FORCE

, Vis, or Power, in Mechanics, Philosophy, &c, denotes the cause of the change in the state of a body, with respect to motion, rest, pressure, &c; as well as its endeavour to oppose or resist any change made in such its state. Thus, whenever a body, which was at rest, begins to move; or when its motion is either not uniform, or not direct; the cause of this change in the state of the body, is what is called Force, and is an external Force. Or, while a body remains in the same state, either of rest, or of uniform and rectilinear motion, the cause of its remaining in such state, is in the nature of the body, being an innate internal Force, and is called its Inertia.

Mechanical Forces may be reduced to two sorts; one of a body at rest, the other of a body in motion.

The Force of a body at rest, is that which we conceive to be in a body lying still on a table, or hanging by a rope, or supported by a spring, &c; and this is called by the names of Pressure, Tension, Force, or Vis Mortua, Solicitatio, Conatus Movendi, Conamen, &c; which kind of Force may be always measured by a weight, viz, the weight that sustains it. To this class of Forces may also be referred Centripetal and Centrifugal Forces, though they reside in a body in motion; because these Forces are homogeneous to weights, pressures, or tensions of any kind. The pressure, or Force of gravity in any body, is proportional to the quantity of matter in it.

The Force of a body in motion, is a power residing in that body, so long as it continues its motion; by means of which, it is able to remove obstacles lying in its way; to lessen, destroy, or overcome the Force of any other moving body, which meets it in an opposite direction; or to surmount any the largest dead pressure or resistance, as tension, gravity, friction, &c, for some time; but which will be lessened or destroyed by such resistance as lessens or destroys the motion of the body. This is called Vis Motrix, Moving Force, or Motive Force, and by some late writers Vis Viva, to distinguish | it from the Vis Mortua spoken of before; and by these appellations, however different, the same thing is understood by all mathematicians; namely, that power of displacing, of withstanding opposite moving Forces, or of overcoming any dead resistance, which resides in a moving body, and which, in whole or in part, continues to accompany it, as long as the body moves; and may be otherwise called Percussive Force, or Momentum.

But concerning the measure of this kind of Force, mathematicians have been divided into two parties. It is allowed by both sides, that the measure of this Force depends partly upon the mass of matter in the body, or its weight, and partly upon the velocity of its motion; so that upon any increase of either weight or velocity, the moving Force will become greater. It is also agreed, that the velocity being given, or being the same in two moving bodies, their Forces will be in proportion to their masses or weights. But, when two bodies are equal, and the velocities with which they move are different, the two parties no longer agree about the measure of the moving Force.

The Cartesians and Newtonians maintain, that, in this case, the moving Force is in proportion simply as the velocity with which a body moves; so that with a double velocity it has a double Force, &c: But the Leibnitians assert, that the moving Force is proportional to the square of the velocity; so as, with a double velocity to have a quadruple Force, &c. Or, when the bodies are different, the former hold, that the momentum or moving Force of bodies, is in the compound ratio of their weights and velocities: But the Leibnitians hold, that it is in the compound ratio of the weights and squares of the velocities.

Though Leibnitz was the first who expressly asserted, that the Force of a body in motion is as the square of its velocity, which was in a paper inserted in the Leipsic Acts for the year 1686, yet it is thought that Huygens led him into that notion, by some demonstrations in the 4th part of his book De Horologio Oscillatorio, relating to the centre of oscillation, and by his dissertations, in answer to the objections of the abbot Catalan, one of which was published in 1684. This eminent mathematician had demonstrated, that in the collision of two bodies that are perfectly elastic, the sum of the products of each body multiplied by the square of its velocity, was the same after the shock as before; (though the same thing is true of the sums of the products of the bodies multiplied simply by their velocities). Now that proposition is so far general as to obtain in all collisions of bodies that are perfectly elastic: and it is also true, when bodies of a perfect elasticity strike any immoveable obstacle, as well as when they strike one another; or when they are constrained by any power or resistance to move in directions different from those in which they impel one another. These considerations might have induced Huygens to lay it down as a general rule, that bodies constantly preserve their Ascensional Force, i. e. the product of their mass by the height to which their centre of gravity can ascend, which is as the square of the velocity; and therefore, in a given system of bodies, the sum of the squares of their velocities will remain the same, and not be altered by the action of the bodies among themselves, nor against immoveable obstacles. Leibnitz's metaphysical system led him to think that the same quantity of action or Force subsisted in the universe; and finding this impossible, if Force were estimated by the quantity of motion, he adopted Huygens's principle of the preservation of the Ascensional Force, and made it the measure of moving Forces. But it is to be observed that Huygens's principle, above-mentioned, is general only when bodies are perfectly elastic; and in some other cases which Maclaurin has endeavoured to distinguish: shewing at the same time that no useful conclusion in mechanics is affected by the disputes concerning the measure of the Force of bodies in motion, which have been objected to mathematicians. Analyst, Query 9. See Maclaurin's Fluxions, vol. 2, art. 533; Huygens Oper. tom. 1, pa. 248; &c.

Leibnitz's principle was adopted by several persons; as Wolfius, the Bernoullis, &c. Mr. Dan. Bernoulli, in his Treatise, has assumed the preservation of the Vis Ascendens of Huygens, or, as others express it, the Conservatio Virium Vivarum; and, in Bernoulli's own expression, æqualitas inter descensum actualem ascensumque potentialem, as an hypothesis of wonderful use in mechanics. But a late author contends, that the conclusions drawn from this principle are oftener false than true. See De Conservat. Virium Vivarum Dissert. Lond. 1744.

Catalan and Papin answered Leibnitz's paper published in 1686; and from that time the controversy became more general, and was carried on for several years by Leibnitz, John and Daniel Bernoulli, Poleni, Wolfius, s'Gravesande, Camus, Muschenbroek, &c, on one side; and Pemberton, Eames, Desaguliers, Dr. S. Clark, M. de Mairan, Jurin, Maclaurin, Robins, &c, on the other. See Act. Erud. 1686, 1690, 1691, 1695; Nouv. de la Rep. des Let. Sept. 1686, 1687, art. 2; Comm. Epist. inter Leibn. et Bern. Ep. 24, p. 143; Discourses sur les Loix de la Comm. du Mouvement, Oper. tom. 3, & Diss. de vera Notione Virium Vivarum, ib.; Act. Petropol. tom. 1, p. 131, &c; Hydronamica, sect. 1; Herman, in Act. Petrop. tom. 1, p. 2, &c; Polen. de Castellis; Wolf. in Act. Petrop. tom. 1, p. 217, &c, and in Cosmol. Gener.; Graves. in Journ. Lit. and Phys. Elem. Math. 1742, lib. 2, cap. 2 and 3; Memoir. de l'Acad. des Sciences 1728; Muschenbr. Int. ad Phil. Nat. 1762, vol. 1, p. 83 &c; Pemb. &c, in Phil. Trans. number 371, 375, 376, 396, 400, 401, or Abridg. vol. 6, p. 216 &c. Mairan in Memo. de l'Acad. des Sc. 1728, Phil. Trans. numb. 459, or Abridg. vol. 8, p. 236, Philos. Trans. vol. 43, p. 423 &c; Maclaurin's Acc. of Newton's Discoveries, p. 117 &c, Flux. ubi supr. & Recueil des Pieces qui ont emporté le Prix &c. tom. 1 : Desagul. Course Exp. Philos. vol. 1, p. 393 &c, vol. 2, p. 49 &c; and Robins's Tracts, vol. 2, p. 135.

The nature and limits of this work will not admit of a full account of the arguments and experiments that have been urged on both sides of this question; but they may be found chiefly in the preceding references. A few of them however may be considered, as follows.

The defenders of Leibnitz's principle, beside the arguments above-mentioned, refer to the spaces that bodies ascend to, when thrown upwards, or the penetrations of bodies let fall into soft wax, tallow, clay, snow, | and other soft substances, which spaces are always as the squares of the velocities of the bodies. On the other hand, their opponents retort, that such spaces are not the measures of the force in question, which is rather percussive and momentary, as those above are passed over in unequal times, and are indeed the joint effect of the Forces and times.

Desaguliers brings an argument from the familiar experiment of the balance, and the other simple mechanic powers, shewing that the effect is in proportion to the velocity multiplied by the weight; for example, 4 pounds being placed at the distance of 6 inches from the centre of motion of a balance, and 2 pounds at the distance of 12 inches; these will have a Vis Viva if the balance be put into a swinging motion. Now it appears that these Forces are equal, because, with contrary directions, they soon destroy each other; and they are to each other in the simple ratio of the velocity multiplied by the mass, viz , and also.

Mr. Robins, in his remarks on J. Bernoulli's treatise, entitled, Discours sur les Loix de la Communication du Mouvement, informs us, that Leibnitz adopted this opinion through mistake; for though he maintained that the quantity of Force is always the same in the universe, he endeavours to expose the error of Des Cartes, who also asserted, that the quantity of motion is always the same; and in his discourse on this subject in the Acta Eruditorum for 1686, he says that it is agreed on by the Cartesians, and all other philosophers and mathematicians, that there is the same Force requisite to raise a body of 1 pound to the height of 4 yards, as to raise a body of 4 pounds to the height of 1 yard; but being shewn how much he was mistaken in taking that for the common opinion, which would, if allowed, prove the force of the body to be as the square of the velocity it moved with, he afterwards, rather than own himself capable of such a mistake, endeavoured to defend it as true; since he found it was the necessary consequence of what he had once asserted; and maintained, that the force of a body in motion was proportional to the height from which it must fall, to acquire that velocity; and the heights being as the squares of the velocities, the Forces would be as the masses multiplied by them; whereas, when a body descends by its gravity, or is projected perpendicularly upwards, its motion may be considered as the sum of the uniform and continual impulses of the power of gravity, during its falling in the former case, and till they extinguish it in the latter. Thus when a body is projected upwards with a double velocity, these uniform impulses must be continued for a double time, in order to destroy the motion of the body; and hence it follows, that the body, by setting out with a double velocity, and ascending for a double time, must arise to a quadruple height, before its motion is exhausted. But this proves that a body with a double velocity moves with a double Force, since it is produced or destroyed by the same uniform power continued for a double time, and not with a quadruple force, though it rises to a quadruple height; so that the error of Leibnitz consisted in his not considering the time, since the velocities alone are not the causes of the spaces described, but the times and the velocities together; yet this is the fallacious argument on which he first built his new doctrine; and those which have been since much insisted on, and derived from the indentings or hollows produced in soft bodies by others falling into them, are much of the same kind. Robins's Tracts, vol. 2, p. 178.

But many of the experiments and reasonings, that have been urged on both sides in this controversy, have been founded in the different senses applied to the term Force. The English and French philosophers, by the word Force, mean the same thing as they do by momentum, motion, quantity of motion, percussion, or instantaneous pressure, which is measured by the mass drawn into the velocity, and may be known by its effect; and when they consider bodies as moving through a certain space, they allow for the time in which that space is described: whereas the Dutch, Italian, and German philosophers, who have espoused the new opinion, mean by the word Force, or Force inherent in a body in motion, that which it is able to produce; or, in other words, the Force is always measured by the whole effect produced by the body in motion; until its whole Force be entirely communicated or destroyed, without any regard to the time employed in producing this total effect. Thus, say they, if a point runs through a determinate space, and presses with a certain given Force, or intensity of pressure, it will perform the same action whether it move fast or slow, and therefore the time of the action in this case ought not to be regarded. s'Gravesande, Phys. El. Math. § 723—728.

Mr. Euler observes, with regard to this dispute concerning the measure of vivid Force, or living Force, as it is sometimes called, that we cannot absolutely ascribe any Force to a body in motion, whether we suppose this Force proportional to the velocity, or to the square of the velocity: for the Force exerted by a body, striking another at rest, is different from that which it exerts in striking the same body in motion; so that this Force cannot be ascribed to any body considered in itself, but only relatively to the other bodies it meets with. There is no force in a body absolutely considered, but its inertia, which is always the same, whether the body be in motion or at rest. But if this body be forced by others to change its state, its inertia then exerts itself as a Force, properly so called, which is not absolutely determinable; because it depends on the changes that happen in the state of the body.

A second observation which has been made by some eminent writers, is, that the effect of a shock of two or more bodies, is not produced in an instant, but requires a certain interval of time. If this be so, the heterogeneity between the Vis Viva and Mortua, or Living and Dead Force, will vanish; since a pressure may always be assigned, which in the same time, however little, shall produce the same effect. If then the Vis Viva be homogeneous to the Vis Mortua, and having a perfect measure and knowledge of the latter, we need require no other measure of the former than that which is derived from the Vis Mortua equivalent to it.

Now that the change in the state of two bodies, by their shock, does not happen in an instant, appears evidently from the experiments made on soft bodies: in these, percussion forms a small cavity, visible after the shock, if the bodies have no elasticity. Such a cavity cannot certainly be made in an instant. And if the | shock of soft bodies require a determinate time, we must certainly say as much of the hardest, though this time may be so small as to be beyond all our ideas. Neither can an instantaneous shock agree with that constant law of nature, by virtue of which nothing is performed per saltum. But it is needless to insist farther upon this, since the duration of any shock may be determined from the most certain principles.

There can be no shock or collision of bodies, without their making mutual impressions on each other: these impressions will be greater or less, according as the bodies are more or less soft, other circumstances being the same. In bodies called hard, the impressions are small; but a perfect hardness, which admits of no impression, seems inconsistent with the laws of nature; so that while the collision lasts, the action of bodies is the result of their mutually pressing each other. This pressure changes their state; and the Forces exerted in percussion are really pressures, and truly Vires Mortuæ, if we will use this expression, which is no longer proper, since the pretended infinite difference between the Vires Vivæ and Mortuæ ceases.

The Force of percussion, resulting from the pressures that bodies exert on each other, while the collision lasts, may be perfectly known, if these pressures be determined for every instant of the shock. The mutual action of the bodies begins the first moment of their contact; and is then least; after which this action increases, and becomes greatest when the reciprocal impressions are strongest. If the bodies have no elasticity, and the impressions they have received remain, the Forces will then cease. But if the bodies be elastic, and the parts compressed restore themselves to their former state, then will the bodies continue to press each other till they separate. To comprehend therefore perfectly the Force of percussion, it is requisite first to define the time the shock lasts, and then to assign the pressure corresponding to each instant of this time; and as the effect of pressures in changing the state of any body may be known, we may thence come at the true cause of the change of motion arising from collision. The Force of percussion therefore is no more than the operation of a variable pressure during a given time; and to measure this Force, we must have regard to the time, and to the variations according to which the pressure increases and decreases.

Mr. Euler has given some calculations relative to these particulars; and he illustrates their tendency by this instance: Suppose that the hardness of the two bodies, A and B, is equal; and such, that being pressed together with the force of 100lb, the impression made on each is of the depth of (1/1000)th part of a foot. Suppose also that B is fixed, and that A strikes it with the velocity of 100 feet in a second; according to Mr. Euler, the greatest Force of compression will be equivalent to 400lb, and this Force will produce in each of these bodies an impression equal to 1/25 of a foot; and the duration of the collision, that is, till the bodies arrive at their greatest compression, will be about 1/800 of a second. Mr. Euler, in his calculations, supposes the hardness of a body to be proportional to the Force or pressure requisite to make a given impression on it; so that the Force by which a given impression is made on a body, is in a compound ratio of the hardness of the body and of the quantity of the impression. But he observes, that regard must be had to the magnitude of the bodies, as the same impression cannot be made on the least bodies as on the greatest, from the defect of space through which their component particles must be driven: he considers therefore only the least impressions, and supposes the bodies of such magnitudes, that with respect to them the impressions may be looked upon as nothing. What he supposes concerning the hardness of bodies, neither implies elasticity nor the want of it, as elasticity only produces a restitution of figure and impression when the pressing Force ceases; but this restitution need not be here considered. It is also supposed, that the bodies which strike each other, have plane and equal bases, by which they touch each other in the collision; so that the impression hereby made diminishes the length of each body. It is farther to be observed, that in Mr. Euler's calculations, bodies are supposed so constituted, that they may not only receive impressions from the Forces pressing them, but that a greater Force is requisite to make a greater impression. This excludes all bodies, fluid or solid, in which the same Force may penetrate farther and farther, provided it have time, without ever being in equilibrio with the resistance: thus a body may continually penetrate farther into soft wax, although the force impelling it be not increased: in these, and the like cases, nothing is required but to surmount the first obstacles; which being once done, and the connection of parts broken, the penetrating body always advances, meeting with the same obstacles as before, and destroying them by an equal Force. But Mr. Euler only considers the first obstacles which exist before any separation of parts, and which are doubtless such, that a greater impression requires a greater Force. Indeed this chiefly takes place in elastic bodies; but it seems likewise to obtain in all bodies when the impressions made on them are small, and the contexture of their parts is not altered.

These things being premised, let the mass or weight of the body A be expressed in general by A, and let its velocity before the shock be that which it might acquire by falling from the height a. Farther, let the harduess of A be expressed by M, and that of B by N, and let the area of the base, on which the impression is made, be cc; then will the greatest compression be made with the Force Therefore if the hardness of the two bodies, and the plane of their contact during the whole time of their collision be the same, this Force will be as √Aa, that is, as the square root of the Vis Viva of the striking body A. And as √a is proportional to the velocity of the body A, the Force of percussion will be in a compound ratio of the velocity and of the subduplicate ratio of the mass of the body striking; so that in this case neither the Leibnitian nor the Cartesian propositions take place. But this Force of percussion depends chiefly on the hardness of the bodies; the greater this is, the greater will the Force of percussion be. If M = N, this Force will be as √(Mcc X Aa,) that is, in a compound subduplicate ratio of the Vis Viva of the striking body, of the hardness, and of the plane of contact. But if | M, the hardness of one of the bodies, be infinite, the Force of percussion will be as √(Ncc X Aa;) at the same time, if M =N, this Force will be as √((1/2)Ncc X Aa.) Therefore, all other things being equal, the Force of percussion, if the striking body be infinitely hard, will be to the Force of percussion when both the bodies are equally hard, as √2 to 1.

Mr. Euler farther deduces from his calculation, that the impression received by the bodies A and B will be as follows; viz, as and respectively. If therefore the hardness of A, that is M, be infinite, it will suffer no impression; whereas that on B will extend to the depth of √Aa/Ncc. But if the hardness of the two bodies be the same, or M = N, they will each receive equal impressions of the depth √Aa/2Ncc. So that the impression received by the body B in this case, will be to the impression it receives in the former, as 1 to √2.

Mr. Euler has likewise considered and computed the case where the striking body has its anterior surface convex, with which it strikes an immoveable body whose surface is plane. He has also examined the case when both bodies are supposed immoveable; and from his formulæ he deduces the known laws of the collision of elastic and non-elastic bodies. He has also determined the greatest pressures the bodies receive in these cases; and likewise the impressions made on them. In particular he shews, that the impressions received by the body struck, or B, if moveable, is to the impression received by the same body when immoveable, as √B to √(A+B).

There are several curious as well as useful observations in Desaguliers's Experimental Philosophy, concerning the comparative Forces of men and horses, and the best way of applying them. A horse draws with the greatest advantage when the line of direction is level with his breast; in such a situation, he is able to draw 200lb for 8 hours a day, walking about 2 1/2 miles an hour. But if the same horse be made to draw 240lb, he can work only 6 hours a day, and cannot go quite so fast. On a carriage, indeed, where friction alone is to be overcome, a middling horse will draw 1000lb. But the best way to try the Force of a horse, is to make him draw up out of a well, over a single pulley or roller; and in that case, an ordinary horse will draw about 200lb, as besore observed.

It is found that 5 men are of equal Force with one horse, and can with equal ease push round the horizontal beam of a mill, in a walk 40 feet wide; whereas 3 men will do it in a walk only 19 feet wide.

The worst way of applying the Force of a horse is to make him carry or draw up hill: for if the hill be steep, 3 men will do more than a horse, each man climbing up faster with a burden of 100lb weight, than a horse that is loaded with 300lb: a difference which is owing to the position of the parts of the human body being better adapted to climb, than those of a horse.

On the other hand, the best way of applying the Force of a horse, is the horizontal direction, in which a man can exert the least Force: thus, a man that weighs 140lb, when drawing a boat along by means of a rope coming over his shoulders, cannot draw above 27lb, or exert above 1-7th part of the Force of a horse employed to the same purpose; so that in this way the Force of a horse is equal to that of 7 men.

The best and most effectual posture in a man, is that of rowing; when he not only acts with more muscles at once for overcoming the resistance, than in any other position; but also as he pulls backwards, the weight of his body assists by way of lever. See Desaguliers's Exp. Philos. vol. 1, p. 241, where several other observations are made relative to Force acquired by certain positions of the body; from which that author accounts for most feats of strength and activity. See also a Memoir on this subject by M. De la Hire, in the Mem. Roy. Acad. 1729; or in Desaguliers's Exp. &c. p. 267 &c, who has published a translation of part of it with remarks.

Force is distinguished into Motive and Accelerative or Retardive.

Motive Force, otherwise called Momentum, or Force of Percussion, is the absolute Force of a body in motion, &c; and is expressed by the product of the weight or mass of matter in the body multiplied by the velocity with which it moves. But

Accelerative Force, or Retardive Force, is that which respects the velocity of the motion only, accelerating or retarding it; and it is denoted by the quotient of the motive Force divided by the mass or weight of the body. So, if m denote the motive Force, and b the body, or its weight, and f the accelerating or retarding Force, then is f as m/b.

Again, Forces are either Constant or Variable.

Constant Forces are such as remain and act continually the same for some determinate time. Such, for example, is the Force of gravity, which acts constantly the same upon a body while it continues at the same distance from the centre of the earth, or from the centre of Force, wherever that may be. In the case of a constant Force F acting upon a body b, for any time t, we have these following theorems; putting f = the constant accelerating Force = F ÷ b, v = the velocity at the end of the time t, s = the space passed over in that time, by the constant action of that Force on the body: and g = 16 1/12 feet, the space generated by gravity in 1 second, and calling the accelerating Force of gravity 1; then is

Variable Forces are such as are continually chang- | ing in their effect and intensity; such as the Force of gravity at different distances from the centre of the earth, which decreases in proportion as the square of the distance increases. In variable Forces, theorems similar to those above may be exhibited by using the fluxions of quantities, and afterwards taking the fluents of the given fluxional equations. And herein consists one of the great excellencies of the Newtonian or modern analysis, by which we are enabled to manage, and compute the effects of all kinds of variable Forces, whether accelerating or retarding. Thus, using the same notation as above for constant forces, viz, f the accelerating Force at any instant, t the time a body has been in motion by the action of the variable Force, v the velocity generated in that time, s the space run over in that time, and g = 16 1/12 feet; then is

In these four theorems, the Force f, though variable, is supposed to be constant for the indefinitely small time t^.; and they are to be used in all cases of variable Forces, as the former ones in constant Forces; viz, from the circumstances of the problem under consideration, deduce a general expression for the value of the force f, at any indefinite time t; then substitute it in one of these theorems, which shall be proper to the case in hand; and the equation thence resulting will determine the corresponding values of the other quantities in the problem.

It is also to be observed, that the foregoing theorems equally hold good for the destruction of motion and velocity, by means of retarding or resisting Forces, as for the generation of the same by means of accelerating Forces.

Many applications of these theorems may be seen in my Select Exercises, p. 172 &c.

There are many other denominations and kinds of Forces; such as attractive, central, centrifugal, &c, &c; for which see the respective words.