Mechanical equilibrium

“ It is easy to conceive, that when forces applied to mechanical instruments are in equilibrium, if the least addition be made to one of them, it will preponderate and overcome the effort. But the want of a perfect polish or smoothness in the parts of all instruments, and the rigidity of all ropes, which increases with the tension, are great impediments to motion, and in compounded engines are found to diminish about one fourth of the effect of the power. ”

“ But the cases which we have more particularly in view are mechanical causes, the causes of the motion and of the equilibrium of bodies. In these cases, forces are conceived as capable of addition; and what has been said of the measure of causes in such cases, applies peculiarly to mechanical forces. Two weights, placed together, may be considered as a single weight, equal to the sum of the two. Two pressures, pushing a body in the same direction at the same point, are identical in all respects with some single pressure, their sum, pushing in like manner ”

“ Also the law of the lever is proven in ordinary mechanics on the basis of proportionality of force and acceleration, thus it can impossibly be applied in the case considered at the beginning. Indeed, we can show by application of a mechanical principle, which doesn't use the force concept (for example Gauss' principle of least constraint) , that equilibrium must exist also in the case of the broken lever, even though the forces doesn't satisfy the law of the lever. ”

“ 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. ”

“ If the mechanical inertia of the moving loop carry it slightly past its position of equilibrium, and at the same moment the current through the loop be reversed, it will be pulled around by the attraction of the magnetic poles to a new position of equilibrium ”

“ As little as one examines the conditions of equilibrium in the lever and in other machines, it is easy to recognize this law, that the weight and the force are always in inverse ratio to the spaces that the one and the other can travel through in the same time. ”

“ This brought the inertia of the reciprocating parts of the engine, at every point in the revolution, into equilibrium with the horizontal component of the centrifugal force of the revolving counterweight. ”

“ Using this principle the method of finding the balance weights to be added to a given system of reciprocating weights in order to produce a system of forces on the frame continuously in equilibrium is exactly the same as that just explained for a system of revolving weights, because for the purpose of finding the balance weights each reciprocating weight may be supposed attached to the crank pin which operates it, thus forming an equivalent revolving system. ”

“ When any elastic object has been strained, and is suddenly released, it regains its position of equilibrium by a series of oscillations or vibrations. Thus, for instance, if we take a strip of steel and fix one end of it in a vice, and pull the other end on one side and then release it, the steel regains its position of equilibrium only after having executed a series of diminishing swings to and fro. ”

“ The condition that each piece of the structure is to be maintained in equilibrium by having its gross load, consisting of its own weight and of the external pressure applied to it, balanced by the resistances or pressures exerted between it and the contiguous pieces, furnishes the means of determining the magnitude, position and direction of the resistances required at each joint in order to produce equilibrium ”

“ Moreover, if i.e., if the pressure of the interior mass increases less rapidly (or decreases more rapidly) with increasing radius than is necessary to preserve neutral equilibrium, the equilibrium is stable. But if the equilibrium is unstable. In the remaining case, when farther conditions are of course necessary to determine absolutely whether the equilibrium is stable or unstable, but in general the equilibrium will be stable in respect to change in one direction and unstable in respect to change in the opposite direction, and is therefore to be considered unstable. ”

“ We must therefore first define the equality of two forces. When shall we say two forces are equal? It is, we are told, when, applied to the same mass, they impress upon it the same acceleration, or when, opposed directly one to the other, they produce equilibrium. This definition is only a sham. A force applied to a body can not be uncoupled to hook it up to another body, as one uncouples a locomotive to attach it to another train. It is therefore impossible to know what acceleration such a force, applied to such a body, would impress upon such another body, if it were applied to it. ”

“ 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. ”

“ That the dynamical equilibrium of the systems in the stationary stages is governed by the ordinary laws of mechanics, while these laws do not hold for the passing of the systems between the different stationary states. ”

“ 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. ”

“ If the machine were frictionless, its mechanical efficiency would be of course equal to its velocity ratio; owing to the presence of friction the mechanical efficiency is less than the velocity ratio, and it will be necessary to make experiments to determine the exact value. I attach to the load hook a weight of 280 lbs., and insert a few small hooks into the links of the power chain in order to receive weights: 56 lbs. is sufficient to produce motion, hence the mechanical efficiency is 5. ”

“ If any disturbing force act sideways upon the wheel, then (if we suppose the material of the wheel, shaft, and bearings to be completely rigid) no alteration of the circular motion occurs; and this is true equally whether be great or small, continuous or intermittent, constant or changing in direction. There is nevertheless continuous equilibrium here, but in another way than before. So soon as the force begins to act it calls forth in the interior of the wheel, the shaft and the supports, internal molecular forces, opposite in direction and exactly equal to it. ”

“ It may be added that the thermodynamic equilibrium which subsists between two such solid states of the same substance differs from that which subsists when one of the states is fluid, very much as in statics an equilibrium which is maintained by friction differs from that of a frictionless machine in which the active forces are so balanced, that the slightest change of force will produce motion in either direction. ”

“ After so many great men have worked on this subject, I almost do not dare to say that I have discovered the universal principle upon which all these laws are based, a principle that covers both elastic and inelastic collisions and describes the motion and equilibrium of all material bodies. ”

“ Because in making a turn, the momentum of the plane sets up a centrifugal force, acting horizontally outward. To counteract this, the force of lift must be inclined until it has a horizontal component equal to the centrifugal force. The machine is therefore tilted to one side, or banked, thus maintaining a state of equilibrium in which it will turn steadily. ”

“ At every point of the two equipotential surfaces and , let a tension = be applied, and at every point of the surface of no induction let a pressure = be applied. These forces will keep the figure in equilibrium. ”

“ If the point of support of a body lies in a vertical line below the centre of gravity, it is in unstable equilibrium. If the centre of gravity lies below the point of support; the state of equilibrium is very stable, and a vibration may take place about this position of stable equilibrium, as in a pendulum or balance, when finally the body assumes a position of stable equilibrium. ”

“ Considering a rigid body in any given position, we may contemplate the whole group of infinitesimal displacements which might be given to it. If the extraneous forces are in equilibrium the total work which they would perform in any such displacement would be zero, since they reduce to a zero force and a zero couple. ”

“ These laws which rule the equilibrium of two weights suspended at the arms of a lever have been drawn from hypotheses special to this problem; when the mechanicist will have to treat another problem of equilibrium, distinct from that one, it will be necessary for him to invoke new hypotheses, heterogenous to the first, and the analysis of the first hypotheses will not give him any indication which can guide him in the choice of seconds. ”

“ If there are intersections of the nodal line not on opposite points of the sphere, then is at the intersection of three or more lines of equilibrium. For the equipotential surface through must cut itself in each line of equilibrium. ”

“ We may infer therefrom that if a couple is applied to the first wheel it will balance a ten times greater couple applied to the last one. This information is obtained without our having to consider the mechanism whereby this state of equilibrium is realised and without our having to know how the various forces will balance one another inside the machine. ”

“ As special cases, the system may reduce to a couple, or it may be in equilibrium.
In general, however, a three-dimensional system of forces cannot be replaced by a single resultant force. But it may be reduced to simpler elements in a variety of ways. For example, it may be reduced to two forces in perpendicular skew lines. For consider any plane, and let each force, at its intersection with the plane, be resolved into two components, one (P) normal to the plane, the other (Q) in the plane. ”

“ It is assumed that the form can be sufficiently represented by a plane curve, that the stress (tension) at any point P of the curve, between the two portions which meet there, is in the direction of the tangent at P, and that the forces on any linear element δs must satisfy the conditions of equilibrium laid down in § 1. It follows that the forces on any finite portion will satisfy the conditions of equilibrium which apply to the case of a rigid body (§ 4) . ”

“ But now, curiously enough, if you were to apply a sidewise pressure at A, pushing to the left (as we view the diagram) to help on the motion of precession, the obstinate apparatus will cease altogether to move in that direction and the point A will begin to rise instead, the frame B A C rotating on its axis B C. This rise of the axis O A will take place even though the downward pressure is continued. You have disturbed the equilibrium of the top—unbalanced it—and it must seek a new position. Contrariwise, if you would have the point A moved to the right, you must push it upward ”

“ We may in this way reduce any real structure to the case of a system of points with attractive or repulsive forces acting between certain pairs of these points, and keeping them in equilibrium. The direction of each of these forces is sufficiently indicated by that of the line joining the points, so that we have only to determine its magnitude. We might do this by calculation, and then write down on each link the pressure or the tension which acts in it. ”

“ To discover whether two weights be equal, we must oppose them to each other in a machine where they act in a similar manner, and then see if they maintain an equilibrium; for example, we fulfil this condition if we suspend them at the two extremities of a lever, supported at its centre, and whose arms are equal. Such is the general idea of a balance. The beam of a good balance ought to be a bar of well-tempered steel, of such form as to secure perfect inflexibility under any load which may be fitly applied to its extremities. ”

“ Respecting the structural state simultaneously reached, it must obviously be one presenting an arrangement of forces that counterbalance all the forces to which the aggregate is subject. So long as there remains a residual force in any direction—be it excess of a force exercised by the aggregate on its environment, or of a force exercised by its environment on the aggregate, equilibrium does not exist ”

“ And the fact which we have here particularly to observe, is, that as a corollary from the general law of equilibration above set forth, the evolution of every aggregate must go on until this equilibrium mobile is established; since, as we have seen, an excess of force which the aggregate possesses in any direction, must eventually be expended in overcoming resistances to change in that 448direction: leaving behind only those movements which compensate each other, and so form a moving equilibrium. ”

“ Now it has been proved mathematically by Professor Maxwell, in connexion with the kinetic theory of gases, that a self-acting adjustment goes on among a system of bodies or particles in free collision, such that the particles are caused to move equally towards all directions, this being the condition requisite to produce equilibrium of pressure. ”

“ Static Equilibrium of Mechanisms.—The principle stated in the preceding section, namely, that the energy exerted is equal to the work performed, enables the ratio of the components of the forces acting in the respective directions of motion at two points of a mechanism, one being the point of application of the effort, and the other the point of application of the resistance, to be readily found. ”