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Seventh Edition VECTOR MECHANICS FOR ENGINEERS: CHAPTER DYNAMICS Ferdinand P. Beer Systems of Particles E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Texas Tech University


  1. Seventh Edition VECTOR MECHANICS FOR ENGINEERS: CHAPTER �� DYNAMICS Ferdinand P. Beer Systems of Particles E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Texas Tech University � ������������������������������������������������������� ��

  2. Edition Seventh Vector Mechanics for Engineers: Dynamics Contents Introduction Principle of Impulse and Momentum Application of Newton’s Laws: Sample Problem 14.4 Effective Forces Sample Problem 14.5 Linear and Angular Momentum Variable Systems of Particles Motion of Mass Center of System of Steady Stream of Particles Particles Steady Stream of Particles. Applications Angular Momentum About Mass Center Streams Gaining or Losing Mass Conservation of Momentum Sample Problem 14.6 Sample Problem 14.2 Kinetic Energy Work-Energy Principle. Conservation of Energy � ������������������������������������������������������� �� 14 - 2

  3. Edition Seventh Vector Mechanics for Engineers: Dynamics Introduction • In the current chapter, you will study the motion of systems of particles . • The effective force of a particle is defined as the product of it mass and acceleration. It will be shown that the system of external forces acting on a system of particles is equipollent with the system of effective forces of the system. • The mass center of a system of particles will be defined and its motion described. • Application of the work-energy principle and the impulse- momentum principle to a system of particles will be described. Result obtained are also applicable to a system of rigidly connected particles, i.e., a rigid body . • Analysis methods will be presented for variable systems of particles , i.e., systems in which the particles included in the system change. � ������������������������������������������������������� �� 14 - 3

  4. Edition Seventh Vector Mechanics for Engineers: Dynamics Application of Newton’s Laws. Effective Forces • Newton’s second law for each particle P i in a system of n particles, � � � n � + = F f m a i ij i i = j 1 � � � n � � � ( ) � × + × = × r F r f r m a i i i ij i i i = j 1 � � = = F external force f internal forces i ij � = m a effective force i i • The system of external and internal forces on a particle is equivalent to the effective force of the particle. • The system of external and internal forces acting on the entire system of particles is equivalent to the system of effective forces. � ������������������������������������������������������� �� 14 - 4

  5. Edition Seventh Vector Mechanics for Engineers: Dynamics Application of Newton’s Laws. Effective Forces • Summing over all the elements, � � n n n n � � � � � + = F f m a i ij i i = 1 = 1 = 1 = 1 i i j i � � n � n n � n � � ( ) ( ) � � � � ( ) × + × = × r F r f r m a i i i ij i i i = = = = i 1 i 1 j 1 i 1 • Since the internal forces occur in equal and opposite collinear pairs, the resultant force and couple due to the internal forces are zero, � � � � = F m a i i i � � � � ( ) � � ( ) × = × r F r m a i i i i i • The system of external forces and the system of effective forces are equipollent by not equivalent . � ������������������������������������������������������� �� 14 - 5

  6. Edition Seventh Vector Mechanics for Engineers: Dynamics Linear & Angular Momentum • Linear momentum of the system of • Angular momentum about fixed point O of particles, system of particles, � � n � n � � � � ( ) = = × L m v H r m v O i i i i i = = i 1 i 1 � � n � n � n � � n � � � � � � � � � � � ( ) ( ) = × + × = = H r m v r m v L m v m a i i i i O i i i i i i = = = 1 = 1 i 1 i 1 i i n � � � ( ) = × r m a • Resultant of the external forces is i i i = i 1 equal to rate of change of linear momentum of the system of • Moment resultant about fixed point O of particles, the external forces is equal to the rate of � � � � change of angular momentum of the system = F L of particles, � � � � = M H O O � ������������������������������������������������������� �� 14 - 6

  7. Edition Seventh Vector Mechanics for Engineers: Dynamics Motion of the Mass Center of a System of Particles • Mass center G of system of particles is defined by � position vector which satisfies r G � n � � = m r m r G i i = i 1 • Differentiating twice, � n � � � � = m r m r G i i = i 1 � � n � � = = m v m v L G i i = i 1 � � � � � = = m a L F G • The mass center moves as if the entire mass and all of the external forces were concentrated at that point. � ������������������������������������������������������� �� 14 - 7

  8. Edition Seventh Vector Mechanics for Engineers: Dynamics Angular Momentum About the Mass Center • The angular momentum of the system of particles about the mass center, � n � � � ( ) ′ ′ ′ = × H r m v G i i i = i 1 � n � � n � � � � � � ( ) ( ( ) ) ′ ′ ′ ′ = × = × − H r m a r m a a G i i i i i i G = = i 1 i 1 � � � � � n � � n � � ′ = + a a a � � � � ( ) i G i ′ ′ = × − × r m a m r a � � i i i i G � � i = 1 i = 1 • Consider the centroidal frame � n � � n � ( ) � � ( ) ′ ′ = × = × r m a r F of reference Gx’y’z’, which i i i i i = = i 1 i 1 translates with respect to the � � = M Newtonian frame Oxyz. G • The moment resultant about G of the external forces • The centroidal frame is not, in is equal to the rate of change of angular momentum general, a Newtonian frame. about G of the system of particles. � ������������������������������������������������������� �� 14 - 8

  9. Edition Seventh Vector Mechanics for Engineers: Dynamics Angular Momentum About the Mass Center • Angular momentum about G of particles in their absolute motion relative to the Newtonian Oxyz frame of reference. � n � � � ( ) ′ = × H r m v G i i i = i 1 n � � � � ( ( ) ) ′ ′ = × + r m v v i i G i � � � = i 1 ′ = + v v v i G G � � n � � n � � � � � � ( ) ′ ′ = × + × m r v r m v � � i i G i i i � � • Angular momentum about G of the i = 1 i = 1 � � � � particles in their motion relative to ′ = = H H M G G G the centroidal Gx’y’z’ frame of reference, • Angular momentum about G of the particle � n � � momenta can be calculated with respect to either � ( ) ′ ′ ′ = × H r m v G i i i the Newtonian or centroidal frames of reference. = i 1 � ������������������������������������������������������� �� 14 - 9

  10. Edition Seventh Vector Mechanics for Engineers: Dynamics Conservation of Momentum • If no external forces act on the • Concept of conservation of momentum also particles of a system, then the linear applies to the analysis of the mass center momentum and angular momentum motion, � � � � about the fixed point O are conserved. � � � � = = = = L F 0 H M 0 G G � � � � � � = = L m v constant � � G � � � = = = = � L F 0 H M 0 O O � � = = v constant H constant G G = = L constant H constant O • In some applications, such as problems involving central forces, � � � � � � � � = ≠ = = L F 0 H M 0 O O � � ≠ = L constant H constant O � ������������������������������������������������������� �� 14 - 10

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