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2869 results in Solid mechanics and materials

Page 55 of 144

  • 4 - Momentum

  • Daniel Kleppner, Massachusetts Institute of Technology, Robert Kolenkow
  • Book:
    An Introduction to Mechanics
    Published online:
    28 May 2018
    Print publication:
    18 November 2013, pp 115-160

  • Summary

    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.

  • 2 - Newton's Laws

  • Daniel Kleppner, Massachusetts Institute of Technology, Robert Kolenkow
  • Book:
    An Introduction to Mechanics
    Published online:
    28 May 2018
    Print publication:
    18 November 2013, pp 47-80

  • Summary

    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.

  • To the Teacher

  • Daniel Kleppner, Massachusetts Institute of Technology, Robert Kolenkow
  • Book:
    An Introduction to Mechanics
    Published online:
    28 May 2018
    Print publication:
    18 November 2013, pp xv-xvi

  • Summary

    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.

  • 1 - Vectors and Kinematics

  • Daniel Kleppner, Massachusetts Institute of Technology, Robert Kolenkow
  • Book:
    An Introduction to Mechanics
    Published online:
    28 May 2018
    Print publication:
    18 November 2013, pp 1-46

  • Summary

    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.

  • 3 - Forces and Equations of Motion

  • Daniel Kleppner, Massachusetts Institute of Technology, Robert Kolenkow
  • Book:
    An Introduction to Mechanics
    Published online:
    28 May 2018
    Print publication:
    18 November 2013, pp 81-114

  • Summary

    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.

  • 12 - The Special Theory of Relativity

  • Daniel Kleppner, Massachusetts Institute of Technology, Robert Kolenkow
  • Book:
    An Introduction to Mechanics
    Published online:
    28 May 2018
    Print publication:
    18 November 2013, pp 439-476

  • Summary

    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.

  • 10 - Central Force Motion

  • Daniel Kleppner, Massachusetts Institute of Technology, Robert Kolenkow
  • Book:
    An Introduction to Mechanics
    Published online:
    28 May 2018
    Print publication:
    18 November 2013, pp 373-410

  • Summary

    Introduction

    Johannes Kepler was the assistant of the sixteenth-century Danish astronomer Tycho Brahe. They had an ideal combination of talents. Brahe had the ingenuity and skill to measure planetary positions to better than

    0.01°, all made by naked eye because the telescope was not invented until a few years after his death. Kepler had the mathematical genius and fortitude to discover that Brahe's measurements could be fitted by three simple empirical laws. The task was formidable. It took Kepler 18 years of laborious calculation to obtain the following three laws of planetary motion, which he stated early in the seventeenth century:

  • Every planet moves in an ellipse with the Sun at one focus.

  • The radius vector from the Sun to a planet sweeps out equal areas in equal times.

  • The period of revolution T of a planet about the Sun is related to the major axis A of the ellipse by T2 = kA3, where k is the same for all the planets.

    • Kepler's empirical laws went unexplained until the latter half of the seventeenth century, when Newton's fascination with the problem of planetary motion inspired him to formulate his laws of motion and the law of universal gravitation. Using these mathematical laws, Newton explained Kepler's empirical laws, giving an overwhelming argument in favor of the new mechanics and marking the beginning of modern mathematical physics. Planetary motion and the more general problem of motion under a central force continue to play an important role in many branches of physics and turn up in such topics as particle scattering, atomic structure, and space navigation.

  • Preface

  • Daniel Kleppner, Massachusetts Institute of Technology, Robert Kolenkow
  • Book:
    An Introduction to Mechanics
    Published online:
    28 May 2018
    Print publication:
    18 November 2013, pp xi-xiv

  • Summary

    An Introduction to Mechanics grew out of a one-semester course at the Massachusetts Institute of Technology—Physics 8.012—intended for students who seek to understand physics more deeply than the usual freshman level. In the four decades since this text was written physics has moved forward on many fronts but mechanics continues to be a bedrock for concepts such as inertia, momentum, and energy; fluency in the physicist's approach to problem-solving—an underlying theme of this book—remains priceless. The positive comments we have received over the years from students, some of whom are now well advanced in their careers, as well as from faculty at M.I.T. and elsewhere, reassures us that the approach of the text is fundamentally sound. We have received many suggestions from colleagues and we have taken this opportunity to incorporate their ideas and to update some of the discussions.

    We assume that our readers know enough elementary calculus to differentiate and integrate simple polynomials and trigonometric functions. We do not assume any familiarity with differential equations. Our experience is that the principal challenge for most students is not with understanding mathematical concepts but in learning how to apply them to physical problems. This comes with practice and there is no substitute for solving challenging problems. Consequently problem-solving takes high priority. We have provided numerous worked examples to help provide guidance. Where possible we try to tie the examples to interesting physical phenomena but we are unapologetic about totally pedagogical problems.

Page 55 of 144