Summer School 2007B. Rossetto1 7. System of particles  Center of mass 1 – Definition. If O is any point in the plane, the center of mass G of n particles.

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Summer School 2007B. Rossetto1 7. System of particles  Center of mass 1 – Definition. If O is any point in the plane, the center of mass G of n particles A i, coefficiented by m i, is defined by: A 1 (m 1 ). O. A 2 (m 2 ). A 3 (m 3 ).. 2 – Characteristic property G. A n (m n )

Summer School 2007B. Rossetto2 7. System of particles  Momentum 1 – Definition. Momentum of the sytem referred to any origin O: 2 – Property: from the definition of CM: A 1 (m 1 ). O. A 2 (m 2 ). A 3 (m 3 ).. G A n (m n ) Particularly, the momentum of the system referred to G is null:

Summer School 2007B. Rossetto3 7. System of particles  Fundamental theorem (Newton 2 nd law) 1 – Action-reaction « principle ». Consider a system of n particles. From the inertia principle applied to the system, the sum of internal forces is null. 2 – Theorem of the center of mass: System or The CM moves as if it were a particle of mass equal to the total mass and if all the external forces were applied to it..G(M)

Summer School 2007B. Rossetto4 7. System of particles  Angular momentum theorems (1) Theorem 1

Summer School 2007B. Rossetto5 7. System of particles  Angular momentum theorems (2) Lemma The CM moves as if the only external torques were applied to it. Theorem 2. From the precedent results:

Summer School 2007B. Rossetto6 7. System of particles  Theorems r CM : location of the center of mass referred to an inertial frame /Oxyz v CM : velocity of the center of mass /Oxyz P, L : total momentum, angular momentum of the system /Oxyz P/ CM, L/ CM : momentum, angular momentum of the system referred to the center of mass F ext,  ext : external force, torque, M: total mass of the system

Summer School 2007B. Rossetto7 7. System of particles  Relative motion and reduced mass Proof: The relative motion of two particles subject only to their mutual interaction is equivalent to the motion, relative to an inertial observer, of a particle of mass equal to to the reduced mass under a force equal to their interaction. Example: sun and eath isolated or earth and moon (isolated…)