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chapter4 - CHAPTER 4 DYNAMICS OF A SYSTEM OF PARTICLES We...

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1 CHAPTER 4 DYNAMICS OF A SYSTEM OF PARTICLES We consider a system consisting of n particles One can treat individual particles, as before; i.e.,one can draw FBD for each particle, define a coordinate system and obtain an expression of the absolute acceleration for the particle. One can then use Newton’s second law and proceed to get n second-order coupled ODEs. Focus here is on overall motion of the system - also a precursor to rigid body dynamics.
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2 4.1 Equations of Motion: Consider a system with: n particles masses - positions - There are two types of forces acting : External forces - Internal forces - r i F i ; m i f ij Z X Y O r 1 F 1 r C r i F 2 F i m i m 2 m 1 m C f 12 f 21 f 2i f i2 i
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3 - force on the particle due to its interaction with the particle Newton’s 3rd law Also Newton’s 2nd law for particle: f ij i th j th i th internalforcesareequalandopposi ( ) 0when , . . te 0 ij ji ij ii f f f i j i e f 1 , 1,2,3, . . . . , n i i i ij j m r F f i n 
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4 Now, for 3-dimensional motions, the position of each particle (in Cartesian coordinates) is: Thus, each equation in Newton’s second law has 3 scalar second-order ordinary diff. equations. 3 n scalar second- order o.d.e.’s for the system In order to solve for the motion, one needs to know: external forces on each of the particles nature of internal forces F i f ij , 1,2,3, . . . , i i i i r x i y j z k i n
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5 e.g., Newton’s law of gravitation: We also need: initial conditions: The general solutions to these nonlinear ODEs are unknown; they are difficult to solve except for in some very simple cases and small n . 2 3 ( ) , ( ) / j i i j ij j i j i ij i j i j j i r r m m f G r r r r or f Gm m r r r r (0), (0), 1,2, . . . . , i i r r i n
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6 Suppose we would like to get overall motion of the system , not those of individual particles . Adding the n equations : Now, (net interaction force is zero) - total mass - defines center of mass ; note that it is a function of time since the particles move. 1 1 1 1 n n n n i i i ij i i i j m r F f  1 1 0 n n ij i j f 1 n i i m m 1 ( ) ( ) n C i i i mr t m r t
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7 - total external force Equation of motion for the center of mass Internal forces do not affect the motion of the center of mass . 1 1 Thus,additionof Eqns. n n i i i C i i F m r mr   1 Let n i i F F 1 n i C i F F mr 
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8 4.2 Work and Kinetic Energy The motion of individual particle is defined by The motion of center of mass C is defined by where the total mass is 1 , 1,2,3, . . . . , n i i i ij j m r F f i n  1 n i C i F F mr  1 n i i m m
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9 Consider a motion of the system. The initial state is A, and the final state is B. Let A C and B C denote the positions of the CM. Now, for the CM work-energy statement for the CM 2 ( / 2) C C C C C C C B B B C C C C A A A F mr F dr mr dr mv   Y Z X O r C r i m i m A i B i A C B C
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10 Note that is only the work done by external forces, and it is related to the change in translational kinetic energy associated with the CM Let work done on the particle by all the forces acting on it in moving from W i i th A to B i i C C B C A F dr 1 ( ) i i B n i i ij i j A W F f dr
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