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(72.) Motion and pressure are terms too familiar to need explanation. It may be observed, generally, that definitions in the first rudiments of a science are seldom, if ever, comprehended. The force of words is learned by their application; and it is not until a definition becomes useless, that we are taught the meaning of the terms in which it is expressed. Moreover, we are perhaps justified in saying, that in the mathematical sciences the fundamental notions are of so uncompounded a character, that definitions, when developed and enlarged upon, often draw us into metaphysical subtleties and distinctions, which, whatever be their merit or importance, would be here altogether misplaced. We shall, therefore, at once take it for granted, that the words motion and pressure express phenomena or effects which are the subjects of constant experience and hourly observation; and if the scientific use of these words be more precise than their general and popular application, that precision will soon be learned by their frequent use in the present treatise.

(73.) Force is the name given in mechanics to whatever produces motion or pressure. This word is also often used to express the motion or pressure itself; and when the cause of the motion or pressure is not known, this is the only correct use of the word. Thus, when a piece of iron moves toward a magnet, it is usual to say that the cause of the motion is “the attraction of the magnet;” but in effect we are ignorant of the cause of this phenomenon; and the name attraction would be better applied to the effect of which we have experience. In like manner the attraction and repulsion of electrified bodies should be understood, not as names for unknown causes, but as words expressing observed appearances or effects.

When a certain phraseology has, however, gotten into general use, it is neither easy nor convenient to supersede it. We shall, therefore, be compelled, in speaking of motion and pressure, to use the language of causation; but must advise the student that it is effects and not causes which will be expressed.

(74.) If two forces act upon the same point of a body in different directions, a single force may be assigned, which, acting on that point, will produce the same result as the united effects of the other two.

Let P, fig.7., be the point on which the two forces act, and let their directions be PA and PB. From the point P, upon the line PA, take a length P a, consisting of as many inches as there are ounces in the force PA; and, in like manner, take P b, in the direction PB, consisting of as many inches as there are ounces in the force PB. Through a draw a line parallel to PB, and through b draw a line parallel to PA, and suppose that these lines meet at c. Then draw PC. A single force, acting in the direction PC, and consisting of as many ounces as the line Pc consists of inches, will produce upon the point P the same effect as the two forces PA and PB produce acting together.

(75.) The figure P acb is called in GEOMETRY a parallelogram; the lines P a, P b, are called its sides, and the line P c is called its diagonal. Thus the method of finding an equivalent for two forces, which we have just explained, is generally called “the parallelogram of forces,” and is usually expressed thus: “If two forces be represented in quantity and direction by the sides of a parallelogram, an equivalent force will be represented in quantity and direction by its diagonal.”

(76.) A single force, which is thus mechanically equivalent to two or more other forces, is called their resultant, and relatively to it they are called its components. In any mechanical investigation, when the resultant is used for the components, which it always may be, the process is called “the composition of force.” It is, however, frequently expedient to substitute for a single force two or more forces, to which it is mechanically equivalent, or of which it is the resultant. This process is called “the resolution of force.”

(77.) To verify experimentally the theorem of the parallelogram of forces is not difficult. Let two small wheels, MN, fig.8., with grooves in their edges to receive a thread, be attached to an upright board, or to a wall. Let a thread be passed over them, having weights A and B, hooked upon loops at its extremities. From any part P of the thread between the wheels let a weight C be suspended: it will draw the thread downwards, so as to form an angle MPN, and the apparatus will settle itself at rest in some determinate position. In this state it is evident that since the weight C, acting in the direction PC, balances the weights A and B, acting in the directions PM and PN, these two forces must be mechanically equivalent to a force equal to the weight C, and acting directly upwards from P. The weight C is therefore the quantity of the resultant of the forces PM and PN; and the direction of the resultant is that of a line drawn directly upwards from P.

To ascertain how far this is consistent with the theorem of “the parallelogram of forces,” let a line PO be drawn upon the upright board to which the wheels are attached, from the point P upward, in the direction of the thread CP. Also, let lines be drawn upon the board immediately under the threads PM and PN. From the point P, on the line PO, take as many inches as there are ounces in the weight C. Let the part of PO thus measured be P c, and from c draw ca parallel to PN, and cb parallel to PM. If the sides P a and P b of the parallelogram thus formed be measured, it will be found that P a will consist of as many inches as there are ounces in the weight A, and P b of as many inches as there are ounces in the weight B.

In this illustration, ounces and inches have been used as the subdivisions of weight and length. It is scarcely necessary to state, that any other measures of these quantities would serve as well, only observing that the same denominations must be preserved in all parts of the same investigation.

(78.) Among the philosophical apparatus of the University of London, is a very simple and convenient instrument which I constructed for the experimental illustration of this important theorem. The wheels MN are attached to the tops of two tall stands, the heights of which may be varied at pleasure by an adjusting screw. A jointed parallelogram, ABCD, fig.9., is formed, whose sides are divided into inches, and the joints at A and B are moveable, so as to vary the lengths of the sides at pleasure. The joint C is fixed at the extremity of a ruler, also divided into inches, while the opposite joint A is attached to a brass loop, which surrounds the diagonal ruler loosely, so as to slide freely along it. An adjusting screw is provided in this loop so as to clamp it in any required position.

In making the experiment, the sides AB and AD, CB and CD are adjusted by the joints B and A to the same number of inches respectively as there are ounces in the weights A and B, fig.8. Then the diagonal AC is adjusted by the loop and screw at A, to as many inches as there are ounces in the weight C. This done, the point A is placed behind P, fig.8., and the parallelogram is held upright, so that the diagonal AC shall be in the direction of the vertical thread PC. The sides AB and AD will then be found to take the direction of the threads PM and PN. By changing the weights and the lengths of the diagonal and sides of the parallelogram, the experiment may be easily varied at pleasure.

(79.) In the examples of the composition of forces which we have here given, the effects of the forces are the production of pressures, or, to speak more correctly, the theorem which we have illustrated, is “the composition of pressures.” For the point P is supposed to be at rest, and to be drawn or pressed in the directions PM and PN. In the definition which has been given of the word force, it is declared to include motions as well as pressures. In fact, if motion be resisted, the effect is converted into pressure. The same cause acting upon a body, will either produce motion or pressure, according as the body is free or restrained. If the body be free, motion ensues; if restrained, pressure, or both these effects together. It is therefore consistent with analogy to expect that the same theorems which regulate pressures, will also be applicable to motions; and we find accordingly a most exact correspondence.

(80.) If a body have a motion in the direction AB, and at the point P it receive another motion, such as would carry it in the direction PC, fig.10., were it previously quiescent at P, it is required to determine the direction which the body will take, and the speed with which it will move, under these circumstances.

Let the velocity with which the body is moving from A to B be such, that it would move through a certain space, suppose PN, in one second of time, and let the velocity of the motion impressed upon it at P be such, that if it had no previous motion it would move from P to M in one second. From the point M draw a line parallel to PB, and from N draw a line parallel to PC, and suppose these lines to meet at some point, as O. Then draw the line PO. In consequence of the two motions, which are at the same time impressed upon the body at P, it will move in the straight line from P to O.

Thus the two motions, which are expressed in quantity and direction by the sides of a parallelogram, will, when given to the same body, produce a single motion, expressed in quantity and direction by its diagonal; a theorem which is to motions exactly what the former was to pressures.

There are various methods of illustrating experimentally the composition of motion. An ivory ball, being placed upon a perfectly level square table, at one of the corners, and receiving two equal impulses, in the directions of the sides of the table, will move along the diagonal. Apparatus for this experiment differ from each other only in the way of communicating the impulses to the ball.

(81.) As two motions simultaneously communicated to a body are equivalent to a single motion in an intermediate direction, so also a single motion may be mechanically replaced, by two motions in directions expressed by the sides of any parallelogram, whose diagonal represents the single motion. This process is “the resolution of motion,” and gives considerable clearness and facility to many mechanical investigations.

(82.) It is frequently necessary to express the portion of a given force, which acts in some given direction different from the immediate direction of the force itself. Thus, if a force act from A, fig.11., in the direction AC, we may require to estimate what part of that force acts in the direction AB. If the force be a pressure, take as many inches AP from A, on the line AC, as there are ounces in the force, and from P draw PM perpendicular to AB; then the part of the force which acts along AB will be as many ounces as there are inches in AM. The force AB is mechanically equivalent to two forces, expressed by the sides AM and AN of the parallelogram; but AN, being perpendicular to AB, can have no effect on a body at A, in the direction of AB, and therefore the effective part of the force AP in the direction AB is expressed by AM.

(83.) Any number of forces acting on the same point of a body may be replaced by a single force, which is mechanically equivalent to them, and which is, therefore, their resultant. This composition may be effected by the successive application of the parallelogram of forces. Let the several forces be called A, B, C, D, E, &c. Draw the parallelogram whose sides express the forces A and B, and let its diagonal be A′. The force expressed by A′ will be equivalent to A and B. Then draw the parallelogram whose sides express the forces A′ and C, and let its diagonal be B′. This diagonal will express a force mechanically equivalent to A′ and C. But A′ is mechanically equivalent to A and B, and therefore B′ is mechanically equivalent to A, B, and C. Next construct a parallelogram, whose sides express the forces B′ and D, and let its diagonal be C′. The force expressed by C′ will be mechanically equivalent to the forces B′ and D; but the force B′ is equivalent to A, B, C, and therefore C′ is equivalent to A, B, C, and D. By continuing this process it is evident, that a single force may be found, which will be equivalent to, and may be always substituted for, any number of forces which act upon the same point.

If the forces which act upon the point neutralise each other, so that no motion can ensue, they are said to be in equilibrium.

(84.) Examples of the composition of motion and pressure are continually presenting themselves. They occur in almost every instance of motion or force which falls under our observation. The difficulty is to find an example which, strictly speaking, is a simple motion.

When a boat is rowed across a river, in which there is a current, it will not move in the direction in which it is impelled by the oars. Neither will it take the direction of the stream, but will proceed exactly in that intermediate direction which is determined by the composition of force.

Let A, fig.12., be the place of the boat at starting; and suppose that the oars are so worked as to impel the boat towards B with a force which would carry it to B in one hour, if there were no current in the river. But, on the other hand, suppose the rapidity of the current is such, that without any exertion of the rowers the boat would float down the stream in one hour to C. From C draw CD parallel to AB, and draw the straight line AD diagonally. The combined effect of the oars and the current will be, that the boat will be carried along AD, and will arrive at the opposite bank in one hour, at the point D.

If the object be, therefore, to reach the point B, starting from A, the rowers must calculate, as nearly as possible, the velocity of the current. They must imagine a certain point E at such a distance above B that the boat would be floated by the stream from E to B in the time taken in crossing the river in the direction AE, if there were no current. If they row towards the point E, the boat will arrive at the point B, moving in the line AB.

In this case the boat is impelled by two forces, that of the oars in the direction AE, and that of the current in the direction AC. The result will be, according to the parallelogram of forces, a motion in the diagonal AB.

The wind and tide acting upon a vessel is a case of a similar kind. Suppose that the wind is made to impel the vessel in the direction of the keel; while the tide may be acting in any direction oblique to that of the keel. The course of the vessel is determined exactly in the same manner as that of the boat in the last example.

The action of the oars themselves, in impelling the boat, is an example of the composition of force. Let A, fig.13., be the head, and B the stern of the boat. The boatman presents his face towards B, and places the oars so that their blades press against the water in the directions CE, DF. The resistance of the water produces forces on the side of the boat, in the directions GL and HL, which, by the composition of force, are equivalent to die diagonal force KL, in the direction of the keel.

Similar observations will apply to almost every body impelled by instruments projecting from its sides, and acting against a fluid. The motions of fishes, the act of swimming, the flight of birds, are all instances of the same kind.

(85.) The action of wind upon the sails of a vessel, and the force thereby transmitted to the keel, modified by the rudder, is a problem which is solved by the principles of the composition and resolution of force; but it is of too complicated and difficult a nature to be introduced with all its necessary conditions and limitations in this place. The question may, however, be simplified, if we consider the canvass of the sails to be stretched so completely as to form a plane surface. Let AB, fig.14., be the position of the sail, and let the wind blow in the direction CD. If the line CD be taken to express the force of the wind, let DECF be a parallelogram, of which it is the diagonal. The force CD is equivalent to two forces, one in the direction FD of the plane of the canvass, and the other ED perpendicular to the sail. The effect, therefore, is the same as if there were two winds, one blowing in the direction of FD or BA, that is against the edge of the sail, and the other, ED, blowing full against its face. It is evident that the former will produce no effect whatever upon the sail, and that the latter will urge the vessel in the direction DG.

Let us now consider this force DG as acting in the diagonal of the parallelogram DHGI. It will be equivalent to two forces, DH and DI, acting along the sides. One of these forces, DH, is in the direction of the keel, and the other, DI, at right angles to the length of the vessel, so as to urge it sideways. The form of the vessel is evidently such as to offer a great resistance to the latter force, and very little to the former. It consequently proceeds with considerable velocity in the direction DH of its keel, and makes way very slowly in the sideward direction DI. The latter effect is called lee-way.

From this explanation it will be easily understood, how a wind which is nearly opposed to the course of a vessel may, nevertheless, be made to impel it by the effect of sails. The angle BDV, formed by the sail and the direction of the keel, may be very oblique, as may also be the angle CDB formed by the direction of the wind and that of the sail. Therefore the angle CDV, made up of these two, and which is that formed by the direction of the wind and that of the keel, may be very oblique. In fig.15. the wind is nearly contrary to the direction of the keel, and yet there is an impelling force expressed by the line DH, the line CD expressing, as before, the whole force of the wind.

In this example there are two successive decompositions of force. First, the original force of the wind CD is resolved into two, ED and FD; and next the element ED, or its equal DG, is resolved into DI and DH; so that the original force is resolved into three, viz. FD, DI, DH, which, taken together, are mechanically equivalent to it. The part FD is entirely ineffectual; it glides off on the surface of the canvass without producing any effect upon the vessel. The part DI produces lee-way, and the part DH impels.

H. Adlard, sc.

London, Pubd. by Longman & Co.

(86.) If the wind, however, be directly contrary to the course which it is required that the vessel should take, there is no position which can be given to the sails which will impel the vessel. In this case the required course itself is resolved into two, in which the vessel sails alternately, a process which is called tacking. Thus, suppose the vessel is required to move from A to E, fig.16., the wind setting from E to A. The motion AB being resolved into two, by being assumed as the diagonal of a parallelogram, the sides A a, aB of the parallelogram are successively sailed over, and the vessel by this means arrives at B, instead of moving along the diagonal AB. In the same manner she moves along B b, b C, C c, c D, D d, d E, and arrives at E. She thus sails continually at a sufficient angle with the wind to obtain an impelling force, yet at a sufficiently small angle to make way in her proposed course.

The consideration of the effect of the rudder, which we have omitted in the preceding illustration, affords another instance of the resolution of force. We shall not, however, pursue this example further.

(87.) A body falling from the top of the mast when the vessel is in full sail, is an example of the composition of motion. It might be expected, that during the descent of the body, the vessel having sailed forward, would leave it behind, and that, therefore, it would fall in the water behind the stern, or at least on the deck, considerably behind the mast. On the other hand, it is found to fall at the foot of the mast, exactly as it would if the vessel were not in motion. To account for this, let AB, fig.17., be the position of the mast when the body at the top is disengaged. The mast is moving onwards with the vessel in the direction AC, so that in the time which the body would take to fall to the deck, the top of the mast would move from A to C. But the body being on the mast at the moment it is disengaged, has this motion AC in common with the mast; and therefore in its descent it is affected by two motions, viz. that of the vessel expressed by AC, and its descending motion expressed by AB. Hence, by the composition of motion, it will be found at the opposite angle D of the parallelogram, at the end of the fall. During the fall, however, the mast has moved with the vessel, and has advanced to CD, so that the body falls at the foot of the mast.

(88.) An instance of the composition of motion, which is worthy of some attention, as it affords a proof of the diurnal motion of the earth, is derived from observing the descent of a body from a very high tower. To render the explanation of this more simple, we shall suppose the tower to be on the equator of the earth. Let EPQ, fig.18., be a section of the earth through the equator, and let PT be the tower. Let us suppose that the earth moves on its axis in the direction EPQ. The foot P of the tower will, therefore, in one day move over the circle EPQ, while the top T moves over the greater circle TT′R. Hence it is evident, that the top of the tower moves with greater speed than the foot, and therefore in the same time moves through a greater space. Now suppose a body placed at the top; it participates in the motion which the top of the tower has in common with the earth. If it be disengaged, it also receives the descending motion TP. Let us suppose that the body would take five seconds to fall from T to P, and that in the same time the top T is moved by the rotation of the earth from T to T′, the foot being moved from P to P′. The falling body is therefore endued with two motions, one expressed by TT′, and the other by TP. The combined effect of these will be found in the usual way by the parallelogram. Take T p equal to TT′; the body will move from T to p in the time of the fall, and will meet the ground at p. But since TT′ is greater than PP′, it follows that the point p must be at a distance from P′ equal to the excess of TT′ above PP′. Hence the body will not fall exactly at the foot of the tower, but at a certain distance from it, in the direction of the earth’s motion, that is, eastward. This is found, by experiment, to be actually the case; and the distance from the foot of the tower, at which the body is observed to fall, agrees with that which is computed from the motion of the earth, to as great a degree of exactness as could be expected from the nature of the experiment.

(89.) The properties of compounded motions cause some of the equestrian feats exhibited at public spectacles to be performed by a kind of exertion very different from that which the spectators generally attribute to the performer. For example, the horseman standing on the saddle leaps over a garter extended over the horse at right angles to his motion; the horse passing under the garter, the rider lights upon the saddle at the opposite side. The exertion of the performer, in this case, is not that which he would use were he to leap from the ground over a garter at the same height. In the latter case, he would make an exertion to rise, and, at the same time, to project his body forward. In the case, however, of the horseman, he merely makes that exertion which is necessary to rise directly upwards to a sufficient height to clear the garter. The motion which he has in common with the horse, compounded with the elevation acquired by his muscular power, accomplishes the leap.

To explain this more fully, let ABC, fig.19., be the direction in which the horse moves, A being the point at which the rider quits the saddle, and C the point at which he returns to it. Let D be the highest point which is to be cleared in the leap. At A the rider makes a leap towards the point E, and this must be done at such a distance from B, that he would rise from B to E in the time in which the horse moves from A to B. On departing from A, the rider has, therefore, two motions, represented by the lines AE and AB, by which he will move from the point A to the opposite angle D of the parallelogram. At D, the exertion of the leap being overcome by the weight of his body, he begins to return downward, and would fall from D to B in the time in which the horse moves from B to C. But at D he still retains the motion which he had in common with the horse; and therefore, in leaving the point D, he has two motions, expressed by the lines DF and DB. The compounded effects of these motions carry him from D to C. Strictly speaking, his motion from A to D, and from D to C, is not in straight lines, but in a curve. It is not necessary here, however, to attend to this circumstance.

(90.) If a billiard-ball strike the cushion of the table obliquely, it will be reflected from it in a certain direction, forming an angle with the direction in which it struck it. This affords an example of the resolution and composition of motion. We shall first consider the effect which would ensue if the ball struck the cushion perpendicularly.

Let AB, fig.20., be the cushion, and CD the direction in which the ball moves towards it. If the ball and the cushion were perfectly inelastic, the resistance of the cushion would destroy the motion of the ball, and it would be reduced to a state of rest at D. If, on the other hand, the ball were perfectly elastic, it would be reflected from the cushion, and would receive as much motion from D to C after the impact, as it had from C to D before it. Perfect elasticity, however, is a quality which is never found in these bodies. They are always elastic, but imperfectly so. Consequently the ball after the impact will be reflected from D towards C, but with a less motion than that with which it approached from C to D.

Now let us suppose that the ball, instead of moving from C to D, moves from E to D. The force with which it strikes D being expressed by DE′, equal to ED, may be resolved into two, DF and DC′. The resistance of the cushion destroys DC′, and the elasticity produces a contrary force in the direction DC, but less than DC or DC′, because that elasticity is imperfect. The line DC expressing the force in the direction CD, let DG (less than DC) express the reflective force in the direction DC. The other element DF, into which the force DE′ is resolved by the impact, is not destroyed or modified by the cushion, and therefore, on leaving the cushion at D, the ball is influenced by two forces, DF (which is equal to CE) and DG. Consequently it will move in the diagonal DH.

(91.) The angle EDC is in this case called the “angle of incidence,” and CDH is called “the angle of reflection.” It is evident, from what has been just inferred, that the ball, being imperfectly elastic, the angle of incidence must always be less than the angle of reflection, and with the same obliquity of incidence, the more imperfect the elasticity is, the less will be the angle of reflection.

In the impact of a perfectly elastic body, the angle of reflection would be equal to the angle of incidence. For then the line DG, expressing the reflective force, would be taken equal to CD, and the angle CDH would be equal to CDE. This is found by experiment to be the case when light is reflected from a polished surface of glass or metal.

Motion is sometimes distinguished into absolute and relative. What “relative motion” means is easily explained. If a man walk upon the deck of a ship from stem to stern, he has a relative motion which is measured by the space upon the deck over which he walks in a given time. But while he is thus walking from stem to stern, the ship and its contents, including himself, are impelled through the deep in the opposite direction. If it so happen that the motion of the man, from stem to stern, be exactly equal to the motion of the ship in the contrary way, the man will be, relatively to the surface of the sea and that of the earth, at rest. Thus, relatively to the ship, he is in motion, while, relatively to the surface of the earth, he is at rest. But still this is not absolute rest. The surface itself is moving by the diurnal rotation of the earth upon its axis, as well as by the annual motion in its orbit round the sun. These motions, and others to which the earth is subject, must be all compounded by the theorem of the parallelogram of forces before we can obtain the absolute state of the body with respect to motion or rest.