Modern Physics. Unit 1: Classical Models and the Birth of Modern Physics Lecture 1.2: Classical Concepts Review of Particles and Waves

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1 Modern Physics Unit 1: Classical Models and the Birth of Modern Physics Lecture 1.: Classical Concepts Reiew of Particles and Waes Ron Reifenberger Professor of Physics Purdue Uniersity 1

2 Equations of Motion for Particle with Mass m Newton (1687): Strong emphasis on predicting the future trajectory of a particle! m = Fr () dt dr

3 Equations of Motion for Particles Linear Motion Rotational Motion Displacement d = o t+½ at θ = ω o t + ½ αt Velocity = o + at ω = ω o + αt Inertia m I=mr Newton s nd Law Momentum Newton s nd Law Conseration of momentum F = m a p = m F = dp/dt If F et =0, then p = constant τ =I α l = r p τ = dl/dt If τ et =0, then l = constant Kinetic Energy (K) ½ m ½ (mr ) ω = l /I 3

4 Equations of Motion for Particles cont. Linear Motion Rotational Motion Work W= F d W= τdθ Potential Energy, V V()=-W V(θ)=-W If Force is conseratie (not a function of time), then Potential Energy for linear motion can be defined as: VP ( ) F( ) d P ref P ref Likewise, if Torque is conseratie (not a function of time), then Potential Energy for rotational motion is: V Q Q ( θ) τθ ( ) dθ θ o Q θ o 4

5 Important Consequences: Constants of Motion Work-Energy Theorem: W net = K f K i If no work, then K=constant of motion If no eternal forces acting at point of contact, then momentum=constant (Newton s 3 rd Law) Virial Theorem: if V(r) 1/r, then <K> time aer. =-½<V> time aer. 5

6 Important Consequences: Constants of Motion Noether s Theorem (1915): Any conseration law implies an underlying symmetry: Conseration of energy implies time translation symmetry (inariance of EoM under the transformation t t+ T o ) Conseration of linear momentum implies linear translation symmetry (inariance of EoM under the transformation r r + R ) Conseration of angular momentum implies rotational translation symmetry (inariance of EoM under the transformation ϑ ϑ+θ ) o EoM = Equation of Motion 6

7 Waes are Really different A wae is a traelling disturbance in a medium Disturbance can be either transerse or longitudinal Not so interested in trajectories No bulk transport of matter A wae transfers energy Two waes can be at the same place at the same time Constructie or destructie interference possible At a fied time t Schematic Plot of a Transerse Harmonic Disturbance h(,t) PEAK VALUE Waes satisfy the following differential equation: h 1 = h 7

8 Deriation of wae equation is based on Newton s laws of motion ( ) ( ) m = ρ + h T(+,t) h 1 h = T(,t) θ(,t) h(,t) h 1. Broadly applicable: T Y = wae on string = wae in solid bar ρ ρ B γ RT = wae in liquid = wae in gas ρ M. Since the wae equation is of second order with respect to time, knowledge of the first time deriatie of the solution is needed. 8

9 Solutions to Wae Equation? Solution I (sine): h 1 = h t h (,) t = Asin π Asin[ k ωt] λ T = π π k ω = π f λ T Traelling wae solution in + direction. (How do you know the wae is moing in + direction?) Does proposed solution satisfy wae equation? h = Ak cos ( k ωt) = A( k ) sin ( k ωt) h = A( ω) cos ( k ωt) = A( ω ) sin ( k ωt) 1 A( k ) sin ( k ωt) = ( A)( ω ) sin ( k ωt) ω ω = = ± k k 9

10 The dispersion relation ω If the dispersion relation is non-linear, it typically leads to spreading of a pulse with time. (Pulse spreading is also known as dispersion.) ω = ± k π ω T λ = = = = λ f k π T λ k 10

11 Solution II: t ht (, ) = A'sin π + ϕ cos π + δ λ T [ ϕ] cos[ ω δ] = A'sin k + t + π π k ω = π f λ T ϕδ, adjusted to match boundary conditions kl = nπ ; n = Any other solutions to Wae Equation? integer Standing wae solution (confined wae) Does proposed solution satisfy wae equation? L n= Allowed k alues are now quantized! h = A' k cos ( k + ϕ) = A' k sin ( k + ϕ) h = A' ω sin ( ωt + δ) = A' ω cos ( ωt + δ) 1 A' k sin ( k + ϕ) cos ( ωt + δ) = ' sin A ω ( k + ϕ) cos( ωt + δ) ω = k 11

12 A standing wae is the sum of two traeling waes moing in opposite direction Working it Out t h1 (, t) = A'sin π = A'sin[ k ωt] moing in + direction λ T t h (, t) = A'sin π + A'sin[ k ωt] moing in direction λ T = + Recall sinθ cosγ = sin( θ + γ) + sin( θ γ ) trig identity Let θ = k and γ = ωt [ ] [ ω ] A'sin k cos t = h ( t, ) + h ( t, ) = A'sin k+ ω t + A'sin k ωt 1 [ ] [ ] standing wae wae moing toward - wae moing toward + 1

13 In pictures: black = red + blue E F G H 13

14 Up Net Mawell s Electromagnetic Waes 14

15 APPENDIX A: Cosine Solution to Wae Equation Solution III (cosine): h 1 = h t h (,) t = Acos π Acos[ k ωt] λ T = π π k ω = π f λ T Linear dispersion ω k Traelling wae solution in + direction Does proposed solution satisfy wae equation? h = Ak sin ( k ωt) = A( k ) cos ( k ωt) h = A( ω) sin ( k ωt) = A( ω) cos ( k ωt) 1 A( k ) cos ( k ωt) = ( A)( ω) cos ( k ωt) π ω ω = = ± = T λ = = λ f ( same result) k k π T λ 15

APPENDIX B: Representing Classical Waes Using Eponential Notation: Comple notation used for CONVENIENCE Euler s Formula: iθ e = cosθ + isinθ i 1 ik Ae = Acos k + iasin k ( ωt) i k Ae = Acos( k ωt) +

16 APPENDIX B: Representing Classical Waes Using Eponential Notation: Comple notation used for CONVENIENCE Euler s Formula: iθ e = cosθ + isinθ i 1 ik Ae = Acos k + iasin k ( ωt) i k Ae = Acos( k ωt) + iasin( k ωt) Stationary (standing) wae; does not depend on time Traeling wae i( k ωt) i( k ωt) [ ω ] = [ ω ] = Acos k t Re Ae and Asin k t Im Ae The adantage of Euler s formula is that the algebra of eponentials is usually easier than algebra of sines and cosines. 16

17 Euler's wae: Orthogonal D projection of 3D Heli Imag. Note that the -components of Euler s wae are out of phase by 90 degrees. Real fundamental_relationship_to_circle_%8and_heli%9.gif 17

10. Yes. Any function of (x - vt) will represent wave motion because it will satisfy the wave equation, Eq

10. Yes. Any function of (x - vt) will represent wave motion because it will satisfy the wave equation, Eq CHAPER 5: Wae Motion Responses to Questions 5. he speed of sound in air obeys the equation B. If the bulk modulus is approximately constant and the density of air decreases with temperature, then the speed