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
Equations of Motion for Particle with Mass m Newton (1687): Strong emphasis on predicting the future trajectory of a particle! m = Fr () dt dr
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
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
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
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
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
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
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
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
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
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
In pictures: black = red + blue E F G H http://www.wwnorton.com/college/physics/om/_tutorials/chap16/waes/inde.htm 13
Up Net Mawell s Electromagnetic Waes 14
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) + 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
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 http://upload.wikimedia.org/wikipedia/commons/8/8e/sine_and_cosine_ fundamental_relationship_to_circle_%8and_heli%9.gif 17