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net2 - DYNAMICS Dynamics is composed of 2 major topics: (1)...

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DYNAMICS Dynamics is composed of 2 major topics: (1) Kinematics Mathematical description of motion without regards to Newton’s Second Law. (2) Kinetics Basically Newton’s Second Law or Euler’s First Law. Basic Definitions of “Particle” Kinematics The following definitions are made in abstract algebra notation, therefore, they are valid in all coordinate systems. Displacement The displacement of a particle is referred to as the location of a particle with respect to the origin. Usually, the “displacement” is represented by the position vector ~ r ( t ) . Velocity The average velocity over a time period Δ t is defined as ~v avg = ~r ( t t ) - ~ r ( t ) Δ t . while the instantaneous velocity at a given time t is defined as = lim Δ t 0 avg = d~r dt P ˆ e 2 ˆ e 3 ˆ e 1 Acceleration The average acceleration over a time period Δ t is defined as ~a avg = ( t t ) - ~ v ( t ) Δ t . while the instantaneous acceleration at a given time t is defined as = lim Δ t 0 avg = d~v dt = d 2 ~ r dt 2 Integral Equations of Motion From the definitions of instantaneous velocity and acceleration, we can write the integral definitions as Z = Z ~ adt and Z = Z ~ vdt . Clearly, the above integrals preserve the vector properties. There is another integral form which is commonly used for particles traveling along a path, i.e., Z · = Z · d~ v, but this formulation retains only the tangential component. The nt-coordinate system is the best for this particular form. –2/1–
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Kinematic Formulas in Frequently Used Bases After defining a basis, the kinematic quantities can be expressed easily in matrix algebra notations. Cartesian Coordinates The basis vectors used are ˆ ı , ˆ and ˆ k . The direction of these unit vectors are fixed so their time derivatives vanish because they have constant magnitudes and no rotation. In abstract algebra notation, the vectors written in component form can be expressed as ~ r ( t )= x ( t ı + y ( t + z ( t ) ˆ k ~ v ( t )= ˙ x ( t ı y ( t z ( t ) ˆ k ~ a ( t )=¨ x ( t ı y ( t z ( t ) ˆ k where x ( t ) , y ( t ) , z ( t ) , ˙ x ( t ) , ˙ y ( t ) , ˙ z ( t ) , ¨ x ( t ) , ¨ y ( t ) and ¨ z ( t ) are scalar functions of time. At each instant of time, these scalar functions represents the magnitude of the components. x y z P ~r ˆ ı ˆ ˆ k The same expressions in matrix algebra notation can be written as ~ r ( t x ( t ) y ( t ) z ( t ) ,~ v ( t ˙ x ( t ) ˙ y ( t ) ˙ z ( t ) and ~a ( t ¨ x ( t ) ¨ y ( t ) ¨ z ( t ) Cylindrical (Polar) Coordinates The basis vectors used are ˆ e r , ˆ e θ and ˆ e z . The direction of ˆ e r and ˆ e θ changes with the angle θ but ˆ e z do not change. In abstract algebra notation, the position vector written in component form is ~ r ( t r ( t e r + z ( t e z where r ( t ) and z ( t ) are scalar functions of time. There is no component in the ˆ e θ direction for the displacement vector but do remember that ˆ e r is a function of θ .T o obtain the velocity vector, take time derivative of the displacement vector ~ r ( t ) to yield ~v ( t d~r dt r ( t e r z ( t e z + r ( t ) d ˆ e r dt + z ( t ) d ˆ e z dt , x y z P θ ˆ e z ˆ e θ ˆ e r
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This note was uploaded on 04/16/2008 for the course CE 325 taught by Professor Docwong during the Spring '08 term at USC.

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net2 - DYNAMICS Dynamics is composed of 2 major topics: (1)...

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