bajaj562chpt2

bajaj562chpt2 - CHAPTER 2 KINEMATICS OF A PARTICLE...

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1 CHAPTER 2 KINEMATICS OF A PARTICLE Kinematics : It is the study of the geometry of motion of particles, rigid bodies, etc., disregarding the forces associated with these motions. Kinematics of a particle motion of a point in space
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2 Interest is on defining quantities such as position , velocity , and acceleration . Need to specify a reference frame (and a coordinate system in it to actually write the vector expressions). Velocity and acceleration depend on the choice of the reference frame . Only when we go to laws of motion, the reference frame needs to be the inertial frame .
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3 From the point of view of kinematics, no reference frame is more fundamental or absolute. 2.1 Position, velocity, acceleration path followed by the object. O - origin of a coordinate system in the reference frame. x y z o r OP P v P
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4 r OP - position vector (specifies position, given the choice of the origin O). Clearly, r OP changes with time r OP (t) velocity vector: acceleration vector: 0 ( ) lim . OP P OP t r d v r t dt t   2 2 ( ) ( ). P P OP dd a v t r t dt dt x y z P P’ r r OP’ O r OP
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5 speed: magnitude of acceleration: Important: the time derivatives or changes in time have been considered relative to (or with respect to) a reference frame. Description in various coordinate systems (slightly different from the text) Cartesian coordinates, cylindrical coordinates etc. P P P P v v v v P P P P a a a a
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6 Let be the unit vectors Cartesian coordinate system: The reference frame is - it is fixed. are an orthogonal set . Then, position of P is: r OP = x (t) i + y (t) j + z (t) k i j k , , i j k j k i k i j etc , , . i j k , , i j k z O x y P P’ y x z
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7 The time derivative of position is velocity : If considering rate of change in a frame in which are fixed, velocity vector Similarily, acceleration vector i j k , , 0 dj di dk dt dt dt     ( ) ( ) ( ) ( ) ( ) ( ) OP P dr dx t dy t dz t v i j k dt dt dt dt di d k x t y t z t dt dt dt ( ) ( ) ( ) P dx t dy t dz t v i j k dt ( ) ( ) ( ) P a x t i y t j z t k
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8 Cylindrical Coordinates: e r - unit vector in xy plane in radial direction. e - unit vector in xy plane to e r in the direction of increasing k - unit vector in z. Then, by definition r x y z e r e x y z r P O P’ k 2 2 1/2 1 ( ) ; tan ( / ). r x y y x
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9 The position is: r OP = x (t) i + y (t) j + z (t) k Also, r OP = r ( ) e r + z (t) k and r OP = r ( ) cos i + r ( ) sin j + z (t) k Also, Imp. to Note : change with position ( ). e and e r cos sin r rr e i j   but ( sin cos ) and sin cos r e r i j r r e i j  
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10 position : r OP = r ( ) e r + z (t) k or r OP = r ( ) cos i + r ( ) sin j + z (t) k velocity : z- direction fixed Thus or =radial comp+transverse comp+axial comp ( ) k 0 dk dt P OP r r v dr dt re r de dt zk zdk dt ( )( ) ( / ) r r r de dt de d d dt d e   Pr v r e zk
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11 acceleration : ; rr e
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bajaj562chpt2 - CHAPTER 2 KINEMATICS OF A PARTICLE...

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