Chapter IV: Part 4

If a plate of glass be placed upon the surface of water without being permitted to sink, it will require more force to raise it from the water than is sufficient merely to balance the weight of the glass. This shows the adhesion of the water and glass, and also the cohesive force with which the particles of the water resist separation.

If a needle be dipped in certain liquids, a drop will remain suspended at its point when withdrawn from them: and, in general, when a solid body has been immersed in a liquid and withdrawn, it is _wet_; that is, some of the liquid has adhered to its surfaces. If no attraction existed between the solid and liquid, the solid would be in the same state after immersion as before. This is proved by liquids and solids between which no attraction exists. If a piece of glass be immersed in mercury, it will be in the same state when withdrawn as before it was immersed. No mercury will adhere to it; it will not be _wet_.

When it rains, the person and vesture are affected only because this attraction exists between them and water. If it rained mercury, none would adhere to them.

(101.) When molecular attraction is exhibited by liquids pervading the interstices of porous bodies, ascending in crevices or in the bores of small tubes, it is called _capillary attraction_. Instances of this are innumerable. Liquids are thus drawn into the pores of sponge, sugar, lamp-wick, &c. The animal and vegetable kingdom furnish numerous examples of this class of effects.

A weight being suspended by a dry rope, will be drawn upwards through a considerable height, if the rope be moistened with a wet sponge. The attraction of the particles composing the rope for the water is in this case so powerful, that the tension produced by several hundred weight cannot expel them.

A glass tube, of small bore, being dipped in water tinged by mixture with a little ink, will retain a quantity of the liquid suspended when withdrawn. The height of the liquid in the tube will be seen by looking through it. It is found that the less the bore of the tube is, the greater will be the height of the column sustained. A series of such tubes fixed in the same frame, with their lower orifices at the same level, and with bores gradually decreasing, being dipped in the liquid, will exhibit columns gradually increasing.

A _capillary syphon_ is formed of a hank of cotton threads, one end of which is immersed in the vessel containing the liquid, and the other is carried into the vessel into which the liquid is to be transferred. The liquid may be thus drawn from the one vessel into the other. The same effect may be produced by a glass syphon with a small bore.

(102.) It frequently happens that a _molecular repulsion_ is exhibited between a solid and a liquid. If a piece of wood be immersed in quicksilver, the liquid will be depressed at that part of the surface which is near the wood; and in like manner, if it be contained in a glass vessel, it will be depressed at the edges. In a barometer tube, the surface of the mercury is convex, owing partly to the repulsion between the glass and mercury.

All solids, however, do not repel mercury. If any golden trinket be dipped in that liquid, or even be exposed for a moment to contact with it, the gold will be instantly intermingled with particles of quicksilver, the metal changes its colour, and becomes white like silver, and the mercury can only be extricated by a difficult process. Chains, seals, rings, &c. should always be laid aside by those engaged in experiments or other processes in which mercury is used.

(103.) Of all the forms under which molecular force is exhibited, that in which it takes the name of _affinity_ is attended with the most conspicuous effects. Affinity is in chemistry what inertia is in mechanics, the basis of the science. The present treatise is not the proper place for any detailed account of this important class of natural phenomena. Those who seek such knowledge are referred to our treatise on CHEMISTRY. Since, however, affinity sometimes influences the mechanical state of bodies, and affects their mechanical properties, it will be necessary here to state so much respecting it as to render intelligible those references which we may have occasion to make to such effects.

When the particles of different bodies are brought into close contact, and more especially when, being in a fluid state, they are mixed together, their union is frequently observed to produce a compound body, differing in its qualities from either of the component bodies. Thus the bulk of the compound is often greater or less than the united volumes of the component bodies. The component bodies may be of the ordinary temperature of the atmosphere, and yet the compound may be of a much higher or lower temperature. The components may be liquid, and the compound solid. The colour of the compound may bear no resemblance whatever to that of the components. The species of molecular action between the components, which produce these and similar, effects, is called _affinity_.

(104.) We shall limit ourselves here to the statement of a few examples of these phenomena.

If a pint of water and a pint of sulphuric acid be mixed, the compound will be considerably less than a quart. The density of the mixture is, therefore, greater than that which would result from the mere diffusion of the particles of the one fluid through those of the other. The particles have assumed a greater proximity, and therefore exhibit a mutual attraction.

In this experiment, although the liquids before being mixed be of the temperature of the surrounding air, the mixture will be so intensely hot, that the vessel which contains it cannot be touched without pain.

If the two aeriform fluids, called oxygen and hydrogen, be mixed together in a certain proportion, the compound will be water. In this case, the components are different from the compound, not merely in the one being _air_ and the other _liquid_, but in other respects not less striking. The compound water extinguishes fire, and yet of the components, hydrogen is one of the most inflammable substances in nature, and the presence of oxygen is indispensably necessary to sustain the phenomenon of combustion.

Oxygen gas, united with quicksilver, produces a compound of a black colour, the quicksilver being white and the gas colourless. When these substances are combined in another proportion, they give a red compound.

(105.) Having noticed the principal molecular forces, we shall now proceed to the consideration of those attractions which are exhibited between bodies existing in masses. The influence of molecular attractions is limited to insensible distances. On the contrary, the forces which are now to be noticed act at considerable distances, and to the influence of some there is no limit, the effect, however, decreasing as the distance increases.

The effect of the loadstone on iron is well known, and is one of this class of forces. For a detailed account of this force, and the various phenomena of which it is the cause, the reader is referred to our treatise on MAGNETISM.

When glass, wax, amber, and other substances are submitted to friction with silken or woollen cloth, they are observed to attract feathers, and other light bodies placed near them. A like effect is produced in several other ways, and is attended with other phenomena, the discussion of which forms a principal part of physical science. The force thus exhibited is called electricity. For details respecting it, and for its connection with magnetism, the reader is referred to our treatises on ELECTRICITY and ELECTRO-MAGNETISM.

(106.) These attractions exist either between bodies of particular kinds, or are developed by reducing the bodies which manifest them to a certain state by friction, or some other means. There is, however, an attraction, which is manifested between bodies of all species, and under all circumstances whatever; an attraction, the intensity of which is wholly independent of the nature of the bodies, and only depends on their masses and mutual distances. Thus, if a mass of metal and a mass of clay be placed in the vast abyss of space, at a mile asunder, they will instantly commence to approach each other with certain velocities. Again, if a mass of stone and of wood respectively equal to the former, be placed at a like distance, they will also commence to approach each other with the same velocities as the former. This universal attraction, which only depends on the quantity of the masses and their mutual distances, is called the “attraction of gravitation.” We shall first explain the “law” of this attraction, and shall then point out some of the principal phenomena by which its existence and its laws are known.

(107.) The “law of gravitation” sometimes from its universality called the “law of nature,” may be explained as follows:

Let us suppose two masses, A and B, placed beyond the influence or attraction of any other bodies, in a state of rest, and at any proposed distance from each other. By their mutual attraction they will approach each other, but not with the same velocity. The velocity of A will be greater than that of B, in the same proportion as its mass is less than that of B. Thus, if the mass of B be twice that of A, while A approaches B through a space of two feet, B will approach A through a space of one foot. Hence it follows, that the force with which A moves towards B is equal to the force with which B moves towards A (68). This is only a consequence of the property of inertia, and is an example of the equality of action and reaction, as explained in Chapter IV. The velocity with which A and B approach each other is estimated by the diminution of their distance, A B, by their mutual approach in a given time. Thus, if in one second A move towards B through a space of two feet, and in the same time B moves towards A through the space of one foot, they will approach each other through a space of three feet in a second, which will be their relative velocity (91).

If the mass of B be doubled, it will attract A with double the former force, or, what is the same, will cause A to approach B with double the former velocity. If the mass of B be trebled, it will attract A with treble the first force, and, in general, while the distance A B remains the same, the attractive force of B upon A will increase or diminish in exactly the same proportion as the mass of B is increased or diminished.

In the same manner, if the mass A be doubled, it will be attracted by B with a double force, because B exerts the same degree of attraction on every part of the mass A, and any addition which it may receive will not diminish or otherwise affect the influence of B on its former mass.

To express this in general arithmetical symbols let _a_ and _b_ express the space through which A and B respectively would be moved towards each other by their mutual attraction. We would then have

A × _a_ = B × _b_.

Thus, it is a general law of gravitation, that so long as the distance between two bodies remains the same, each will attract and be attracted by the other, in proportion to its mass; and any increase or decrease of the mass will cause a corresponding increase or decrease in the amount of the attraction.

(108.) We shall now explain the law, according to which the attraction is changed, by changing the distance between the bodies. At the distance of one mile the body B attracts A with a certain force. At the distance of two miles, the masses not being changed, the attraction of B upon A will be one-fourth of its amount at the distance of one mile. At the distance of three miles, it will be one-ninth of its original amount; at four miles, it is reduced to a sixteenth, and so on. The following table exhibits the diminution of the attraction corresponding to the successive increase of distance:

+-----------+---+----+----+----+----+----+----+----+----+
|Distance | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | &c.|
+-----------+---+----+----+----+----+----+----+----+----+
|Attraction | 1 | 1/4| 1/9|1/16|1/25|1/36|1/49|1/64| &c.|
+-----------+---+----+----+----+----+----+----+----+----+

In ARITHMETIC, that number which is found by multiplying any proposed number by itself, is called its _square_. Thus 4, that is, 2 multiplied by 2, is the square of 2; 9 that is, 3 times 3, is the square of 3, and so on. On inspecting the above table, it will be apparent, therefore, that the attraction of gravitation decreases in the same proportion as the square of the distance from the attracting body increases, the mass of both bodies in this case being supposed to remain the same; but if the mass of either be increased or diminished, the attraction will be increased or diminished in the same proportion.

(109.) Hence the _law of gravitation_ may be thus expressed: “The mutual attraction of two bodies increases in the same proportion as their masses are increased, and as the square of their distance is decreased; and it decreases in proportion as their masses are decreased, and as the square of their distance is increased.”

This law may be more clearly expressed by means of general symbols. Let _f_ express the force with which a mass weighing 1 lb. will attract another mass weighing 1 lb., at the distance of 1 foot. The force with which they will mutually attract, when removed to the distance expressed in feet by D, will be

_f_/D^2

that is, the force _f_ divided by the square of the number D.

If one of the bodies, instead of weighing 1 lb., weigh the number of pounds expressed by A, their mutual attraction will be increased A times, and will therefore be expressed by

(A × _f_)/D^2

In fine, if the other be also the number of pounds expressed by B, their mutual attraction will be

(A × B × _f_)/D^2

(110.) Having explained the law of gravitation, we shall now proceed to show how the existence of this force is proved, and its law discovered.

The earth is known to be a globular mass of matter, incomparably greater than any of the detached bodies which are found upon its surface. If one of these bodies suspended at any proposed height above the surface of the earth be disengaged, it will be observed to descend perpendicularly to the earth, that is, in the direction of the earth’s centre. The force with which it descends will also be found to be in proportion to the mass, without any regard to the species of the body. These circumstances are consistent with the account which we have given of gravitation. But by that account we should expect, that as the falling body is attracted with a certain force towards the earth, the earth itself should be attracted towards it by the same force; and instead of the falling body moving towards the earth, which is the phenomenon observed, the earth and it should move towards each other, and meet at some intermediate point. This, in fact, is the case, although it is impossible to render the motion of the earth observable, for reasons which will easily be understood.

Since all the bodies around us participate in this motion, it would not be directly observable, even though its quantity were sufficiently great to be perceived under other circumstances. But setting aside this consideration, the space through which the earth moves in such a case is too minute to be the subject of sensible observation. It has been stated (107), that when two bodies attract each other, the space through which the greater approaches the lesser, bears to that through which the lesser approaches the greater, the same proportion as the mass of the lesser bears to the mass of the greater. Now the mass of the earth is more than 1000,000,000,000,000 times the mass of any body which is observed to fall on its surface; and therefore if even the largest body which can come under observation were to fall through an height of 500 feet, the corresponding motion of the earth would be through a space less than the 1000,000,000,000,000th part of 500 feet, which is less than the 100,000,000,000th part of an inch.

The attraction between the earth and detached bodies on its surface is not only exhibited by the descent of these bodies when unsupported, but by their pressure when supported. This pressure is what is called _weight_. The phenomena of weight, and the descent of heavy bodies, will be fully investigated in the next chapter.

(111.) It is not alone by the direct fall of bodies that the gravitation of the earth is manifested. The curvilinear motion of bodies projected in directions different from the perpendicular, is a combination of the effects of the uniform velocity which has been given to the projectile by the impulse which it has received, and the accelerated velocity which it receives from the earth’s attraction. Suppose a body placed at any point P, _fig. 21._, above the surface of the earth, and let P C be the direction of the earth’s centre. If the body were allowed to move without receiving any impulse, it would descend to the earth in the direction P A, with an accelerated motion. But suppose that at the moment of its departure from P, it receives an impulse in the direction P B, which would carry it to B in the time the body would fall from P to A, then, by the composition of motion, the body must at the end of that time be found in the line B D, parallel to P A. If the motion in the direction of P A were uniform, the body P would in this case move in the straight line from P to D. But this is not the case. The velocity of the body in the direction P A is at first so small as to produce very little deflection of its motion from the line P B. As the velocity, however, increases, this deflection increases, so that it moves from P to D in a curve, which is convex, towards P B.

The greater the velocity of the projectile in the direction P A, the greater sweep the curve will take. Thus it will successively take the forms P D, P E, P F, &c., and that velocity can be computed, which (setting aside the resistance of the air) would cause the projectile to go completely round the earth, and return to the point P from which it departed. In this case, the body P would continue to revolve round the earth like the moon. Hence it is obvious, that the phenomenon of the revolution of the moon round the earth, is nothing more than the combined effects of the earth’s attraction, and the impulse which it received when launched into space by the hand of its Creator.

(112.) This is a great step in the analysis of the phenomenon of gravitation. We have thus reduced to the same class two effects apparently very dissimilar, the rectilinear descent of a heavy body, and the nearly circular revolution of the moon round the earth. Hence we are conducted to a generalisation still more extensive.

As the moon’s revolution round the earth, in an orbit nearly circular, is caused by the combination of the earth’s attraction, and an original projectile impulse, so also the singular phenomena of the planets’ revolution round the sun in orbits nearly circular, must be considered an effect of the same class, as well as the revolution of the satellites of those planets which are attended by such bodies. Although the orbits in which the comets move deviate very much from circles, yet this does not hinder the application of the same principle to them, their deviation from circles not depending on the sun’s attraction, but only on the direction and force of the original impulse which put them in motion.

(113.) We therefore conclude that gravitation is the principle which, as it were, animates the universe. All the great changes and revolutions of the bodies which compose our system, can be traced to or derived from this principle. It still remains to show how that remarkable law, by which this force is declared to increase or decrease in the same proportion as the square of the distance from the attracting body is decreased or increased, may be verified and established.

It has been shown, that the curvilinear path of a projectile depends on, and can be derived, by mathematical reasoning, from the consideration of the intensity of the earth’s attraction, and the force of the original impulse, or the velocity of projection. In the same manner, by a reverse process, when we know the curve in which a projectile moves, we can infer the amount of the attracting force which gives the curvature to its path. In this way, from our knowledge of the curvature of the moon’s orbit, and the velocity with which she moves, the intensity of the attraction which the earth exerts upon her can be exactly ascertained. Upon comparing this with the force of gravitation at the earth’s surface, it is found that the latter is as many times greater than the former, as the square of the moon’s distance is greater than the square of the distance of a body on the surface of the earth from its centre.

(114.) If this were the only fact which could be brought to establish the law of gravitation, it might be thought to be an accidental relation, not necessarily characterising the attraction of gravitation. Upon examining the orbits and velocities of the several planets, the same result is, however, obtained. It is found that the forces with which they are severally attracted by the sun are great, in exactly the same proportion as the squares of the several numbers expressing their distances are small. The mutual gravitation of bodies on the surface of the earth towards each other is lost in the predominating force exerted by the earth upon all of them. Nevertheless, in some cases, this effect has not only been observed, but actually measured.

A plumb-line, under ordinary circumstances, hangs in a direction truly vertical; but if it be near a large mass of matter, as a mountain, it has been observed to be deflected from the true vertical, towards the mountain. This effect was observed by Dr. Maskeline near the mountain called Skehallien, in Scotland, and by French astronomers near Chimboraco. For particulars of these observations, see our treatise on GEODÆSY.

Cavendish succeeded in exhibiting the effects of the mutual gravitation of metallic spheres. Two globes of lead A, B, each about a foot in diameter, were placed at a certain distance asunder. A light rod, to the ends of which were attached small metallic balls C, D, was suspended at its centre E from a fine wire, and the rod was placed as in _fig. 22._, so that the attractions of each of the leaden globes had a tendency to turn the rod round the centre E in the same direction. A manifest effect was produced upon the balls C, D, by the gravitation of the spheres. In this experiment, care must be taken that no magnetic substance is intermixed with the materials of the balls.

Having so far stated the principles on which the law of gravitation is established, we shall dismiss this subject without further details, since it more properly belongs to the subject of PHYSICAL ASTRONOMY; to which we refer the reader for a complete demonstration of the law, and for the detailed development of its various and important consequences.

CHAP. VII.

TERRESTRIAL GRAVITY.

(115.) GRAVITATION is the general name given to this attraction, by whatever masses of matter it may be manifested. As exhibited in the effects produced by the earth upon surrounding bodies, it is called “terrestrial gravity.”

As the attraction of the earth is directed towards its centre, it might be expected that two plumb-lines should appear not to be parallel, but so inclined to each other as to converge to a point under the surface of the earth. Thus, if A B and C D, _fig. 23._, be two plumb-lines, each will be directed to the centre O, where, if their directions were continued, they would meet. In like manner, if two bodies were allowed to fall from A and C, they would descend in the directions A B and C D, which converge to O. Observation, on the contrary, shows, that plumb-lines suspended in places not far distant from each other are truly parallel; and that bodies allowed to fall descend in parallel lines. This apparent parallelism of the direction of terrestrial gravity is accounted for by the enormous proportion which the magnitude of the earth bears to the distance between the two plumb-lines or the two falling bodies which are compared. If the distance between the places B, D, were 1200 feet, the inclination of the lines A B and C D would not amount to a quarter of a minute, or the 240th part of a degree. But the distance, in cases where the parallelism is assumed, is never greater than, and seldom so great as, a few yards; and hence the inclination of the directions A B and C D is too small to be appreciated by any practical measure. In the investigation of the phenomena of falling bodies, we shall, therefore, assume, that all the particles of the same body are attracted in parallel directions, perpendicular to an horizontal plane.

(116.) Since the intensity of terrestrial gravity increases as the square of the distance decreases, it might be expected that, as a falling body approaches the earth, the force which accelerates it should be continually increasing, and, strictly speaking, it is so. But any height through which we observe falling bodies to descend bears so very small a proportion to the whole distance from the centre, that the change of intensity of the force of gravity is quite beyond any practical means of estimating it. The radius, or the distance from the surface of the earth to its centre, is 4000 miles. Now, suppose a body descended through the height of half a mile, a distance very much beyond those used in experimental enquiries, the distances from the centre, at the beginning and end of the fall, are then in the proportion of 8000 to 8001, and therefore the proportion of the force of attraction at the commencement to the force at the end, being that of the squares of these numbers, is 64,000,000 to 64,016,001, which, in the whole descent, is an increase of about one part in 4000; a quantity practically insignificant. We shall, therefore, in explaining the laws of falling bodies, assume that, in the entire descent, the body is urged by a force of uniform intensity.

Although the force which attracts all parts of the same body during its descent in a given place is the same, yet the force of gravity, at different parts of the earth’s surface, has different intensities. The intensity diminishes with the latitude, so that it is greater towards the poles, and lesser towards the equator. The causes of this variation, its law, and the experimental proofs of it, will be explained, when we shall treat of centrifugal force, and the motion of pendulums. It is sufficient merely to advert to it in this place.

(117.) Since the earth’s attraction acts separately and equally on every particle of matter, without regard to the nature or species of the body, it follows that all bodies, of whatever kind, or whatever be their masses, must be moved with the same velocity. If two equal particles of matter be placed at a certain distance above the surface of the earth, they will fall in parallel lines, and with exactly the same speed, because the earth attracts them equally. In the same manner, a thousand particles would fall with equal velocities. Now, these circumstances will in no wise be changed if those 1000 particles, instead of existing separately, be aggregated into two solid masses, one consisting of 990 particles, and the other of 10. We shall thus have a heavy body and a light one, and, according to our reasoning, they must fall to the earth with the same speed.

Common experience, however, is not always consistent with this doctrine. What are called light substances, as feathers, gold-leaf, paper, &c., are observed to fall slowly and irregularly, while heavier masses, as solid pieces of metal, stones, &c., fall rapidly. Nay, there are not a few instances in which the earth, instead of attracting bodies, seems to repel them, as in the case of smoke, vapours, balloons, and other substances which actually ascend. We are to consider that the mass of the earth is not the only agent engaged in these phenomena. The earth is surrounded by an atmosphere composed of an elastic or aeriform fluid. This atmosphere has certain properties, which will be explained in our treatise on PNEUMATICS, and which are the causes of the anomalous circumstances alluded to. Light bodies rise in the atmosphere, for the same reason that a piece of cork rises from the bottom of a vessel of water; and other light bodies fall more slowly than heavy ones, for the same reason that an egg in water falls to the bottom more slowly than a leaden bullet. This treatise is not the place to give a direct explanation of these phenomena. It will be sufficient for our present purpose to show, that if there were no atmosphere, all bodies, heavy and light, would fall at the same rate. This may easily be accomplished by the aid of an air-pump. Having by that instrument abstracted the air from a tall glass vessel, we are enabled, by means of a wire passing air-tight through a hole in the top, to let fall several bodies from the top of the vessel to the bottom. These, whether they be feathers, paper, gold-leaf, pieces of money, &c. all descend with the same speed, and strike the bottom at the same moment.

(118.) Every one who has seen a heavy body fall from a height, has witnessed the fact, that its velocity increases as it approaches the ground. But if this were not observable by the eye, it would be betrayed by the effects. It is well known, that the force with which a body strikes the ground increases with the height from whence it has fallen. This force, however, is proportional to the velocity which it has at the moment it meets the ground, and therefore this velocity increases with the height.

When the observations on attraction in the last chapter are well understood, it will be evident that the velocity which a body has acquired in falling from any height, is the accumulated effects of the attraction of terrestrial gravity during the whole time of the fall. Each instant of the fall a new impulse is given to the body, from which it receives additional velocity; and its final velocity is composed of the aggregation of all the small increments of velocity which are thus communicated. As we are at present to suppose the intensity of the attraction invariable, it will follow that the velocity communicated to the body in each instant of time will be the same, and therefore that the whole quantity of velocity produced or accumulated at the end of any time is proportional to the length of that time. Thus, if a certain velocity be produced in a body having fallen for one second, twice that velocity will be produced when it has fallen for two seconds, thrice that velocity in three seconds, and so on. Such is the fundamental principle or characteristic of _uniformly accelerated motion_.

(119.) In examining the circumstances of the descent of a body, the time of the fall and the velocity at each instant of that time are not the only things to be attended to. The spaces through which it falls in given intervals of time, counted either from the commencement of its fall, or from any proposed epoch of the descent, are equally important objects of enquiry. To estimate the space in reference to the time and the final velocity, we must consider that this space has been moved through with varying speed. From a state of rest at the beginning of the fall, the speed gradually increases with the time, and the final velocity is greater still than that which the body had at any preceding instant during its descent. We cannot, therefore, _directly_ appreciate the space moved through in this case by the time and final velocity. But as the velocity increases uniformly with the time, we shall obtain the average speed, by finding that which the body had in the middle of the interval which elapsed between the beginning and end of the fall, and thus the space through which the body has actually fallen is that through which it would move in the same time with this average velocity uniformly continued.

But since the velocity which the body receives in any time, counted from the beginning of its descent, is in the proportion of that time, it follows that the velocity of the body after half the whole time of descent is half the final velocity. From whence it appears, that the height from which a body falls in any proposed time is equal to the space through which a body would move in the same time with half the final velocity, and it is therefore equal to half the space which would be moved through in the same time with the final velocity.

(120.) It follows from this reasoning, that between the three quantities, the height, the time, and the final velocity, which enter into the investigation of the phenomena of falling bodies, there are two fixed relations: _First_, the time, counted from the beginning of the fall and the final velocity, are proportional the one to the other; so that as one increases, the other increases in the same proportion. _Secondly_, the height being equal to half the space which would be moved through in the _time_ of the fall, with the _final velocity_, must have a fixed proportion to these two quantities, viz. the _time_ and the _final velocity_, or must be proportional to the product of the two numbers which express them.

But since the time is always proportional to the final velocity, they may be expressed by equal numbers, and the product of equal numbers is the square of either of them. Hence, the product of the numbers expressing the time and final velocity is equivalent to the square of the number expressing the time, or to the square of the number expressing the final velocity. Hence we infer, that the height is always proportional to the square of the time of the fall, or to the square of the final velocity.

(121.) The use of a few mathematical characters will render these results more distinct, even to students not conversant with mathematical science.

Let S = the height from which the body falls, expressed in feet.

V = the velocity at the end of the fall in feet per second.

T = the number of seconds in the time of the fall.

_g_ = the number of feet through which a body would fall in one
second.

It will therefore follow that the velocity acquired in one second will be 2_g_, and the velocity acquired in T seconds will therefore be 2_g_ × T; so that

V = 2_g_ × T [1]

Since the space which a body falls through in T seconds is found by multiplying the space it falls through in one second by T^2, we shall have

S = _g_ × T^2 [2]

from which, combined with [1] we deduce

S = V^2/(4_g_) [3]

S = (1/2)V × T [4]

By these formularies, if the height through which a body falls freely in one second be known, the height through which it will fall in any proposed time may be computed. For since the height is proportional to the square of the time, the height through which it will fall in _two_ seconds will be _four_ times that which it falls through in _one_ second. In _three_ seconds it will fall through _nine_ times that space; in _four_ seconds, _sixteen_ times; in _five_ seconds, _twenty-five_ times, and so on. The following, therefore, is a general rule to find the height through which a body will fall in any given time: “Reduce the given time to seconds, take the square of the number of seconds in it, and multiply the height through which a body falls in one second by that number; the result will be the height sought.”

The following table exhibits the heights and corresponding times as far as 10 seconds:

+-------+---+---+---+----+----+----+----+----+----+-----+
|Time | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
+-------+---+---+---+----+----+----+----+----+----+-----+
|Height | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 |
+-------+---+---+---+----+----+----+----+----+----+-----+

Each unit in the numbers of the first row expresses a second of time, and each unit in those of the second row expresses the height through which a body falls freely in a second.

(122.) If a body fall continually for several successive seconds, the spaces which it falls through in each succeeding second have a remarkable relation among each other, which may be easily deduced from the preceding table. Taking the space moved through in the first second still as our unit, four times that space will be moved through in the first two seconds. Subtract from this 1, the space moved through in the first second, and the remainder 3 is the space through which the body falls in the _second_ second. In like manner if 4, the height fallen through in the first two seconds, be subtracted from 9, the height fallen through in the first three seconds, the remainder 5 will be the space fallen through in the third second. To find the space fallen through in the fourth second, subtract 9, the space fallen through in the first three seconds, from 16, the space fallen through in the first four seconds, and the result is 7, and so on. It thus appears that if the space fallen through in the first second be called 1, the spaces described in the second, third, fourth, fifth, &c. seconds, will be expressed by the odd numbers respectively, 3, 5, 7, 9, &c. This places in a striking point of view the accelerated motion of a falling body, the spaces moved through in each succeeding second being continually increased.

(123.) If velocity be estimated by the space through which the body would move uniformly in one second, then the final velocity of a body falling for one second will be 2; for with that final velocity the body would in one second move through twice the height through which it has fallen.

(124.) Since the final velocity increases in the same proportion as the time, it follows that after two seconds it is twice its amount after one, and after three seconds thrice that, and so on. Thus, the following table exhibits the final velocities corresponding to the times of descent:

+---------------+---+---+---+---+----+----+----+----+----+----+
|Time | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
+---------------+---+---+---+---+----+----+----+----+----+----+
|Final velocity | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
+---------------+---+---+---+---+----+----+----+----+----+----+

The numbers in the second row express the spaces through which a body with the final velocity would move in one second, the unit being, as usual, the space through which a body falls freely in one second.

(125.) Having thus developed theoretically the laws which characterise the descent of bodies, falling freely by the force of gravity, or by any other uniform force of the same kind, it is necessary that we should show how these laws can be exhibited by actual experiment. There are some circumstances attending the fall of heavy bodies which would render it difficult, if not impossible, to illustrate, by the direct observation of this phenomenon, the properties which have been explained in this chapter. A body falling freely by the force of gravity, as we shall hereafter prove, descends in one second of time through a height of about 16 feet[1]; in two seconds, it would, therefore, fall through four times that space, or 64 feet; in three seconds, through 9 times the height, or 144 feet; and in four seconds, through 256 feet. In order, therefore, to be enabled to observe the phenomena for only four seconds, we should command an height of at least 256 feet. But further; the velocity at the end of the first second would be at the rate of 32 feet per second; at the end of the second second, it would be 64 feet per second; and towards the end of the fall it would be about 120 feet per second. It is evident that this great degree of rapidity would be a serious impediment to accurate observation, even though we should be able to command the requisite height. It appears therefore that the number expressed by _g_ in the preceding formulæ is 16·083.

[1] More exactly through 16-1/12 feet, or 193 inches.

It occurred to Mr. George Attwood, a mathematician and natural philosopher of the last century, that all the phenomena of falling bodies might be experimentally exhibited and accurately observed, if a force of the same kind as gravity, viz. an uniformly accelerating force, be used, but of a much less intensity; so that while the motion continues to be governed by the same laws, its quantity may be so much diminished, that the final velocity, even after a descent of many seconds, shall be so moderated as to admit of most deliberate and exact observation. This being once accomplished, nothing more would remain but to find the height through which a body would fall in one second, or, what is the same, the proportion of the force of gravity to the mitigated but uniform accelerating force thus substituted for it.

(126.) To realise this notion, Attwood constructed a wheel turning on its axle with very little friction, and having a groove on its edge to receive a string. Over this wheel, and in the groove, he placed a fine silken cord, to the ends of which were attached equal cylindrical weights. Thus placed, the weights perfectly balance each other, and no motion ensues. To one of the weights he then added a small quantity, so as to give it a slight preponderance. The loaded weight now began to descend, drawing up on the other side the unloaded weight. The descent of the loaded weight, under these circumstances, is a motion exactly of the _same kind_ as the descent of a heavy body falling freely by the force of gravity; that is, it increases according to the same laws, though at a very diminished rate. To explain this, suppose that the loaded weight descends from a state of rest through one inch in a second, it will descend through 4 inches in two seconds, through 9 in three, through 16 in four, and so on. Thus in 20 seconds, it would descend through 400 inches, or 33 feet 4 inches, a height which, if it were necessary, could easily be commanded.

It might, perhaps, be thought, that since the weights suspended at the ends of the thread are in equilibrium, and therefore have no tendency either to move or to resist motion, the additional weight placed upon one of them ought to descend as rapidly as it would if it were allowed to fall freely and unconnected with them. It is very true that this weight will receive from the attraction of the earth the same force when placed upon one of the suspended weights, as it would if it were disengaged from them; but in the consequences which ensue, there is this difference. If it were unconnected with the suspended weights, the whole force impressed upon it would be expended in accelerating its descent; but being connected with the equal weights which sustain each other in equilibrium, by the silken cord passing over the wheel, the force which is impressed upon the added weight is expended, not as before, in giving velocity to the added weight alone, but to it together with the two equal weights appended to the string, one of which descends with the added weight, and the other rises on the opposite side of the wheel. Hence, setting aside any effect which the wheel itself produces, the velocity of the descent must be lessened just in proportion as the mass among which the impressed force is to be distributed is increased; and therefore the _rate_ of the fall bears to that of a body falling freely the same proportion as the added weight bears to the sum of the masses of the equal suspended weights and the added weight. Thus the smaller the added weight is, and the greater the equal suspended weights are, the slower will the rate of descent be.

To render the circumstances of the fall conveniently observable, a vertical shaft (see _fig. 24._) is usually provided, which is placed behind the descending weight. This pillar is divided to inches and halves, and of course may be still more minutely graduated, if necessary. A stage to receive the falling weight is moveable on this pillar, and capable of being fixed in any proposed position by an adjusting screw. A pendulum vibrating seconds, the beat of which ought to be very audible, is placed near the observer. The loaded weight being thus allowed to descend for any proposed time, or from any required height, all the circumstances of the descent may be accurately observed, and the several laws already explained in this chapter may be experimentally verified.

(127.) The laws which govern the descent of bodies by gravity, being reversed, will be applicable to the ascent of bodies projected upwards. If a body be projected directly upwards with any given velocity, it will rise to the height from which it should have fallen to acquire that velocity. The earth’s attraction will, in this case, gradually deprive the body of the velocity which is communicated to it at the moment at which it is projected. Consequently, the phenomenon will be that of _retarded motion_. At each part of its ascent it will have the same velocity which it would have if it descended to the same place from the highest point to which it rises. Hence it is clear, that all the particulars relative to the ascent of bodies may be immediately inferred from those of their descent, and therefore this subject demands no further notice.

To complete the investigation of the phenomena of falling bodies, it would now only remain to explain the method of ascertaining the exact height through which a body would descend in one second, if unresisted by the atmosphere, or any other disturbing cause. As the solution of this problem, however, requires the aid of principles not yet explained, it must for the present be postponed.

CHAP. VIII.

OF THE MOTION OF BODIES ON INCLINED PLANES AND CURVES.

(128.) In the last chapter, we investigated the phenomena of bodies descending freely in the vertical direction, and determined the laws which govern, not their motion alone, but that of bodies urged by any uniformly accelerating force whatever. We shall now consider some of the most ordinary cases in which the free descent of bodies is impeded, and the effects of their gravitation modified.

(129.) If a body, urged by any forces whatever, be placed upon a hard unyielding surface, it will evidently remain at rest, if the resultant (76) of all the forces which are applied to it be directed perpendicularly against the surface. In this case, the effect produced is pressure, but no motion ensues. If only one force act upon the body, it will remain at rest, provided the direction of that force be perpendicular to the surface.

But the effect will be different, if the resultant of the forces which are applied to the body be oblique to the surface. In that case this resultant, which, for simplicity, may be taken as a single force, may be considered as mechanically equivalent to two forces (76), one in the direction of the surface, and the other perpendicular to it. The latter element will be resisted, and will produce a pressure; the former will cause the body to move. This will perhaps be more clearly apprehended by the aid of a diagram.

Let A B, _fig. 25._, be the surface, and let P be a particle of matter placed upon it, and urged by a force in the direction P D, perpendicular to A B. It is manifest, that this force can only press the particle P against A B, but cannot give it any motion.

But let us suppose, that the force which urges P is in a direction P F, oblique to A B. Taking P F as the diagonal of a parallelogram, whose sides are P D and P C (74), the force P F is mechanically equivalent to two forces, expressed by the lines P D and P C. But P D, being perpendicular to A B, produces pressure without motion, and P C, being in the direction of A B, produces motion without pressure. Thus the effect of the force P F is distributed between motion and pressure in a certain proportion, which depends on the obliquity of its direction to that of the surface. The two extreme cases are, 1. When it is in the direction of the surface; it then produces motion without pressure: and, 2. When it is perpendicular to the surface; it then produces pressure without motion. In all intermediate directions, however, it will produce both these effects.

(130.) It will be very apparent, that the more oblique the direction of the force P F is to A B, the greater will be that part of it which produces motion, and the less will be that which produces pressure. This will be evident by inspecting _fig. 26._ In this figure the line P F, which represents the force, is equal to P F in _fig. 25._ But P D, which expresses the pressure, is less in _fig. 26._ than in _fig. 25._, while P C, which expresses the motion, is greater. So long, then, as the obliquity of the directions of the surface and the force remain unchanged, so long will the distribution of the force between motion and pressure remain the same; and therefore, if the force itself remain the same, the parts of it which produce motion and pressure will be respectively equal.

(131.) These general principles being understood, no difficulty can arise in applying them to the motion of bodies urged on inclined planes or curves by the force of gravity. If a body be placed on an unyielding horizontal plane, it will remain at rest, producing a pressure on the plane equal to the total amount of its weight. For in this case the force which urges the body, being that of terrestrial gravity, its direction is vertical, and therefore perpendicular to the horizontal plane.

But if the body P, _fig. 25._, be placed upon a plane A B, oblique to the direction of the force of gravity, then, according to what has been proved (129), the weight of the body will be distributed into two parts, P C and P D; one, P D, producing a pressure on the plane A B, and the other, P C, producing motion down the plane. Since the obliquity of the perpendicular direction P F of the weight to that of the plane A B must be the same on whatever part of the plane the weight may be placed, it follows (130), that the proportion P C of the weight which urges the body down the plane must be the same throughout its whole descent.

(132.) Hence it may easily be inferred, that the force down the plane is uniform; for since the weight of the body P is always the same, and since its proportion to that part which urges it down the plane is the same, it follows that the quantity of this part cannot vary. The motion of a heavy body down an inclined plane is therefore an uniformly-accelerated motion, and is characterised by all the properties of uniformly-accelerated motion, explained in the last chapter.