l4anno - Rectilinear Motion x(t) F O rOP (t ) = x(t ) i...

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Rectilinear Motion F OP x ( t) () OP P P tx t t t = = = a ri vi i & && P moves along a straight path Suppress dependence on F
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Define Constraint Equations Pulley problems: Rope length is constant Relations between and Relations between and e.g. () OA x r OB y r OA r OB r || A BO B O A = rr r
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Differentiate constraint equations w.r.t. time Constraint equations may be nonlinear in position components Derivative equations are linear in velocity and acceleration components
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Rectangular Cartesian Coordinates () OP P P tx t y t z t t y t z t t y t z t =+ + =++ ri j k vi j k ai j k && & Suppress dependence on F
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Recipe () 2 ( () ) ) () ( ( ' '' )) ( ) ( (() )() ) ' OP PO P P t t tx t y t xt t xt xt t xt x t y y yy =+ = = + ri j ij vr ar & && & G iven , ()o r Find remaining functions ( ), ( ) or ( ) , (() )a n ' d ( ( )) Express ( ), ) Find ()and OP P P x x t x t t y tt t rv a & & Different colors denote difference functions (of t and x ) O OP r P x = x y
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Chain rule () ( ) '( ) '( ( )) x x xg z yf x y hz ff g z f g z gz 
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Case 2 : acceleration is a function of velocity If possible invert v=v(t) to obtain t=t(v). Now  x ( t ) g (
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This note was uploaded on 01/26/2012 for the course TAM 212 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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l4anno - Rectilinear Motion x(t) F O rOP (t ) = x(t ) i...

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