Tractive force

“ 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. ”

“ As it contracted its velocity would, in obedience to one of the fundamental laws of mechanics, constantly increase, so that a time would arrive when, at the outer boundary of the mass, the centrifugal force due to the rotation would counterbalance the attractive force of the central mass. ”

“ To derive the play of internal force, this energy function must be varied with respect to all these independent quantities; the result is elastic tractional stress in the medium across every ideal interface, together with forcive tending to displace each strain-centre, which we can consider either as resisted by extraneous constraint preventing displacement of the strain-centre, or as compensated by the reaction of the inertia of the strain-form against acceleration. ”

“ First we calculate the forces exerted upon the resting condenser. Attraction is the by far strongest force-action exerted by the plates upon each other; it has (as far as the field in the interior can be considered as homogeneous) the strength per unit area, when is the field strength. If we neglect the marginal effects, then these forces produce the resultants of strength
whose points of contact are the midpoints of the plates. Far less strong, but still useful for our purposes, are the traction-forces acting at the rim of the plates. ”

“ Matter possessing only inertia and motion (whose product is momentum) would very speedily arrive at a state of stable and inert equilibrium, without having ever exhibited a single phenomenon of force, and without the possibility of any dynamic potential. ”

“ When we conceive that inertia is resistance to deflection, it becomes a proposition, easily comprehended, that rigidity is resistance to the differential deflection of the molecular parts of a body. Every one of its minute parts must be moved if the body is moved, and the regional parts as distinguished from the molecular parts cannot be moved without fracturing the body. Thus we see that rigidity can be explained simply as a mode of motion without resort to occult force. ”

“ For in those Propositions, instead of the uniform resistance made to an ascending body arising from its gravity, one may substitute the uniform resistance which arises from the tenacity of the medium, when the body moves by its vis insita alone; and when the body ascends in a right line, add this uniform resistance to the force of gravity, and subduct it when the body descends in a right line. One might also go on to the motion of bodies which are resisted in part uniformly, in part in the ratio of the velocity, and in part in the duplicate ratio of the same velocity. ”

“ Let us, recurring to that only safe guide experience, recall as the necessary outcome of the fourth, fifth, and sixth propositions, that " gravitation is a property immutable and inconvertible." Let us, to avoid confusion of idea by the unconscious double entendre of the word " force," limit the term for the present to that innate and primitive tendency or tension which appears as the last result of dynamic analysis, and which obviously differs as much from the action of the falling stone, as the flying arrow differs in function from the elasticity of the bow which has impelled it. ”

“ So that if the disparity of velocities is so great as to overcome all that resistance which commonly arises either from the attrition of contiguous bodies as they slide by one another, or from the cohesion of continuous bodies that are to be separated, or from the weights of bodies to be raised, the excess of the force remaining, after all those resistances are overcome, will produce acceleration of motion proportional thereto, as well in the parts of the machine as in the resisting body. ”

“ It becomes evident that bodies do not owe their gravity to an intrinsic force, but to the pressure of the medium in which they are immersed. The motion of heavy bodies no longer appears to us other than as a transformation of the aetherial motions ”

“ But if force be a virtue resident in the particle, it must at its origin be either attractive or repulsive, and it seems impossible to conceive how by emanation to a distance it can change its quality. This difficulty, as will be shown, is not encountered in a theory of molecular forces which deduces their laws from the dynamical action of an elastic medium. ”

“ This power of resistance, or power of annular elasticity, is the most striking characteristic of these singular forms, real magazines of energy, in which all the living primitive force collects itself under the influence of external friction. Theoretically, the vortical movement, in a perfect fluid, can neither be created nor destroyed; eternal as matter itself, it has neither beginning nor end. ”

“ It may be objected that it is inconsistent to represent a line of force, which is essentially dipolar, by an axis of stress, which is necessarily isotropic; but we know that every phenomenon of action and reaction is isotropic in its results, because the effects of the force on the bodies between which it acts are equal and opposite, while the nature and origin of the force may be dipolar, as in the attraction between a north and a south pole. ”

“ But if the force of gravity does not sustain any loss as work is performed by it, what then is it that is supposed to sustain the loss? Some form of energy must diminish as work is performed ; and if it be not gravity it must be something else. The generally received explanation is this : when a body is projected upward, the potential form of energy into which the upward motion of the body is transformed does not consist in the simple force of gravity or tendency of the body to descend, but consists in this force or tendency multiplied by the distance through which it is capable of descending. ”

“ In defining atoms as centers of pressure, they are thus no less distinguished on the one hand from centers of force than from the little hard bodies of the ordinary theories; for such centers of force are just as absolute and self-existent in the ordinary conception of them as those little bodies ; and in a scientific theory there can (except as temporary conveniences) be no absolute existences, entities. Hence (mechanical) force, or the cause of motion, is conceived not as an entity but as a condition, — the condition namely of a difference of pressure ”

“ So also, as we have seen in quarries, when several men have endeavoured to move a large mass of stone with a lever, they have not used one long and powerful effort, but a succession of slighter ones, until a tremulous motion has been seen in the mass, when by one exertion of force they have hurled it from its place.
The degree of pressure exercised by any given persons will be in the inverse ratio of the degree of control which they can exercise over the muscular system, and over their ideas ”

“ This momentum carries the prong beyond the position of rest, nearly as far as it was originally drawn in the reverse direction; until at length, being gradually used up in producing an opposing tension among the particles, it is all lost. The opposing tension into which the expended momentum has been transformed, then generates a second recoil; and so on continually—the vibration eventually ceasing only because at each movement a certain amount of force goes in creating atmospheric and etherial undulations. ”

“ But it is not the influence of these external forces we are considering, but only the energy due to the position of the raised weight. Assuming, therefore, the disturbing conditions absolutely eliminated, it is not difficult to see that no downward force or pressure, however small, could be applied for ever so short a time, to the upper weight without setting the system in motion. The motion would be an accelerated one so long as the force was applied, it would become uniform when the force ceased to act; it would have a velocity proportionate to the force. ”

“ If the two forces act away from each other, as in the breaking of a string, the stress is called a tension, but if they act toward each other as in crushing anything, the stress is called a pressure. In order for a body to exert force it must meet with resistance. The force exerted is never greater than the resistance encountered. Thus one can exert but little force upon a feather floating in the air or upon other light objects. A fast moving shot exerts no force unless it encounters some resistance. ”

“ This central mass agitated with intestine motions of extreme violence in all directions, as in a boiling caldron, produces an intense light and heat, while detached particles at the surface, continually more and more broken up by the frequency and impetuosity of their collisions, acquire a subtilty transcending all conception of the imagination, and are finally thrown off in all directions with explosive force to immense distances, and with the velocity known as that of radiant light. ”

“ The momentum, which is in the simple reciprocal ratio of the bodies, might consequently be increased at pleasure by the means proposed, in the subduplicate ratio of the bodies employed; and if momentum were an efficient force capable of reproducing itself, and of overcoming friction in proportion to its estimated magnitude, the additional force acquired by such a means of increase, might be employed for counteracting the usual resistances, and perpetual motion would be easily effected. ”

“ If the magnitude of b exceeds a certain limit, then a is pulled along the table and b descends; but if b be smaller than this limit, both a and b remain at rest. When b is not heavy enough to produce motion it is supported by the tension of the cord, which is itself neutralized by the friction produced by a certain coherence between a and the table. Friction is by this experiment proved to be a force, because it prevents the motion of b. Indeed friction is generally manifested as a force by destroying motion, though sometimes indirectly producing it. ”

“ If there is the slightest displacement of the centre of gravity of the system from the axis of revolution a force acts on the shaft tending to deflect it, and varies as the deflexion and as the square of the speed. If the shaft is therefore to revolve stably, this force must be balanced at any instant by the elastic resistance of the shaft to deflexion. ”

“ Here all kinetic theory fails: and there is then something in the world beside " inertia and motion." Until some rational analysis be discovered, we must accept the fact (in the language of Sir John Herschel) as an "ultimate phenomenon." And as we find that the molecules of all material substance resist compression with a force proportional in some high ratio thereto, we can only conceive that they are separated by an interval through which this repellant resistance is exercised. ”

“ Thus collisions which might result simply in deflection if the particles acted severally, produce fracture when they act conjointly. Particles may produce pressure when they act separately, but when they act conjointly pressures may lead to rupture. By the device of the lever forces are multiplied in effect without increase or diminution of force as such ”

“ Commencing with the generalization that " a mechanical force, or the cause of a mechanical motion, we know to be in general the condition of a difference of pressure," he infers, "hence it appears that if a general mechanical theory is possible, the ultimate property of matter must be conceived to be a mutual repulsion of its parts, and the indubitable Newtonian law of universal attraction be deduced herefrom, under the actual conditions of the world. ”

“ The effect of an instantaneous force is produced in an infinitely short time. If the body which sustains such an action be previously quiescent and free, it will move with a uniform velocity in the direction of the impressed force. (93.) If, on the other hand, the body be not free, but so restrained that the impulse cannot put it in motion, then the fixed points or lines which resist the motion sustain a corresponding shock at the moment of the impulse. This effect, which is called percussion, is, like the force which causes it, instantaneous. ”

“ Impenetrability, which we must not regard as a force, but as a resistance essential to matter, not permitting two bodies to occupy the same place, what must happen when two molecules, which attract the more powerfully as they approach nearer, suddenly strike against each other? Does not then this invincible resistance of impenetrability, become an active force, which, in the contact, drives the bodies with as much velocity, as they had acquired at the moment they touched? ”

“ Hence, we learn that two equal and directly opposed forces counteract each other, and each may be regarded [Pg 6] as destroying the motion which the other is striving to produce. If I make the weights unequal by adding to one of them, the knot is no longer at rest; it instantly begins to move in the direction of the larger force. ”

“ But when the atoms strike and recoil, the motion of translation is converted into a motion of vibration, which is heat. The vibration, however, so far from causing the extinction of the original attraction, is in part carried on by that attraction. The atoms recoil, in virtue of the elastic force which opposes actual contact, and in the recoil they are driven too far back. The original attraction then triumphs over the force of recoil, and urges the atoms once more together. Thus, like a pendulum, they oscillate, until their motion is imparted to the surrounding aether ”