MECHANICS a machine), or DYNAMICS, the science that. treats of forces (OfrvaPtc, force), and of the motions produced by them. The notion of a force, as evinced to the senses as a push or pull, is common to all, but the notion of force as that which produces or destroys motion, which is the proper defini tion of force, is modern, and is to he ascribed to Newton, the chief founder of the science. It is a familiar fact that two opposite pushes or pulls may neutralize each other's effects, and thus fail to produce motion; we then speak of them as forces in equilibrium. The portion of Mechanics that treats of forces in equilibrium is denoted by the term Statics (root era, stand). contrasted with which we have the subject of Kinetics (limb.), to set moving), which deals with effects of forces in acting to produce motion in bodies. Since we can distinguish motion only in matter, the laws of motion in volve the essential properties of matter, so that Dynamics is a 'branch of Physics —indeed, its most fundamental branch, for, until recently, it was the effort of physicists to reduce all ex planations of physical phenomena to descrip tions of matter in motion.• Nowadays the tend ency of interpreting mechanics in electromag netic terms is making itself felt. As in geom etry, instead of dealing with actual substance, we make abstraction and conceive of points, lines and surfaces apart from the substance in which they lie, so we may make abstraction and consider the motion of points, lines or geomet rical configurations, quite apart from any matter or physical properties. This geometry of moving configurations or geometry of space and time is generally distinguished by the name of Kine matics ( Kivva, motion), and is included under treatments of Dynamics only for convenience, as it is impossible to make dynamical investi gations except in kinematical terms.
The ancients knew but little of Mechanics, and what they did know belonged exclusively to Statics. Archimedes was familiar with the principles of the lever and of the pulley. Leo nardo da Vinci generalized the principle of the lever, and Stevinus (1548-1620) demonstrated the principle of the inclined plane and of the composition of forces. Varignon clearly enun ciated the principle of moments, and also of the composition of forces. Galileo, in the course of his investigations on the inclined plane, came to a recognition of a particular case of the Principle of Virtual Work, which was made general by Daniel Bernoulli. These are the chief names in the development of Statics. The beginnings of Kinematics were made by Galileo, who determined the laws of falling bodies, and introduced the fundamental idea of acceleration. Huygens, in his (Horologium Oscillatorium,) published in 1673, examined in detail the laws of the pendulum, introduced the ideas of moment of inertia, of the centre of i oscillation and of kinetic energy. Most m portant of all was the work of Newton, who in his
1. STATICS.—Although the principles of Statics may be logically deduced from those of Kinetics by assuming all velocities to be zero, it is simpler to follow the historical method and treat statics first, since we may dispense with the idea of time, and thus with the preliminary study of kinematics. We begin by assuming the identical nature of all forces. For instance, the effect of any force may be neutralized by a pull on a string fastened to the point at which the force is applied. The tension on a string is produced by equal and opposite pulls on its two ends, and it may be cut anywhere, if at the cut end is applied a force equal to the one previously applied to the end. Such a force may be produced by the weight of any body hanging from the end of the string. But as a weight always acts vertically downward, while forces may act in any direction, we may suppose the string carried over a smooth pulley with horizontal axis, whose effect is assumed to be merely to change the direction of the string without changing its tension. Thus any force in any direction may be equilibrated by the tension of a string produced by a certain weight. Two forces are equal when they are
equilibrated by the same weight. Two weights found to be equal (by equilibration), when hung from the same string, produce double the tension produced by one, and thus forces may be measured in terms of a single weight. A force having magnitude and direction may be geometrically represented by a line parallel to it, and of a length proportional to its magni tude. To this line an arrow-head may be at tached to indicate the sense of the direction of the force. We may now enunciate the prin ciple, capable of experimental verification, that when two forces, represented by AB, AC, Fig. 1, are applied at the same material pOint A, they may be replaced by a single force whose direction and magnitude are represented by the diagonal AD of the parallelogram formed on the sides AB, AC. (The direction of the arrows must be observed). This is the princi ple of the Parallelogram of Forces. Obviously it may be replaced by the equivalent statement that if we form a triangle by placing at the extremity B of one of the lines representing the forces the initial point of the line repre senting the other force, BD, and complete the triangle, the line drawn from the initial point of the first to the terminal point of the second line will represent the resultant of the two forces, that being the term applied to the single force which replaces their effect. The original forces AB, AC, are said to be the components of AD. From the properties of the parallelo gram, AB sin (BAD)— AC sin (CAD), so that the magnitudes of the components are in versely proportional to the sines of the angles they make with the resultant. Obviously the two forces may be equilibrated by a force equal but opposite to the resultant, so that if we draw AE equal and opposite to AD (Fig. 2) the three forces AB, AC, AE will be in equilibrium. As the angles BAD and BAE are tary, their sines are equal, similarly DAC and CAE; consequently we have AB sin (CAE) AC sin and in turn considering each of the three forces as equilibrating the other two we get the theorem that the magnitudes of three forces in equilibrium are proportional to the sines of the angles lying opposite them re spectively. This may be experimentally veri fied as in Fig. 3, where weights P, Q are hung from strings passing over pulleys, and united at 0 to a string carrying a weight R. If a parallelo gram be drawn on lengths proportional to P, Q. its diagonal will be vertical, and proportional to the weight R. A convenient form of the experiment is one in which the three strings are horizontal, and 0 is the centre of a hori zontal circular table, on whose rim the three pulleys may be placed, their relative positions being read off on a graduation of the edge of the table.
Obviously, by a reversal of the previous process, a given force may be resolved into components in any two given directions, as only one parallelogram can be drawn on a given diagonal, whose sides have given directions. If these directions are at right angles to each other, as in Fig. 4, OP is the component of OF in the direction OA and OQ the component in the direction perpendicular thereto. The length OP is called the projection of OF in the direction OA, and we have OP= OF cos (POF). Having found the resultant of two forces applied at a common point, we may compound this resultant with another force, and so on, the simplest rule of procedure being by an extension of the triangle method above, that is, apply the initial point of each line representing a force to the terminal point of the preceding line; then the line drawn from the first initial point to the last terminal point will represent the resultant. The slight est consideration will show that the resultant is independent of the order in which the forces are compounded. This construction is known as the polygon of forces (Fig. 5). It is ob ant F, the components respectively X, Y, Z, we have X = cos (Fx), (1) Y = F cos (Fy), Z= F cos (Fz); (2) X' + Y' + —P[cos'(Fx)+coe(Fy)+cos'(Fz)]=P, since the sum of the squares of the direction cosines of any line is identically equal to unity. Since the projection in any direction of any broken line is the same as that of a straight line with the same ends, it is evident that the projection of any resultant is the same as the algebraic sum of the projections of all its com ponents. Thus the analytical expression of the principle of the parallelogram or polygon of forces is, if F is the resultant, X, Y, Z its components along the axes,