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net4 - Work-Energy Principle Every kinetics problem in...

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Work-Energy Principle Every kinetics problem in CE-325 is governed by Newton’s Second Law, i.e., X ~ F = m~ a, but sometimes alternate forms of this law is more convenient. One form which is heavily used can be derived by integrating over distance along the path Newton’s Second Law, i.e., Z 2 1 X ~ F · d~r = Z 2 1 a · d~ r. This will result in the “work-energy formulation” and the above equation is a scalar equation. This formulation is extremely useful when the path configuration is complicated but the details of how the particle moved between points 1 and 2 is not important. Clearly, this method destroys information. Consider now the derivation of the work-energy principle. For the left-side of the equation, define W as the work done by external forces, i.e., W = Z ~ r 2 ~ r 1 X ~ F · r, and for the right-side of equation, use the fact that ~a · = d~v dt · = · dt = · ~v = · to evaluate the integral as Z ~ r 2 ~ r 1 m~a · = m Z v 2 v 1 · = 1 2 mv 2 2 - 1 2 mv 2 1 . Define now the kinetic energy T as T = 1 2 mv 2 , then the integrated form of Newton’s Second Law can be written as W T = T 2 - T 1 . The term R ~ r 2 ~ r 1 ~ F · can be simplified a little further, we can classify the external forces into (1) Conservative Forces, ~ F c and (2) Non-conservative Forces, ~ F n . Conservative forces include gravitational forces, elastic spring forces, electromagnetic forces, etc while non-conservative forces include friction forces and randomly applied forces. Conservative forces yield an integral that is independent of the path, i.e., the integral I ~ F c · =0 for any closed path. With this property, the conservative forces can be derived from a potential, i.e., –4/1–
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X ~ F c = - dV d~r , where V = - Z X ~ F c · is a scalar potential. Two types of conservative forces which are most common in dynamics problems are the gravitational force and the elastic spring force. For their frequent appearances, it is therefore convenient to write the total external forces as X ~ F = X ~ F n + X ± ~ F e + ~ F g ² in which ~ F n is the sum of all non-conservative forces, ~ F e is the sum of all elastic spring forces and ~ F g is the gravitational force on particle m . The work term can now be separated as W = Z ~ r 2 ~ r 1 X ~ F n · + Z ~r 2 1 X ~ F e · + Z 2 1 X ~ F g · = U - Δ V e - Δ V g where U is the work done by all non-conservative forces, - Δ V e is the work done by the elastic forces and - Δ V g is the work done by the gravitational force, all between the interval from ~ r 1 to ~ r 2 . While - Δ V is referred to as the work done, Δ V is referred to as the potential energy. With these new definitions, the most frequently used work-energy relationship can be written as U T V e V g The Derivation of Δ V g To obtain a more convenient form for the gravitational potenial energy Δ V g = - Z 2 1 ~ F g · d~ r, use the cylindrical coordinates. Since the gravita-
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net4 - Work-Energy Principle Every kinetics problem in...

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