Physics 319 Classical Mechanics. G. A. Krafft Old Dominion University Jefferson Lab Lecture 2

Transcription

1 Physics 319 Clssicl Mechnics G. A. Krfft Old Dominion University Jefferson Lb Lecture Undergrdute Clssicl Mechnics Spring 017

2 Sclr Vector or Dot Product Tkes two vectors s inputs nd yields number (sclr) b bcos b b b b b b x x y y z z Liner in both inputs nd symmetricl For norml (perpendiculr) vectors vnishes Mximum when vectors prllel (co-liner) Get the vector norm by x y z v v v v v v v Undergrdute Clssicl Mechnics Spring 017

3 Vector Cross Product Tkes two vectors s inputs nd yields norml vector b b Liner in both inputs nd nti-symmetricl For co-liner vectors vnishes Mximum when vectors perpendiculr Norm of the vector product is the re of prllelogrm spnned by the input vectors b bsin Importnt when rottions involved b b eˆ i1 j1 k 1 xˆ yˆ zˆ det x y z bx by b z ijk i j k Undergrdute Clssicl Mechnics Spring 017

4 3D Velocity nd Accelertion Consider prticle orbit The velocity is v t By product rule from clculus Accelertion is dr lim t 0 3 x teˆ t r t i1 r t t r t t dxit ˆ ˆ d deˆ i t xi t ei t ei t xi t i1 i1 i1 3 3 ˆ 3 ˆi dv d x t dx t de t d eˆ t t e t x t i i i i i i i i1 i1 i1 Undergrdute Clssicl Mechnics Spring 017

5 Simplest Cse In frmes where the unit vectors do not depend on time For constnt forces hve deˆ t i t 3 i1 Directions solved seprtely with the freshmn physics result. Simple expressions nd equtions only in inertil frme. ˆ d ei t d xi t 0 eˆ d xi t F eˆ i Fi m m 1 Fi xi t xi 0 vi 0t t m i Undergrdute Clssicl Mechnics Spring 017

6 Incline Plne Problem Undergrdute Clssicl Mechnics Spring 017

7 Inertil frme Frme of reference where the frme unit vectors do not depend on time (there is no rottion) nd where there re no externl forces nd ccelertions. Opertionl definition: frme where observer feels no inertil forces. By the previous result: only if the frme is t rest or uniformly trnslting t constnt velocity Newton s first lw: in the bsence of forces prticle moves t constnt velocity. A frme of reference tied to the prticle motion is n inertil frme. Newton s second lw: prticle of mss m cted on by vector force ccelertes with the vector ccelertion F m Undergrdute Clssicl Mechnics Spring 017

8 Wht Newton Relly Sid He defined something tht we cll momentum tody p mv nd sid the totl vector force ws the time derivtive of the momentum F dp This formul ctully works in reltivistic situtions if include proper reltivistic mss in the momentum In clssicl mechnics the two formultions re obviously equivlent becuse dp m Undergrdute Clssicl Mechnics Spring 017

9 Newton s Third Lw Essentilly wht we now cll conservtion of totl momentum: if object 1 exerts force F 1 on object, then object exerts n equl nd opposite force on object 1: F 1 = F 1 dp p tot dp tot tot b b ext, ext, Fb b F p dp F 0 F F F tot dptot 0 0 F ext, Undergrdute Clssicl Mechnics Spring 017

10 Polr coordintes First experience with rotting coordinte system. Unit vectors cn now hve time derivtives. x r cos r x y You show 1 y y rsin tn x r r t t x r t t yˆ rˆ t cost xˆ sint yˆ r cos ˆ sin r t sin cos ˆ t t x ˆ t y ˆ Undergrdute Clssicl Mechnics Spring 017

11 Derivtives nd Second Derivtives Derivtives drˆ t d ˆ t Second Derivtives ˆ t xˆ cos t yˆ ˆt sin t xˆ sin t yˆ rˆ t cos d r t d ˆ t t t t r t ˆ ˆ ˆ ˆ ˆ rˆ t rˆ t rˆ t t Undergrdute Clssicl Mechnics Spring 017

12 Accelertion Accelertion clculted from functions r(t) nd θ(t) d d r t dr t t r t rˆ t rˆ t ˆ t r t t rˆ t Resolved into cylindricl components F rˆ t dr r r m m r d r m m ˆ d z F z z m m F ˆ t F F ˆ F Undergrdute Clssicl Mechnics Spring 017

13 In this Rotting Coordinte System The form of the equtions of motion is no longer the sme. It hs extr terms Centrifugl force pushing wy from the rottion xis So-clled Coriolis ccelertion (more generl expression lter) When F θ vnishes d mr mr This is the conserved ngulr momentum in the z direction. Importnt for the plnetry motion problem (Kepler s second lw) mr 0 Undergrdute Clssicl Mechnics Spring 017

14 Sktebord Problem Undergrdute Clssicl Mechnics Spring 017