Mechanics

MECHANICS. - Mechanics has as its object the study of the phenomena of material bodies that take place without altering either their physical or their chemical nature. It will therefore study, for example, the flow of water in channels or pipes, the transfer of its kinetic energy to devices capable of absorbing and utilizing it, and any other phenomenon that leaves its nature as water unaltered. It will not, on the other hand, concern itself with its possible evaporation or transformation into water vapor, nor, even less, with its possible decomposition into its constituents oxygen and hydrogen. Already from the most ancient times it had been recognized that in mechanical phenomena what is at play are relations between quantities inherent in motion, in forces, and in some quid intrinsic to the material bodies that participate in them. It is therefore useful, before fully confronting the problem of determining those relations, to undertake a preliminary general study both of motion (kinematics, or the geometry of movement) and of forces.

I. KINEMATICS

Observation teaches that every displacement is always accompanied by an act of motion that unfolds in space with a certain continuity and in a close correlation with time or, as is often said, as a function of time.

The study of possible acts of motion is the chief aim of kinematics, which thereby also justifies its name of geometry of motion. As is intuitive, in order to follow and possibly to express formally the acts of motion one must have recourse to some system of reference, which however can in general be chosen with wide latitude. But already at this point an essential difference appears between kinematics and geometry. The latter always chooses the system of reference that it intends to adopt from among infinitely many supposed systems, even when this is not explicitly stated, in relative repose, that is, fixed with respect to one another. For example, the position of a point in a room can be referred in a simple manner to the system of Cartesian coordinates whose axes are constituted by three edges of the walls converging at one of the vertices, or to any system of polar coordinates with the corresponding pole at that vertex. In any case the position of the point is always understood relative to the chosen system of reference. Now, if the point is supposed to be in a certain motion with respect to the chosen system of reference, it will be equally so with respect to all the others. In the case just considered, motion would therefore have an absolute character. But, once the notion of motion has been introduced, it is logical to apply it also to systems of reference and therefore to consider also systems of reference in relative motion with respect to one another. In that case, however, the notion of the motion of a body loses any absolute meaning and remains a notion relative to the chosen system of reference, which it will always be necessary to indicate, unless it can be understood without ambiguity.

Among all the systems in relative motion that it is possible to conceive, will there not exist one that is truly fixed? To this question it is logical to oppose at once another question: fixed with respect to what? At this point, however, it suffices to recall the evolutions of the notion of a fixed body in order to understand that it is not possible to answer. For a very long period the Earth was believed fixed, then people passed to considering the Sun fixed; subsequently new astronomical observations demonstrated that the Sun too, with its whole system of planets, moves with a velocity of fully 10 km/sec toward the distant constellation of Hercules, and that the whole immense galactic system is likewise in motion relative to the most distant stars, which by now no one any longer dares to suppose fixed. The wave theories of optics and Maxwell's electromagnetic theory likewise presupposed the existence of a fixed ether, but here too experience brought out the absolute untenability of that hypothesis (v. RELATIVITY). In conclusion: physics, and in particular kinematics, must by now cease to make the slightest use of the equivocal and fallacious notion of an absolute system.

Motion of a point. — In geometry and also in daily life the notion of a point and of the position of a point is often employed. This is an abstract notion, insufficient to define completely the position of a real body, because it prescinds from every consideration of orientation, which has no meaning for a point but which has instead a well-determined sense for any real body, however small. Yet even in many real cases that incomplete indication is practically sufficient. For example, the position of a ship in the ocean is generally indicated by means of its longitude and its latitude, prescinding from its orientation. Hence physics often considers the bodies that enter into its considerations as concentrated in points, sometimes attributing to them also some non-conflicting material property, and distinguishing them from purely geometric points by the qualification of material points. In such cases the notion of the motion of a body is reduced to that of the succession of points of space occupied by that representative point as time passes, points of space that cannot be isolated but must constitute a line, the trajectory of the moving point. It may be either straight or curved. But knowledge of the trajectory alone is not yet sufficient to define completely the motion of the point. Evidently it is still necessary to establish in some way an exact correspondence between the positions of the point, that is, between the points of the trajectory and time. This can be done in various graphical and analytical forms. An essential element in the consideration of motions is velocity. Of this there is by now a sufficiently exact notion, because modern life continually puts us in contact with the most varied velocities. In the last analysis the velocities of any object always turn out to be expressed by the ratio of a distance l to the time taken t: v = l/t. This is what leads one to attribute to velocity the physical magnitude v = LT^{-1}. But a point can move with constant velocity along its whole trajectory or also with variable velocity from place to place. In the first case the length of the distance s for every time t measured from the instant for which t = 0 will be given by the very simple equation s = s₀ + vt, where s₀ is the position of the point at the initial instant. But in the case of variable velocities, in order to describe them better, it is useful to introduce the very useful notion of acceleration a, that is, of the increment of velocity in the unit of time, hence a = Δ v/Δ t, denoting by Δ v the variation of velocity that occurred in the time interval Δ t. Consequently one attributes to acceleration the physical dimension a = vT^{-1} and, taking into account the dimension of v, a = LT^{-2}. Here too, as previously for velocity, the acceleration may turn out to be constant along an entire path, and then one has a motion with constant acceleration, generally called uniformly accelerated motion; but it may also turn out to be variable from place to place, and then one has variously accelerated motion. In perfect analogy with what was said a moment ago, in the case of constant acceleration the variable velocity at every instant will be given by v = v₀ + at, where v₀ indicates the velocity that one had at the initial instant. In the case of retarded motion, that is, of a decrease in velocity, a negative sign is attributed to the acceleration. At this point it is interesting to ask what the expression is for the distance s of the material point in the case of uniformly accelerated motion. Simple mathematical considerations lead to the expression s = s₀ + v₀ t + frac{1}{2}at, where s₀, v₀ are respectively the position and the velocity at the initial instant. This expression, as will be seen later, has a notable importance in the history of m.

With considerations fundamentally analogous from the physical point of view, but necessarily more complex, kinematics studies all the other types of motions not only of points but of any bodies. The difficulties that can arise in such studies are only of a more mathematical nature.

II. GEOMETRY OF FORCES

Even of the notion of force everyone has an intuition sufficient to make a first partial analysis of it. Already the force most familiar to us, the one that manifests itself as the weight of material bodies, was from remote times recognized as variable from body to body, or more precisely as endowed with its own intensity, a direction and a sense, the latter a priori respectively vertical and directed downward, but capable of being deflected by means of suitable devices. For example, the force to raise a bucket of water evidently has an intensity equal to their total weight, and normally the vertical direction and the downward sense; but with the rope-and-pulley device applied to wells, direction and sense can be arbitrarily deflected, while the intensity remains the same. From the evident fact that the effect on a rope stretched in a straight line of a force of equal direction is independent of the point of application, the conclusion was reached that forces can in general be arbitrarily displaced along their line of action.

Composition of forces. — Already in the 13th century G. Nemorarius had qualitatively recognized the possibility of the decomposition of forces, considering the case of the vertical downward force corresponding to the weight of a body suspended from a rope more or less taut between two fixed points, which force was in fact balanced by the two tensions, hence by the two forces, directed along the two branches of the rope. But despite this recognition, no one dared to pass to the assertion of a similar possibility of the composition of forces. Only Leonardo da Vinci, about two centuries later, having determined those two tensions and demonstrated that their rational composition was also quantitatively equivalent to a force equal and opposite to the weight of the suspended body, rose to the general notion of composition and decomposition of forces. On the basis of this notion and of its law, later formulated with the well-known parallelogram rule, all the most interesting problems concerning forces in themselves were solved, from Galileo onward. Considering first coplanar forces, that is, all endowed with directions lying in one and the same plane, let us recall that any number of them is in general still composable into a single resultant in that plane, except when, in the penultimate operation of this attempted composition, there should result two forces of equal intensity F, parallel, of opposite sense and having lines of action at any non-zero distance d, because in that case every further composition is impossible. In other words, the particular system just described represents a new physical quantity, which is customarily called a couple. An easy analysis shows that couples are characterized by their moment, that is, by the product Fd. Two couples of equal moment are equivalent. It is very important to bear in mind that if a force is transported parallel to itself rather than along its own line of action (as has been supposed up to now), a couple is always generated. This premised, with respect to the plane one can conclude: any number of coplanar forces always compose either into a single resultant force, or into a couple.

In space, analogous considerations lead to the conclusion: two skew forces are in general equivalent to a force and a couple; it follows that any number of forces in space is likewise equivalent (is reducible) to a force and a couple.

III. GENERAL DYNAMICS

Of the three elements which, through their mutual relations, determine the laws of the mechanics of material bodies — motion, forces considered as extrinsic to the bodies, and that quid which was instead said to be intrinsic to them — only the first two could be studied beforehand individually, on account of their easy intuition, and give rise to the so-called geometries of motion and of forces. Of the third element, on the other hand, wholly implicit in the theory of motion itself, no reliable preliminary study was possible, also because in the broad field of the mechanics of material bodies there exist, as will be seen, various quantities that individually would be capable of representing the aforesaid intrinsic quid. This is perhaps the most recondite reason which, aggravated by the lack, or worse by the fallaciousness, of easy intuitions, made it so difficult to extract from the observation of the phenomena of mechanics a coherent system of laws capable of interpreting them all.

1. The principle of inertia

The first real result in the various attempts at this extraction of the fundamental laws of mechanics was the attainment of the notion of the inertia of material bodies and the formulation of its law, achieved first by Leonardo da Vinci and then, independently of him, by Galileo. In Newton's subsequent systematic formulation of the fundamental laws of mechanics, it constitutes LEX I: «Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur illum mutare». And without doubt the ignorance of this fundamental basis was the chief cause of Aristotle's erroneous interpretation of many of the phenomena of mechanics, and in particular of that of the fall of material bodies near the earth's surface, which, as will be seen, was later the touchstone between the Aristotelian views and those of Galileo.

At this point it deserves to be noted that it was precisely the founders of the experimental criterion who raised to the rank of a fundamental principle a phenomenon which, within the field of our experimental possibilities, can never be completely realized and can only be recognized as a limiting case, because it is not possible for us to observe an act of motion absolutely removed from the action of some external force. The supporters, on the other hand, of the contrary idea — that bodies could not remain in motion except under the action of some force — could delude themselves into thinking they were in agreement with everyday experience, which shows that every body not kept in motion by ascertainable external forces comes to a stop.

2. The fundamental principle of dynamics

Having attained the law of the motion of bodies in the absence of external forces, it is logical to ask what effects the possible presence of such forces must produce on the bodies themselves. But here too our pure intuition is insufficient to confront the question, and recourse to well-understood and rigorously interpreted experience appears necessary. The first question that arises is whether it is legitimate to speak of the action of a force on a given body, independently of the state of motion in which it may happen to be. The question is extremely delicate because, since direct experience is not possible, it will have to be resolved indirectly and also through abstractions and idealizations to be carried out with great prudence. Only Galileo succeeded in posing this question in its proper terms and in resolving it, summing up its essence in the principle that a force acts on a body in the same manner always, whether the body is in motion or at rest, whether it is subject or not to the action of other forces. Only on the basis of this principle can experience on bodies initially at rest with respect to the observer be conclusive. The law that he obtained from the results of Galileo's experiments was formulated by Newton as LEX II: «Mutationem motus proportionalem esse vi motrici impressae et fieri secundum lineam rectam qua vis illa imprimatur».

Let it now be borne in mind that the asserted proportionality between the force and the mutatio motus (which is nothing other than a positive or negative acceleration) refers to the effects of various forces on one and the same body: in other words, that the ratio between the impressed force F and the acceleration obtained a is a constant of the body considered. But since the accelerations obtained by applying equal forces to different bodies are also different from one another, or, what amounts to the same thing, since different bodies offer different resistances to the variation of their state of motion, the idea arises quite spontaneously of ascribing to each body its own inert or dynamic mass m. This is precisely what Newton did, assuming moreover as its measure, without further ado, the aforesaid ratio F/a; that is, expressing his II law in the form F=m.a.

To this fundamental law various other forms may also be given, very useful in different cases. Observing that the (average) acceleration during a time interval Δt can be written a=Δv/Δt, the preceding law takes the form FΔt=mΔv, or, given the constancy of m, also FΔt=Δ(mv). This form becomes very expressive if two new physical concepts are introduced: that of the product of a force by the time during which it acts, that is, the impulse of that force, and that of the product of a mass by its velocity, that is, the momentum of that mass (or of the body possessing it); the equation then expresses that the impulse of a force is always equal to the variation of the momentum of the body to which it is applied. Often the terms are inverted and one says: the variation of the momentum of a body is always equal to the impulse that produced it. Again, dividing by Δt, the same equation tells us: the variation of momentum per unit time is equal to the acting force. All forms of the fundamental law that can be useful.

The quantity mass thus introduced is eminently physical in nature, independent of the fundamental quantities length L and time T; it must therefore be considered as a new fundamental quantity proper to dynamics, whose physical dimension is customarily denoted by M. From the preceding equation there follows, taking into account the dimensions of acceleration, the dimensional characterization of force F=LT-2M.

3. The principle of action and reaction

The preceding two principles are already sufficient to solve satisfactorily numerous dynamical questions, but not all of them and, in any case, never in their fullness. Indeed, to apply a force to a body it is necessary to resort to the employment of another body. And then it cannot be said that a dynamical phenomenon is known in a complete manner unless account is also taken of what happens to that second body.

This necessity is provided for by the principle due to Newton, which he states as LEX III: actioni contrariam semper et aequalem esse reactionem: sive corporum duorum actiones in se mutuo semper esse aequales et in partes contrarias dirigi. This, like the two preceding principles, cannot be demonstrated, but only illustrated and clarified with suitable examples. A body which, for whatever reason, exerts for a certain time on another body a force of a certain intensity and of a certain direction and sense is always inevitably subjected, for the same time, to a force of equal intensity and direction, but of opposite sense. Even when the force is exerted only for a very brief time and therefore its effect is reduced to an impulse on the body that undergoes it, the acting body inevitably receives an equal and opposite impulse. Let it be observed, to avoid misunderstandings, that the action and the corresponding reaction are always, as Newton clearly says, on two different bodies.

4. Developments of dynamics

The preceding principles are sufficient to solve conceptually any mechanical problem that may present itself in reality. The reservation is meant to signify that our ability to derive those solutions formally does not always correspond to that theoretical possibility. Hence from the time of Newton onward, many expressions perfectly equivalent to those principles were sought and discovered, but formally better suited to the practical resolution of the various types of dynamical problems. Among them, D'Alembert's principle, Lagrange's equations, Hamilton's principle with its various formal transformations, etc., are of the greatest importance. Given the impossibility of considering in detail these lofty formulations of dynamics and the ingenious criteria devised for their application to the various problems, which together constitute one of the most magnificent theoretical disciplines, analytical dynamics, let us consider only that the greatest difficulty that almost always arises is that of duly taking into account the so-called constraints. These are in general intimately bound up with the conditions to which the various masses participating in the phenomenon may be subjected, conditions that can sometimes be extremely complex and possibly even variable with time. For example, a railway car is generally constrained by the condition of having to remain on its own rails; such a constraint, when the path is straight, can be considered practically nonexistent, but when it is curved it can give rise to even very considerable reactions because the car, endowed with a certain velocity, will tend by its inertia to maintain it and to proceed in a straight line, whereas one of the rails will instead force it onto a curved path by exerting on it (through the wheels) an action, against which the car will react, by Newton's principle, with an equal and opposite reaction on the rail. These and other more complex constraint reactions, exactly predicted by the III principle, and of which it may be difficult to take account solely on the basis of the fundamental principles, can instead be taken into account almost automatically by the aforementioned analytical formulations, apart naturally from any difficulties of calculation.

A further necessary observation is the following. Although it is by now known that every motion in practice always encounters various passive resistances that tend to annul it, for the sake of simplicity, in what precedes, we have tacitly abstracted from them. However, even their consideration, which can be necessary in various important cases, can in principle be provided for by the II and III laws of dynamics; but a more practical consideration of them is still reached through the aforementioned analytical dynamics.

Let us recall finally that the expression dynamic equilibrium is often used, which evidently does not coincide with the well-known static equilibrium. Evidently a body assimilable to a material point is in static equilibrium when it is acted upon by two equal and opposite forces; a stationary balance is in static equilibrium, because then the resultant of all the parallel forces (weights) acting on it and directed vertically downward, passing through the fulcrum, is balanced by the equal and opposite reaction of its support. On the other hand, a train moving in a straight line with uniform velocity will be said to be in dynamic equilibrium, for example, because evidently in that case the propulsive force of the locomotive and the resultant of all the passive resistances balance each other, as do its weight and the reaction of the rails, so that the train moves, in accordance with the I principle, as if no force acted upon it.

IV. DYNAMICS OF MATERIAL POINTS

The particular case of the dynamics of bodies which, for whatever reason, can rigorously, or at least with great approximation, be assimilated to material points according to their previous conception, is very useful because it allows one to see the dynamical principles in action in the simplest possible form, as well as to establish certain concepts and to demonstrate certain laws consequent upon the fundamental principles which may later be used to advantage in their place. In other words, what happened here is what has happened since the most ancient times in the field of geometry; namely, that although its fundamental axioms and postulates contained all that was needed to resolve every geometrical question, it almost always proved more convenient in those treatments to have recourse to theorems that were a logical consequence of them and, in a certain sense, equivalent to them.

1. Motion of a free point

If no force acts on a free material point of mass m, then by the I principle its acceleration a will be zero, and its velocity v constant, therefore equal to the initial velocity v₀; its path s after a time t will be given by the equation s = s₀ + vt, where s₀ denotes its position at the initial time t = 0. But if a constant force F acts on the point, then by the I and II principles: its acceleration will be a = F/m, its velocity v = v₀ + F/m.t and its path at time t, s = s₀ + v₀t + 1/a.F/m.t². In the case where the force F has acted for the period of time t starting from the state of rest v = 0, the impulse theorem gives us Ft = mv, and therefore allows us to calculate the velocity v reached by the moving body without passing through the consideration of acceleration.

2. Uniform circular motion

Passing from the simplest case of the motion of a free point to the still particularly simple one of the motion of a point constrained to move uniformly along a circle of radius r, we observe: 1) that elementary kinematical considerations show that by the mere fact of circular rather than rectilinear motion, the point is subject to a continuous and constant acceleration toward the center, which is therefore called centripetal acceleration, whose value is calculated without difficulty as a = v²/r, where v denotes the velocity of the point on the said circle; 2) that by the II principle that centripetal acceleration can only be produced by a force acting on the point and at every instant directed radially toward the center of rotation, which we shall therefore denote by F; its magnitude, by the same principle, is given by F = ma = mv²/r, and this independently of any particular concerning its application to the point. The centripetal force may be constituted by the reaction of the constraint, as happens e.g. for the ball launched in a roulette, or it may be a force directly applied to the point, such as the tension of a thread stretched between the point and the center, or even a force of gravitational attraction such as that exerted by the Sun on a planet (v. GRAVITATION).

3. Centripetal force and centrifugal force

Already this very simple case of circular motion leads us to consider important concepts which by now also have importance in ordinary life, and which it is therefore necessary to understand well in order to avoid easy misunderstandings. Consider again the ball that moves in the circular groove of the roulette. It is continuously subjected by this to a centripetal force against which it reacts with an equal but oppositely directed force, which it applies to its successive points of contact with the groove. If for an instant this disappeared, the ball would escape along the tangent, proceeding with rectilinear and uniform motion, and, no longer subject to any force, it would naturally no longer in turn exert any reaction. Suppose now that one places an identical ball on a rotating horizontal platform. One would immediately observe that the ball, like any other material body, would be subject to a centrifugal force, which would tend to move it away from that center unless it were located exactly at it, until it encountered a support to which to apply its force, undergoing the consequent reaction in the opposite direction, and thus being able to remain fixed relative to the platform. This centrifugal force, which is also ascertained and undergone by anyone who participates in the rotation, is, for that person, a real force, whereas the same force does not exist at all for one who does not participate in it. In other words: a motor vehicle which, attempting a tight turn, skids on the ground, in the judgment of one who is on the vehicle has skidded as a result of its centrifugal force provoked by the turn and not sufficiently compensated by the friction between wheels and ground; in the judgment of one who is on the ground, the vehicle has skidded because the centripetal force automatically imparted to it by the friction between its wheels and the ground has not proved sufficient to overcome its natural tendency to continue its rectilinear motion by inertia (I principle). Every misunderstanding can however be avoided by imposing on oneself the rule of not speaking and treating of centrifugal forces except by placing oneself rigorously from the point of view of one who participates in the rotation, and vice versa of not speaking and treating of centripetal forces except from the point of view of one who does not participate in it.

4. Theorem of the conservation of momentum and of impulse

On the basis of the correlation between momentum and impulse of forces deduced previously one can arrive at other theorems of great utility. On the masses m₁ and m₂ let the external forces F₁ and F₂ act for a time Δt respectively. Applying the previous theorem separately and summing, one has (F₁ + F₂) Δt = Δ (m₁v₁ + m₂v₂), an equation which tells us that the sum of the impulse of the external forces is equal to the variation of the momentum of the masses. In the absence of external forces the variation of momentum will be zero, therefore the momentum itself must be constant. This holds for however many masses one wishes, so that one can affirm that the momentum of an isolated system does not vary, or as is often said is conserved. This is the first theorem or law of conservation which brings out a useful and general dynamical law undoubtedly virtually contained in the fundamental principles, but in a form very useful for applications. When external forces act, one naturally always has the more general impulse theorem.

5. Work and energy

A brief definition of very general notions, even when they appear to us fairly intuitive, is almost always difficult. For the notion of work Maxwell proposed the following: WORK is the act by which a modification is brought about in the configuration of a system against the forces that oppose such modification. If one considers that for the performance of work the necessary elements are therefore the modification of the system, which will generally reduce to one or more displacements, and the forces that oppose it, one will immediately understand that the usual definition WORK = Force × Displacement falls as a particular case within the more general Maxwellian definition, naturally taking into account that of the displacement only the component in the direction of the force must be considered, or vice versa the force in the direction of the displacement. Bearing in mind that both the force and the displacement are vector quantities while work is a scalar quantity, it will suffice to consider that product as the scalar product of vector theory, equivalently the product of the magnitudes of the force and of the displacement by the cosine of the angle between their directions: L = F.s.cos (F.s). Since cos (F.s) can take positive or negative values depending on the directions of F and of s, it can make the value of L positive or negative; but it is easy to realize that the two cases can conventionally be made to correspond to the two notions, full of physical meaning and intuitive, of driving work and resisting work. From the dimensions of force and of displacement there follow for work the dimensions: Work = |L| = |LT⁻² M| · |L| = L²T⁻² M) which always pertain, and in every field of physics, to all energies, independently of the particular forms in which they manifest themselves.

6. Work of acceleration. Living force

Let us now consider a force F acting on a mass m that can move freely without encountering frictional resistances. Then at every instant the application of the force will give rise to the corresponding acceleration a on the basis of the II principle, and its work will essentially be a so-called work of acceleration. For an infinitesimal displacement ds the corresponding work dL, according to what was said previously, will be dL = Fds = m a ds = m dv/dt.ds = m ds/dt.dv = m v dv. This equation shows that the work is positive, that is, driving, when the variation of velocity dv is positive, otherwise negative, that is, resisting; applied to a variation of velocity between zero and a given finite value v, as the elements of infinitesimal calculus teach, it takes the form: T = 1/2 m v². The second member of this equation is given the name (quite improper) of living force, while in fact it is one of the forms of mechanical energy, as also results from the fact that its physical dimensions are identical to those of work. The theorem therefore holds: the living force acquired by a body is equal to the work of the force that set it in motion.

7. Principle of the conservation of energy

The preceding theorem and various other considerations led thinkers, as early as the 18th century, to think that, when certain conditions occur, such as, for example, the absence of friction, the laws of dynamics imply that in the unfolding of its phenomena certain equivalences must occur between quantities that are sometimes even apparently heterogeneous, as if they were only different aspects of a single more general quantity that is conserved. But the necessity of the aforesaid restrictive conditions made them hesitant about the advisability of fully adopting such a conception. The situation was made even more complex by the fact that, in the very numerous cases in which those restrictive conditions could not be realized, experience revealed the concomitance of some thermal phenomenon with the dynamic phenomena under consideration. It was only around the middle of the 19th century, when the views of J. R. Mayer were generally adopted (v. TERMODINAMICA), that the question could be resolved.

Nowadays energy is considered generically as the physical quantity that is rigorously conserved through all its infinite transformations from one to another of its innumerable forms. Experience has largely verified, without a single exception, the reliability of this presumption, now elevated to the rank of a fundamental principle for all of physics. Heat is the form of energy into which all the others can be transformed, and in particular work and kinetic energy, generally through phenomena of friction. It has therefore been possible to establish experimentally the corresponding law of equivalence: one large calorie is equivalent to 427 kilogram-meters.

8. Dynamics and gravitation

Still without leaving the dynamics of material points, it is possible to give an account of several interesting phenomena that Newtonian gravitation can produce on them.

a) Motions near the earth's surface. - Let us consider a material point of mass m near the surface of the earth. Its weight is m g, denoting by g the acceleration of gravity, which there is approximately constant; it follows that the force tending to drag the point downward is likewise constant. The point left to itself therefore tends to fall with constant acceleration, that is, with uniformly accelerated velocity. Recalling therefore the corresponding preceding kinematic laws, one arrives directly at Galileo's law of free fall v = gt and s = 1/2 gt². If the point were constrained to remain on an inclined plane at an angle β to the horizontal plane, the point would fall along it as if the acceleration were reduced in proportion to sen β, and therefore the law of fall would be a = g.senβ, v = g.senβ.t, s = 1/2 g.senβ.t². This is the law which, verified experimentally by Galileo, led to the definitive refutation of Aristotelian dynamics. Similar considerations allowed the somewhat more general study of the motion of a material point launched, and therefore of projectiles.

Another example worthy of remembrance is that of the oscillations of a material point suspended from a thread of negligible mass, that is, of the so-called simple pendulum. Galileo, observing the classic lamp, had concluded that such oscillations must be isochronous and therefore that their duration must be a function only of the length of the thread l and of the local acceleration of gravity g. The law of that motion must naturally have followed from the first two principles known to Galileo; but difficulties of calculation did not permit him its exact determination, which C. Huyghens later achieved. Now the theory of physical dimensions, which has been repeatedly mentioned, allows one to conclude directly that, since time has the dimension T and with the length of dimension L and the acceleration of dimension (LT⁻²) one can compose only a single quantity of dimension homogeneous with a time, namely the quantity |½g⁻¹/₂|, whose dimensions are in fact |L½|·|LT⁻²|·½ = |T|, the law of the simple pendulum cannot but be of the form t = const. √{t/g}. The constant is naturally determined by a single experiment and is found to be equal to 2π.

b) The concept of potential energy. - The necessary consideration of this important form of energy is arrived at in the simplest way by applying the general principle of conservation to the phenomenon of the fall of masses in fields where the acceleration is constant, hence near the Earth's surface where g is approximately equal to 10 m/sec. - Consider a body of mass m fixed at height h and let it fall freely. From the preceding laws of free fall it is known that its velocity v at the end of the fall will be v = gt and that the time taken is related to h by the equation h = 1/2 gt². The vis viva acquired in the fall, that is its kinetic energy at the moment of its impact with the ground, is E = 1/2 m v² = 1/2 m gt², which is transformed into heat. But where was this energy before? Let the body of mass m be brought back to its original height h: since its weight is mg, the work required will be E = m g.h = m g.1/2 gt² = 1/2 m gt² = E. The idea therefore arises spontaneously of attributing to the body the energy of position E = mgh, which must be supplied to it in order to raise it to that height, an energy that is also called potential energy because it confers on it the power to acquire the said kinetic energy. The water accumulated in mountain reservoirs possesses the potential energy which, suitably transformed into electrical energy in the valleys below, is used in distant cities.

9. Celestial mechanics

Celestial bodies, despite their great dimensions, can, at least to a first approximation, be assimilated to simple material points by reason of their enormous distances. Only the forces resulting from their mutual gravitation according to Newton's law act upon them; and it is these that determine the character of their motions. The solar system is considered to be constituted precisely in this way, with its central point (the Sun) having a mass almost a thousand times greater than the combined mass of all the other material points (the planets). It is interesting to recall that the particular case of a large mass with a single planet would correspond to a motion of rotation of the latter on an elliptical orbit satisfying the well-known laws that J. Kepler (1571-1630) had enunciated about a century before the discovery of universal gravitation, as a law that (approximately) each planet of the solar system followed. Modern celestial mechanics too considers the stars predominantly as point-like, except for introducing as corrections the possible consequences of some phenomenon incompatible with the concept of a point, such as, for example, rotations.

V. STATICS AND DYNAMICS OF SOLID BODIES

Basing ourselves on the modern scientific definitions of a solid body, which transcend the limits of ordinary mechanics and imply notions from the theory of the constitution of matter (v. QUANTI), we shall here consider the mechanics of solid bodies in the ordinary sense, that is, according to the intuition that we all have of them. Their masses, rather than being abstractly concentrated at points, are considered more realistically as distributed throughout the space they occupy. But under such conditions mechanical phenomena manifest themselves in general as much more complex, and their theoretical treatment is almost always extremely difficult, although it can always still be based on the fundamental principles of Galileo-Newton.

For the treatment of these more complex questions it is advisable to have recourse to suitable concepts, in part new, but more often natural extensions of others already considered in the mechanics of material points. Among the former, for example, we should recall the important concept of density (the ratio of mass to the volume occupied) at the various places within bodies, a concept that would have no meaning in the case of material points; and the related concept of specific weight when the body is located in a field of acceleration. As an example of the latter, one may consider the transition from the notion of the center of gravity of a system of points to that of the center of gravity of a body with masses distributed at any density; as also the transition from the notions of the various moments of point masses with respect to given points or given straight lines to the analogous moments of distributed masses.

This being premised, it is possible to give an account, for example, of the relatively simple problem of the motion of a body when several forces act upon it. It is known, on the basis of the laws for the composition and transposition of forces, that the acting system can always be reduced to a force acting on the center of gravity of the body and to a couple. It is convenient to consider the two simplest extreme cases: that the couple turns out to be zero and therefore only a force acts on the center of gravity, or that, conversely, the force turns out to be zero and therefore only the couple acts on the body. In the first case the body would move like a material point in which the entire mass of the body were concentrated, if those given forces acted upon it; that is, there would be a pure translation. In the second case, on the other hand, there would be a pure rotation, for whose better description the theory introduced a whole series of special concepts that lead to an interesting parallelism between the formulations of the phenomena of rotation and those of the phenomena of pure translation. For the phenomena of rotation it was therefore sufficient to refer, instead of to the usual masses, velocities and accelerations, to the quantities moment of inertia, angular velocity and angular acceleration, all of which can be defined without difficulty on the basis of the usual dynamical quantities. The moment of inertia of a body with respect to a given axis is nothing but the extension of the analogous concept for isolated points of mass m₁, m₂, ..., mₙ respectively situated at distances d₁, d₂, ..., dₙ from the axis, which is notoriously expressed as J = m₁ d₁² + m₂ d₂² + ..., mₙ dₙ². Integral calculus makes it possible to evaluate these moments of inertia for any body with respect to any axis. By angular velocity is meant the angle traversed, in the unit of time, by the radius vector perpendicular to the axis of any point of the solid; it is denoted by ω and is related to the intuitive notion of revolutions per second n by the equation ω = 2πn. The angular acceleration η is nothing other than the variation of ω over time. Once these quantities have been introduced, it is immediately seen that the moment of inertia J appears in rotations as the analogue of the mass w in translations, with properties of inertia with respect to changes in angular velocity analogous to those of masses with respect to changes in translational velocity, and that the following parallel formulas hold:

But in general, when forces are applied to a solid body, there will be neither a pure translational motion nor a pure rotational motion, but rather a motion in which the two preceding ones are in a certain sense concomitant and give rise to a numerous series of interesting phenomena, sometimes harmful, but sometimes also usable. By way of example, let us consider the case of a gyroscope, that is, of a flywheel rapidly rotating about its own axis. As long as this remains fixed, there is a phenomenon of pure rotation; but as soon as one attempts to modify the direction of the axis by imparting to it any slight rotation in a direction normal to the axis itself, the entire rotating system will react with another slight rotation of its own, normal both to the axis of the flywheel and to that of the preceding rotation. It is known that certain large ships use analogous gyroscopes with a vertical axis to counteract their possible rolling.

Static and dynamic equilibrium. A material point is in equilibrium when only forces whose resultant is zero act upon it: the equilibrium will be static if the point is fixed, dynamic if it is in motion (necessarily uniform). In the case of a solid body, analogous laws hold, suitably extended; for example, a ladder resting on the ground and against a wall is in static equilibrium because its weight, supposed transported parallel to itself from its center of gravity to the lower end, and the couple and hence the moment of rotation that this transport produces, are balanced by the reactions of the two supports. On the other hand, a flywheel rotating about its horizontal axis is in dynamic equilibrium when, disregarding friction, it is not subject to any impulse of rotation; in fact, then, since its weight is balanced by the reactions of the supports of the axis, it can move only by inertia, rotating with constant velocity.

For building constructions, the conditions of equilibrium of bodies (towers, columns, etc.) resting on the ground are of interest. Among other things, it is essential that the verticals dropped from the center of gravity of these bodies pass within the zone of their contact with the ground, in order to avoid the production of moments of rotation not compensated by the corresponding reactions. For the same reason a homogeneous sphere cannot be in equilibrium on an inclined plane, however slight the incline, because the said condition is not realizable there. Moreover, for the equilibrium to be stable it is necessary and sufficient that any small displacement of the bodies correspond to a raising of their center of gravity; if instead there is a lowering, the bodies are in unstable equilibrium and, once slightly moved, they will tend not to return to their previous position but rather to move further away from it; if finally the height of the center of gravity does not vary, as in the case of the sphere on a horizontal plane, there is neutral equilibrium.

VI. MECHANICS OF FLUIDS

The phenomena considered in this chapter are of the greatest importance, both theoretical and practical. They are still always mechanical phenomena, and therefore ultimately subject to the fundamental laws of Galileo-Newton; but their treatment must be developed taking into account the particular characteristic of the bodies considered, that is, their extension and fluidity, just as previously account was taken of the extension and solidity of solid bodies.

In order to set up this study it is not necessary to descend to the hypothesis of the molecular and atomic constitution of matter; it is enough to admit that every tiny element of fluid matter is not rigidly connected with the surrounding elements and that therefore it can move, more or less freely, with respect to them. Moreover, simple cursory observation of phenomena had led, already from the most remote times, to a fairly exact consideration of that great category of fluids which, like solid bodies, offer a very great resistance to any variation of their volume, while they are arbitrarily and easily deformable (e.g. by means of simple pouring from one vessel into another of different shape), fluids that today we briefly call liquids. On the other hand, a similar knowledge of fluids which, besides being deformable, are also easily compressible and also have a clear tendency to expand and therefore to occupy entirely the spaces offered to them, fluids that we now call aeriform, could not be attained, because of inveterate prejudices, before the establishment of Galileo's school.

Knowledge of the mechanics of fluid bodies, of evident theoretical interest, is also very important from the point of view of its by now innumerable practical applications. It suffices to recall that all hydraulic and naval constructions are essentially based on the laws of the mechanics of liquids and that all pneumatic constructions, such as ventilation and compressed-air installations, and all aeronautical constructions are essentially based on the laws of the mechanics of aeriforms. It cannot therefore be surprising that for the exposition of the mechanics of fluids very different paths have been followed and are still followed, particularly suited to the various applications. But given the nature and the narrow limits of this exposition, we shall follow the path that seems simplest and most expressive, keeping distinct the remarks on the statics and on the dynamics of fluids and recalling for each of them first the most general notions and laws, which hold for both categories of fluids, and then, complementarily, the more particular ones proper to liquids or to aeriforms.

1. Statics of fluids - Considering any fluid in a vessel one arrives quite spontaneously at the notion of the pressure that it exerts on the walls. This pressure, in general defined as a force per unit area, can be measured in the case in question by making, e.g., a small hole in the wall and determining what minimum force must be exerted on a plug applied externally to the hole so that the fluid does not escape. That force (expressed, e.g., in kilograms) divided by the area of the hole (expressed, e.g., in square centimeters) will give the pressure of the fluid at that place on the wall, expressed in kilograms per square centimeter (Kg/cm²). Experience shows that in every fluid this pressure is constant at equal heights. Evidently this pressure is not exerted only on the walls, but everywhere within the fluid; it is therefore a pressure in the fluid itself. Moreover it is easy to realize that the differences at various heights are due to the gravitational field of the earth in the vicinity of its surface and that therefore, denoting by p₀ the pressure of the fluid at the level assumed as zero level, its pressure in the same vessel, at the level h, must be equal to that at the zero level, increased by the weight of the overlying column of fluid, therefore, denoting by p the density of the fluid and by g the acceleration of gravity, p = p₀ + p g h. Evidently if the gravitational field were absent the pressure would be uniform throughout the whole extent of the fluid (Pascal).

2. Complements relating to liquids

The consequences of the properties of all fluids just noted sometimes have different forms for liquids and for aeriforms.

The preceding law on the distribution of pressures has without further ado the immediate consequence that the maximum level of a liquid at rest in communicating vessels is always equal, independently of their particular shapes. Likewise on the uniformity of pressure (apart from the differences due to height differences that are almost always negligible) throughout the whole liquid mass enclosed in two communicating vessels is based the construction of the by now very numerous types of hydraulic compressors capable of developing practically unlimited forces by simple multiplication of even very slight efforts.

Another important and immediate consequence of the behavior of pressure in fluids is the famous law discovered by Archimedes: any solid body immersed in a fluid at rest loses as much of its own weight as is the weight of the fluid that it has displaced; to it, because of its great importance, one customarily gives the improper name of principle rather than the one that would be its due, of theorem. Even today, after almost sixteen centuries, this law is resorted to in order to determine by means of a simple weighing the specific weight of solid bodies when one has available a liquid of known density, or conversely to determine the density of a liquid when one has available a solid body (sample) of known specific weight.

3. Complements relating to aeriforms

The fact of the compressibility and above all of the expansibility of bodies in the aeriform state makes the study of their particular laws of equilibrium and of their applications more varied, but also more difficult. Almost always aeriform fluids must be considered placed in closed vessels, in which they undoubtedly have pressures not only subject to the preceding general laws, but also dependent on their volumes. Because of these difficulties, but above all because of a series of strange prejudices over which for brevity we must pass, ideas on the matter began to become clear only from the 17th century, after Galileo had pointed out that air (and therefore aeriforms) had a weight like all other material bodies, and after his school, especially through the work of E. Torricelli, the inventor of the barometer, had clarified that the mercury in barometric tubes and the water in the tubes of suction pumps rose not because of the horror vacui of ancient physics, but because of the pressure of atmospheric air; the one and the other, however, rose only up to a determined height strictly linked to that pressure.

The relation between the volume of an aeriform (in particular of air) and its pressure, by now known as Mariotte's law or Boyle's law, was discovered independently by the said physicists shortly after 1660. Formally it is expressed, for a given quantity of air maintained at the same temperature, by the well-known formula p.w = const, where p denotes the pressure and v the volume occupied by that quantity of air. Today it is known that the law, substantially valid for all aeriforms, is not rigorously exact, but only approximate; the more approximate the lower the pressure. Aeriforms that are found under the conditions for which this law is valid are briefly called perfect.

4. Dynamics of fluids

The motion of fluids in reality is almost always very complex and practically unobservable in its details. To be convinced of this, one need only glance at the course of a river or even just a small stream. But there is no lack of cases in which the motion appears notably regular, such as the slow overflow of water from a wide basin in which it has had time to settle, the tranquil vertical fall of a thread (vein) of water from a suitably regulated tap, etc. In these cases the flowing mass appears tranquil and transparent, almost as if it were still, whereas in others it appears disturbed and diaphanous. By slowly injecting, without disturbing the course of the flowing mass, a little of the same fluid previously colored, one sees that, in the particular cases mentioned above, it then proceeds in continuous filaments without mixing with the rest of the fluid, as instead happens in general. Contiguous filaments constitute sheets, whence the name laminar for that regular motion. More often, however, there occur tubulent motions. In the study of numerous cases of fluid motions, motions of fluids in conduits of every type, of objects immersed in fluids, of objects floating on liquids, etc., it is necessary to take due account of the possible laminarity or turbulence of the motion, as many and profound recent studies have demonstrated. In any case it may be affirmed that the present astonishing results of aeronautics are, at least in large part, a consequence of the deepening of the questions of the motion of air according to the views outlined above.

The first and most elementary laws of the motion of fluids, immediate consequences of the notions of pressure and of the fundamental laws of dynamics, date from the second half of the 17th century. Torricelli gave the law of the velocity v of efflux of water from a hole in a vessel as a function of the height h of the free surface above the said hole, in the simple form v = √[3]{2gh}, independently of the direction of efflux. A few decades later D. Bernoulli gave the law of the motion of fluids in pipes which is still used today for both liquids and aeriforms.

In the 18th century, then, especially through the work of D'Alembert, a complete formal theory of the motion of fluids was constituted on the basis of the notions considered up to that time. But while in several cases it corresponded to reality, in one essential point it was in complete contrast with it. Everyone knows that any body encounters a resistance when it moves in a fluid or, what is equivalent, that it undergoes a thrust when a fluid strikes it. A parachute falling in the air encounters a resistance that greatly limits its speed; a tree can be shattered by the wind that strikes it. But D'Alembert's theory, although fully based on the fundamental dynamic laws, reached the conclusion that neither the said resistances nor the said thrusts should exist. And this is what was called and is still called the paradox of D'Alembert. In other words, the theory excluded among other things every possibility of sustaining forces and therefore of flight, whereas birds have always flown and since 1903 mechanical devices have been flying.

The introduction into the theory of the consideration of the effects of the viscosity of the fluid, at first neglected to avoid serious formal complications, did not eliminate but aggravated the difficulty, because instead of predicting useful sustaining forces it led only to the prediction of even more deleterious passive resistances. Finally, in the second decade of this century the situation was better understood: as long as the fluid can be considered in laminar motion relative to the obstacle, both in the case of pure translational motion and in that of pure rotational motion, the gaps between theoretical predictions and reality are fairly slight; they are instead serious when, as almost always occurs, the fluid is in turbulent motion, or even when many laminar translational and rotational motions are superimposed, because such a superposition gives rise to a strong thrust in the direction perpendicular to the translation, a thrust which, by a suitable choice of the direction of rotation, can be sustaining.

Once this was understood, the most important problem for the improvement of flight was that of the search for the shapes and other characteristics of wings which, by enhancing the production of suitable circulations around them, bring about ever greater sustaining forces. And thus one arrived at the imposing payloads of modern flying machines. More generally, one also arrived at perfecting the shapes of all mechanical devices that must move rapidly in the air, not excluding locomotives and automobiles, to which the so-called aerodynamic profile is now generally given, which reduces to a minimum the passive resistances they encounter.

Also in the field of the motion of liquids, a better knowledge of the fundamental phenomena led to notable improvements in the construction of the numerous types of hydraulic machines.

Paolo Straneo