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Unformatted text preview: 905 REVIEW AND SUMMARY In this chapter we analyzed the motion of systems of particles, i.e., the motion of a large number of particles considered together. In the first part of the chapter we considered systems consisting of well- defined particles, while in the second part we analyzed systems which are continually gaining or losing particles, or doing both at the same time. We first defined the effective force of a particle P i of a given system as the product m i a i of its mass m i and its acceleration a i with respect to a newtonian frame of reference centered at O [Sec. 14.2]. We then showed that the system of the external forces acting on the particles and the system of the effective forces of the particles are equipollent; i.e., both systems have the same resultant and the same moment resultant about O : O n i 5 1 F i 5 O n i 5 1 m i a i (14.4) O n i 5 1 ( r i 3 F i ) 5 O n i 5 1 ( r i 3 m i a i ) (14.5) Defining the linear momentum L and the angular momentum H O about point O of the system of particles [Sec. 14.3] as L 5 O n i 5 1 m i v i H O 5 O n i 5 1 ( r i 3 m i v i ) (14.6, 14.7) we showed that Eqs. (14.4) and (14.5) can be replaced by the equations o F 5 L . o M O 5 H . O (14.10, 14.11) which express that the resultant and the moment resultant about O of the external forces are, respectively, equal to the rates of change of the linear momentum and of the angular momentum about O of the system of particles . In Sec. 14.4, we defined the mass center of a system of particles as the point G whose position vector r satisfies the equation m r 5 O n i 5 1 m i r i (14.12) Effective forces Effective forces Linear and angular momentum of a system of particles Linear and angular momentum of a system of particles Motion of the mass center of a system of particles Motion of the mass center of a system of particles 906 Systems of Particles where m represents the total mass O n i 5 1 m i of the particles. Differ- entiating both members of Eq. (14.12) twice with respect to t , we obtained the relations L 5 m v ˙ L 5 m a (14.14, 14.15) where v and a represent, respectively, the velocity and the accelera- tion of the mass center G . Substituting for ....
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- Fall '10