Mechanics

MECHANICS. - Mechanics aims to study the phenomena of material bodies that occur without altering either their physical or chemical nature. It will therefore examine, 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 unchanged. It will not, however, concern itself with its possible evaporation or transformation into water vapor, nor even less with its possible decomposition into its constituent elements, oxygen and hydrogen. From the earliest times, it was recognized that mechanical phenomena involve relationships between quantities pertaining to motion, forces, and some intrinsic quality of the material bodies participating in them. It is therefore useful, before fully addressing the problem of determining those relationships, to conduct a preliminary generic study of both motion (kinematics or geometry of movement) and 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 strict correlation with time or, as is often said, as a function of time.

The study of possible acts of motion is the primary purpose of kinematics, which thereby also justifies its designation as the geometry of motion. As is intuitive, to follow and, if necessary, to formally express acts of motion, one must resort to some reference system, which can generally be chosen with considerable freedom. Yet at this point, an essential difference between kinematics and geometry already emerges. The latter always selects the reference system it intends to adopt from among infinite assumed systems, even when 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 simply referred to a Cartesian coordinate system whose axes are formed by three edges of the walls converging at one of the vertices, or to any polar coordinate system with its pole at that vertex. In any case, the position of the point is always understood relative to the chosen reference system. Now, if one supposes that the point is in a certain motion with respect to the chosen reference system, it will be equally so with respect to all others. In the case just considered, the motion would therefore have an absolute character. However, once the notion of motion is introduced, it is logical to apply it also to reference systems and thus to consider reference systems in relative motion with respect to one another. In this case, however, the notion of a body's motion loses any absolute meaning and remains a notion relative to the chosen reference system, which must always be indicated unless it can be implied 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 immediately oppose another: fixed with respect to what? At this point, however, it suffices to recall the evolution of the notion of a fixed body to understand that no answer is possible. For a very long period, the Earth was believed to be fixed, then the Sun was considered fixed; subsequently, new astronomical observations demonstrated that even the Sun, with its entire planetary system, moves at a speed of no less than 10 km/sec toward the distant constellation of Hercules, and that even the immense galactic system is itself in motion relative to the most distant stars, which no one now dares to suppose fixed. Optical wave theories and Maxwell's electromagnetic theory also presupposed the existence of a fixed ether, but here too, experience demonstrated the absolute untenability of that hypothesis (v. RELATIVISMO). In conclusion: physics, and kinematics in particular, must henceforth refrain from making 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 frequently employed. This is an abstract notion, insufficient to completely define the position of a real body, because it disregards any consideration of orientation, which has no meaning for a point but has a well-defined 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 at sea is generally indicated by means of its longitude and latitude, disregarding its orientation. Therefore, physics often considers the bodies involved in its considerations as concentrated in points, sometimes attributing to them also some non-contradictory material property, and distinguishing them from purely geometric points with the designation of material points. In these cases, the notion of the motion of a body reduces to that of the succession of points in space occupied by that representative point as time passes, points in space that cannot be isolated but must constitute a line, the trajectory of the moving point. This may be straight or curved. However, the mere knowledge of the trajectory is not yet sufficient to completely define the motion of the point. It is evidently 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 graphic and analytic forms. An essential element in the consideration of motions is velocity. By now, we have a fairly precise notion of this because modern life continually puts us in contact with the most varied velocities. Ultimately, the velocities of any object are always expressed by the ratio of a distance traveled l to the time taken t: v = l/t. This leads to attributing to velocity the physical dimension v = LT⁻¹. However, a point may move with constant velocity along its entire trajectory or also with variable velocity from place to place. In the first case, the length of the path s for any time t measured from the instant when 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, to better describe them, it is useful to introduce the very useful notion of acceleration a, that is, of the increase in velocity per unit of time, thus a = Δv/Δt, where Δv denotes the change in velocity occurring in the time interval Δt. Consequently, acceleration is attributed the physical dimension a = vT⁻¹, and taking into account the dimension of v, a = LT⁻². Here too, as previously for velocity, acceleration may be constant along an entire path, and then one has motion with constant acceleration, generally called uniformly accelerated motion; but it may also vary from place to place, and then one has variably accelerated motion. In perfect analogy to what was said earlier, in the case of constant acceleration, the variable velocity at any instant will be given by v = v₀ + at, where v₀ denotes the velocity at the initial instant. In the case of decelerated motion, that is, of a decrease in velocity, acceleration is assigned a negative sign. At this point, it is interesting to ask what the expression for the path s of the material point is in the case of uniformly accelerated motion. Simple mathematical considerations lead to the expression s = s₀ + v₀t + ½at², where s₀ and v₀ are, respectively, the position and velocity at the initial instant. This expression, as will be seen later, has considerable importance in the history of mechanics.

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

II. GEOMETRY OF FORCES

Everyone also possesses a sufficient intuition of the notion of force to allow for its initial partial analysis. Even the most familiar force, that which manifests as the weight of material bodies, was recognized from remote times as varying from body to body, or more precisely, as possessing its own intensity, direction, and sense, the latter two a priori respectively vertical and directed downward, though capable of being deviated by suitable devices. For example, the force required to lift a bucket of water clearly has an intensity equal to their combined weight, and normally a vertical direction and downward sense; but with the device of a rope and pulley applied to wells, direction and sense can be arbitrarily deviated, while the intensity remains the same. From the evident fact that the effect on a taut, rectilinear rope of a force of equal direction is independent of the point of application, it was concluded that forces can in general be arbitrarily displaced along their line of action.

Composition of forces. — As early as the 13th century, G. Nemorario had qualitatively recognized the possibility of the decomposition of forces, considering the case of the downward vertical 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. Despite this recognition, however, the assertion of an analogous possibility of composition of forces was not dared. 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, elevated himself to the general notion of composition and decomposition of forces. On the basis of this notion and its law, later formulated with the well-known rule of the parallelogram, all the most interesting problems concerning forces in themselves were resolved, from Galileo onward. Considering first coplanar forces, that is, those all having directions lying on the same plane, it should be recalled that any number of them can in general still be composed into a single resultant on that plane, except when, in the penultimate operation of this attempted composition, two forces of equal intensity F, parallel, of opposite sense, and having lines of action at any non-zero distance d result, because in such a case any further composition is impossible. In other words, the particular system just described represents a new physical magnitude, which is customarily called a couple. A simple analysis demonstrates 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 assumed thus far), a couple always originates. That said, 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 (reducible) to a force and a couple.

III. GENERAL DYNAMICS

Of the three elements that, through their mutual relations, determine the laws of the mechanics of material bodies—motion, forces (considered extrinsic to bodies), and that quid which is instead intrinsic to them—only the first two could be studied individually beforehand due to their easy intuition, giving rise to the so-called geometries of movement and forces. The third element, however, entirely implicit in the theory of motion itself, allowed no reliable preliminary study, also because in the broad field of the mechanics of material bodies, as will be seen, there exist various quantities that individually could represent the aforementioned intrinsic quid. This is perhaps the most recondite reason which, compounded by the lack—or worse, the fallacy—of easy intuitions, made it so difficult to extract from the observation of mechanical phenomena a coherent system of laws capable of interpreting them all.

1. The Principle of Inertia

The first real result in the various attempts to extract the fundamental laws of mechanics was the attainment of the notion of the inertia of material bodies and the formulation of its law, first by Leonardo da Vinci and then, independently of him, by Galileo. In Newton’s subsequent systematic formulation of the fundamental laws of mechanics, this constitutes LEX I: «Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur illum mutare». Undoubtedly, the ignorance of this fundamental basis was the primary cause of Aristotle’s erroneous interpretation of many mechanical phenomena, particularly that of the fall of material bodies near the Earth’s surface, which, as will be seen, later became the touchstone between Aristotelian and Galilean views.

At this point, it is worth noting that it was precisely the founders of the experimental method who elevated to the rank of a fundamental principle a phenomenon that, within the realm of our experimental possibilities, can never be fully realized and can only be recognized as a limiting case, since we cannot observe an act of motion absolutely free from the action of any external force. The proponents of the contrary idea—that bodies could not remain in motion unless under the action of some force—could delude themselves into believing they were in agreement with daily experience, which shows that every body not kept in motion by observable external forces comes to rest.

2. The Fundamental Principle of Dynamics

Having established the law of 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 even here, our pure intuition is insufficient to address the question, and it appears necessary to resort to properly understood and rigorously interpreted experience. The first question that arises is whether it is permissible to speak of the action of a force on a given body, independently of the state of motion in which it may find itself. The question is extremely delicate because, since direct experience is impossible, it must be resolved indirectly and also through abstractions and idealizations to be conducted with great caution. Only Galileo succeeded in posing this question in the proper terms and resolving it, summarizing its essence in the principle that a force acts on a body in the same manner always, whether the body is in motion or subject to the action of other forces. Only on the basis of this principle can experience with bodies initially at rest relative to the observer be conclusive. The law that followed from Galileo’s experimental results was formulated by Newton as LEX II: «Mutationem motus proportionalem esse vi motrici impressae et fieri secundum lineam rectam qua vis illa imprimatur».

It should now be noted that the asserted proportionality between force and mutatio motus (which is nothing other than a positive or negative acceleration) refers to the effects of various forces on the same body: in other words, the ratio between the impressed force F and the resulting acceleration a is a constant of the body considered. But since the accelerations obtained by applying equal forces to different bodies also differ from one another—or, equivalently, since different bodies offer different resistances to the variation of their state of motion—the idea naturally arises of ascribing to each body its own inertial or dynamic mass m. This is precisely what Newton did, further assuming as its measure, without further qualification, the aforementioned ratio F/a; that is, expressing his second law in the form F = m·a.

This fundamental law can also be given various other forms, very useful in different cases. Observing that the (average) acceleration during a time interval Δt can be written as 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, i.e., the impulse of that force, and that of the product of a mass by its own velocity, i.e., the quantity of motion (or momentum) of that mass (or of the body possessing it). The equation then states 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 it is said: the variation of the momentum of a body is always equal to the impulse that caused II. Again, dividing by Δt, the same equation tells us: the variation of momentum per unit time is equal to the acting force. All these are forms of the fundamental law that may be useful.

The mass thus introduced is eminently of a physical 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, taking into account the dimensions of acceleration, follows the dimensional characterization of force: F = LT⁻²M.

3. The Principle of Action and Reaction

The two preceding principles are already sufficient to satisfactorily resolve numerous dynamic questions, but not all, and in any case never in their fullness. Indeed, to apply a force to a body, it is necessary to employ another body. And then, one cannot claim to know a dynamic phenomenon completely unless account is also taken of what occurs on that second body.

This necessity is addressed by the principle due to Newton, which he enunciates 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». Like the two preceding principles, this cannot be demonstrated but only illustrated and clarified with appropriate examples. A body that, for any reason, exerts a force of a certain intensity and direction on another body for a certain time inevitably finds itself subjected, for the same time, to a force of equal intensity and direction but opposite sense. Even when the force is exerted only for a very short time, and thus its effect reduces to an impulse on the body that undergoes it, the acting body inevitably receives an equal and opposite impulse. To avoid misunderstandings, it should be noted that the action and the corresponding reaction are always, as Newton clearly states, on two different bodies.

4. Developments in Dynamics

The foregoing principles are sufficient to resolve, in conceptual terms, any mechanical problem that may arise in reality. The caveat signifies that this theoretical possibility does not always correspond to our ability to formally derive those solutions. Thus, from the time of Newton onward,

many expressions perfectly equivalent to those principles but formally more suited to the practical resolution of various types of dynamic problems were sought and discovered. Among these, the principle of D'Alembert, 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 elevated formulations of dynamics and the ingenious criteria devised for their application to various problems—which collectively constitute one of the most grandiose theoretical disciplines, analytical dynamics—it suffices to note that the greatest difficulty almost always encountered is that of duly accounting for so-called constraints. These are generally intimately linked to the conditions to which the various masses participating in the phenomenon may be subjected, conditions that can sometimes be extremely complex and even variable over time. For example, a railway carriage is generally constrained by the condition that it must remain on its tracks; when the path is straight, this constraint may be considered practically nonexistent, but when it is curved, it can provoke even considerable reactions because the carriage, possessing a certain velocity, due to its inertia, will tend to maintain it and proceed in a straight line, while one of the rails will instead force it into a curved path, exerting upon it (through the wheels) an action, against which the carriage will react, by Newton’s principle, with an equal and opposite reaction on the rail. Of this or other more complex constraint reactions, precisely foreseen by the third principle and which may be difficult to account for based solely on the fundamental principles, the aforementioned analytical formulations can instead account almost automatically, save, of course, any computational difficulties.

A further necessary observation is the following. Although it is now known that every motion in practice always encounters various passive resistances that tend to nullify it, for simplicity, in the foregoing, these have been tacitly abstracted. However, even their consideration, which may be necessary in various important cases, can in principle be addressed by the second and third laws of dynamics; but a more practical approach to their consideration is again achieved through the aforementioned analytical dynamics.

Finally, we recall that the expression dynamic equilibrium is frequently used, which evidently does not coincide with the well-known static equilibrium. A body assimilable to a material point is evidently in static equilibrium when acted upon by two equal and opposite forces; a stationary balance is in static equilibrium because the resultant of all the parallel forces (weights) acting upon 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 is said to be in dynamic equilibrium because, in this case, the propulsive force of the locomotive and the resultant of all passive resistances are in equilibrium, as are its weight and the reaction of the rails, so that the train moves, according to the first principle, as if no force were acting upon II.

IV. DYNAMICS OF MATERIAL POINTS

The particular case of the dynamics of bodies that, for any reason, may be rigorously—or at least with great approximation—assimilated to material points according to their preceding conception, is highly useful because it allows the dynamic principles to be observed in their simplest possible form, as well as to establish certain concepts and demonstrate certain laws derived from the fundamental principles, which may subsequently be advantageously employed in their stead. In other words, what occurred in geometry since ancient times is verified here: namely, that although its fundamental axioms and postulates contained all that was necessary to resolve every geometric question, it was almost always more convenient to resort, for those treatments, to theorems that were their logical consequence and, in a certain sense, equivalent to them.

5. Motion of a Free Point

If no force acts upon a free material point of mass m, by the first principle its acceleration a will be null, and its velocity v constant, thus 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 upon the point, by the first and second principles, its acceleration will be a = F/m, its velocity v = v₀ + F/m·t, and its path at time t, s = s₀ + v₀t + ½·F/m·t². In the case where the force F has acted for a time t starting from a state of rest (v = 0), the impulse theorem gives Ft = mv, thus allowing the velocity v reached by the moving body to be calculated without passing through the consideration of acceleration.

6. Uniform Circular Motion

Moving from the simplest case of the motion of a free point to the still particularly simple case of the motion of a point constrained to move uniformly on a circumference of radius r, we observe: 1) that elementary kinematic considerations show that, by the mere fact of circular motion 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 readily calculated as a = v²/r, where v denotes the velocity of the point on the said circumference; 2) that by the second principle, that centripetal acceleration can only be produced by a force acting upon 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, independently of any particular circumstance regarding its application to the point. The centripetal force may consist of the reaction of the constraint, as occurs, for example, with a ball launched in a roulette wheel, 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 gravitational force of attraction, such as that exerted by the Sun on a planet (v. GRAVITAZIONE).

7. Centripetal Force and Centrifugal Force

Even this simplest case of circular motion leads to the consideration of important concepts that now also hold significance in ordinary life, and which it is therefore necessary to understand well in order to avoid easy misunderstandings. Consider again the ball moving in the circular groove of the roulette wheel. It is continuously subjected by the latter to a centripetal force, against which it reacts with an equal but opposite force, applied to its successive points of contact with the groove. If, for an instant, the groove were to disappear, the ball would escape along the tangent, proceeding with uniform rectilinear motion, and, no longer subject to any force, would naturally no longer exert any reaction in turn. Now suppose one places an identical ball on a rotating horizontal platform. It would immediately be observed 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 precisely at the center, until it encountered a support to which it could apply its force, undergoing the consequent opposite reaction, and thus remain fixed relative to the platform. This centrifugal force, which one also experiences and undergoes when participating 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 other words: a motor vehicle that, attempting a sharp turn, skids on the ground, in the judgment of one aboard the vehicle, has skidded due to its centrifugal force caused by the turn and not sufficiently compensated by the friction between wheels and ground; in the judgment of one standing on the ground, the vehicle has skidded because the centripetal force automatically impressed upon it by the friction between its wheels and the ground was insufficient to overcome its natural tendency to continue its rectilinear motion by inertia (first principle). Every misunderstanding can, however, be avoided by imposing the rule of speaking and treating of centrifugal forces only when rigorously adopting the point of view of one participating in the rotation, and conversely, of speaking and treating of centripetal forces only from the point of view of one who does not participate.

8. Theorem of the Conservation of Momentum and Impulse

Based on the correlation between momentum and impulse of forces previously deduced, one may arrive at other highly useful theorems. Let external forces F₁ and F₂ act upon masses m₁ and m₂ for a time Δt. Applying the preceding theorem separately and summing, one obtains (F₁ + F₂) Δt = Δ (m₁v₁ + m₂v₂), an equation stating 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 null, and thus the momentum itself must be constant. This holds for any number of masses, so one may affirm that the momentum of an isolated system does not vary, or, as is often said, is conserved. This is the first conservation theorem or law, which highlights a useful and general dynamic law undoubtedly virtually contained in the fundamental principles, but in a form highly advantageous for applications. When external forces act, one naturally always has the more general impulse theorem.

9. Work and Energy

Providing a concise definition of highly general notions, even when they appear quite intuitive, is almost always challenging. For the concept of work, Maxwell proposed the following: WORK is the act by which a modification in the configuration of a system is realized against the forces opposing such modification. Considering that the execution of work thus requires both the modification of the system—which generally reduces to one or more displacements—and the forces opposing it, one immediately understands that the common definition WORK = Force × Displacement is a particular case of Maxwell’s more general definition, naturally accounting for the fact that only the component of displacement in the direction of the force (or vice versa, the force in the direction of displacement) must be considered. Bearing in mind that both force and displacement are vector quantities while work is a scalar quantity, it suffices to regard this product as the scalar product of vector theory, equivalently expressed as the product of the magnitudes of force and displacement multiplied by the cosine of the angle between their directions: L = F·s·cos(F,s). Since cos(F,s) can assume positive or negative values depending on the directions of F and s, the value of L may be positive or negative; however, it is easy to recognize that these two cases correspond conventionally to the physically meaningful and intuitive notions of motive work and resistive work. From the dimensions of force and displacement, the dimensions of work follow as: Work = |L| = |LT⁻² M| · |L| = L²T⁻² M, which are always, in every field of physics, those of all forms of energy, regardless of their particular manifestations.

6. Acceleration Work. Vis Viva. - Consider now a force F acting on a mass m that can move freely without encountering frictional resistance. At every instant, the application of the force will produce the corresponding acceleration a according to the second principle, and its work will essentially be what is called acceleration work. For an infinitesimal displacement ds, the corresponding work dL, as previously stated, will be dL = F ds = m a ds = m dv/dt · ds = m ds/dt · dv = m v dv. This equation shows that the work is positive—that is, motive—when the variation in velocity dv is positive; otherwise, it is negative—that is, resistive. Applied to a variation in velocity from zero to a given finite value v, as taught by the elements of infinitesimal calculus, it takes the form: T = ½ m v². The right-hand side of this equation is given the (rather improper) name of vis viva, though it is actually one of the forms of mechanical energy, as is also evident from the fact that its physical dimensions are identical to those of work. The following theorem thus holds: the vis viva acquired by a body is equal to the work of the force that set it in motion.

10. Principle of the Conservation of Energy

The preceding theorem and various other considerations led thinkers as early as the 18th century to suppose that, under certain conditions—such as the absence of friction—the laws of dynamics imply that in the unfolding of phenomena, certain equivalences must occur among quantities sometimes even apparently heterogeneous, as if they were merely different aspects of a single, more general quantity that is conserved. However, the necessity of the aforementioned restrictive conditions made scholars hesitant to fully adopt such a conception. The situation was further complicated by the fact that, in the vast majority of cases where those restrictive conditions could not be met, experience revealed the concomitant occurrence of thermal phenomena alongside the dynamical phenomena under consideration. It was only around the mid-19th century, when the views of J. R. Mayer were generally adopted (v. TERMODINAMICA), that the question could be resolved.

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

11. Dynamics and Gravitation

Without yet leaving the dynamics of material points, it is possible to account for several interesting phenomena that Newtonian gravitation can produce upon them.

a) Motions near the Earth’s Surface. - Consider a material point of mass m near the Earth’s surface. Its weight is m g, where g denotes the acceleration due to gravity, which is approximately constant there; it follows that the force tending to draw the point downward is also constant. A point left to itself thus tends to fall with constant acceleration—that is, with uniformly accelerated velocity. Recalling the corresponding preceding kinematic laws, one immediately arrives at Galileo’s law of free fall: v = gt and s = ½ gt². If the point were constrained to remain on an inclined plane forming an angle β with the horizontal plane, the point would fall along it as if the acceleration were reduced in proportion to sin β, and thus the law of fall would be a = g·sin β, v = g·sin β·t, s = ½ g·sin β·t². This is the law that, experimentally verified by Galileo, led to the definitive refutation of Aristotelian dynamics. Analogous considerations allowed for the somewhat more general study of the motion of a material point projected and thus of projectiles.

Another noteworthy example is that of the oscillations of a material point suspended by a massless thread—that is, the so-called simple pendulum. Galileo, observing the classic lamp, had concluded that such oscillations must be isochronous and that their duration must therefore depend only on the length of the thread l and the local acceleration due to gravity g. The law of that motion naturally followed from the two first principles known to Galileo; however, computational difficulties prevented him from determining it exactly, a task later accomplished by C. Huyghens. Now, the theory of physical dimensions, to which repeated reference has been made, allows one to conclude immediately that since time has the dimension T, and length the dimension L, and acceleration the dimension LT⁻², one can compose only one quantity with dimensions homogeneous to time—namely, the quantity |½ g⁻¹/²|, whose dimensions are indeed |L½|·|LT⁻²|⁻½ = |T|—so that the law of the simple pendulum can only be of the form t = cost. √(l/g). The constant is naturally determined by a single experiment and is found to be equal to 2π.

b) The concept of potential energy. - Consideration of this important form of energy is most simply arrived at by applying the general principle of conservation to the phenomenon of falling masses in fields where acceleration is constant, thus in the vicinity of the Earth's surface where g is approximately equal to 10 m/sec². Let us 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 = ½gt². The vis viva acquired during the fall, that is, its kinetic energy at the moment of impact with the ground, is E = ½mv² = ½mgt², which is transformed into heat. But where was this energy before? Let us return the body of mass m to its original height h: since its weight is mg, the work required will be E = mgh = mg·½gt² = ½mgt² = E. The idea thus spontaneously arises to attribute to the body the positional energy E = mgh, which must be supplied to raise it to that height, an energy also called potential energy because it confers upon the body the capacity to acquire the aforementioned kinetic energy. The water accumulated in mountain reservoirs possesses potential energy which, appropriately transformed into electrical energy in the valleys below, is utilized in distant cities.

12. Celestial mechanics

Celestial bodies, despite their great dimensions, can, at least in a first approximation, be likened to simple material points due to 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 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 (planets). It is interesting to recall that the particular case of a large mass with a single planet would correspond to a rotational motion of the latter in 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 also predominantly considers the stars as point-like, except to introduce corrections for any consequences of phenomena incompatible with the concept of a point, such as, for example, rotations.

V. STATICS AND DYNAMICS OF SOLID BODIES

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

For the treatment of these more complex questions, it is convenient to resort to appropriate concepts, partly new but more often natural extensions of others already considered in the mechanics of material points. Among the former, for example, is the important concept of density (the ratio of mass to the volume occupied) at various locations in 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 in an acceleration field. 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; likewise, the transition from the notions of the various moments of point masses with respect to given points or lines to the analogous moments of distributed masses.

With this premise, it is possible to understand, for example, the relatively simple problem of the motion of a body when various forces act upon II. It is known, based on the laws for the composition and transfer 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 is null and thus only a force acts on the center of gravity, or that, conversely, the force is null and thus only the couple acts on the body. In the first case, the body would move as a material point in which the entire mass of the body were concentrated, when those given forces acted upon it; that is, there would be pure translation. In the second case, instead, there would be pure rotation, for whose better description the theory introduced a whole series of special concepts leading to an interesting parallelism between the formulations of rotational phenomena and those of pure translational phenomena. It was sufficient, for rotational phenomena, to refer not to masses, velocities, and customary accelerations, but instead to the quantities of moments of inertia, angular velocities, and angular accelerations, all definable without difficulty based on customary dynamic 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 allows the evaluation of 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. Angular acceleration η is nothing other than the variation in time of ω. Having introduced these quantities, it is immediately seen that the moment of inertia J presents itself in rotations as the analogue of 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, one does not have either pure translational motion or pure rotational motion, but rather a motion in which the two previous types are in a certain sense concomitant and give rise to a numerous series of interesting phenomena, sometimes harmful but sometimes also useful. By way of example, consider the case of a gyroscope, that is, a flywheel rapidly rotating around 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 normal to the axis itself, the entire rotating system will react with another slight rotation normal both to the axis of the flywheel and to that of the previous rotation. It is known that some 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 act upon it whose resultant is null: 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 leaning against the ground and a wall is in static equilibrium because its weight, assumed to be transferred parallel from its center of gravity to the lower end, and the couple and thus the rotational moment caused by said transfer, are balanced by the reactions of the two supports. Instead, a flywheel rotating around its horizontal axis is in dynamic equilibrium when, neglecting friction, it is not subject to any rotational impulse; in fact, then, since its weight is balanced by the reactions of the axis supports, 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 area of their contact with the ground, so as to avoid the production of rotational moments not compensated by corresponding reactions. For the same reason, a homogeneous sphere cannot be in equilibrium on an inclined plane, even slightly, because the aforementioned condition cannot be realized there. Moreover, for equilibrium to be stable, it is necessary and sufficient that any small displacement of the bodies corresponds to a raising of their center of gravity; if, instead, there is a lowering, the bodies are in unstable equilibrium and, once moved, will not tend to return to their previous position but rather to move further away; if, finally, the height of the center of gravity does not vary, as in the case of a sphere on a horizontal plane, there is indifferent equilibrium.

BIBL.:
Per la statica e dinamica dei corpi solidi, vedi razionale, p. es. P. Appell, Traité de mécanique rationnelle, Parigi 1909-1931; T. Levi-Civita e U. Amaldi, Lezioni di meccanica razionale, Bologna 1923-1927; G. Krall, Meccanica tecnica, Roma 1940. Per il giroscopio, V. GIROSCOPIO. Per la teoria della costituzione della materia, V. QUANTITÀ.

VI. MECHANICS OF FLUIDS

The phenomena considered in this chapter are of the greatest theoretical and practical importance. They remain, in the final analysis, mechanical phenomena and thus subject to the fundamental laws of Galilei-Newton; however, their treatment must take into account the particular characteristics of the bodies under consideration, namely their extension and fluidity, just as the extension and solidity of solid bodies were previously considered.

To undertake this study, it is not necessary to descend to the hypothesis of the molecular and atomic constitution of matter; it suffices to admit that every infinitesimal element of fluid matter is not rigidly connected to the surrounding elements and can therefore move, more or less freely, relative to them. Furthermore, simple summary observation of phenomena had already led, from the most remote times, to a fairly accurate consideration of that vast category of fluids which, like solid bodies, offer great resistance to any variation in their volume, while being arbitrarily and easily deformable (e.g., by simply pouring them from one container into another of different shape)—fluids that we now briefly call liquids. Conversely, an analogous understanding of fluids that, in addition to being deformable, are also easily compressible and have a marked tendency to expand and thus occupy entirely the spaces offered to them—fluids we now call aeriforms—could not be achieved due to inveterate prejudices before the establishment of Galileo’s school.

The knowledge of the mechanics of fluid bodies, of evident theoretical interest, is also highly important from the perspective of its now countless 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 systems, as well as all aeronautical constructions, are essentially based on the laws of the mechanics of aeriforms. It should therefore come as no surprise that the exposition of the mechanics of fluids has followed, and continues to follow, vastly different paths particularly suited to various applications. However, given the nature and limited scope of this exposition, we shall follow the path that appears simplest and most expressive, distinguishing between the outlines of the statics and dynamics of fluids, and recalling for each first the most general notions and laws applicable to both categories of fluids, and then, complementarily, those more particular to liquids or aeriforms.

1. Statics of fluids - Considering any fluid in a container, one quite naturally arrives at the notion of the pressure it exerts on the walls. This pressure, generally defined as a force per unit area, can be measured in the case at hand by, for example, making a small hole in the wall and determining the minimum force required to exert on a plug applied externally to the hole to prevent the fluid from escaping. That force (expressed, for example, in kilograms) divided by the area of the hole (expressed, for example, in square centimeters) will give the pressure of the fluid at that point 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 at every point within the fluid; it is therefore a pressure within the fluid itself. Furthermore, it is easy to realize that the differences at various heights are due to the Earth’s gravitational field near its surface, and that, therefore, denoting by p₀ the pressure of the fluid at the level taken as zero, its pressure in the same container at height h must equal that at zero level, increased by the weight of the overlying column of fluid; thus, denoting by ρ the density of the fluid and by g the acceleration due to gravity, p = p₀ + ρ g h. Evidently, where the gravitational field is absent, the pressure would be uniform throughout the fluid (Pascal).

13. Complements concerning liquids

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

The preceding law on the distribution of pressures has the immediate consequence that the maximum level of a liquid at rest in communicating vessels is always equal, regardless of their particular shapes. The principle of the uniformity of pressure (apart from differences due to height variations, which are almost always negligible) throughout the liquid mass enclosed in two communicating vessels underlies the construction of the now countless types of hydraulic presses capable of developing practically unlimited forces through the 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 the weight of the fluid it has displaced; due to its great importance, this law is customarily given the improper name of principle rather than the one it properly deserves, that of theorem. Even today, after nearly sixteen centuries, this law is used to determine, by means of a simple weighing, the specific weight of solid bodies when a liquid of known density is available, or conversely to determine the density of a liquid when a solid body (sample) of known specific weight is available.

14. Complements concerning aeriforms

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

The relationship between the volume of an aeriform (particularly air) and its pressure, now known as Mariotte’s law or Boyle’s law, was discovered independently by these physicists shortly after 1660. It is formally expressed, for a given quantity of air maintained at the same temperature, by the well-known formula p v = 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, and increasingly so as the pressure decreases. Aeriforms that are in conditions for which this law holds are briefly called perfect.

4. Fluid Dynamics. - The motion of fluids in reality is almost always highly complex and practically unobservable in detail. To be convinced of this, one need only glance at the course of a river or even a small stream. However, there are cases in which the motion appears notably regular, such as the slow overflow of water from a large basin where it has had time to settle, the tranquil vertical fall of a stream (vein) of water from a properly regulated tap, etc. In these cases, the flowing mass appears calm and transparent, almost as if it were still, whereas in others it appears turbulent and opaque. By slowly injecting, without disturbing the flow of the mass, a small amount of the same fluid previously dyed, one observes that, in the aforementioned particular cases, it then proceeds in continuous filaments without mixing with the rest of the fluid, as generally occurs instead. Contiguous filaments form laminae, hence the term laminar for that regular motion. More often, however, turbulent motions occur. In the study of numerous cases of fluid motion—motion of fluids in conduits of all types, objects immersed in fluids, objects floating on liquids, etc.—it is necessary to duly account for the possible laminarity or turbulence of the motion, as demonstrated by many recent and profound studies. In any case, it can be affirmed that the current astonishing results of aeronautics are, at least in large part, a consequence of the deepening of questions regarding the motion of air according to the aforementioned perspectives.

The first and most elementary laws of fluid motion, immediate consequences of the notions of pressure and the fundamental laws of dynamics, date back to the second half of the 17th century. Torricelli provided the law of the velocity v of water efflux from an orifice of a container as a function of the height h of the free surface above said orifice, in the simple form v = √(2*g**h*), independently of the direction of efflux. A few decades later, D. Bernoulli gave the law of fluid motion in pipes, which is still used today for both liquids and gases.

In the 18th century, particularly through the work of D'Alembert, a complete formal theory of fluid motion was established based on the notions considered up to that time. However, while it corresponded to reality in many cases, in one essential point it was in complete contrast with II. Everyone knows that any body encounters resistance when moving in a fluid or, equivalently, that it undergoes a thrust when a fluid strikes II. A parachute falling through the air encounters resistance that greatly limits its velocity; a tree can be shattered by the wind that strikes II. Yet D'Alembert's theory, though fully based on the fundamental laws of dynamics, reached the conclusion that neither such resistances nor such thrusts should exist. This is what was and still is called the D'Alembert paradox. In other words, the theory excluded, among other things, any possibility of sustaining forces and thus of flight, while birds have always flown and since 1903 mechanical devices have flown.

The introduction into the theory of the consideration of the effects of fluid viscosity, initially neglected to avoid serious formal complications, did not eliminate but rather aggravated the difficulty, because instead of leading to the prediction of useful sustaining forces, it only led to that 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 discrepancies between theoretical predictions and reality are fairly slight;

they are instead severe when, as almost always occurs, the fluid is in turbulent motion, or even when only many laminar translational and rotational motions overlap, because such a superposition gives rise to a strong thrust in the direction perpendicular to the translation, a thrust that, with an appropriate 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 researching the shapes and other characteristics of wings that, by enhancing the production of appropriate circulations around them, provoke ever greater sustaining forces. And thus, the impressive capacities of modern aircraft were achieved. More generally, the forms of all mechanical devices that must move rapidly through the air were also perfected, including locomotives and automobiles, to which the so-called aerodynamic profile is now generally given to minimize the passive resistances they encounter.

Even in the field of liquid motion, the better understanding of fundamental phenomena led to notable improvements in the construction of the numerous types of hydraulic machines.