Geometry and Motion of Isotropic Harmonic Oscillators (Ch. 9 and Ch. 11 of Unit 1)

Transcription

1 Lecture Revised.. from..0 Geometr and Motion of Isotropic Harmonic Oscillators (Ch. and Ch. of Unit ) Geometr of idealized Sophomore-phsics Earth Coulomb field outside Isotropic Harmonic Oscillator (IHO) field inside Contact-geometr of potential curve(s) Crushed-Earth models: ke energ steps and ke energ levels Earth matter vs nuclear matter: Introducing the neutron starlet and Black-Hole-Earth Isotropic harmonic oscillator dnamics in D, D, and D Sinusoidal -time dnamics derived b geometr Isotropic harmonic oscillator orbits in D and D (You get D for free!) Constructing D Isotropic harmonic oscillator orbits using plots Eamples with - phase lag : α-= α-α =, 0, and ±

2 Geometr of idealized Sophomore-phsics Earth Coulomb field outside Isotropic Harmonic Oscillator (IHO) field inside Contact-geometr of potential curve(s) Crushed-Earth models: ke energ steps and ke energ levels Earth matter vs nuclear matter: Introducing the neutron starlet and Black-Hole-Earth

3 Coulomb force vanishes inside-spherical shell (Gauss-law) Gravit at r due to shell mass elements G M - G m D d D - d ( )A = 0 D d = Shell mass element m =(solid-angle factor A) d B dω A= sinθ Cancellation of Shell mass element M =(solid-angle factor A)D Θ Θ d M m D You are Here! O r You are Here! (...and weightless!) Coulomb force inside-spherical bod due to stuff below ou, onl. Gravitational force at r < is just that of planet M < below r < Unit Fig.. M < r <

4 Coulomb force vanishes inside-spherical shell (Gauss-law) Gravit at r due to shell mass elements G M - G m D d D - d ( )A = 0 D d = Shell mass element m =(solid-angle factor A) d B dω A= sinθ Cancellation of Shell mass element M =(solid-angle factor A)D Θ Θ d M m D You are Here! O r You are Here! (...and weightless!) Coulomb force inside-spherical bod due to stuff below ou, onl. M < r < Gravitational force at r < is just that of planet M < below r < Note: Hooke s (linear) force law for r< inside uniform bod Unit Fig.. F inside (r < ) = G mm < r < = Gm π M < π r < r < = Gm π ρ r < = mg r < R mg

5 Coulomb force vanishes inside-spherical shell (Gauss-law) Gravit at r due to shell mass elements G M - G m D d D - d ( )A = 0 D d = Shell mass element m =(solid-angle factor A) d B dω A= sinθ Cancellation of Shell mass element M =(solid-angle factor A)D Θ Θ d M m D You are Here! O r You are Here! (...and weightless!) Coulomb force inside-spherical bod due to stuff below ou, onl. M < r < Gravitational force at r < is just that of planet M < below r < Note: Hooke s (linear) force law for r< inside uniform bod Unit Fig.. Earth surface gravit acceleration: g = G M G=.(0) 0 - Nm /C ~(/)0-0 F inside (r < ) = G mm < r < = Gm π R = G M R R = G π M < π r < M π R r < = Gm π R = G π ρ r < = mg r < R mg ρ R =.m / s

6 Coulomb force vanishes inside-spherical shell (Gauss-law) Gravit at r due to shell mass elements G M - G m D d D - d ( )A = 0 D d = Shell mass element m =(solid-angle factor A) d B dω A= sinθ Cancellation of Shell mass element M =(solid-angle factor A)D Θ Θ d M m D You are Here! O r You are Here! (...and weightless!) Coulomb force inside-spherical bod due to stuff below ou, onl. M < r < Earth surface gravit acceleration: g = G M G=.(0) 0 - Nm /C ~(/)0-0 Earthradius : R =. 0 m. 0 m Gravitational force at r < is just that of planet M < below r < F inside (r < ) = G mm < r < = Gm π R Earth mass : M =. 0 kg..0 0 kg. = G M R R = G π M < π r < M π R r < = Gm π R = G π Note: Hooke s (linear) force law for r< inside uniform bod ρ r < = mg r < R mg ρ R =.m / s Unit Fig.. Solar radius : R =. 0 m..0 0 m. Solar mass : M =. 0 0 kg kg.

7 Geometr of idealized Sophomore-phsics Earth Coulomb field outside Isotropic Harmonic Oscillator (IHO) field inside Contact-geometr of potential curve(s) Crushed-Earth models: ke energ steps and ke energ levels Earth matter vs nuclear matter: Introducing the neutron starlet and Black-Hole-Earth

8 The ideal Sophomore-Phsics-Earth model of geo-gravit F()=- F()=-/ (inside Earth) (outside Earth) U()=( -)/ U()=-/ (0,0) 0. =.0 F(-.0) F(-.) F(.0) F(.) F(-.0) - F(-0.) F(.0) F(0.) F(.) Unit Fig.. Focus Latus rectum λ F(-0.) - F(0.) Parabolic potential inside Earth Sub-directri Eample of contacting line and contact point focal distance = directri distance Directri Directri

9 ...conventional parabolic geometr...carried to etremes (Review of Lect. p.) (a) Parabola p ==λ Circle of curvature radius=λ p p (b) Parabola p ==λ p p Latus rectum λ =p radius p p p p tangent slope=- p=λ/ 0 =0 p p p p p slope= directri =-p tangent slope=-/ p 0 =0 p p p p p slope=/ =-p Unit Fig..

10 Geometr of idealized Sophomore-phsics Earth Coulomb field outside Isotropic Harmonic Oscillator (IHO) field inside Contact-geometr of potential curve(s) Crushed-Earth models: ke energ steps and ke energ levels Earth matter vs nuclear matter: Introducing the neutron starlet and Black-Hole-Earth 0

11 The Three (equal) steps from Hell surface gravit: g = G M R Dissociation threshold : PE= 0 -orbit energ: E = G M R -orbit speed: v = G M R Orbit level : PE= G M R Surface level : PE= G M R Ground level : PE= G M R surface potential: PE= G M R surface escape speed: v e = G M R KE = PE relation: mv e =G mm R

12 If ou could crush Earth radius -times-smaller... Crushed Earth / radius times as dense / focal distance or λ / minimum radius of curvature times maimum curvature times -orbit energ: E = G M R times -orbit speed: v = G M R times the surface potential: PE= G M R times surface escape speed: v e = G M R times the surface gravit: g = G M R

13 Geometr of idealized Sophomore-phsics Earth Coulomb field outside Isotropic Harmonic Oscillator (IHO) field inside Contact-geometr of potential curve(s) Crushed-Earth models: ke energ steps and ke energ levels Earth matter vs nuclear matter: Introducing the neutron starlet and Black-Hole-Earth

14 Eamples of crushed matter Earth matter Earth mass : M Densit ~ ~ 0 kg/m =. 0 kg..0 0 kg. Earthradius : R =. 0 m. 0 m Earthvolume :(π / )R 0 0 ~ 0 m ρ

15 Eamples of crushed matter Earth matter Earth mass : M Densit ~ ~ 0 kg/m =. 0 kg..0 0 kg. Earthradius : R =. 0 m. 0 m Earthvolume :(π / )R 0 0 ~ 0 m Nuclear matter Nucleon mass =. 0 - kg.~ 0 - kg. Sa a nucleus of atomic weight 0 has a radius of fm, or 0 nucleons each with a mass 0 - kg. That s =0 - kg packed into a volume of π /r = π / ( 0 - ) m or about 0 - m. ρ

16 Eamples of crushed matter Earth matter Earth mass : M Densit ~ ~ 0 kg/m =. 0 kg..0 0 kg. Earthradius : R =. 0 m. 0 m Earthvolume :(π / )R 0 0 ~ 0 m Nuclear matter Nucleon mass =. 0 - kg.~ 0 - kg. Sa a nucleus of atomic weight 0 has a radius of fm, or 0 nucleons each with a mass 0 - kg. That s =0 - kg packed into a volume of π /r = π / ( 0 - ) m or about 0 - m. Nuclear densit is 0 -+ = 0 kg /m or a trillion (0 ) kilograms in the size of a fingertip. ρ Earth radius crushed b a factor of to R crush 00m would approach neutron-star densit.

17 Eamples of crushed matter Earth matter Earth mass : M Densit ~ ~ 0 kg/m =. 0 kg..0 0 kg. Earthradius : R =. 0 m. 0 m Earthvolume :(π / )R 0 0 ~ 0 m Nuclear matter Nucleon mass =. 0 - kg.~ 0 - kg. Sa a nucleus of atomic weight 0 has a radius of fm, or 0 nucleons each with a mass 0 - kg. That s =0 - kg packed into a volume of π /r = π / ( 0 - ) m or about 0 - m. Nuclear densit is 0 -+ = 0 kg /m or a trillion (0 ) kilograms in the size of a fingertip. ρ Earth radius crushed b a factor of to R crush 00m would approach neutron-star densit. Introducing the Neutron starlet cm of nuclear matter: mass= 0 kg.

18 Eamples of crushed matter Earth matter Earth mass : M Densit ~ ~ 0 kg/m =. 0 kg..0 0 kg. Earthradius : R =. 0 m. 0 m Earthvolume :(π / )R 0 0 ~ 0 m Nuclear matter Nucleon mass =. 0 - kg.~ 0 - kg. Sa a nucleus of atomic weight 0 has a radius of fm, or 0 nucleons each with a mass 0 - kg. That s =0 - kg packed into a volume of π /r = π / ( 0 - ) m or about 0 - m. Nuclear densit is 0 -+ = 0 kg /m or a trillion (0 ) kilograms in the size of a fingertip. ρ Earth radius crushed b a factor of to R crush 00m would approach neutron-star densit. Introducing the Neutron starlet cm of nuclear matter: mass= 0 kg. Introducing the Black Hole Earth Suppose Earth is crushed so that its surface escape velocit is the speed of light c =.0 0 m/s. c = (GM/R ) R = GM/c = mm ~cm (fingertip size!)

19 Isotropic harmonic oscillator dnamics in D, D, and D Sinusoidal -time dnamics derived b geometr Isotropic harmonic oscillator orbits in D and D (You get D for free!) Constructing D Isotropic harmonic oscillator orbits using plots Eamples with - phase lag : α-= α-α =, 0, and ±

20 Isotropic Harmonic Oscillator phase dnamics in uniform-bod I.H.O. Force law F =- (-Dimension) (a) F =-r ( or -Dimensions) Each dimension,, or z obes the following: F=-r Total E = KE + PE = Equations for -motion mv +U() = mv + k = const. [(t) and v=v(t)] are given first. The appl as well to dimensions [(t) and v=v(t)] and [z(t) and vz=v(t)] in the ideal isotropic case. F F (b) v 0 Unit Fig..0 -D -D (Paths are alwas -D ellipses if viewed right!) 0

21 Isotropic Harmonic Oscillator phase dnamics in uniform-bod I.H.O. Force law F =- (-Dimension) (a) F =-r ( or -Dimensions) Each dimension,, or z obes the following: F=-r Total E = KE + PE = mv +U() = mv + k = const. Equations for -motion [(t) and v=v(t)] are given first. The appl as well to dimensions [(t) and v=v(t)] and [z(t) and vz=v(t)] in the ideal isotropic case. = mv E + k E = = mv E + k E = v F E /m F + (b) E /k ( cosθ ) + ( sinθ ) Let : v = E /m cosθ, and : = E /k sinθ () () v 0 Unit Fig..0 -D -D (Paths are alwas -D ellipses if viewed right!) Another eample of the old scale-a-circle trick...

22 Isotropic Harmonic Oscillator phase dnamics in uniform-bod I.H.O. Force law F =- (-Dimension) (a) F =-r ( or -Dimensions) Each dimension,, or z obes the following: F=-r Total E = KE + PE = mv +U() = mv + k = const. Equations for -motion [(t) and v=v(t)] are given first. The appl as well to dimensions [(t) and v=v(t)] and [z(t) and vz=v(t)] in the ideal isotropic case. = mv E + k E = = mv E + k E = v F E /m F + (b) E /k ( cosθ ) + ( sinθ ) Let : v = E /m cosθ, and : = E /k sinθ () () v 0 Unit Fig..0 -D -D (Paths are alwas -D ellipses if viewed right!) Another eample of the old scale-a-circle trick... E m d cosθ = v= dt = dθ d d =ω dt dθ dθ =ω b () b def. () b () E k cosθ

23 Isotropic Harmonic Oscillator phase dnamics in uniform-bod I.H.O. Force law F =- (-Dimension) (a) F =-r ( or -Dimensions) Each dimension,, or z obes the following: F=-r Total E = KE + PE = mv +U() = mv + k = const. Equations for -motion [(t) and v=v(t)] are given first. The appl as well to dimensions [(t) and v=v(t)] and [z(t) and vz=v(t)] in the ideal isotropic case. = mv E + k E = = mv E + k E = v F E /m F + (b) E /k ( cosθ ) + ( sinθ ) Let : v = E /m cosθ, and : = E /k sinθ () () v 0 Unit Fig..0 -D -D (Paths are alwas -D ellipses if viewed right!) Another eample of the old scale-a-circle trick... E m d cosθ = v= dt = dθ d d =ω dt dθ dθ =ω E k b () b def. () cosθ ω = dθ dt = b def. () b () b ()/() k m

24 Isotropic Harmonic Oscillator phase dnamics in uniform-bod I.H.O. Force law F =- (-Dimension) (a) F =-r ( or -Dimensions) Each dimension,, or z obes the following: F=-r Total E = KE + PE = mv +U() = mv + k = const. Equations for -motion [(t) and v=v(t)] are given first. The appl as well to dimensions [(t) and v=v(t)] and [z(t) and vz=v(t)] in the ideal isotropic case. = mv E + k E = = mv E + k E = v F E /m F + (b) E /k ( cosθ ) + ( sinθ ) Let : v = E /m cosθ, and : = E /k sinθ () () v 0 Unit Fig..0 -D -D (Paths are alwas -D ellipses if viewed right!) Another eample of the old scale-a-circle trick... E m d cosθ = v= dt = dθ d d =ω dt dθ dθ =ω E k b () b def. () cosθ ω = dθ dt = b def. () b () b ()/() k m b integration given constant ω: θ = ω dt = ω t + α

25 Isotropic harmonic oscillator dnamics in D, D, and D Sinusoidal -time dnamics derived b geometr Isotropic harmonic oscillator orbits in D and D (You get D for free!) Constructing D Isotropic harmonic oscillator orbits using plots Eamples with - phase lag : α-= α-α =, 0, and ±

26 Isotropic Harmonic Oscillator phase dnamics in uniform-bod (a) -D Oscillator Phasor Plot velocit v ωt position Phasor goes clockwise b angle ωt (b) -D Oscillator Phasor Plot -position - (-Phase behind the -Phase) (0,) -velocit v (,) (,) (,) -position (,) clockwise orbit if is behind Lefthanded Unit Fig..0 -velocit 0 φ (,) counter-clockwise (,) if is behind (,-) (,-) (,-) Righthanded - (a) F=-r (b) v 0 F F

27 (a) Phasor Plots for -D Oscillator or Two D Oscillators (-Phase 0 behind the -Phase) velocit v - 0 a a - -position b Unit Fig.. 0 -velocit - - -position - - b (0,) (,)(,) b a (,) (,) (,) (0,) (,) (,) (,) b (b) -Phase 0 behind the -Phase - - -velocit v a a -position These are more generic eamples with radius of - differing from that of the -. (In-phase case) velocit -position b (,) (0,0) (,) (,) (,) (,) b a Fig..

28 Isotropic harmonic oscillator dnamics in D, D, and D Sinusoidal -time dnamics derived b geometr Isotropic harmonic oscillator orbits in D and D (You get D for free!) Constructing D Isotropic harmonic oscillator orbits using plots Eamples with - phase lag : α-= α-α =, 0, and ±

29 v 0 0

30 (t) = A cos(ω t α ) = A cos(α ω t) v v (t) ω = A sin(ω t α ) = A sin(α ω t) 0 α (t) = A cos(ω t α ) = A cos(α ω t) (t) ω = A sin(ω t α ) = A sin(α ω t) - α 0 Initial phases at t=0 0

31 (t) = A cos(ω t α ) = A cos(α ω t) v v (t) ω = A sin(ω t α ) = A sin(α ω t) 0 α ω t (t) = A cos(ω t α ) = A cos(α ω t) (t) ω = A sin(ω t α ) = A sin(α ω t) - α ω t 0 Later phases at time t

32 v 0 Ellipsometr Contact Plots vs. Relative phase Δα=α -α Δα =+ (Left-polarized clockwise case) Δα

33 +0 case Ellipsometr Contact Plots vs. Relative phase Δα=α -α Δα=α -α =+0 Left-polarized (clockwise) orbit 0 v α =+ α lags α b +0 α =+ Initial phase (α =,α =+ ) α =+ α =+ α lags α b r(t=0 ) Δα

34 +0 case Ellipsometr Contact Plots vs. Relative phase Δα=α -α Δα=α -α =+0 Left-polarized (clockwise) orbit 0 v Initial phase (α =,α =+ )...after time advances b 0 : (α = -0 =-0,α =+ -0 =- ) Δα r(ωt=0 ) r(ωt= )

35 case Ellipsometr Contact Plots vs. Relative phase Δα=α -α Δα=α -α = Right-polarized (anti-clockwise) orbit 0 α =+ α lags α b - v α =+0 α leads α b + Initial phase (α =0,α =+ ) Δα α = α lags α b - α leads α b + - r( ωt=0 ) α =

36 case Ellipsometr Contact Plots vs. Relative phase Δα=α -α Δα=α -α = Right-polarized (anti-clockwise) orbit Initial phase (α =0,α =+ )...after time advances b : (α =0 - =-,α =+ - =+0 ) α =+0 α lags α b α leads α b + α = α = v α lags α b α leads α b + -0 α = Δα r( ωt= ) r( ωt=0 )

37 v - v(0 ) 0 v 0 Ellipsometr Contact Plots vs. Relative phase Δα=α -α Δα = (Right-polarized anti-clockwise case) v v Initial phase (α =0,α =+ ) Δα r(0 )

38 v - v(0 ) v 0 0 Ellipsometr Contact Plots vs. Relative phase Δα=α -α Δα = (Right-polarized anti-clockwise case) v v Initial phase (α =0,α =+ )...after time advances b 0 : (α =0-0 =0,α = + -0 =0 ) Δα r(0 )

39 v - v(0 ) v 0 0 Ellipsometr Contact Plots vs. Relative phase Δα=α -α Δα = (Right-polarized anti-clockwise case) v v Initial phase (α =0,α =+ )...after time advances b 0 : (α =0-0 =0,α = + -0 = ) Δα r(0 )

40 v - v(0 ) 0 v 0 Ellipsometr Contact Plots vs. Relative phase Δα=α -α Δα = (Right-polarized anti-clockwise case) v v Initial phase (α =0,α =+ )...after time advances b 0 : (α =0-0 =0,α = + -0 = ) Δα r(0 )

41 v - v 0 Ellipsometr Contact Plots vs. Relative phase Δα=α -α Δα = (Right-polarized anti-clockwise case) v v(0 ) v 0 Initial phase (α =0,α =+ )...after time advances b 0 : (α =0-0 =0,α = + -0 = ) r(0 ) Δα

42 v - v 0 Ellipsometr Contact Plots vs. v v Relative phase Δα=α -α Δα = (Right-polarized anti-clockwise case) Initial phase (α =0,α =+ )...after time advances b 0 : (α =0-0 =-0,α = + -0 =- ) v(0 ) r(0 ) -0 Δα

43 v - v 0 Ellipsometr Contact Plots vs. v v Relative phase Δα=α -α Δα = (Right-polarized anti-clockwise case) ) Initial phase (α =0,α =+ )...after time advances b 0 : (α =0-0 =-0,α = + -0 =- ) v(0 ) r(0 ) -0 Δα

44 v - v 0 Ellipsometr Contact Plots vs. v v Relative phase Δα=α -α Δα = (Right-polarized anti-clockwise case) Initial phase (α =0,α =+ )...after time advances b 0 : (α =0-0 =-0,α = + -0 =- ) v(0 ) Δα r(0 )

45 v - v 0 Ellipsometr Contact Plots vs. v v Relative phase Δα=α -α Δα = (Right-polarized anti-clockwise case) Initial phase (α =0,α =+ )...after time advances b 0 : (α =0-0 =-0,α = + -0 =-0 ) v(0 ) Δα r(0 )

46 v - v 0 Ellipsometr Contact Plots vs. Relative phase Δα=α -α Δα = (Right-polarized anti-clockwise case) v v(0 ) v Initial phase (α =0,α =+ )...after time advances b 0 : (α =0-0 =-0,α = + -0 =- ) Δα r(0 )

47 v - v 0 Ellipsometr Contact Plots vs. v v(00 ) v Relative phase Δα=α -α Δα = (Right-polarized anti-clockwise case) Initial phase (α =0,α =+ )...after time advances b 00 : (α =0-00 =-0,α = =- ) Δα r(00 )

48 v - v v(0 ) Ellipsometr Contact Plots vs. v v Relative phase Δα=α -α Δα = (Right-polarized anti-clockwise case) Initial phase (α =0,α =+ )...after time advances b 0 : (α =0-00 =-0 =+0,α = + -0 =- =+ ) Δα r(0 )

49 v - v(0 )=v(0 ) v 0 Ellipsometr Contact Plots vs. v v Relative phase Δα=α -α Δα = (Right-polarized anti-clockwise case) Initial phase (α =0,α =+ )...after time advances b 0 : (α =0-0 =-0 =+0,α = + -0 =- =+ ) Δα r(0 )=r(0 )