lecture-ch8 - Contents 8 Conservation of Energy 1 8.1 Path...

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Unformatted text preview: Contents 8 Conservation of Energy 1 8.1 Path Independence and Conservative Forces . . . . . . . . . . 1 8.1.1 Counter Examples . . . . . . . . . . . . . . . . . . . . 3 8.2 Work and Potential Energy . . . . . . . . . . . . . . . . . . . 4 8.3 Determining Potential Energy Values . . . . . . . . . . . . . . 5 8.3.1 Gravitational Potential Energy . . . . . . . . . . . . . 5 8.3.2 Elastic Potential Energy . . . . . . . . . . . . . . . . . 6 8.3.3 Spherically Symmetric Central Force . . . . . . . . . . 7 8.4 Determining Force from Potential Energy . . . . . . . . . . . . 8 8.4.1 One-dimensional U ( x ) . . . . . . . . . . . . . . . . . . 9 8.4.2 Three-dimensional U ( vectorx ) . . . . . . . . . . . . . . . . . 9 8.5 Conservation of Mechanical Energy . . . . . . . . . . . . . . . 10 8.6 Reading a Potential Energy Curve . . . . . . . . . . . . . . . . 12 8.6.1 Turning Points . . . . . . . . . . . . . . . . . . . . . . 12 8.6.2 Equilibrium Points . . . . . . . . . . . . . . . . . . . . 13 8.7 Work Done on a System by an External Force . . . . . . . . . 13 8.7.1 Conservative System . . . . . . . . . . . . . . . . . . . 13 8.7.2 Friction Involved . . . . . . . . . . . . . . . . . . . . . 15 8.8 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . 16 8.8.1 Isolated System . . . . . . . . . . . . . . . . . . . . . . 17 8.8.2 External Force and Internal Energy Transfer . . . . . . 17 8.8.3 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 8 Conservation of Energy One general type of energy is potential energy U , which can be associated with the configuration. 8.1 Path Independence and Conservative Forces In Chapter 7, we discussed the relation between work and a change in kinetic energy. The work-kinetic energy theorem says that Δ K = Δ W 1 Here, we discuss the relation between work and a change in potential energy. A force field vector F ( x, y, z ) is called conservative if the net work Δ W parenleftBig vector A → vector B parenrightBig done on a particle in moving between any two points vector A to vector B does not depend on the path from vector A to vector B taken by the particle. Otherwise, vector F is called non- conservative. A necessary and sufficient condition for a force field to be conservative is: The net work done by a conservative force on a particle moving around any closed path is zero. A B c 1 c 2 To show that the necessary condition follows, for the closed path in the above, the work done can be split into two parts: one along the path c 1 and the other along c 2 . Δ W parenleftBig vector A → vector A parenrightBig = Δ W parenleftBig vector A → vector B parenrightBig c 1 + Δ W parenleftBig vector B → vector A parenrightBig c 2 Since reversing the path taken by the particle between any two prescribed points also reverses the work done on the particle, we have Δ W parenleftBig vector B → vector A parenrightBig c 2 = − Δ W parenleftBig vector A → vector B parenrightBig c 2 Thus Δ W parenleftBig...
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This note was uploaded on 05/14/2011 for the course ECON 101 taught by Professor Asdaf during the Spring '11 term at Universidad de San Buenaventura Bogota.

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lecture-ch8 - Contents 8 Conservation of Energy 1 8.1 Path...

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