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PHYS 172: Modern Mechanics Lecture 4 – Physical Models, Fundamental Interactions Fall 2011 Read 2.7–2.8, 3.1-3.4 Predictions using the Momentum Principle The Momentum Principle Dp = Fnet Dt Update form of the momentum principle p f - pi = Fnet Dt p f = pi + Fnet Dt p fx , p fy , p fz pix , piy , piz Fnet , x , Fnet , y , Fnet , z t For components: p fx pix Fnet , x t p fy piy Fnet , y t p fz piz Fnet , z t Short enough, F~const Example Dp = Fnet Dt p f = pi + Fnet Dt Force: provided by a spring stretched by L=4 cm interaction duration: 1 s ? Find momentum pf if pi=<0,0,0> kg.m/s 1. Force: Fspring = kS DL Fspring = 500(N/m)0.04(m)=20 N NB: force must not change during t Fspring = 20,0,0 N 2. Momentum: p f = pi + Fnet Dt ( ) p f =< 0,0,0 > kg × m/s + 20,0,0 N × 1 s p f =< 20,0,0 > kg × m/s N.s = kg.m/s2.s = kg.m/s Physical models “Spherical cow” Ideal model: ignore factors that have no significant effect on the outcome Example: colliding students Two students are late for class and run into each other head-on. Q: Estimate the force that one student exerts on the other during collision Simplest model: F Ffloor , N Fair Ffloor , P FEarth System: one spherical student Surroundings: earth, floor, air, second spherical student Force: Earth, floor, air, other student – unknown! Example: colliding students y Ffloor , N F Strategy: p f pi Fnet t p mv Fair Ffloor , P FEarth rf ri vavg t x p f pi Fnet t 0, 0 pix , 0 Ffloor , P Fair F , Ffloor , N FEarth t pix ,0 F ,0 t pix F t Example: colliding students y Ffloor , N F Strategy: p f pi Fnet t p mv Fair Ffloor , P FEarth pix F t What is the collision time? vavg x t x t vavg What is the initial momentum? Find F: rf ri vavg t x Assume: vi =5 m/s, x=0.05m x t vi v f / 2 Assume: m=60 kg pix 300 kg m/s F 15000 N t 0.02 s t 0.02 s pix mvix 300 kg m/s Newton’s Great Insight: The force that attracts things toward the earth (e.g. a falling apple) is the same force that keeps planets orbiting about the sun The gravitational force law m2 Newton r21 Fgrav on 2by1 G r̂21 m1 r2 m2 r21 r2 r1 r1 m1 m2 m1 r21 2 rˆ2 1 Cavendish G 6.7 10 11 N×m2 kg 2 Gravitational constant Predicting motion of a planet Where will the planet be after one month? Use position update formula: p rf ri vavg t If we assume that velocity is constant F Does not work because the force is changing the velocity! The force changes with position. The momentum changes with position. In general, there is no algebraic equation to predict motion of more than 2 interacting objects. Iterative prediction of a motion of one planet Simple case: one planet star is fixed in space 1. Calculate gravitational force: Fgrav on 2by1 G p m2 m1 r21 2. Update momentum 2 rˆ2 1 p f pi Fnet t Choose t short enough (F & p do not change much) F 3. Calculate v and update position rf ri vavg t 4. Repeat Critical parameter: t Iterative prediction of motion Real case: many objects objects are free to move 1. Calculate net force on each mass: Fon m = å Fm i i¹ j j on mi 2. Update momentum of each mass p f = pi + Fnet Dt Choose t short enough (F & v do not change much) 3. Calculate v and update position of each mass Iterative approach: works for any kind of force, not just gravity! rf = ri + vavg Dt 4. Repeat t is a critical parameter!