chapter3

chapter3 - CHAPTER 3 DYNAMICS OF A PARTICLE Newtons Second...

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1 Newton‟s Second Law: It is an experimentally derived law, valid in a reference frame – Inertial reference frame . XYZ - inertial reference frame Let m be mass, r OP - position vector. Then CHAPTER 3 DYNAMICS OF A PARTICLE path X Z Y O P F r OP e t e n e b ( , , ),and ( , , ) ( ) PP F F r r t F r r t d mr dt ma
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2 3.1 Direct Integration of Equations of Motion Newton‟s Law gives: This needs to be solved, subject to initial conditions: Case 1: The external force is a constant . In Cartesian coordinate system ( , , ) P F r r t ma 0 0 0 0 ( ) ; ( ) r t t r r t t r  ,, Consider thesystemin x-direction : / constant (say' ') ( ) x y z x mx F my F mz F x F m a d x dt a
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3 Integrating first by “separation of variables”: or, ( speed vs. time ) Integrating again: or ( position vs. time ) One can also approach the integration with position as the independent variable: 0 0 0 ( ) ( ) vt d x ad v t v at 0 () v t v at 0 0 0 ( ) ( ) xt d u v a d 2 00 ( ) / 2 x t x v t at
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4 Let ( position vs. speed ) Reading Assignment: Motion of a particle in a uniform gravitational field. 00 2 2 2 22 ( ) ( ) ( ) ( / 2) ( ) ( ) ( / 2) Then, Newton' c s2nd Law Separation of variables ( / 2) , hain rule 2 ( ) vx d d x d x d x d x xx dt dx dt dx dx dx a dx d x adu or v v a x x 
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5 Then, in Cartesian coordinates, Case 2: The external force is a function of time . 0 0 0 0 0 0 00 0 0 ( ) ( ( )/ ) ( ) ( Similarly, ( ) { ( } ( ) ( ( ) v t t xx v t t t x t t x t d x F m d v t v F m d d x v F m d d x t x v t F s m ds d ( ), ( ), ( ). x y z mx F t my F t mz F t 
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6 Special case: e.g. (example 3.3) Case 3: The force is a function of position. 00 2 2 2 2 2 (linear,separablefunction) Equation of motion : Integration : ( / 2) ( ) / 2 ( ) / 2 x vx F kx mx kx m d x kudu m v v k x x  m k x ( ) ( ) ( ) ( ) x y z F r F x i F y j F z k
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7 Then, to integrate again, we write as or or ( position vs. speed ) 2 2 2 2 00 ( )/ v v k x x m 2 2 2 1/2 [ ( )/ ] dx dt v k x x m 0 2 2 2 0 11 0 2 2 2 2 0 0 0 0 22 -1 2 2 0 0 0 [ ( / [sin sin ] ( / ) ( / ) or ( ) ( / ) sin( / ) where =sin ( / ( / ) ) tx x d v k u x m du x x t k m m k v x m k v x x t m k v x k m t x m k v x ( position vs. time )
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8 Case 4: The force is a function of velocity. Special case: Ex. 3.4: Projectile with air drag drag force ~ velocity ( ) ( ) ( ) ( ) x y z F r F x i F y j F z k X Y P -cv mg v P O where F mg j cv v xi yj  : (1) : (2) i cx mx j cy mg my
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9 x-motion : initial conditions: Integrating: cx mx   00 ( 0) , ( x t x x t x  0 0 0 ( / ) ( / ) 0 0 ( ) ( / ) ( ) / ( / ) ( / ) ln( / ) ( / ) ( ) Integratingagain : xt x c m t cm x d x dt c m x d u u c m d c m t x x c m t x t x e du x e d  
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10 or y-motion : initial conditions: Integrating: ( / ) ( / ) 0 0 0 0 ( / ) 00 ( ) ( / ) ( / ) [1 ] ( ) ] t c m t c m t c m t x t x m c x e m c x e x t x m c x e  my cy mg   ( 0) , ( y t y y t y 0 0 0 0 () ( ) ( / ) ( / ) ( / )ln[ ] ( / ) ( / ) y t y y y md u dt cu mg d u y mg c or t m c m c u mg c y mg c
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11 or Integrating again Summarizing: ( / ) 0 ( ) ( / ) [ ( / )] c m t y t mg c y mg c e  0 ( / ) 0 ( ) ( / ) ( / ){ ( / )}[1 ] c m t y t y mg c t m c y mg c e ( / ) 00 ( ) ( / ) [1 ] c m t x t x m c x e ( / ) 0 0 limiting x-displac () , ( ) 0, ( ) / eme ( n ) 0, ( ) ( tan ) t c m t x t x e as t x t x t x mx c of c x t x remainscons t
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12 x(t) as t ( unbounded x-displacement ) as t , (
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This note was uploaded on 12/27/2011 for the course ME 562 taught by Professor Bajaj during the Fall '10 term at Purdue.

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chapter3 - CHAPTER 3 DYNAMICS OF A PARTICLE Newtons Second...

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