Abstract
Several methods of integration of the Newton-Euler vector equation of motion and its related scalar equations have been studied in a variety of applications in previous chapters. Although it is not possible to integrate these equations in general terms for all types of problems, certain kinds of problems do admit general first integrals that lead to several additional and useful basic principles of mechanics: the impulse-momentum principle, the torque-impulse principle, and the work-energy principle. Moreover, for certain kinds of forces, the work-energy principle may be reduced to a powerful fundamental law known as the principle of conservation of energy. The law of conservation of momentum and the law of conservation of moment of momentum are two more first integral principles that derive from the Newton-Euler law and the moment of momentum principle. This chapter concerns the development and application of these several additional principles.
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Keywords
- Angular Speed
- Potential Energy Function
- Jacobian Elliptic Function
- Energy Principle
- Gravitational Potential Energy
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References
BLANCO, V. M., and MCCUSKEY, S. W., Basic Physics of the Solar System, Addison-Wesley, Reading, Massachusetts, 1961. The principles of celestial and orbital mechanics are presented mainly for engineers, scientists, and other nonspecialists interested in space dynamics, the physical aspects of the solar system, and the motion of artificial satellites.
CAJORI, F., Newton’s Principia, English translation of Mathematical Principles of Natural Philosophy by Isaac Newton, 1687, University of California Press, Berkeley, 1947. The motion of Huygens’s cycloidal pendulum is discussed in Book 1: Motion of Bodies, pp. 153–60.
FERNANDEZ-CHAPOU, J. L., SALAS-BRITO, A. L., and VARGAS, C. A., An elliptic property of parabolic trajectories, American Journal of Physics 72, 1109 (2004). The authors show that the loci of points of maximum height of a projectile is an ellipse, an easy projectile problem described in Exercise 7.12.
LAMB, H., Dynamics, Cambridge University Press, Cambridge, Massachusetts 1929. An excellent old text in which many straightforward and interesting additional problems and examples may be found. The conservation laws are presented in Chapter 4. The circular and cycloidal pendulums are discussed in Chapter 5, orbital motions under central forces in Chapter 11.
LASS, H., Vector and Tensor Analysis, McGraw-Hill, New York, 1950. This text deals mostly with vector analysis. The line integral and potential function are treated in Chapter 4. Additional examples for collateral study may be found there.
MERIAM, J. L., and KRAIGE, L. G., Engineering Mechanics. Vol. 2, Dynamics, 3rd Edition, Wiley, New York, 1992. Work, energy, impulse-momentum, and central force motion are discussed in Chapter 3. The energy method is applied in vibrations problems in Chapter 8.
POLLARD, H., Celestial Mechanics, No. 18 of the Cams Mathematical Monographs, Mathematical Association of America, Washington, D.C., 1976. This small book presents a brief but thorough mathematical introduction to celestial mechanics. Details concerning the central force problem, including analysis of the two body problem, are provided in Chapter 1. Advanced topics in two further chapters include analysis of the n-body problem and use of the Hamilton-Jacobi theory of analytical dynamics.
ROUTH, E. J., Dynamics of a Particle, Dover, New York, 1960; originally published by the Cambridge University Press in 1898. The cycloidal pendulum is discussed in Articles 204–9, the simple pendulum in Articles 213–16.
SHAMES, I. H., Engineering Mechanics. Vol. 2, Dynamics, 2nd Edition, Prentice-Hall, Englewood Cliffs, New Jersey, 1966. This is an often cited useful resource for collateral study and additional problems. Central force motion is covered in Chapter 12, work and energy in Chapter 13, and momentum methods follow in Chapter 14.
SOMMERFELD, A., Mechanics. Lectures on Theoretical Physics, Vol. 1. Academic, New York, 1952. See Chapter III for discussion of the cycloidal pendulum.
SYNGE, J. L., and GRIFFITH, B. A., Principles of Mechanics, 3rd Edition, McGraw-Hill, New York, 1959. Orbital motion, stability, and Kepler’s laws are discussed in Chapter 6. The motion of a spherical pendulum, its solution in terms of Jacobian-elliptic functions, and its connection with potential experimental error in measurements of the Foucault effect are described in Chapter 13. For further discussion of the practical difficulties and errors encountered in measurements of the Foucault effect, see the excellent Foucault biography by Tobin cited in the previous chapter.
TIMOSHENKO, S., Vibration Problems in Engineering, 3rd Edition, Van Nostrand, Princeton, New Jersey, 1955. The energy method is treated briefly in Chapter 1. This is a classical resource for additional mechanical vibrations problems in engineering, including topics on vibrations of continua.
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Beatty, M.F. (2006). Momentum, Work, and Energy. 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_3
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DOI: https://doi.org/10.1007/978-0-387-31255-2_3
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