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# 01_handout - 1 Fundamental Laws of Motion for Particles...

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1 Fundamental Laws of Motion for Particles, Material Volumes, and Control Volumes Ain A. Sonin Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, MA 02139, USA March 2003 © Ain A. Sonin Contents 1 Basic laws for material volumes 3 1.1 Material volumes and material particles 3 1.2 Laws for material particles 4 Mass conservation 4 Newton’s law of (non-relativistic) linear motion 4 Newton’s law applied to angular momentum 5 First law of thermodynamics 5 Second law of thermodynamics 5 1.3 Laws for finite material volumes 6 Mass conservation 6 Motion (linear momentum) 6 Motion (angular momentum) 7 First law of thermodynamics 7 Second law of thermodynamics 8 2 The transformation to control volumes 9 2.1 The control volume 9 2.2 Rate of change over a volume integral over a control volume 9 2.3 Rate of change of a volume integral over a material volume 11 2.4 Reynolds’ material-volume to control-volume transformation 11 3 Basic laws for control volumes 13 3.1 Mass conservation 13 3.2 Linear momentum theorem 14 3.3 Angular momentum theorem 14 3.4 First law of thermodynamics 15 3.5 Second law of thermodynamics 16

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2 4 Procedure for control volume analysis 17 Appendix 1: Summary of fundamental laws 19
3 1 Basic laws for material volumes 1.1 Material volumes and material particles Material systems behave according to universal physical laws. Perhaps the most ubiquitous of these are the law of mass conservation, Newton’s laws of motion, and the first and second laws of thermodynamics, all of which were understood before the nineteenth century ended. In this chapter we review these four laws, starting with their most primitive forms, and show how they can be expressed in forms that apply to control volumes. These turn out to be very powerful tools in engineering analysis 1 . The most fundamental forms of these four laws are stated in terms of a material volume . A material volume contains the same particles of matter at all times 2 . A particular material volume may be defined by the closed bounding surface that envelops its material particles at a certain time. Since every point of a material volume’s bounding surface moves (by definition) with the local material velocity v r (Fig. 1), the shape of the volume at all other times is determined by the laws of dynamics. Fig. 1 A material volume moves with the material particles it encloses. 1.2 Laws for material particles The simplest forms of the four basic laws apply to an infinitesimal material particle v that is so small that the velocity v , density ρ , thermodynamic temperature Τ , and other 1 For a historical note on control volume analysis in engineering, see Chapter 4 of Walter G. Vincenti’s What Engineers Know and How They know It , John Hopkins University Press, 1990.

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01_handout - 1 Fundamental Laws of Motion for Particles...

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