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PHYS 172: Modern Mechanics Lecture 6 – Momentum Conservation, Complex Systems Summer 2012 Read 3.11-3.15 Electric force: the electric charges Charges: property of an object • Two types: positive (+) and negative (-) • Like charges: repel. • Opposite charges: attract • Net charge of a system: algebraic sum of all the charges • Conservation of charge • The force exerted by one point charge on another acts along the line joining the charges Charge: measured in C (Coulomb) Elementary charge: e = 1.602×10-19 C Charge of electron is –e, of a proton +e The electric force law (Coulomb’s law) q1 Felec on1by 2 q2 r21 Felec on 2by1 Coulomb’s law r̂21 Felec on 2by1 1 q2 q1 4 0 r21 2 2 N×m 9 109 4 0 C2 1 rˆ21 Predicting the future of a gravitational system Massive star And small planets fixed Two body: ellipse (or circle) fixed Determinism: If we know the positions and momenta of all particles in the Universe we can predict the future Predicting the future of gravitational system Solar system Binary star Sun, Earth and Moon Problems: Sensitivity Initial condition and t Inability to account for all interactions 1025 molecules in glass of water ! Small particles: quantum mechanics Probability and uncertainty Example: a free neutron decays with ~15 minutes: n p e Probability t Clicker: Can we predict the motion of an electron near a free neutron? A)Yes B) No Probability and uncertainty Clicker poll: The photon will: A) B) Semitransparent mirror photon Reflect Pass through Where is the electron? Classical Quantum The Heisenberg uncertainty principle Werner Karl Heisenberg 1901-1976 Position and momentum of a particle cannot be exactly measured simultaneously 1927 Nobel Prize: 1932 xpx h Planck’s constant h = 6.6×10-34 kg.m2/s E=h 1900 Nobel Prize: 1918 Max Planck 1858-1947 System consisting of two objects F1,surr system p1 Momentum principle: F1,int ( ) Dp1 º p f ,1 - pi,1 = F1,surr + F1,int Dt + ( ) Dp2 º p f ,2 - pi,2 = F2,surr + F2,int Dt F2,int p2 F2,surr ( ) p f ,1 + p f ,2 - pi,1 - pi,2 = F1,surr + F2,surr + F1,int + F2,int Dt ptotal, f - ptotal,i Fnet ,surr Clicker question 2: What is the last term in that equation? A) Zero B) Not zero, directed toward the larger object C) Not zero, directed toward the smaller object D) It is always positive E) It is always negative System consisting of two objects F1,surr system p1 Momentum principle: F1,int ( ) Dp1 º p f ,1 - pi,1 = F1,surr + F1,int Dt + ( ) Dp2 º p f ,2 - pi,2 = F2,surr + F2,int Dt F2,int p2 F2,surr ( ) p f ,1 + p f ,2 - pi,1 - pi,2 = F1,surr + F2,surr + F1,int + F2,int Dt ptotal, f - ptotal,i Fnet ,surr 0 Dptotal º ptotal, f - ptotal,i = Fnet Dt Only from surrounding! System consisting of many objects Because all the forces inside the system come in pairs they cancel out. The only forces left over are forces from the surroundings! System consisting of several objects The Momentum Principle for a system: ptotal ptotal , f ptotal ,i Fnet t Total momentum of the system: ptotal p1 p2 p3 ... The sum of all external forces due to surrounding Conservation of momentum system p1 F1,int + p1 p f ,1 pi ,1 F1,int t p2 p f ,2 pi ,2 F2,int t p f ,1 p f ,2 pi ,1 pi ,2 F1,int F2,int t F2,int ptotal , f ptotal ,i p2 0 In the absence of external forces system p1 p2 0 p1 F1 Conservation of momentum psystem psurrounding 0 F2 p2 surrounding ptotal 0 Reciprocity: Newton’s 3rd law Fspring on mass Fmass on spring p Fnet t Force magnitudes are the same Directions are opposite They act on different objects Fspring on mass Fmass on spring Reciprocity (Newton’s 3rd law): The forces of two objects on each other are always equal and are directed in opposite directions NOTE: Velocity-dependent forces (e.g., magnetic forces) do not obey Newton’s 3rd law! Collisions: negligible external forces p1 f p2 f p1i 1. Sticky ball p1,i p2,i p1, f p2, f Assume =1: m1 p2i Momentum conservation: m1v1i m2v2i m1v f m2v f m1v1i m2v2i vf m1 m2 m2 What if balls bounce? m1v1i m2v2i m1v1 f m2v2 f Two unknowns, one equation