Introduction

In discussing the principles of dynamics in Chapter 2, we stressed that Newton's second law F = ma holds true only in inertial coordinate systems. We have so far avoided non-inertial systems in order not to obscure our goal of understanding the physical nature of forces and accelerations. Because that goal has largely been realized, in this chapter we turn to the use of non-inertial systems with a twofold purpose. By introducing non-inertial systems we can simplify many problems; from this point of view, the use of non-inertial systems represents one more computational tool. However, consideration of non-inertial systems also enables us to explore some of the conceptual difficulties of classical mechanics. Consequently, the second goal of this chapter is to gain deeper insight into Newton's laws, the properties of space, and the meaning of inertia. We start by developing a formal procedure for relating observations in different inertial systems.

Galilean Transformation

In this section we shall show that any coordinate system moving uniformly with respect to an inertial system is also inertial. This result is so transparent that it hardly warrants formal proof. However, the argument will be helpful in the next section when we analyze non-inertial systems.

Suppose that two physicists, Alice and Bob, set out to observe a series of events such as the position of a body of mass m as a function of time. Each has their own set of measuring instruments and each works in their own laboratory. Alice has confirmed by experiments that Newton’s laws hold accurately in her laboratory, and she concludes that her reference frame is therefore inertial.

Introduction

So far we have viewed nature as if it were composed of ideal particles rather than real bodies. Sometimes such a simplification is justified—for instance in the study of planetary motion, where the size of the planets is of little consequence compared with the vast distances of our solar system, or in the case of elementary particles moving through an accelerator, where the size of the particles, about 10−15 m, is minute compared with the size of the machine. However, most of the time we deal with large bodies that may have elaborate structure. For example, consider the landing of an explorer vehicle on Mars. Even if we could calculate the gravitational field of such an irregular and inhomogeneous body as Mars, the explorer itself hardly resembles a particle—it has wheels, gawky antennas, extended solar panels, and a lumpy body.

Furthermore, the methods of the last chapter fail when we try to analyze systems such as rockets in which there is a flow of mass. Rockets accelerate forward by ejecting mass backward; it is not obvious how we can apply F = Ma to such a system.

Introduction

Our goal in this chapter is to understand Newton's laws of motion. Newton's laws are simple to state and they are not mathematically complex, so at first glance the task looks modest. As we shall see, Newton's laws combine definitions, observations from nature, partly intuitive concepts, and some unexamined assumptions about space and time. Newton's presentation of his laws of motion in his monumental Principia (1687) left some of these points unclear. However, his methods were so successful that it was not until two hundred years later that the foundations of Newtonian mechanics were carefully examined, principally by the Viennese physicist Ernst Mach. Our treatment is very much in the spirit of Mach.

Newton's laws of motion are by no means self-evident. According to Aristotle, the natural state of bodies is rest: bodies move only when a force is applied. Aristotelian mechanics was accepted for two thousand years because it seemed intuitively correct. Careful reasoning from observation and a great leap of imagination were needed to break out of the Aristotelian mold.

Analyzing physical systems from the Newtonian point of view requires effort, but the payoff is handsome. To launch the effort, this chapter is devoted to presenting Newton's laws and showing how to apply them to elementary problems. In addition to deepening our understanding of dynamics, there is an immediate reward for these exercises—the power to analyze physical phenomena that at first sight might seem incomprehensible.

This edition of An Introduction to Mechanics, like the first edition, is intended for a one-semester course. Like the first edition, there are 14 chapters, though much of the material has been rewritten and two chapters are new. The discussion of Newton's laws, which sets the tone for the course, is now presented in two chapters. Also, the discussion of energy and energy conservation has been augmented and divided into two chapters. Chapter 5 on vector calculus from the first edition has been omitted because the material was not essential and its presence seemed to generate some math anxiety. A portion of the material is in an appendix to Chapter 5.

The discussion of energy has been extended. The idea of heat has been introduced by relating the ideal gas law to the concept of momentum flux. This simultaneously incorporates heat into the principle of energy conservation, and illustrates the fundamental distinction between heat and kinetic energy. At the practical end, some statistics are presented on international energy consumption, a topic that might stimulate thinking about the role of physics in society,

The only other substantive change has been a recasting of the discussion of relativity with more emphasis on the spacetime description. Throughout the book we have attempted to make the math more user friendly by solving problems from a physical point of view before presenting a mathematical solution. In addition, a number of new examples have been provided.

The course is roughly paced to a chapter a week. The first nine chapters are vital for a strong foundation in mechanics: the remainder covers material that can be picked up in the future. The first chapter introduces the language of vectors and provides a background in kinematics that is used throughout the text. Students are likely to return to Chapter 1, using it as a resource for later chapters.

Introduction

Mechanics is at the heart of physics; its concepts are essential for understanding the world around us and phenomena on scales from atomic to cosmic. Concepts such as momentum, angular momentum, and energy play roles in practically every area of physics. The goal of this book is to help you acquire a deep understanding of the principles of mechanics.

The reason we start by discussing vectors and kinematics rather than plunging into dynamics is that we want to use these tools freely in discussing physical principles. Rather than interrupt the flow of discussion later, we are taking time now to ensure they are on hand when required.

Vectors

The topic of vectors provides a natural introduction to the role of mathematics in physics. By using vector notation, physical laws can often be written in compact and simple form. Modern vector notation was invented by a physicist, Willard Gibbs of Yale University, primarily to simplify the appearance of equations. For example, here is how Newton's second law appears in nineteenth century notation:

Fx = max

Fy = may

Fz = maz.

In vector notation, one simply writes

F = ma,

where the bold face symbols F and a stand for vectors.

Our principal motivation for introducing vectors is to simplify the form of equations. However, as we shall see in Chapter 14, vectors have a much deeper significance. Vectors are closely related to the fundamental ideas of symmetry and their use can lead to valuable insights into the possible forms of unknown laws.

Introduction

The concept of force is central in Newtonian physics. This chapter describes the gravitational force and the electrostatic force, two of the fundamental forces of nature. We also discuss several phenomenological forces, for example friction. Such forces are commonly encountered in “everyday” physics and are approximately described by empirical equations. Because the concept of force is meaningful only if one knows how to solve problems involving forces, this chapter includes many examples in which Newton's laws are put into practice.

The problem of calculating motion from known forces frequently occurs in physics. For instance, a physicist who sets out to design a particle accelerator employs the laws of mechanics and knowledge of electric and magnetic forces to calculate how the particles will move in the accelerator. Equally important, however, is the converse process of deducing the physical interaction from observations of the motion, which is how new laws are discovered. The classic example is Newton's deduction of the inverse-square law of gravitation from Kepler's laws of planetary motion. A contemporary example is the effort to elucidate the interactions between elementary particles from high energy scattering experiments at the Large Hadron Collider at CERN in Geneva and at other high energy laboratories.

Unscrambling experimental observations to find the underlying forces can be complicated. In a facetious mood, the British cosmologist Arthur Eddington once said that force is the mathematical expression we put into the left-hand side of Newton's second law to obtain results that agree with observed motions. Fortunately, force has a more concrete physical reality.

Introduction

In the centuries following publication of the Principia, Newtonian dynamics was accepted whole-heartedly not only because of its enormous success in explaining planetary motion but also in accounting for all motions commonly encountered on the Earth. Physicists and mathematicians (often the same people) created elegant reformulations of Newtonian physics and introduced more powerful analytical and calculational techniques, but the foundations of Newtonian physics were assumed to be unassailable. Then, on June 30 1905, Albert Einstein presented his special theory of relativity in his publication The Electrodynamics of Moving Bodies. The English translation, available on the web, is reprinted from Relativity: The Special and General Theory, Albert Einstein, Methuen, London (1920). The original publication is Zur Elektrodynamik bewegter Körper, Annalen der Physik 17 (1905). Einstein's paper transformed our fundamental view of space, time, and measurement.

The reason that Newtonian dynamics went unchallenged for over two centuries is that although we now realize that it is only an approximation to the laws of motion, the approximation is superb for motion with speed much less than the speed of light, c ≈ 3 × 108 m/s. Relativistic modifications to observations of a body moving with speed v typically involve a factor of v2/c2. Most familiar phenomena involve speeds v ≪ c. Even for the high speed of an Earth-orbiting satellite, v2/c2 ≈ 10−10. There is one obvious exception to this generalization about speed: light itself.