Physics 201, Lecture 28

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1 Physics 01, Lecture 8 Today s Topics n Oscillations (Ch 15) n n n More Simple Harmonic Oscillation n Review: Mathematical Representation n Eamples: Simple Pendulum, Physical pendulum Damped Oscillation Driven Oscillation, Resonance

2 Review: Harmonic Oscillation q Motion described by epression (t)=acos(ωt+φ) is called (simple) harmonic oscillation A: amplitude ω: angular frequency (t)=acos(ωt+φ) T= 1/f = π/ω φ: phase constant (0)=Acos(φ)

3 Harmonic Oscillation: Summary Of Properties q Harmonic Oscillation Eq: d dt = ω T= 1/f = π/ω q Solution : =Acos(ωt+φ 0 ) q Amplitude A: set by initial condition q Phase φ 0 : set by initial condition q ω (or f = ω/π or T= 1/f = π/ω ): determined by intrinsic and geometric features Intrinsic frequency. q Total mechanic Energy: E= ½ ma ω (0)=Acos(φ 0 )

4 Practical Technique: Recognize Phase Constant (Method I) q An oscillation is described by =Acos(ωt+φ). Find out φ for each of the following figures: Answer π/ π 3π/ π t φ= 0 Use (0)/A=cos(φ)=1 t φ= π/ Use (0)/A=cos(φ)=0 but π/ or 3/π? t φ= π Use (0)/A=cos(φ)=-1

5 Practical Technique: Recognize Phase Constant q An oscillation is described by =Acos(ωt+φ). Find out φ for each of the following figures: Answer π/ π 3π/ π t φ= 0 (or π, -π..) t φ= π/ (or 5π/, or -3π/ ) t φ= π (or )

6 Quiz/Eercise: Determine φ Angle q A simple harmonic motion in the form (t)=acos(wt+φ) is shown in graph below. Estimate the φ angle from the graph. 0. π, 0.8 π, 1.8 π, -0. π, -0.8 π

7 SHM and Uniform Circular Motion q SHM: =Acos(ωt+φ 0 ) q Uniform circular motion with angular velocity ω and radius A: θ = ωt + φ 0 = Acos(θ) = A cos(ωt+φ 0 ) y = Asin(θ) = A sin(ωt+φ 0 ) Ø So the SHM can be mathematically modeled by a projection of uniform circular motion.

8 SHM and UCM Comparison SHM Angular Frequency ω Period T = π/ω Frequency f=1/t Amplitude A UCM Angular Velocity ω Period T = π/ω Frequency f=1/t Radius A initial phase angle (at=0) φ 0 initial angle (at t=0) φ 0 Displacement in : =Acos(ωt+φ 0 ) Displacement in : =Acos(ωt+φ 0 )

9 Spring-Block Oscillator q A block attached on an ideal spring forms a simple harmonic oscillator Ø Hooke s Law: F = -k Newton s nd Law: ma=md /dt = F = -k è d /dt = -k/m Compare to Harmonic equation: d dt = ω à ω = sqrt(k/m) A and φ set by initial condition

10 Spring-Block Oscillator: Energy Consideration E= ½ ma ω

11 Simple Pendulum q A pendulum swinging near equilibrium also forms a harmonic oscillator: Why? Ø Tangential direction F=-mgsinθ à md s/dt =- mgsinθ s=lθ small angle: sinθ θ è Harmonic Eq: d θ/dt = - g/l θ = -ω θ à ω = sqrt(g/l) (Amplitude and phase set by initial condition.)

12 Another Eample: Torsional Pendulum q A torsional pendulum forms a harmonic oscillator too: Why? Ø Torque τ=-κ θ à τ=αι = Ι d θ/dt è Harmonic Eq: d θ/dt = - κ/ιθ = -ω θ à ω = sqrt(κ/ι) à Amplitude and phase set by initial condition.

13 One More Eample: Physical Pendulum q A phyusical pendulum forms a harmonic oscillator too: Why? Ø Torque τ=-mgd sinθ à τ=αι = Ι d θ/dt è Harmonic Eq: d θ/dt = - (mgd/ι)θ = -ω θ à ω = sqrt(mgd/ι) à Amplitude and phase set by initial condition.

14 Damped Oscillation q If in addition to harmonic force (-k), a retarding (resistive) force (-bv ) also presents, the oscillation equation then becomes: d m dt Ø The solution is: = k b d dt b: damping constant where: = Ae b t m cos( ω t + φ) k m b m ω = ( ) = ω0 b ( ) m

15 Demo: Damped Oscillation q Retarding force F R = -bv = -b d/dt ) cos( φ ω + = t Ae t m b 0 ) ( ) ( m b m b m k = ω = ω

16 Effects of Damping q Solution of damped oscillation: = Ae b t m cos( ω t + φ) b ω = ω ( m 0 ) Ø Lower frequency Ø Reducing amplitude Ø Mechanic energy losing to damping force. a:underdamping b:critical damping c:overdamping small b General

17 Forced (driven) Oscillation q If in addition there is a driving force with its own frequency ω: F 0 cos(ωt), the equation becomes: m d dt = k d dt Ø This equation can be solved analytically. b cos( ωt) At large t, the solution is: = Acos( ω t + φ) with F0 / m A = b ( ω ω0 ) + ( ) m Ø At large t, the frequency is determined by driving ω Ø When ω=ω 0, amplitude is maimum resonance + F 0

18 Resonance Amplitude A = ( ω F 0 ω ) 0 / m + b ( ) m See demo

19 Angers Bridge, Angers, France 1850

20 Resonance: Tacoma Narrows Bridge (Nov 7, 1940)

21 Light and Optics Electro-Magnetism Thermodynamics Heat, Temperature, Pressure, Entropy,.. Oscillation and Waves Classical Mechanics Laws of motion Force, Energy, Momentum, Physics 01 and 0 Cosmology Sub-Sub-Atomic: Elementary Particles Sub-Atomic: Nuclear Physics Many-Atoms: Molecules, solids Atomic Structure Quantum Theory Relativity Classical Modern

22 Special Review Lecture: Thursday December 14 th 9:55-10:45am : Chapters since midterm 3 (Note the special date which is after the last class day) Super Friday: December 15 th. 10am-5pm in the lab room.

x(t)=acos(ωt+φ) d x 2 Review: Harmonic Oscillation Physics 201, Lecture 28 Today s Topics Practical Technique: Recognize Phase Constant (Method I)

x(t)=acos(ωt+φ) d x 2 Review: Harmonic Oscillation Physics 201, Lecture 28 Today s Topics Practical Technique: Recognize Phase Constant (Method I) 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