Balance (physics)

“ But I do not understand why the longest arm of the lever should not be in equilibrium with the other?
Caroline. It is because the momentum in the longest, is greater than in the shortest arm; the centre of gravity, therefore, is no longer supported.
Mrs. B. You are right, the fulcrum is no longer in the centre of gravity; but if we can contrive to make the fulcrum in its present situation become the centre of gravity, the scales will again balance each other; for you recollect that the centre of gravity is that point about which every part of the body is in equilibrium. ”

“ The most complex frame-work on which weights are supported, as the roof of a building, or the cordage of a machine, are still examples of equilibrium. In such cases we may have many forces all combining to balance each other; and the equilibrium will depend on various conditions of direction and magnitude among the forces. And in order to understand what are these conditions, we must ask, in the first place, what 214 we understand by the magnitude of such forces;—what is the measure of statical forces. ”

“ After mass and energy there is one physical quantity which plays a very fundamental part in modern physics, known as Action. Action here is a very technical term, and is not to be confused [Pg 153] with Newton’s ‘Action and Reaction.’ In the relativity theory in particular this seems in many respects to be the most fundamental thing of all. ”

“ The metrical, gravitational and electromagnetic fields are all included. We build the quantities enumerated above; and they obey the great laws of field-physics in virtue of the way in which they have been built. That is the special feature; the field laws—conservation of energy, mass, momentum and of electric [Pg 237] charge, the law of gravitation, Maxwell’s equations—are not controlling laws. [38] They are truisms. Not truisms when approached in the way the mind looks out on the world, but truisms when we encounter them in a building up of the world from a basal structure. ”

“ But gravity is not the only force, and we might measure the energy of a moving body by the extent to which it would bend a powerful spring or resist the attraction of a powerful magnet, or, in fine, we might make use of the force which best suits our purpose. If this force be a constant one, we must measure the energy of the moving body by the space which it is able to traverse against the action of the force—just as, in the case of gravity, we measured the energy of the body by the space through which it was able to raise itself against its own weight. ”

“ When a fluid mass is in equilibrium under the influence of gravity, and has the same independently variable components throughout, the intrinsic potentials for the several components are constant in any given level, and diminish uniformly as the height increases, the difference of the values of the intrinsic potential for any component at two different levels being equal to the work done by the force of gravity when a unit of matter falls from the higher to the lower level. ”

“ We next extend the force concept to systems not in equilibrium, in which there are accelerations, and we must conceive that at first all our experiments are made in an isolated laboratory far out in empty space, where there is no gravitational field. ”

“ There, though the weight of a given mass of matter is different from what it is with us, it is nevertheless determined by the law that determines here—the law of gravitation. There energy is indestructible, and is measured as it is measured among us, by work. Then is there any boundary that we can assign to natural law—is it not omnipresent, universal? ”

“ This is the fundamental conception (not hypothesis) of the theory.... Now in a system of atoms as above defined, let the centers be of equal mass and at equal distances; there will be no difference of pressure on any one center, no moving force will be developed, and the conditions of equilibrium will be satisfied. ”

“ Since the time enters to the second power, the product of mass and acceleration is smaller by the ratio than it would be if the system were at rest. And we conclude, therefore, that the unit of force, or the dyne, in a direction transverse to the line of motion is smaller in a moving system than in one at rest by this same ratio. ”

“ While the factors of kinetic and of gravitational energy, velocity and momentum, on the one hand, and force and distance, on the other, are simply related to each other, the capacity factors of other forms of energy—surface, in the case of surface-energy; volume, in the case of volume-energy; entropy for heat; electric capacity when electric charges are being conveyed by means of ions; atomic weight, when chemical energy is being gained or lost—all these are simply connected with the fundamental chemical capacity, atomic weight or mass. ”

“ As force is thus conceived, not as an absolute entity acting upon matter, but as a condition of the parts of matter itself, and as a condition determined by the relative masses and distances of these parts, any valid hypothesis of a force or of a motion to account for any set of phenomena is thus seen to imply an assertion as to relative masses and distances which can be more or less readily submitted to experiment or observation and analysis. ”

“ A force which belongs to individual atoms, is independent of chemical and physical conditions, and cannot be altered or destroyed by any known means, must be closely related to the fundamental nature of the atoms. One of the most essential parts in our concept of matter is mass, and the force of gravitation of an atom is proportional to its mass. Mass and gravitation stand, therefore, either as co-effects of the same cause or as cause and effect. ”

“ Considered in the most general way, we may define matter as a complex of gravitational, kinetic and chemical energies, which are found to cling together in the same space. Of these energies the capacity factors, namely, weight, mass and elements, are conservative as described, while the intensity factors, potential, velocity and affinity, may change in wide limits. To explain why we find these energies constantly combined one with another, we only have to think of a mass without gravity or a ponderable body without mass. ”

“ In other words, the components of the force at any point are determined from the components of acceleration in all parts of the field by the same process by which (in the theories of gravitation, etc.) the value of the potential at any point is determined from the density of matter in all parts of space, except that the sign is to be reversed. Adopting this law, provisionally at leasts we may express it by saying that the components of electrodynamic force are equal to the potentials taken negatively of the components of acceleration of electrical flux. ”

“ 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. ”

“ We must therefore conclude that to maintain equilibrium the force at a must be less than the force at c in the same ratio. We thus see that in a moving system unit force in the longitudinal direction is smaller than unit transverse force in the ratio , and therefore, by the preceding paragraph, smaller than unit force at rest in the ratio . ”

“ But in a non-resisting medium, the quantity of matter in the pendulous body is as the comparative weight and the square of the time directly, and the length of the pendulum inversely. For the comparative weight is the motive force of the body in any heavy medium, as was shewn above; and therefore does the same thing in such a non-resisting medium, as the absolute weight does in a vacuum. ”

“ And here a physical analogy may fitly be used to give the greatest definiteness to this cardinal truth. A mass of matter, of whatever kind, maintains its state of internal equilibrium, so long as its component particles severally stand toward their neighbors in equi-distant positions. Accepting the conclusions of modern physicists, which imply that each molecule moves rhythmically, then a balanced state implies that each performs its movements within a space bounded by the like spaces required for the movements of those around. ”

“ If our ideas possess a certain degree of artificiality, this is also true of others which have long since been adopted into mechanics. The apparent change in rate of a moving clock, and the apparent change in length and mass of a moving body, are completely analogous to that apparent change in energy of a body in motion, which we have long been accustomed to call its kinetic energy. ”

“ From the standpoint of the principle of relativity, one of the most interesting quantities in mechanics is the so-called kinetic energy, which is the increase in energy attributed to a body when it is set in motion with respect to an arbitrarily chosen point of rest. ”

“ The forces are to be measured according to definition in mechanical terms, either by the strains in members of a framework if the system is in equilibrium, or in terms of accelerations and masses if it is not in equilibrium. ”

“ 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. ”

“ In this way both the unit of force and the unit of mass are made to depend on the value of the force of gravity, which varies from place to place, so that statements involving these quantities are not complete without a knowledge of the force of gravity in the places where these statements were found to be true. ”

“ Since the momentum of the weight and power are as the quantities of matter in each, multiplied by their respective celerities; and the celerities are as the distances from the centre of motion, and also as the spaces passed through in a perpendicular direction in the same time, it must follow, that there will be an equilibrium between the weight and power, when they are to each other reciprocally as the distances from the centre, or as the celerities of the motions, or as the perpendicular ascent or descent in the same time ”

“ We want to get a mass isolated both from gravitation and from friction, but this is just what we cannot do—that is, we cannot do it perfectly. We have, however, a simple appliance which will be sufficiently isolated for our present purpose. ”

“ How are we to conceive of this force growing up in A to a million-fold or more ? And if the surrounding particles be then removed, of its diminution in an equal degree? Or how are we to look upon the power raised up in all these outer x^articles by the action of A on them, or by their action one on another, without admitting (according to the limited definition of gravitation) the facile generation and annihilation of force ? ”

“ And we will extend the principle further, and contend that all the phenomena of nature are consequences of the constant tendency of matter to conform to this principle of equilibrium, when suffering temporary derangement from the operation of other laws. That throughout the system of nature, equal spaces possess equal force. That what we call temperature, is nothing more than the motion of equilibrium or atomic momentum of space; or, in other words, that if all space were fluid, and in a state of equilibrium, the product of each atom of equal volume, by its motion would be a constant quality. ”

“ For purposes of accuracy, balances have occasionally means of raising or depressing the centre of gravity, of regulating the length of the arms, &c., and the whole apparatus is usually enclosed in a glass case, to prevent the heat from expanding the arms unequally, or currents of air from disturbing the equilibrium. ”

“ Now, if one be double the weight of another, the force of gravity or attraction must be double to make it oscillate with the same celerity: if one is thrice the weight or quantity of matter of another, it requires thrice the force of gravity to make it move with the same celerity. Hence it is certain, that the power of gravity is always proportional to the quantity of matter in bodies, whatever their bulks or figures are. ”

“ The powers around the centers give these centers the properties of atoms of matter; and these powers again, when many centers by their conjoint forces are grouped into a mass, give to every part of that mass the properties of matter The view now stated of the constitution of matter would seem to involve necessarily the conclusion that matter fills all space, or at least all space to which gravitation extends ; for gravitation is a property of matter dependent on a certain force, and it is this force which constitutes matter. ”

“ The force may thus be considered as an attribute of space, and it becomes possible, even if itself is inaccessible, to determine its magnitude and location by measuring the direction and magnitude of the force experienced by unit matter in different parts of space. It is not even necessary that the measurements should be made on unity of active matter; any convenient quantity of active matter in the small body will serve. All we need do is to reduce the measured force in the ratio of the actual amount of active matter used to its unit value. ”

“ Now the smaller any bits of matter are, the larger in proportion is their surface to their weight. A certain descending force is required to push the resisting atoms of air apart, and so permit the descent of the gravitating particle. ”

“ By the earth’s attraction, all the particles which compose the mass of a body are solicited by equal forces in parallel directions downwards. If these component particles were placed in mere juxtaposition, without any mechanical connection, the force impressed on any one of them could in nowise affect the others, and the mass would, in such a case, be contemplated as an aggregation of small particles of matter, each urged by an independent force. ”