1.2 Conceptos y principios fundamentales
Aunque el estudio de la mecánica se remonta a la época de Aristóteles (384-322 [a.c.]) y Arquímedes (287-212 [a.c.]), no fue hasta Newton (1642-1727) que se desarrolló una formulación satisfactoria de sus principios fundamentales. Estos principios fueron modificados posteriormente por d’Alembert, Lagrange y Hamilton. Su validez permaneció indiscutible hasta que Einstein formuló su teoría de la relatividad (1905). Aunque ya se han reconocido sus limitaciones, la mecánica newtoniana sigue siendo la base de las ciencias de la ingeniería actuales.
Los conceptos básicos de la mecánica son espacio, tiempo, masa y fuerza. Estos conceptos no pueden definirse con precisión; deben aceptarse con base en nuestra intuición y experiencia, y utilizarse como marco de referencia mental para nuestro estudio de la mecánica.
El concepto de espacio se asocia con la posición de un punto P. Podemos definir la posición de P proporcionando tres longitudes medidas desde un punto de referencia determinado, u origen, en tres direcciones dadas. Estas longitudes se conocen como las coordenadas de P.
Para definir un evento, no basta con indicar su posición en el espacio. También es necesario especificar el tiempo del evento.
We used the concept of mass to characterize and compare bodies on the basis of certain fundamental mechanical experiments. Two bodies of the same mass, for example, are attracted by the earth in the same manner; they also offer the same resistance to a change in translational motion.
A force represents the action of one body on another. A force can be exerted by actual contact, like a push or a pull, or at a distance, as in the case of gravitational or magnetic forces. A force is characterized by its point of application, its magnitude, and its direction; a force is represented by a vector (Sec. 2.1B).
In newtonian mechanics, space, time, and mass are absolute concepts that are independent of each other. (This is not true in relativistic mechanics, where the duration of an event depends upon its position and the mass of a body varies with its velocity.) On the other hand, the concept of force is not independent of the other three. Indeed, one of the fundamental principles of newtonian mechanics listed below is that the resultant force acting on a body is related to the mass of the body and to the manner in which its velocity varies with time.
In this text, you will study the conditions of rest or motion of particles and rigid bodies in terms of the four basic concepts we have introduced. By particle, we mean a very small amount of matter, which we assume occupies a single point in space. A rigid body consists of a large number of particles occupying fixed positions with respect to one another. The study of the mechanics of particles is therefore a prerequisite to that of rigid bodies. Besides, we can use the results obtained for a particle directly in a large number of problems dealing with the conditions of rest or motion of actual bodies.
The study of elementary mechanics rests on six fundamental principles, based on experimental evidence.
The Parallelogram Law for the Addition of Forces. Two forces acting on a particle may be replaced by a single force, called their resultant, obtained by drawing the diagonal of the parallelogram with sides equal to the given forces (Sec. 2.1A).
The Principle of Transmissibility. The conditions of equilibrium or of motion of a rigid body remain unchanged if a force acting at a given point of the rigid body is replaced by a force of the same magnitude and same direction, but acting at a different point, provided that the two forces have the same line of action (Sec. 3.1B).
Newton’s There Laws of Motion. Formulated by Sir Isaac Newton in the late seventeenth century, these laws can be stated as follows:
FIRST LAW. If the resultant force acting on a particle is zero, the particle remains at rest (if originally at rest) or moves with constant speed in a straight line (if originally in motion) (Sec. 2.3B).
SECOND LAW. If the resultant force acting on a particle is not zero, the particle has an acceleration proportional to the magnitude of the resultant and in the direction of this resultant force.
As you will see in Sec. 12.1, this law can be stated as
\[ F = ma \tag{1}\]
where \(F\), \(m\), and \(a\) represent, respectively, the resultant force acting on the particle, the mass of the particle, and the acceleration of the particle expressed in a consistent system of units.
THIRD LAW. The forces of action and reaction between bodies in contact have the same magnitude, same line of action, and opposite sense (Chap. 6, Introduction).
Newton’s Law of Gravitation. Two particles of mass \(M\) and \(m\) are mutually attracted with equal and opposite forces \(F\) and \(-F\) of magnitude \(F\) (Fig. 1.1), given by the formula
\[ F = G \frac{Mm}{r^2} \]
where \(r\) = the distance between the two particles and \(G\) = a universal constant called the constant of gravitation. Newton’s law of gravitation introduces the idea of an action exerted at a distance and extends the range of application of Newton’s third law: the action \(F\) and the reaction \(-F\) in Fig. 1.1 are equal and opposite, and they have the same line of action.
A particular case of great importance