Physics 01, Lecure 8 Today s Topics n Oscillaions (Ch 15) n More Simple Harmonic Oscillaion n Review: Mahemaical Represenaion n Eamples: Simple Pendulum, Physical pendulum n Damped Oscillaion n Driven Oscillaion, Resonance Review: Harmonic Oscillaion q Moion descried y epression ()=Acos(ω+φ) is called (simple) harmonic oscillaion A: ampliude ()=Acos(ω+φ) φ: phase consan ω: angular frequency graphically T= 1/f = π/ω (0)=Acos(φ) Harmonic Oscillaion: Summary Of Properies q Harmonic Oscillaion Eq: d = ω d q Soluion : =Acos(ω+φ 0 ) q Ampliude A: se y iniial condiion q Phase φ 0 : se y iniial condiion q ω (or f = ω/π or T= 1/f = π/ω ): deermined y inrinsic and geomeric feaures Inrinsic frequency. q Toal mechanic Energy: E= ½ ma ω T= 1/f = π/ω (0)=Acos(φ 0 ) Pracical Technique: Recognize Phase Consan (Mehod I) q An oscillaion is descried y =Acos(ω+φ). Find ou φ for each of he following figures: π/ π 3π/ π Answer φ= 0 Use (0)/A=cos(φ)=1 φ= π/ Use (0)/A=cos(φ)=0 u π/ or 3/π? φ= π Use (0)/A=cos(φ)=-1 1
Pracical Technique: Recognize Phase Consan q An oscillaion is descried y =Acos(ω+φ). Find ou φ for each of he following figures: Answer Quiz/Eercise: Deermine φ Angle q A simple harmonic moion in he form ()=Acos(w+φ) is shown in graph elow. Esimae he φ angle from he graph. π/ π 3π/ π φ= 0 (or π, -π..) φ= π/ (or 5π/, or -3π/ ) 0. π, 0.8 π, 1.8 π, -0. π, -0.8 π φ= π (or ) SHM and Uniform Circular Moion q SHM: =Acos(ω+φ 0 ) q Uniform circular moion wih angular velociy ω and radius A: θ = ω + φ 0 = Acos(θ) = A cos(ω+φ 0 ) y = Asin(θ) = A sin(ω+φ 0 ) SHM and UCM Comparison SHM UCM Angular Frequency ω Angular Velociy ω Period T = π/ω Period T = π/ω Frequency f=1/t Frequency f=1/t Ampliude A Radius A iniial phase angle (a=0) φ 0 iniial angle (a =0) φ 0 Displacemen in : =Acos(ω+φ 0 ) Displacemen in : =Acos(ω+φ 0 ) Ø So he SHM can e mahemaically modeled y a projecion of uniform circular moion.
Spring-Block Oscillaor q A lock aached on an ideal spring forms a simple harmonic oscillaor Spring-Block Oscillaor: Energy Consideraion Ø Hooke s Law: F = -k E= ½ ma ω Newon s nd Law: ma=md /d = F = -k è d /d = -k/m Compare o Harmonic equaion: d = ω d à ω = sqr(k/m) A and φ se y iniial condiion Simple Pendulum Anoher Eample: Torsional Pendulum q A pendulum swinging near equilirium also forms a harmonic oscillaor: Why? Ø Tangenial direcion F=-mgsinθ à md s/d =- mgsinθ s=lθ small angle: sinθ θ è Harmonic Eq: d θ/d = - g/l θ = -ω θ à ω = sqr(g/l) (Ampliude and phase se y iniial condiion.) T S q A orsional pendulum forms a harmonic oscillaor oo: Why? Ø Torque τ=-κ θ à τ=αι = Ι d θ/d è Harmonic Eq: d θ/d = - κ/ιθ = -ω θ à ω = sqr(κ/ι) à Ampliude and phase se y iniial condiion. 3
One More Eample: Physical Pendulum q A phyusical pendulum forms a harmonic oscillaor oo: Why? Ø Torque τ=-mgd sinθ à τ=αι = Ι d θ/d è Harmonic Eq: d θ/d = - (mgd/ι)θ = -ω θ à ω = sqr(mgd/ι) à Ampliude and phase se y iniial condiion. Damped Oscillaion q If in addiion o harmonic force (-k), a rearding (resisive) force (-v ) also presens, he oscillaion equaion hen ecomes: Ø The soluion is: where: d d m = k d d = Ae k m m cos( ω + φ) m ω = ( ) = ω0 ( ) m : damping consan deails of formulism no required for his course Demo: Damped Oscillaion q Rearding force F R = -v = - d/d Effecs of Damping q Soluion of damped oscillaion: = Ae k m m cos( ω + φ) m ω = ( ) = ω0 ( ) m = Ae m cos( ω + φ) Ø Lower frequency Ø Reducing ampliude Ø Mechanic energy losing o damping force. ω = ω ( m 0 ) a:underdamping :criical damping c:overdamping small General 4
Forced (driven) Oscillaion Resonance Ampliude q If in addiion here is a driving force wih is own frequency ω: F 0 cos(ω), he equaion ecomes: d d m = k + F0 cos( ω) d d Ø This equaion can e solved analyically. A large, he soluion is: F0 / m A = = Acos( ω + φ) wih ( ω ω0 ) + ( ) m A = F0 / m ( ω ω0 ) + ( ) m Ø A large, he frequency is deermined y driving ω Ø When ω=ω 0, ampliude is maimum resonance See demo Angers Bridge, Angers, France 1850 Resonance: Tacoma Narrows Bridge (Nov 7, 1940) 5
0 49 01 Ligh and Opics Elecro-Magneism Thermodynamics Hea, Temperaure, Pressure, Enropy,.. Oscillaion and Waves Classical Mechanics Laws of moion Force, Energy, Momenum, Physics 01 and 0 Cosmology Su-Su-Aomic: Elemenary Paricles Su-Aomic: Nuclear Physics Many-Aoms: Molecules, solids Aomic Srucure Quanum Theory Relaiviy Special Review Lecure: Thursday Decemer 14 h 9:55-10:45am : Chapers since miderm 3 (Noe he special dae which is afer he las class day) Super Friday: Decemer 15 h. 10am-5pm in he la room. Classical Modern 6