Read Ebook: A Treatise on Mechanics by Kater Henry Lardner Dionysius

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The transparent wings of certain insects are so attenuated in their structure that 50,000 of them placed over each other would not form a pile a quarter of an inch in height.

In the manufacture of embroidery it is necessary to obtain very fine gilt silver threads. To accomplish this, a cylindrical bar of silver, weighing 360 ounces, is covered with about two ounces of gold. This gilt bar is then wire-drawn, as in the first example, until it is reduced to a thread so fine that 3400 feet of it weigh less than an ounce. The wire is then flattened by passing it between rollers under a severe pressure, a process which increases its length, so that about 4000 feet shall weigh one ounce. Hence, one foot will weigh the 4000th part of an ounce. The proportion of the gold to the silver in the original bar was that of 2 to 360, or 1 to 180. Since the same proportion is preserved after the bar has been wire-drawn, it follows that the quantity of gold which covers one foot of the fine wire is the 180th part of the 4000th of an ounce; that is the 720,000th part of an ounce.

The quantity of gold which covers one inch of this wire will be twelve times less than that which covers one foot. Hence, this quantity will be the 8,640,000th part of an ounce. If this inch be again divided into 100 equal parts, every part will be distinctly visible without the aid of microscopes. The gold which covers this small but visible portion is the 864,000,000th part of an ounce. But we may proceed even further; this portion of the wire may be viewed by a microscope which magnifies 500 times, so that the 500th part of it will thus become visible. In this manner, therefore, an ounce of gold may be divided into 432,000,000,000 visible parts, each of which will possess all the characters and qualities found in the largest masses of the metal. It will retain its solidity, texture, and colour; it will resist the same agents, and enter into combination with the same substances. If the gilt wire be dipped in nitric acid, the silver within the coating will be dissolved, but the hollow tube of gold which surrounded it will still cohere and remain suspended.

The organised world offers still more remarkable examples of the inconceivable subtilty of matter.

Small as these discs are, the animal kingdom presents beings whose whole bodies are still more minute. Animalcules have been discovered, whose magnitude is such, that a million of them do not exceed the bulk of a grain of sand; and yet each of these creatures is composed of members as curiously organised as those of the largest species; they have life and spontaneous motion, and are endued with sense and instinct. In the liquids in which they live, they are observed to move with astonishing speed and activity; nor are their motions blind and fortuitous, but evidently governed by choice, and directed to an end. They use food and drink, from which they derive nutrition, and are therefore furnished with a digestive apparatus. They have great muscular power, and are furnished with limbs and muscles of strength and flexibility. They are susceptible of the same appetites, and obnoxious to the same passions, the gratification of which is attended with the same results as in our own species. Spallanzani observes, that certain animalcules devour others so voraciously, that they fatten and become indolent and sluggish by over-feeding. After a meal of this kind, if they be confined in distilled water, so as to be deprived of all food, their condition becomes reduced; they regain their spirit and activity, and amuse themselves in the pursuit of the more minute animals, which are supplied to them; they swallow these without depriving them of life, for, by the aid of the microscope, the one has been observed moving within the body of the other. These singular appearances are not matters of idle and curious observation. They lead us to enquire what parts are necessary to produce such results. Must we not conclude that these creatures have heart, arteries, veins, muscles, sinews, tendons, nerves, circulating fluids, and all the concomitant apparatus of a living organised body? And if so, how inconceivably minute must those parts be! If a globule of their blood bears the same proportion to their whole bulk as a globule of our blood bears to our magnitude, what powers of calculation can give an adequate notion of its minuteness?

These and many other phenomena observed in the immediate productions of nature, or developed by mechanical and chemical processes, prove that the materials of which bodies are formed are susceptible of minuteness which infinitely exceeds the powers of sensible observation, even when those powers have been extended by all the aids of science. Shall we then conclude that matter is infinitely divisible, and that there are no original constituent atoms of determinate magnitude and figure at which all subdivision must cease? Such an inference would be unwarranted, even had we no other means of judging the question, except those of direct observation; for it would be imposing that limit on the works of nature which she has placed upon our powers of observing them. Aided by reason, however, and a due consideration of certain phenomena which come within our immediate powers of observation, we are frequently able to determine other phenomena which are beyond those powers. The diurnal motion of the earth is not perceived by us, because all things around us participate in it, preserve their relative position, and appear to be at rest. But reason tells us that such a motion must produce the alternations of day and night, and the rising and setting of all the heavenly bodies; appearances which are plainly observable, and which betray the cause from which they arise. Again, we cannot place ourselves at a distance from the earth, and behold the axis on which it revolves, and observe its peculiar obliquity to the orbit in which the earth moves; but we see and feel the vicissitudes of the seasons, an effect which is the immediate consequence of that inclination, and by which we are able to detect it.

If one of them be detached from the others, and the progress of its formation observed, it will be found gradually to increase, always preserving its original figure. Since its increase must be caused by the continued accession of saline molecules, disengaged by the evaporation of the water, it follows that these molecules must be so formed, that by attaching themselves successively to the crystal, they maintain the regularity of its bounding planes, and preserve their mutual inclinations unvaried.

Suppose a crystal to be taken from the liquid during the process of crystallisation, and a piece broken from it so as to destroy the regularity of its form: if the crystal thus broken be restored to the liquid, it will be observed gradually to resume its regular form, the atoms of salt successively dismissed by the vaporising water filling up the irregular cavities produced by the fracture. Hence it follows, that the saline particles which compose the surface of the crystal, and those which form the interior of its mass, are similar, and exert similar attractions on the atoms disengaged by the water.

We may conceive crystallised substances to be regular mechanical structures formed of atoms of a certain figure, on which the figure of the whole structure must depend. The planes of cleavage are parallel to the sides of the constituent atoms; and their directions, therefore, form so many conditions for the determination of their figure. The shape of the atoms being thus determined, it is not difficult to assign all the various ways in which they may be arranged, so as to produce figures which are accordingly found to correspond with the various forms of crystals of the same substance.

When these phenomena are duly considered and compared, little doubt can remain that all substances susceptible of crystallisation, consist of atoms of determinate figure. This is the case with all solid bodies whatever, which have come under scientific observation, for they have been severally found in or reduced to a crystallised form. Liquids crystallise in freezing, and if a?riform fluids could by any means be reduced to the solid form, they would probably also manifest the same effect. Hence it appears reasonable to presume, that all bodies are composed of atoms; that the different qualities with which we find different substances endued, depend on the magnitude and figure of these atoms; that these atoms are indestructible and immutable by any natural process, for we find the qualities which depend on them unchangeably the same under all the influences to which they have been submitted since their creation; that these atoms are so minute in their magnitude, that they cannot be observed by any means which human art has yet contrived; but still that magnitudes can be assigned which they do not exceed.

In bodies which are constituted uniformly throughout their entire dimensions, the component particles and the pores are uniformly distributed through the volume; that is, a given space in one part of the volume will contain the same quantity of matter and the same quantity of pores as an equal space in another part.

The pores of a body are frequently filled with another body of a more subtle nature. If the pores of a body on the surface of the earth, and exposed to the atmosphere, be greater than the atoms of air, then the air may pervade the pores. This is found to be the case with many sorts of wood which have an open grain. If a piece of such wood, or of chalk, or of sugar, be pressed to the bottom of a vessel of water, the air which fills the pores will be observed to escape in bubbles and to rise to the surface, the water entering the pores, and taking its place.

If a tall vessel or tube, having a wooden bottom, be filled with quicksilver, the liquid metal will be forced by its own weight through the pores of the wood, and will be seen escaping in a silver shower from the bottom.

Larger mineral masses exhibit degrees of porosity not less striking. Water percolates through the sides and roofs of caverns and grottoes, and being impregnated with calcareous and other earths, forms stalactites, or pendant protuberances, which present a curious appearance.

All known bodies, whatever be their nature, are capable of having their dimensions reduced without diminishing their mass; and this is one of the most conclusive proofs that all bodies are porous, or that the constituent atoms are not in contact; for the space by which the volume may be diminished must, before the diminution, consist of pores.

The several qualities of bodies which we have noticed in this chapter, when viewed in relation to each other, present many circumstances worthy of attention.

The fact, that the elevation of temperature produces an increase of volume, is manifested by numerous experiments.

If a flaccid bladder be tied at the mouth, so as to stop the escape of air, and be then held before a fire, it will gradually swell, and assume the appearance of being fully inflated. The small quantity of air contained in the bladder is, in this case, so much dilated by the heat, that it occupies a considerably increased space, and fills the bladder, of which it before only occupied a small part. When the bladder is removed from the fire, and allowed to resume its former temperature, the air returns to its former dimensions, and the bladder becomes again flaccid.

Thermometers are constructed on this principle, the rise of the liquid in the tube being used as an indication of the degree of heat which causes it. A particular account of these useful instruments will be found in our treatise on HEAT.

The change of dimension of solids produced by changes of temperature being much less than that of bodies in the liquid or aeriform state, is not so easily observable. A remarkable instance occurs in the process of shoeing the wheels of carriages. The rim of iron with which the wheel is to be bound, is made in the first instance of a diameter somewhat less than that of the wheel; but being raised by the application of fire to a very high temperature, its volume receives such an increase, that it will be sufficient to embrace and surround the wheel. When placed upon the wheel it is cooled, and suddenly contracting its dimensions, binds the parts of the wheel firmly together, and becomes securely seated in its place upon the fellies.

It frequently happens that the stopper of a glass bottle or decanter becomes fixed in its place so firmly, that the exertion of force sufficient to withdraw it would endanger the vessel. In this case, if a cloth wetted with hot-water be applied to the neck of the bottle, the glass will expand, and the neck will be enlarged, so as to allow the stopper to be easily withdrawn.

Since there is a continual change of temperature in all bodies on the surface of the globe, it follows, that there is also a continual change of magnitude. The substances which surround us are constantly swelling and contracting, under the vicissitudes of heat and cold. They grow smaller in winter, and dilate in summer. They swell their bulk on a warm day, and contract it on a cold one. These curious phenomena are not noticed, only because our ordinary means of observation are not sufficiently accurate to appreciate them. Nevertheless, in some familiar instances the effect is very obvious. In warm weather the flesh swells, the vessels appear filled, the hand is plump, and the skin distended. In cold weather, when the body has been exposed to the open air, the flesh appears to contract, the vessels shrink, and the skin shrivels.

The phenomena attending change of temperature are conclusive proofs of the universal porosity of material substances, but they are not the only proofs. Many substances admit of compression by the mere agency of mechanical force.

That this effect is the consequence of the pressure of the liquid will be easily made manifest by showing that, as the pressure is increased, the air is proportionally contracted in its dimensions; and as it is diminished, the dimensions are on the other hand enlarged. If the depth of the goblet in the water be increased, the cork will be seen to rise in it, showing that the increased pressure, at the greater depth, causes the air in the goblet to be more condensed. If, on the other hand, the goblet be raised toward the surface, the cork will be observed to descend toward the edge, showing that as it is relieved from the pressure of the liquid, the air gradually approaches to its primitive dimensions.

That it is the air alone which excludes the water from the goblet, in the preceding experiments, can easily be proved. When the goblet is sunk deep in the vessel of water, let it be inclined a little to one side until its mouth is presented towards the side of the vessel; let this inclination be so regulated, that the surface of the water in the goblet shall just reach its edge. Upon a slight increase of inclination, air will be observed to escape from the goblet, and to rise in bubbles to the surface of the water. If the goblet be then restored to its position, it will be found that the cork will rise higher in it than before the escape of the air. The water in this case rises and fills the space which the air allowed to escape has deserted. The same process may be repeated until all the air has escaped, and then the goblet will be completely filled by the water.

Elasticity does not always accompany compressibility. If lead or iron be submitted to the hammer, it may be hardened and diminished in its volume; but it will not resume its former volume after each stroke of the hammer.

Those who are provided with an air-pump can easily establish this property experimentally. Take a flaccid bladder, such as that already described in , and place it under the glass receiver of an air-pump: by this instrument we shall be able to remove the air which surrounds the bladder under the receiver, so as to relieve the small quantity of air which is inclosed in the bladder from the pressure of the external air: when this is accomplished, the bladder will be observed to swell, as if it were inflated, and will be perfectly distended. The air contained in it, therefore, has a tendency to dilate, which takes effect when it ceases to be resisted by the pressure of surrounding air.

In all the cases where friction or percussion produces heat or fire, it is because they are means of compression. The effects of flints, of pieces of wood rubbed together, the warmth produced by friction on the flesh, are all to be attributed to the same cause.

INERTIA.

The effects and phenomena which hourly fall under our observation afford unnumbered examples of the inability of lifeless matter to put itself into motion, or to increase any motion which may have been communicated to it. But it does not happen that we have the same direct and frequent evidence of its inability to destroy or diminish any motion which it may have received. And hence it arises, that while no one will deny to matter the former effect of inertia, few will at first acknowledge the latter. Indeed, even so late as the time of KEPLER, philosophers themselves held it as a maxim, that "matter is more inclined to rest than to motion;" we ought not, therefore, to be surprised if in the present day those who have not been conversant with physical science are slow to believe that a body once put in motion would continue for ever to move with the same velocity, if it were not stopped by some external cause.

Let us enquire why we are more disposed to admit the inability of matter to produce than to destroy motion in itself. We see most of those motions which take place around us on the surface of the earth subject to gradual decay, and if not renewed from time to time, at length cease. A stone rolled along the ground, a wheel revolving on its axis, the heaving of the deep after a storm, and all other motions produced in bodies by external causes, decay, when the exciting cause is suspended; and if that cause do not renew its action, they ultimately cease.

But is there no exciting cause, on the other hand, which thus gradually deprives those bodies of their motion?--and if that cause were removed, or its intensity diminished, would not the motion continue, or be more slowly retarded? When a stone is rolled along the ground, the inequalities of its shape as well as those of the ground are impediments, which retard and soon destroy its motion. Render the stone round, and the ground level, and the motion will be considerably prolonged. But still small asperities will remain on the stone, and on the surface over which it rolls: substitute for the stone a ball of highly-polished steel, moving on a highly-polished steel plane, truly level, and the motion will continue without sensible diminution for a very long period; but even here, and in every instance of motions produced by art, minute asperities must exist on the surfaces which move in contact with each other, which must resist, gradually diminish, and ultimately destroy the motion.

Independently of the obstructions to the continuation of motion arising from friction, there is another impediment to which all motions on the surface of the earth are liable--the resistance of the air. How much this may affect the continuation of motion appears by many familiar effects. On a calm day carry an open umbrella with its concave side presented in the direction in which you are moving, and a powerful resistance will be opposed to your progress, which will increase with every increase of the speed with which you move.

We are not, however, without direct experience to prove, that motions when unresisted will for ever continue. In the heavens we find an apparatus, which furnishes a sublime verification of this principle. There, removed from all casual obstructions and resistances, the vast bodies of the universe roll on in their appointed paths with unerring regularity, preserving without diminution all that motion which they received at their creation from the hand which launched them into space. This alone, unsupported by other reasons, would be sufficient to establish the quality of inertia; but viewed in connection with the other circumstances previously mentioned, no doubt can remain that this is an universal law of nature.

Again, suppose G H to be a hard plane surface; and let the body be supposed to be perfectly inelastic. When it strikes the surface at B, it will commence to move along it in the direction B H. This change of direction is produced by the resistance of the surface. If the body, instead of meeting the surface in the direction A B, had moved in the direction E B, perpendicular to it, all motion would have been destroyed, and the body reduced to a state of rest.

If a carriage, a horse, or a boat, moving with speed, be suddenly retarded or stopped, by any cause which does not at the same time affect passengers, riders, or any loose bodies which are carried, they will be precipitated in the direction of the motion; because by reason of their inertia, they persevere in the motion which they shared in common with that which transported them, and are not deprived of that motion by the same cause.

If a passenger leap from a carriage in rapid motion, he will fall in the direction in which the carriage is moving at the moment his feet meet the ground; because his body, on quitting the vehicle, retains, by its inertia, the motion which it had in common with it. When he reaches the ground, this motion is destroyed by the resistance of the ground to the feet, but is retained in the upper and heavier part of the body; so that the same effect is produced as if the feet had been tripped.

When a carriage is once put in motion with a determinate speed on a level road, the only force necessary to sustain the motion is that which is sufficient to overcome the friction of the road; but at starting a greater expenditure of force is necessary, inasmuch as not only the friction is to be overcome, but the force with which the vehicle is intended to move must be communicated to it. Hence we see that horses make a much greater exertion at starting than subsequently, when the carriage is in motion; and we may also infer the inexpediency of attempting to start at full speed, especially with heavy carriages.

In racing, the horses shoot far beyond the winning-post before their course can be arrested.

ACTION AND REACTION.

The effects of inertia or inactivity, considered in the last chapter, are such as may be manifested by a single insulated body, without reference to, or connection with, any other body whatever. They might all be recognised if there were but one body existing in the universe. There are, however, other important results of this law, to the development of which two bodies at least are necessary.

A similar result will be obtained, whatever proportion may subsist between the masses A and B. Suppose B to be ten times A; then the whole motion of A must, after the impact, be distributed among the parts of the united masses of A and B: but these united masses are, in this case, eleven times the mass of A. Now, as they all move with a common motion, it follows that A's former motion must be equally distributed among them; so that each part shall have an eleventh part of it. Therefore the velocity after impact will be the eleventh part of the velocity of A before it. Thus A loses by the impact ten-eleventh parts of its motion, which are precisely what B receives.

Again, if the masses of A and B be 5 and 7, then the united mass after impact will be 12. The motion of A before impact will be equally distributed between these twelve parts, so that each part will have a twelfth of it; but five of these parts belong to the mass A, and seven to B. Hence B will receive seven-twelfths, while A retains five-twelfths.

In general, therefore, when a mass A in motion impinges on a mass B at rest, to find the motion of the united mass after impact, "divide the whole motion of A into as many equal parts as there are equal component masses in A and B together, and then B will receive by the impact as many parts of this motion as it has equal component masses."

Since by the quality of inertia a body can neither generate nor destroy motion, it follows that when two bodies act upon each other in any way whatever, the total quantity of motion in a given direction, after the action takes place, must be the same as before it, for otherwise some motion would be produced by the action of the bodies, which would contradict the principle that they are inert. The word "action" is here applied, perhaps improperly, but according to the usage of mechanical writers, to express a certain phenomenon or effect. It is, therefore, not to be understood as implying any active principle in the bodies to which it is attributed.

In the cases of collision of which we have spoken, one of the masses B was supposed to be quiescent before the impact. We shall now suppose it to be moving in the same direction as A, that is, towards C, but with a less velocity, so that A shall overtake it, and impinge upon it. After the impact, the two masses will move towards C with a common velocity, the amount of which we now propose to determine.

If the masses A and B be equal, then their motions or velocities added together must be the motion of the united mass after impact, since no motion can either be created or destroyed by that event. But as A and B move with a common motion, this sum must be equally distributed between them, and therefore each will move with a velocity equal to half the sum of their velocities before the impact. Thus, if A have the velocity 7, and B have 5, the velocity of the united mass after impact is 6, being the half of 12, the sum of 7 and 5.

If A and B be not equal, suppose them divided into equal component parts, and let A consist of 8, and B of 6, equal masses: let the velocity of A be 17, so that the motion of each of the 8 parts being 17, the motion of the whole will be 136. In the same manner, let the velocity of B be 10, the motion of each part being 10, the whole motion of the 6 parts will be 60. The sum of the two motions, therefore, towards C is 196; and since none of this can be lost by the impact, nor any motion added to it, this must also be the whole motion of the united masses after impact. Being equally distributed among the 14 component parts of which these united masses consist, each part will have a fourteenth of the whole motion. Hence, 196 being divided by 14, we obtain the quotient 14, which is the velocity with which the whole moves.

In general, therefore, when two masses moving in the same direction impinge one upon the other, and after impact move together, their common velocity may be determined by the following rule: "Express the masses and velocities by numbers in the usual way, and multiply the numbers expressing the masses by the numbers which express the velocities; the two products thus obtained being added together, and their sum divided by the sum of the numbers expressing the masses, the quotient will be the number expressing the required velocity."

The true estimate, then, of the quantity of motion is found by multiplying together the numbers which express the mass and the velocity. Thus, in the example which has been last given of the impact of masses, the quantities of motion before and after impact appear to be as follow:

These results may be generalised and more clearly and concisely expressed by the aid of the symbols of arithmetic.

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