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Unformatted text preview: J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L11 Conservation Laws for Systems of Particles In this lecture, we will revisit the application of Newton’s second law to a system of particles and derive some useful relationships expressing the conservation of angular momentum. We also specialize these results to twodimensional rigid bodies. Center of Mass Consider a system made up of n particles. A typical particle, i , has mass m i , and, at the instant considered, occupies the position r i relative to a frame xyz . We can then define the center of mass, G , as the point whose position vector, r G , is such that, 1 n r G = ( m i r i ) . (1) m i =1 n Here, m is the total mass of the system given by m = m i . i =1 It is important to note that the center of mass is a property of the system and does not depend on the reference frame used. In particular, if we change the location of the origin O , r G will change, but the absolute position of the point G within the system will not. Often, it will be convenient to describe the motion of particle i as the motion of G plus the motion of i relative to G . To this end, we introduce the relative position vector, r i , and write, r i = r G + r i . (2) It follows immediately, from the definition of the center of mass (1) and the definition of the relative vector r i (2), that, n n n m i r = m i ( r i − r G ) = m i r i − m r G = 0 . (3) i i =1 i =1 i =1 This result will simplify our later analysis. 1 Forces In order to derive conservation laws for our system, we isolate it a little more carefully, identify what mass particles it contains and what forces act upon the individual particles. We will consider two types of forces acting on the particles : External forces arising outside the system. We will denote the resultant of all the external forces N acting on the system as i =1 F Ei = F . Internal forces due to pairwise particle interactions. Let f ij denote the force that particle j exerts on particle i directed along the line joining the two particles i and j . This force could arise from gravitation attraction or from internal force due to the connections between particles. It could also arise from collisions between individual particles that, as we have seen, produce equal and opposite impulsive forces that conserve momentum. By Newton’s third law, these internal forces act in pairwise equal and opposite directions. Therefore, f ij = − f ji , where f ji is the force that particle i exerts on particle j along the line joining the particles The total internal force on particle i is then n , f ij j =1 , j = i and, if we sum over all particles, we have n n f ij = , i =1 j =1 j = i since, for every force, f ij , there is an equal and opposite force, f ji for both normal and tangential forces.(The notation j = i means to exclude j=i in the sum since of course the particle induces no net force upon itself.) 2 Conservation of Linear Momentum The linear momentum of the system is defined...
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This note was uploaded on 11/06/2011 for the course AERO 112 taught by Professor Widnall during the Fall '09 term at MIT.
 Fall '09
 widnall
 Dynamics

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