南开大学：《力学》课程教学资源（PPT课件，英文讲稿）Chapter 6 Vibration and wave

Chapter6Vibration and wave

Chapter 6 Vibration and wave

IntroductionOscillatorOne kind of motionApplication of Newton's LawWaveElectromagnetism.Optics,ModernBut its result isvery important.physics(Quantum Mechanics)energyWhytransportinformationAll starts from

Introduction: One kind of motion. Oscillator Application of Newton’s Law. Wave But its result is very important. Why information energy transport All starts from.  Electromagnetism,Optics,Modern physics(Quantum Mechanics)

S1.Harmonic oscillation1-1 Simple harmonic motionMass on a springf =-kxmx=-kxkx+-x=0mX+ 0?x =0 一 Differential e.q.intrinsic

f = −kx §1. Harmonic oscillation m x = −kx •• + = 0 •• x m k x 0 2 + = •• x  x 1-1 Simple harmonic motion Mass on a spring. Differential e.q. intrinsic x o

Simple pendulumwPendulumCompound pendulum- mg sin ll = IβO!0-mgsinθ=moe.C-mg0=10Xmgl0+0=0mgIsin 0~～00+0°0=0山

− mg sinl c = I •• − mg sin = m •• − mgl c = I + = 0 ••   I mgl c Pendulum Simple pendulum Compound pendulum  sin  x o  mg L C O C 0 2 + = ••   

Pure rotationm DefinitionSPhysical quantity:+02g=0OExample:mmglcmgrc02c.11I0RI, =mR2=I +m(R-r)0I =I +rm=mR2-m(R-r)-rm

0 2 + = ••    Physical quantity:   Definition. Pure rotation Example: I mgr I mglc c = = 2  2 2 0 ( ) ' c c I = mR = I + m R − r I o I c r m mR m R r c r c m 2 2 2 2 = + = − ( − ) − R o' o c m rc 

= mR-m(R2 -2Rr +r)+r'm=2mR0m independent ofgW=2RAnother interestingproblem:What is the behavior of lQas water leak off?Your classmates areg20your good teacher!1

mR m R Rr c r c r c m 2 2 2 2 = − ( − 2 + ) + 2 = 2mR R g w 2 2 = Another interesting problem:   independent of ••  What is the behavior of as water leak off? l l g = 2  Your classmates are your good teacher!

Related example about simple harmonic motionF(1)(2).mg(3).Potentialf =-kEu'(E) ocEu(5) ==kx? + Bx3 ..口2

Related example about simple harmonic motion k mg I o F (1). (2). (3). Potential f = −k u'()    = +  2 3 2 1 u( ) k x x

1-2 Solution E(t)= = Acos(ot + Po)5+025=0Guess!= = Bsin( t + Po) A →amplitude (Period T=_ 2元 )0の frequencyPo initial phaseCiAcos(ot + Po) + C2 B sin( ot + Po)Complex number: real,imagineAei(ot+Po)Not physical quantity

0 2 + = ••    cos( )  =  +0 A t sin( )  =  +0 B t cos( ) sin( ) 1  +0 + 2  +0 C A t C B t  2 T = ( )  0 i t + Ae 1-2 Solution (Period )  frequency Guess! A amplitude 0 initial phase  Complex number: real,imagine. Not physical quantity.  (t)

i(ot+PoAt beginning it was used for convenience inAequantum mechanics ,it is necessary!Normal mode(2)

( )  0 i t + Ae At beginning it was used for convenience in quantum mechanics ,it is necessary! Normal mode: (1). (2)

intrinsicA, @。 alter mined by initial condition50tan Po =t=05.=?一5.=AcosP050t=05.=? → 50 =-Aosin oEnergy:1 kA? cos?(ot + Po)1Ep22mA?? sin ?(ot + Po)Ekmy22

t = 0 ?  0 = 0 0  = Acos 0 0 0 tan    • = − 0 2 0 0    • A = + 0 0  = −A sin  • 0 = ? •  t = 0  intrinsic. A, alter mined by initial condition. Energy: cos ( ) 2 1 2 1 0 2 2 2 Ep = k x = k A t + sin ( ) 2 1 2 1 0 2 2 2 2 Ek = m v = m A  t +