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UB PHY 107 - Center of Mass

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PHY 107 1st Edition Lecture 18 Outline of Last Lecture I. Conservation of Mechanical Energy (cont.) Outline of Current LectureII. Center of MassIII. Linear Momentum Current LectureCenter of Mass – COM- Consider 2 particles of mass m1 and m2 at positions x1 and x2  Position of the COM is xcom = m1x1+m2x2m1+m2 In general, xcom = m1x1+m2x2+m3x3+…+mnxnm1+m2+m3+…+mn = 1M (M is the total mass ofall the particles) - Position vector ´r´rcom = 1M∑i=1nmi´ri ´rcom = xcom^i + ycom^j + zcom^k xcom = 1M∑i=1nmixi, ycom = 1M∑i=1nmiyi, zcom = 1M∑i=1nmiziThe COM of a system of particles moves as though all the mass were concentrated at the COM, and the vector sum of all the external forces were applied at the COM. These notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.- COM for solid bodies (solid bodies can be considered as systems with continuous distribution of matter) xcom = 1M∫xdm, ycom = 1M∫ydm, zcom = 1M∫zdm- Special case: uniform objects with mass density ρ = dmdv is constant and equal tomv xcom = 1V∫xdv, ycom = 1V∫ydv, zcom = 1V∫zdv-m ´vcom = m1´v1 + m2´v2 + m3´v3 + … + mn´vn-m ´acom = m1´a1 + m2´a2 + m3´a3 + … + mn´an- Total force can be decomposed into 2 components: applied and internal  = mi´aiM ´acom = ´F1 + ´F2 + ´F3 + … + ´Fn-´Fnet is the net externally applied force ´Fnet , x = Macom,x, ´Fnet , y = Macom,y, ´Fnet , z = Macom,zM ´acom = ´F1app + ∫¿´F1¿ + ´F2app + ∫¿´F2¿ + … + ´Fnapp + ∫¿´Fn¿ → (´F1app + ´F2app + … + ´Fnapp) + (∫¿´F1¿ +∫¿´F2¿ + … + ∫¿´Fn¿)M ´acom = ´FnetLinear Momentum:-´p (kg · m/s) → ´p = m´v- The time rate of change of the linear momentum of a particle is equal to the magnitude of the net force acting on the particle and has the direction of the force´Fnet = d ´pdt- Linear momentum of a particle can be changed ONLY by an external force -´p of a system of particles  i-th particle of mass mi, velocity ´vi, and linear momentum ´pi´p = ´p1 + ´p2 + p + … + ´pn = m1´v1 + m2´v2 + m3´v3 + … +mn´vn = M


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