached to some part of the building.
(270.) A _single moveable pulley_ is represented in _fig. 116._ A cord is carried from a fixed point F, and passing through a block B, attached to a weight W, passes over a fixed pulley C, the power being applied at P. We shall first suppose the parts of the cord on each side the wheel B to be parallel; in this case, the whole weight W being sustained by the parts of the cords B C and B F, and these parts being equally stretched (268.), each must sustain half the weight, which is therefore the tension of the cord. This tension is resisted by the power at P, which must, therefore, be equal to half the weight. In this machine, therefore, the weight is twice the power.
(271.) If the parts of the cord B C and B F be not parallel, as in _fig. 117._, a greater power than half the weight is therefore necessary to sustain it. To determine the power necessary to support a given weight, in this case take the line B A in the vertical direction, consisting of as many inches as the weight consists of ounces; from A draw A D parallel to B C, and A E parallel to B F; the force of the weight represented by A B will be equivalent to two forces represented by B D and B E. (74.) The number of inches in these lines respectively will represent the number of ounces which are equivalent to the tensions of the parts B F and B C of the cord. But as these tensions are equal, B D and B E must be equal, and each will express the amount of the power P, which stretches the cord at P C.
It is evident that the four lines, A E, E B, B D, and D A, are equal. And as each of them represents the power, the weight which is represented by A B must be less than twice the power which is represented by A E and E B taken together. It follows, therefore, that as parts of the ropes which support the weight depart from parallelism the machine becomes less and less efficacious; and there are certain obliquities at which the equilibrating power would be much greater than the weight.
(272.) The mechanical power of pulleys admits of being almost indefinitely increased by combination. Systems of pulleys may be divided into two classes; those in which a single rope is used, and those which consist of several distinct ropes. _Fig. 118._ and _119._ represent two systems of pulleys, each having a single rope. The weight is in each case attached to a moveable block, B, in which are fixed two or more wheels; A is a fixed block, and the rope is successively passed over the wheels above and below, and, after passing over the last wheel above, is attached to the power. The tension of that part of the cord to which the power is attached is produced by the power, and therefore equivalent to it, and the same tension must extend throughout its whole length. The weight is sustained by all those parts of the cord which pass from the lower block, and as the force which stretches them all is the same, viz. that of the power, the effect of the weight must be equally distributed among them, their directions being supposed to be parallel. It will be evident, from this reasoning, that the weight will be as many times greater than the power as the number of cords which support the lower block. Thus, if there be six cords, each cord will support a sixth part of the weight, that is, the weight will be six times the tension of the cord, or six times the power. In _fig. 118._ the cord is represented as being finally attached to a hook on the upper block. But it may be carried over an additional wheel fixed in that block, and finally attached to a hook in the lower block, as in _fig. 119._, by which one will be added to the power of the machine, the number of cords at the lower block being increased by one. In the system represented in _fig. 118._ the wheels are placed in the blocks one above the other; in _fig. 119._ they are placed side by side. In all systems of pulleys of this class, the weight of the lower block is to be considered as a part of the weight to be raised, and in estimating the power of the machine, this should always be attended to.
(273.) When the power of the machine, and therefore the number of wheels, is considerable, some difficulty arises in the arrangement of the wheels and cords. The celebrated Smeaton contrived a tackle, which takes its name from him, in which there are ten wheels in each block: five large wheels placed side by side, and five smaller ones similarly placed above them in the lower block, and below them in the upper. _Fig. 120._ represents Smeaton’s blocks without the rope. The wheels are marked with the numbers 1, 2, 3, &c., in the order in which the rope is to be passed over them. As in this pulley 20 distinct parts of the rope support the lower block, the weight, including the lower block, will be 20 times the equilibrating power.
(274.) In all these systems of pulleys, every wheel has a separate axle, and there is a distinct wheel for every turn of the rope at each block. Each wheel is attended with friction on its axle, and also with friction between the sheave and block. The machine is by this means robbed of a great part of its efficacy, since, to overcome the friction alone, a considerable power is in most cases necessary.
An ingenious contrivance has been suggested, by which all the advantage of a large number of wheels may be obtained without the multiplied friction of distinct sheaves and axles. To comprehend the excellence of this contrivance, it will be necessary to consider the rate at which the rope passes over the several wheels of such a system, as _fig. 118._ If one foot of the rope G F pass over the pulley F, two feet must pass over the pulley E, because the distance between F and E being shortened one foot, the total length of the rope G F E must be shortened two feet. These two feet of rope must pass in the direction E D, and the wheel D, rising one foot, three feet of rope must consequently pass over it. These three feet of rope passing in the direction D C, and the rope D C being also shortened one foot by the ascent of the lower block, four feet of rope must pass over the wheel C. In the same way it may be shown that five feet must pass over B, and six feet over A. Thus, whatever be the number of wheels in the upper and lower blocks, the parts of the rope which pass in the same time over the wheels in the lower block are in the proportion of the odd numbers 1, 3, 5, &c.; and those which pass over the wheels in the upper block in the same time, are as the even numbers 2, 4, 6, &c. If the wheels were all of equal size, as in _fig. 119._, they would revolve with velocities proportional to the rate at which the rope passes over them. So that, while the first wheel below revolves once, the first wheel above will revolve twice; the second wheel below three times; the second wheel above, four times, and so on. If, however, the wheels differed in size in proportion to the quantity of rope which must pass over them, they would evidently revolve in the same time. Thus, if the first wheel above were twice the size of the first wheel below, one revolution would throw off twice the quantity of rope. Again, if the second wheel below were thrice the size of the first wheel below, it would throw off in one revolution thrice the quantity of rope, and so on. Wheels thus proportioned, revolving in exactly the same time, might be all placed on one axle, and would partake of one common motion, or, what is to the same effect, several grooves might be cut upon the face of one solid wheel, with diameters in the proportion of the odd numbers 1, 3, and 5, &c., for the lower pulley, and corresponding grooves on the face of another solid wheel represented by the even numbers 2, 4, 6, &c., for the upper pulley. The rope being passed successively over the grooves of such wheels, would be thrown off exactly in the same manner as if every groove were upon a separate wheel, and every wheel revolved independently of the others. Such is White’s pulley, represented in _fig. 121._
The advantage of this machine, when accurately constructed, is very considerable. The friction, even when great resistances are to be opposed, is very trifling; but, on the other hand, it has corresponding disadvantages which greatly circumscribe its practical utility. In the workmanship of the grooves great difficulty is found in giving them the exact proportions. In doing which, the thickness of the rope must be accurately allowed for; and consequently it follows, that the same pulley can never act except with a rope of a particular diameter. A very slight deviation from the true proportion of the grooves will cause the rope to be unequally stretched, and will throw on some parts of it an undue proportion of the weight, while other parts become nearly, and sometimes altogether slack. Besides these defects, the rope is so liable to derangement by being thrown out of the grooves, that the pulley can scarcely be considered portable.
For these and other reasons, this machine, ingenious as it unquestionably is, has never been extensively used.
(275.) In the several systems of pulleys just explained, the hook to which the fixed block is attached supports the entire of both the power and weight. When the machine is in equilibrium, the power only supports so much of the weight as is equal to the tension of the cord, all the remainder of the weight being thrown on the fixed point, according to what was observed in (225.)
If the power be moved so as to raise the weight, it will move with a velocity as many times greater than that of the weight as the weight itself is greater than the power. Thus in _fig. 118._ if the weight attached to the lower block ascend one foot, six feet of line will pass over the pulley A, according to what has been already proved. Thus, the power will descend through six feet, while the weight rises one foot. But, in this case, the weight is six times the power. All the observations in (226.) will therefore be applicable to the cases of great weights raised by small powers by means of the system of pulleys just described.
(276.) When two or more ropes are used, pulleys may be combined in various ways so as to produce any degree of mechanical effect. If to any of the systems already described a single moveable pulley be added, the power of the machine would be doubled. In this case, the second rope is attached to the hook of the lower block, as in _fig. 122._, and being carried through a moveable pulley attached to the weight, it is finally brought up to a fixed point. The tension of the second cord is equal to half the weight (270.); and therefore the power P, by means of the first cord, will have only half the tension which it would have if the weight were attached to the lower block. A moveable pulley thus applied is called a _runner_.
(277.) Two systems of pulleys, called _Spanish bartons_, having each two ropes, are represented in _fig. 123._ The tension of the rope P A B C in the first system is equal to the power; and therefore the parts B A and B C support a portion of the weight equal to twice the power. The rope E A supports the tensions of A P and A B; and therefore the tension of A E D is twice the power. Thus, the united tensions of the ropes which support the pulley B is four times the power, which is therefore the amount of the weight. In the second system, the rope P A D is stretched by the power. The rope A E B C acts against the united tensions A P and A D; and therefore the tension of A E or E B is twice the power. Thus, the weight acts against three tensions; two of which are equal to twice the power, and the remaining one is equal to the power. The weight is therefore equal to five times the power.
A single rope may be so arranged with one moveable pulley as to support a weight equal to three times the power. In _fig. 124._ this arrangement is represented, where the numbers sufficiently indicate the tension of the rope, and the proportion of the weight and power. In _fig. 125._ another method of producing the same effect with two ropes is represented.
(278.) If several single moveable pulleys be made successively to act upon each other, the effect is doubled by every additional pulley: such a system as this is represented in _fig. 126._ The tension of the first rope is equal to the power; the second rope acts against twice the tension of the first, and therefore it is stretched with a force equal to twice the power: the third rope acts against twice this tension, and therefore it is stretched with a force equal to four times the power, and so on. In the system represented in _fig. 126._ there are three ropes, and the weight is eight times the power. Another rope would render it sixteen times the power, and so on.
In this system, it is obvious that the ropes will require to have different degrees of strength, since the tension to which they are subject increases in a double proportion from the power to the weight.
(279.) If each of the ropes, instead of being attached to fixed points at the top, are carried over fixed pulleys, and attached to the several moveable pulleys respectively, as in _fig. 127._, the power of the machine will be greatly increased; for in that case the forces which stretch the successive ropes increase in a treble instead of a double proportion, as will be evident by attending to the numbers which express the tensions in the figure. One rope would render the weight three times the power, two ropes nine times, three ropes twenty-seven times, and so on. An arrangement of pulleys is represented in _fig. 128._, by which each rope, instead of being finally attached to a fixed point, as in _fig. 126._, is attached to the weight. The weight is in this case supported by three ropes; one stretched with a force equal to the power; another with a force equal to twice the power; and a third with a force equal to four times the power. The weight is therefore, in this case, seven times the power.
(280.) If the ropes, instead of being attached to the weight, pass through wheels, as in _fig. 129._, and are finally attached to the pulleys above, the power of the machine will be considerably increased. In the system here represented the weight is twenty-six times the power.
(281.) In considering these several combinations of pulleys, we have omitted to estimate the effects produced by the weights of the sheaves and blocks. Without entering into the details of this computation, it may be observed generally, that in the systems represented in _figs. 126._, _127._ the weight of the wheel and blocks acts against the power; but that in _figs. 128._ and _129._ they assist the powers in supporting the weight. In the systems represented in _fig. 123._ the weight of the pulleys, to a certain extent, neutralise each other.
(282.) It will in all cases be found, that that quantity by which the weight exceeds the power is supported by fixed points; and therefore, although it be commonly stated that a small power supports a great weight, yet in the pulley, as in all other machines, the power supports no more of the weight than is exactly equal to its own amount. It will not be necessary to establish this in each of the examples which have been given: having explained it in one instance, the student will find no difficulty in applying the same reasoning to others. In _fig. 126._, the fixed pulley sustains a force equal to twice the power, and by it the power giving tension to the first rope sustains a part of the weight equal to itself. The first hook sustains a portion of the weight equal to the tension of the first string, or to the power. The second hook sustains a force equal to twice the power; and the third hook sustains a force equal to four times the power. The three hooks therefore sustain a portion of the weight equal to seven times the power; and the weight itself being eight times the power, it is evident that the part of the weight which remains to be supported by the power is equal to the power itself.
(283.) When a weight is raised by any of the systems of pulleys which have been last described, the proportion between the velocity of the weight and the velocity of the power, so frequently noticed in other machines, will always be observed. In the system of pulleys represented in _fig. 126._ the weight being eight times the power, the velocity of the power will be eight times that of the weight. If the power be moved through eight feet, that part of the rope between the fixed pulley and the first moveable pulley will be shortened by eight feet. And since the two parts which lie above the first moveable pulley must be equally shortened, each will be diminished by four feet; therefore the first pulley will rise through four feet while the power moves through eight feet. In the same way it may be shown, that while the first pulley moves through four feet, the second moves through two; and while the second moves through two, the third, to which the weight is attached, is raised through one foot. While the power, therefore, is carried through eight feet, the weight is moved through one foot.
By reasoning similar to this, it may be shown that the space through which the power is moved in every case is as many times greater than the height through which the weight is raised, as the weight is greater than the power.
(284.) From its portable form, cheapness of construction, and the facility with which it may be applied in almost every situation, the pulley is one of the most useful of the simple machines. The mechanical advantage, however, which it appears in theory to possess is considerably diminished in practice, owing to the stiffness of the cordage, and the friction of the wheels and blocks. By this means it is computed that in most cases so great a proportion as two thirds of the power is lost. The pulley is much used in building, where weights are to be elevated to great heights. But its most extensive application is found in the rigging of ships, where almost every motion is accomplished by its means.
(285.) In all the examples of pulleys, we have supposed the parts of the rope sustaining the weight and each of the moveable pulleys to be parallel to each other. If they be subject to considerable obliquity, the relative tensions of the different ropes must be estimated according to the principle applied in (271.)
CHAP. XVI.
ON THE INCLINED PLANE, WEDGE, AND SCREW.
(286.) The inclined plane is the most simple of all machines. It is a hard plane surface forming some angle with a horizontal plane, that angle not being a right angle. When a weight is placed on such a plane, a two-fold effect is produced. A part of the effect of the weight is resisted by the plane, and produces a pressure upon it; and the remainder urges the weight down the plane, and would produce a pressure against any surface resisting its motion placed in a direction perpendicular to the plane (131.)
Let A B, _fig. 130._, be such a plane, B C its horizontal base, A C its height, and A B C its angle of elevation. Let W be a weight placed upon it. This weight acts in the vertical direction W D, and is equivalent to two forces, W F perpendicular to the plane, and W E directed down the plane (74.) If a plane be placed at right angles to the inclined plane below W, it will resist the descent of the weight, and sustain a pressure expressed by W E. Thus, the weight W resting in the corner, instead of producing one pressure in the direction W D, will produce two pressures, one expressed by W F upon the inclined plane, and the other expressed by W E upon the resisting plane. These pressures respectively have the same proportion to the entire weight as W F and W E have to W D, or as D E and W E have to W D, because D E is equal to W F. Now the triangle W E D is in all respects similar to the triangle A B C, the one differing from the other only in the scale on which it is constructed. Therefore, the three lines A C, C B, and B A, are in the same proportion to each other as the lines W E, E D, and W D. Hence, A B has to A C the same proportion as the whole weight has to the pressure directed toward B, and A B has to B C the same proportion as the whole weight has to the pressure on the inclined plane.
We have here supposed the weight to be sustained upon the inclined plane by a hard plane fixed at right angles to it. But the power necessary to sustain the weight will be the same in whatever way it is applied, provided it act in the direction of the plane. Thus, a cord may be attached to the weight, and stretched towards A, or the hands of men may be applied to the weight below it, so as to resist its descent towards B. But in whatever way it be applied, the amount of the power will be determined in the same manner. Suppose the weight to consist of as many pounds as there are inches in A B, then the power requisite to sustain it upon the plane will consist of as many pounds as there are inches in A C, and the pressure on the plane will amount to as many pounds as there are inches in B C.
From what has been stated it may easily be inferred that the less the elevation of the plane is, the less will be the power requisite to sustain a given weight upon it, and the greater will be the pressure upon it. Suppose the inclined plane A B to turn upon a hinge at B, and to be depressed so that its angle of elevation shall be diminished, it is evident that as this angle decreases the height of the plane decreases, and its base increases. Thus, when it takes the position B A′, the height A′ C′ is less than the former height A C, while the base B C′ is greater than the former base B C. The power requisite to support the weight upon the plane in the position B A′ is represented by A′ C′, and is as much less than the power requisite to sustain it upon the plane A B, as the height A′ C′ is less than the height A C. On the other hand, the pressure upon the plane in the position B A′ is as much greater than the pressure upon the plane B A, as the base B C′ is greater than the base B C.
(287.) The power of an inclined plane, considered as a machine, is therefore estimated by the proportion which its length bears to its height. This power is always increased by diminishing the elevation of the plane.
Roads which are not level may be regarded as inclined planes, and loads drawn upon them in carriages, considered in reference to the powers which impel them, are subject to all the conditions which have been established for inclined planes. The inclination of the road is estimated by the height corresponding to some proposed length. Thus it is said to rise one foot in fifteen, one foot in twenty, &c., meaning that if fifteen or twenty feet of the road be taken as the length of an inclined plane, such as A B, the corresponding height will be one foot. Or the same may be expressed thus: that if fifteen or twenty feet be measured upon the road, the difference of the levels of the two extremities of the distance measured is one foot. According to this method of estimating the inclination of roads, the power requisite to sustain a load upon them (setting aside the effect of friction), is always proportional to that elevation. Thus, if a road rise one foot in twenty, a power of one ton will be sufficient to sustain twenty tons, and so on.
On a horizontal plane the only resistance which the power has to overcome is the friction of the load with the plane, and the consideration of this being for the present omitted, a weight once put in motion would continue moving for ever, without any further action of the power. But if the plane be inclined, the power will be expended in raising the weight through the perpendicular height of the plane. Thus, in a road which rises one foot in ten, the power is expended in raising the weight through one perpendicular foot for every ten feet of the road over which it is moved. As the expenditure of power depends upon the rate at which the weight is raised perpendicularly, it is evident that the greater the inclination of the road is, the slower the motion must be with the same force. If the energy of the power be such as to raise the weight at the rate of one foot per minute, the weight may be moved in each minute through that length of the road which corresponds to a rise of one foot. Thus, if two roads rise one at the rate of a foot in fifteen feet, and the other at the rate of one foot in twenty feet, the same expenditure of power will move the weight through fifteen feet of the one, and twenty feet of the other at the same rate.
From such considerations as these, it will readily appear that it may often be more expedient to carry a road through a circuitous route than to continue it in the most direct course; for though the measured length of road may be considerably greater than in the former case, yet more may be gained in speed with the same expenditure of power than is lost by the increase of distance. By attending to these circumstances, modern road-makers have greatly facilitated and expedited the intercourse between distant places.
(288.) If the power act obliquely to the plane, it will have a twofold effect; a part being expended in supporting or drawing the weight, and a part in diminishing or increasing the pressure upon the plane. Let W P, _fig. 130._, be the power. This will be equivalent to two forces, W F′, perpendicular to the plane, and W E′ in the direction of the plane. (74.) In order that the power should sustain the weight, it is necessary that that part W E′ of the power which acts in the direction of the plane should be equal to that part W E, _fig. 130._, of the weight which acts down the plane. The other part W F′ of the power acting perpendicular to the plane is immediately opposed to that part W F of the weight which produces pressure. The pressure upon the plane will therefore be diminished by the amount of W F′. The amount of the power which will equilibrate with the weight may, in this case, be found as follows. Take W E′ equal to W E, and draw E′ P perpendicular to the plane, and meeting the direction of the power. The proportion of the power to the weight will be that of W P to W D. And the proportion of the pressure to the weight will be that of the difference between W F and W F′ to W D. If the amount of the power have a less proportion to the weight than W P has to W D, it will not support the body on the plane, but will allow it to descend. And if it have a greater proportion, it will draw the weight up the plane towards A.
(289.) It sometimes happens that a weight upon one inclined plane is raised or supported by another weight upon another inclined plane. Thus, if A B and A B′, _fig. 131._, be two inclined planes forming an angle at A, and W W′ be two weights placed upon these planes, and connected by a cord passing over a pulley at A, the one weight will either sustain the other, or one will descend, drawing the other up. To determine the circumstances under which these effects will ensue, draw the lines W D and W′ D′ in the vertical direction, and take upon them as many inches as there are ounces in the weights respectively. W D and W′ D′ being the lengths thus taken, and therefore representing the weights, the lines W E and W′ E′ will represent the effects of these weights respectively down the planes. If W E and W′ E′ be equal, the weights will sustain each other without motion. But if W E be greater than W′ E′, the weight W will descend, drawing the weight W′ up. And if W′ E′ be greater than W E, the weight W′ will descend, drawing the weight W up. In every case the lines W F and W′ F′ will represent the pressures upon the planes respectively.
It is not necessary, for the effect just described, that the inclined planes should, as represented in the figure, form an angle with each other. They may be parallel, or in any other position, the rope being carried over a sufficient number of wheels placed so as to give it the necessary deflection. This method of moving loads is frequently applied in great public works where rail-roads are used. Loaded waggons descend one inclined plane, while other waggons, either empty or so loaded as to permit the descent of those with which they are connected, are drawn up the other.
(290.) In the application of the inclined plane which we have hitherto noticed, the machine itself is supposed to be fixed in its position, while the weight or load is moved upon it. But it frequently happens that resistances are to be overcome which do not admit of being thus moved. In such cases, instead of moving the load upon the planes, the plane is to be moved under or against the load. Let D E, _fig. 132._, be a heavy beam secured in a vertical position between guides F G and H I, so that it is free to move upwards and downwards, but not laterally. Let A B C be an inclined plane, the extremity of which is placed beneath the end of the beam. A force applied to the back of this plane A C, in the direction C B, will urge the plane under the beam so as to raise the beam to the position represented in _fig. 133._ Thus, while the inclined plane is moved through the distance C B, the beam is raised through the height C A.
(291.) When the inclined plane is applied in this manner, it is called a _wedge_. And if the power applied to the back were a continued pressure, its proportion to the weight would be that of A C to C B. It follows, therefore, that the more acute the angle B is, the more powerful will be the wedge.
In some cases, the wedge is formed of two inclined planes, placed base to base, as represented in _fig. 134._ The theoretical estimation of the power of this machine is not applicable in practice with any degree of accuracy. This is in part owing to the enormous proportion which the friction in most cases bears to the theoretical value of the power, but still more to the nature of the power generally used. The force of a blow is of a nature so wholly different from continued forces, such as the pressure of weights, or the resistance offered by the cohesion of bodies, that it admits of no numerical comparison with them. Hence we cannot properly state the proportion which the force of a blow bears to the amount of a weight or resistance. The wedge is almost invariably urged by percussion; while the resistances which it has to overcome are as constantly forces of the other kind. Although, however, no exact numerical comparison can be made, yet it may be stated in a general way that the wedge is more and more powerful as its angle is more acute.
In the arts and manufactures, wedges are used where enormous force is to be exerted through a very small space. Thus it is resorted to for splitting masses of timber or stone. Ships are raised in docks by wedges driven under their keels. The wedge is the principal agent in the oil-mill. The seeds from which the oil is to be extracted are introduced into hair bags, and placed between planes of hard wood. Wedges inserted between the bags are driven by allowing heavy beams to fall on them. The pressure thus excited is so intense, that the seeds in the bags are formed into a mass nearly as solid as wood. Instances have occurred in which the wedge has been used to restore a tottering edifice to its perpendicular position.
All cutting and piercing instruments, such as knives, razors, scissors, chisels, &c., nails, pins, needles, awls, &c. are wedges. The angle of the wedge, in these cases, is more or less acute, according to the purpose to which it is to be applied. In determining this, two things are to be considered--the mechanical power, which is increased by diminishing the angle of the wedge; and the strength of the tool, which is always diminished by the same cause. There is, therefore, a practical limit to the increase of the power, and that degree of sharpness only is to be given to the tool which is consistent with the strength requisite for the purpose to which it is to be applied. In tools intended for cutting wood, the angle is generally about 30°. For iron it is from 50° to 60°; and for brass, from 80° to 90°. Tools which act by pressure may be made more acute than those which are driven by a blow; and in general the softer and more yielding the substance to be divided is, and the less the power required to act upon it, the more acute the wedge may be constructed.
In many cases the utility of the wedge depends on that which is entirely omitted in its theory, viz. the friction which arises between its surface and the substance which it divides. This is the case when pins, bolts, or nails are used for binding the parts of structures together; in which case, were it not for the friction, they would recoil from their places, and fail to produce the desired effect. Even when the wedge is used as a mechanical engine, the presence of friction is absolutely indispensable to its practical utility. The power, as has already been stated, generally acts by successive blows, and is therefore subject to constant intermission, and but for the friction the wedge would recoil between the intervals of the blows with as much force as it had been driven forward. Thus the object of the labour would be continually frustrated. The friction in this case is of the same use as a ratchet wheel, but is much more necessary, as the power applied to the wedge is more liable to intermission than in the cases where ratchet wheels are generally used.
(292.) When a road directly ascends the side of a hill, it is to be considered as an inclined plane; but it will not lose its mechanical character, if, instead of directly ascending towards the top of the hill, it winds successively round it, and gradually ascends so as after several revolutions to reach the top. In the same manner a path may be conceived to surround a pillar by which the ascent may be facilitated upon the principle of the inclined plane. Winding stairs constructed in the interior of great columns partake of this character; for although the ascent be produced by successive steps, yet if a floor could be made sufficiently rough to prevent the feet from slipping, the ascent would be accomplished with equal facility. In such a case the winding path would be equivalent to an inclined plane, bent into such a form as to accommodate it to the peculiar circumstances in which it would be required to be used. It will not be difficult to trace the resemblance between such an adaptation of the inclined plane and the appearances presented by the thread of a _screw_: and it may hence be easily understood that a screw is nothing more than an inclined plane constructed upon the surface of a cylinder.
This will, perhaps, be more apparent by the following contrivance: Let A B, _fig. 135._, be a common round ruler, and let C D E be a piece of white paper cut in the form of an inclined plane, whose height C D is equal to the length of the ruler A B, and let the edge C E of the paper be marked with a broad black line: let the edge C D be applied to the ruler A B, and being attached thereto, let the paper be rolled round the ruler; the ruler will then present the appearance of a screw, _fig. 136._ the thread of the screw being marked by the black line C E, winding continually round the ruler. Let D F, _fig. 135._, be equal to the circumference of the ruler, and draw F G parallel to D C, and G H parallel to D E, the