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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 2 (m 2 ). A 1 (m 1 ). A 3 (m 3 ). G. O. 2 – Characteristic property. . A n (m n ).
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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 Ai, coefficiented by mi, is defined by: A2 (m2) . A1 (m1) . A3(m3) . G . O . 2 – Characteristic property . An(mn) B. Rossetto
7. System of particles • Momentum A2(m2) . 1 – Definition. Momentum of the sytem referred to any origin O: A1(m1) . A3(m3) . G . O . 2 – Property: from the definition of CM: . An(mn) Particularly, the momentum of the system referred to G is null: B. Rossetto
7. System of particles • Fundamental theorem (Newton 2nd 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. .G(M) System 2 – Theorem of the center of mass: 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. B. Rossetto
7. System of particles • Angular momentum theorems (1) Theorem 1 B. Rossetto
7. System of particles • Angular momentum theorems (2) Lemma Theorem 2. From the precedent results: The CM moves as if the only external torques were applied to it. B. Rossetto
7. System of particles • Theorems rCM : location of the center of mass referred to an inertial frame /Oxyz vCM : 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 Fext,text : external force, torque, M: total mass of the system B. Rossetto
7. System of particles • Relative motion and reduced mass 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. Proof: Example: sun and eath isolated or earth and moon (isolated…) B. Rossetto