homework1.20100120.4b5784f9488059.00941346

homework1.20100120.4b5784f9488059.00941346 - TAM 412...

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TAM 412, Homework 1, due January 27, 2010 Harry Dankowicz Mechanical Science and Engineering University of Illinois at Urbana-Champaign [email protected] 1. Consider the f gure below where α and β are given angles. (a) Explain the sense in which the map C ( x ( P ) ,y ( P )) constitutes a coordinate chart for the family of all allowable con f gurations of a particle represented by the point P . (b) Under what conditions does the same hold for the pair ( q 1 ( P ) ,q 2 ( P )) ?H in t : f nd a transforma- tion between ( x,y ) and ( q 1 2 ) and determine whether it is one-to-one and onto. (c) Under what conditions does the same hold for the pair ( s 1 ( P ) 0 2 ( P )) ? (d) For a given chart, a coordinate curve is obtained by varying one coordinate, while keeping the other(s) f xed. For each of the three coordinate charts, sketch a set of coordinate curves. 2. Consider the f gure below. (a) For what collection of allowable con f gurations C of a particle represented by the point P does the map C ( r 1 ( P ) , sin θ ( P )) constitute a coordinate chart?
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homework1.20100120.4b5784f9488059.00941346 - TAM 412...

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