Abstract
The principles of mechanics for a particle are extended here to a system of n discrete material points. We begin with Newton’s second law for a system of particles and formulate the momentum, impulse-momentum, moment of momentum, work-energy, conservation, and general energy principles for a system of particles. Several of the concepts introduced here are especially useful in the study (in Chapter 11) of Lagrange’s general equations for arbitrary dynamical systems, and the development of the moment of momentum principle for a system of particles provides a foundation for the independent presentation (in Chapter 10) of parallel results for the moment of momentum of a rigid body.
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References
BLANCO, V. M., and MCCUSKEY, S. W., Basic Physics of the Solar System, Addison-Wesley, Reading Massachusetts, 1961. This is a concise treatment of the main physical and dynamical aspects of the solar system, including an introduction to the basic principles of celestial mechanics, written for scientists, engineers, and other nonspecialists with interests in space science. Celestial mechanics and the two body problem are introduced in Chapter 4. The three body problem and the general n-body problem are discussed in Chapter 5.
CAJORI, R, Newton’s Principia. English translation of Mathematical Principles of Natural Philosophy by Isaac Newton, 1687, University of California Press, Berkeley, 1947. The empirical principle of restitution is introduced in the Scholium (pp. 21–5) of Newton’s laws of motion, as a consequence and in support of the third law. In primary work, Wallis, Wren, and Huygens, in the order of their priority according to Newton, “did severally determine the rules of the impact and reflection of hard bodies, and about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiments on pendulums,….” Newton addresses the effects of air resistance on impacting pendula, and further on states: “By the theory of Wren and Huygens, bodies absolutely hard return from one another with the same velocity with which they met. But this may be affirmed with more certainty of bodies perfectly elastic. In bodies imperfectly elastic the velocity of the return is to be diminished together with the elastic force; because that force is certain and determined, and makes the bodies to return one from the other with a relative velocity, which is in a given ratio to that relative velocity with which they met.”
HOUSNER, G. W., and HUDSON, D. E., Applied Mechanics, Vol. II, Dynamics, 2nd Edition, Van-Nostrand, Princeton, New Jersey, 1959. Systems of particles are studied in Chapter 6 and the coefficient of restitution in Chapter 4. A few problems given below are modelled upon those provided in this text.
MARION, J. B., and THORNTON, S. T., Classical Dynamics of Particles and Systems, 3rd Edition, Harcourt Brace Jovanovich, New York, 1988. Central force motion is investigated in Chapter 7. See also Chapter 10 of the earlier 2nd edition by Marion cited in the References to Chapter 6, page 197.
MERIAM, J. L., and KRAIGE, L. G., Engineering Mechanics, Vol. 2 Dynamics, 3rd Edition, Wiley, New York, 1992. Direct and oblique central impact of smooth spheres are investigated in Chapter 3. Here the reader will find many additional examples and exercises for further study.
SHAMES, I. H., Engineering Mechanics. Statics and Dynamics, 4th Edition, Prentice-Hall, New Jersey, 1997. Additional examples and exercises on the central and oblique impact of particles are provided in Chapter 14, and problems on the eccentric impact of bodies by impulsive forces and torques (topics not treated herein) are discussed in Chapter 17.
TIMOSHENKO, S., and YOUNG, D. H., Advanced Dynamics, McGraw-Hill, New York, 1948. A classic, but non-vector treatment of applied topics in dynamics. A few problems provided in the present chapter are modelled upon examples found in Chapters 2 and 3 dealing with a system of particles.
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Beatty, M.F. (2006). Dynamics of a System of Particles. In: Principles of Engineering Mechanics. Mathematical Concepts and Methods in Science and Engineering, vol 33. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-31255-2_4
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DOI: https://doi.org/10.1007/978-0-387-31255-2_4
Publisher Name: Springer, Boston, MA
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