Since, then, we have said that tranquillity is the good condition of a city in respect of the action of its parts, we must in consequence consider what the city is in itself, and what it is for; what and how many are its principal parts; the action appropriate to each of them, their causes, and their ordering in respect of each other. For these considerations are fundamental to the complete delineation of tranquillity and its opposite.

2

However, before we discuss the city – which is the perfect community – and its species or modes, we ought first to introduce the origin of civil communities and their regimes and ways of living. From these, as from the imperfect, men have progressed to perfect communities and regimes and ways of living in them. For nature's path, and that of art which imitates her, is always from the less to the more perfect. And men are not judged to know any particular thing unless they know it together with its primary causes and its first principles right down to its elements.

3

So, proceeding in this fashion, we should be aware that civil communities (according to their different times and places) have begun small, and by gradually receiving an increase have in the end been brought to completion – as we have just said happens in every action of nature or art. For the first and minimum human combination, from which all others have arisen, is that of male and female, as the best of philosophers says in Politics I, chapter 1, and is further apparent from his Economics. This combination produced more human beings, who first of all filled one household; and then as further combinations of this type occurred, the multiplication of human beings was so great that one household was insufficient for them and it was necessary to set up several households. A plurality of these is called a village or neighbourhood, and this (as Aristotle also writes, as above) was the first community.

Someone will raise doubts, however, about what we have said, objecting that the authority to pass or institute laws does not belong to the universal body of the citizens. Firstly because something that is mostly wicked and undiscerning ought not to establish the law; for these two faults, sc. malice and ignorance, must be excluded from the legislator. Indeed it was in order to avoid them in judgements, as well, that we understood the necessity of laws in chapter 11 of this discourse. But the people or the universal body of the citizens is of this nature; for men are visibly wicked and stupid for the most part, since ‘the number of the stupid is infinite’ as it says in Ecclesiastes 1. Again, because it is very hard or impossible to get the opinions of many wicked and foolish individuals to agree, whereas this is not the case with a few who are virtuous. It is therefore more expedient for law to be passed by a few men rather than by the universal body of the citizens or an unnecessary number of them. Again, in any civil community the wise and the learned are few in respect of the rest of the untaught multitude. Since, therefore, it is more expedient for law to be passed by the wise and learned than by the ignorant and the uneducated, it seems that the authority to pass them belongs to the few, and not to many or to all. Further still, it is in vain for something to be done by many if it can be done by fewer. Since, therefore, it is possible for law to be passed by the wise (who are few) – as said before – it would be in vain for the entire multitude or its greater part to be occupied in this business. The authority to legislate does not, therefore, belong to the universal body of the citizens or its prevailing part.

Until now, we have been concerned with the kinematics of particles where the objective has been to determine the motion of a particle or rigid body without regard to the cause of the motion. Clearly in any physical system motion cannot occur without the application of some kind of external stimulus. In particular, in order for a particle to accelerate, it is necessary to apply a force to the particle. In order to study the motion that results from the application of a force (or, in general, the application of multiple forces) to a particle, it is necessary to study the kinetics of the particle. The objective of kinetics is threefold: (1) to describe quantitatively the forces that act on a particle; (2) to determine the motion that results from the application of these forces using postulated laws of physics; and (3) to analyze the motion.

The first topic in this chapter is the development of models for forces that are commonly used in dynamics. In particular, models are developed for contact forces, spring forces, and gravitational forces. These models will be used throughout the remainder of this book when solving problems.

The next topic in this chapter covers Newton's laws, which are the fundamental postulates that govern the nonrelativistic motion of particles.

The first topic in the study of dynamics is kinematics. Kinematics is the study of the geometry of motion without regard to the forces the cause that motion. For any system (which may consist of a particle, a rigid body, or a system of particles and/or rigid bodies) the objectives of kinematics are fourfold: to determine (1) a set of reference frames in which to observe the motion of a system; (2) a set of coordinate systems fixed in the chosen reference frames; (3) the angular velocity and angular acceleration of each reference frame (and/or rigid body) resolved in the chosen coordinate systems; and (4) the position, velocity, and acceleration of each particle in the system. In order to develop a comprehensive and systematic approach, the study of kinematics given in this Chapter is divided into two parts: (1) the study of kinematics of particles and (2) the study of kinematics of rigid bodies.

This Chapter is organized as follows. First, both a qualitative and precise definition of a reference frame is given. In particular, it is discussed that a reference frame provides a perspective from which to observe the motion of a system.

In Chapter 3 we discussed the important principles and methods used in the formulation, solution, and analysis of the motion of a single particle. In this chapter we extend the results of particle kinetics to systems consisting of two or more particles.

The first topic covered in this chapter is the center of mass of a system of particles. Using the definition of the center of mass, the linear momentum of a system of particles is defined. Then, using the definition of linear momentum, the velocity and acceleration of the center of mass of the system are defined.

The second topic covered in this chapter is the angular momentum of a system of particles. In particular, expressions for the angular momentum are derived relative to an arbitrary point, an inertially fixed point, and the center of mass of the system. Then, relationships between these three different forms of angular momentum are derived.

The third and fourth topics covered in this chapter are Newton's 2nd law and the rate of change of angular momentum for a system of particles. In particular, it is shown that the center of mass of the system satisfies Newton's 2nd law. Furthermore, the key results relating the rate of change of angular momentum for a system of particles to moment applied to the system are derived.

Until now we have been concerned with the kinetics of particles, i.e., the kinetics of objects that have nonzero finite mass but do not occupy any physical space. Furthermore, in Section 2.15 of Chapter 2 we studied the kinematics of motion of a rigid body. In this chapter we turn our attention to the kinetics of rigid bodies. To this end, the objectives of this chapter are threefold: (1) to describe quantitatively the forces and moments that act on a rigid body; (2) to determine the motion that results from the application of these forces and moments using postulated laws of physics; and (3) to analyze the motion.

The key difference between a particle and rigid body is that a particle can undergo only translational motion whereas a rigid body can undergo both translational and rotational motion. In general, for motion in ℝ3 it is necessary to specify three variables for the translational motion and to specify another three variables for the rotational motion of the rigid body.

Mechanics is the study of the effect that physical forces have on objects. Dynamics is the particular branch of mechanics that deals with the study of the effect that forces have on the motion of objects. Dynamics is itself divided into two branches called Newtonian dynamics and relativistic dynamics. Newtonian dynamics is the study of the motion of objects that travel with speeds significantly less than the speed of light while relativistic dynamics is the study of the motion of objects that travel with speeds at or near the speed of light. This division in the subject of dynamics arises because the physics associated with the motion of objects that travel with speeds much less than the speed of light can be modeled much more simply than the physics associated with the motion of objects that travel with speeds at or near the speed of light. Moreover, nonrelativistic dynamics deals primarily with the motion of objects on a macroscopic scale while relativistic dynamics deals with the study of the motion of objects on a microscopic or submicroscopic scale.

The subject of dynamics has been taught in engineering curricula for decades, traditionally as a second-semester course as part of a year-long sequence in engineering mechanics. This approach to teaching dynamics has led to a wide array of currently available engineering mechanics books, including Beer and Johnston (1997), Bedford and Fowler (2005), Hibbeler (2001), and Merriam and Kraige (1997). From my experience, the reasons these books are adopted for undergraduate courses in engineering mechanics are threefold. First, they include a wide variety of worked examples and have more than 1000 problems for the students to solve at the end of each chapter. The variety of problems provides instructors with the flexibility to assign different problems every semester for several years. Second, these books are generic enough that they can be used to teach undergraduates in virtually any branch of engineering. Third, they cover both statics and dynamics, thereby making it is possible for a student to purchase a single book for a year-long engineering mechanics course. Using these empirical measures, it is hard to dispute that these books cover a tremendous amount of material and enable an instructor to tailor the material to the needs of a particular course. Given the vast array of undergraduate dynamics books already available, an obvious question that arises is, why write yet another book on the subject of undergraduate engineering dynamics?

Introduction

In marked contrast to the military and political successes of the 1300–1683 era, defeats and territorial withdrawals characterized this long eighteenth century, 1683–1798. The political structure continued to evolve steadily, taking new forms in a process that should be seen as transformation but not decline. Central rule continued in a new and more disguised fashion as negotiation more frequently than command came to assure obedience. Important changes occurred in the Ottoman economy as well: the circulation of goods began to increase; levels of personal consumption probably rose; and the world economy came to play an ever-larger role in the everyday lives of Ottoman subjects.

The wars of contraction, c. 1683–1798

On the international stage, military defeats and territorial contraction marked the era, when the imperial Ottoman state was much less successful than before. At the outset, it seems worthwhile to make several general points.

First, at bottom, the Ottoman defeats are as difficult to explain as the victories of earlier centuries. Sometime during the early sixteenth century, as the wealth of the New World poured into Europe, the military balance shifted away from the Ottomans; they lost their edge in military technology and using similar and then inferior weapons and tactics, battled European enemies. Moreover, the earlier military imbalance between offensive and defensive warfare in favor of the aggressor had worked to the Ottomans' advantage, but now defenses became more sophisticated and vastly more expensive.