Physics 321 Hour 11 Simple and Damped Harmonic Oscillators Bottom Line β’ Equation for a damped oscillator: ππ₯ = βππ₯ β ππ₯ π₯ = βπ02 π₯ β 2π½ π₯ β’ Damping regimes: β’ Underdamped: π½ < π0 β’ Critically damped: π½ = π0 β’ Overdamped: π½ > π0 Exam #1 Reminders (Spring 2015) See Review #1, Sample Test #1, Example Problem Review is Wednesday 5/13 Test is available Thurs. 5/14 β Monday 5/18 Average time will be about 1 hr 15 min Simple Harmonic Oscillator 1) What is the equation of motion? 2) Where is your origin? 3) What is the frequency of oscillation? 4) What are the three ways you can write the general solution? 5) How many constants do you have to fix to get a specific solution? 6) How do you find those constants? x Mass on a Massless Spring 1) 2) 3) 4) 5) 6) Find U and T π+π =0 Equilibrium Equation of motion Ο, T Energy plot y Example VertSpringUWell.nb A Shallow Frictionless Bowl 1) Assume π§ β 0, π§ = πΌπ 2 2) Find U and T 3) π + π = 0 4) πΉ = βπ»π = ππ 5) Ο, T y x z Another Bowl 1) Assume π§ β 0, π§ = πΌπ₯ 2 + π½π¦ 2 2) Find U 3) πΉ = βπ»π = ππ 4) Two frequencies y x z Example Lissajous.nb Damped Oscillator The equation: ππ₯ = βππ₯ β ππ₯ π β π₯ π A little rearranging: π₯ = π₯ = βπ02 π₯ β 2π½ π₯ β π π₯ π A trial solution: π₯(π‘) = π ππ‘ π 2 π₯ π‘ = β π02 π₯ π‘ β 2π½ππ₯ π‘ π 2 + π02 + 2π½π = 0 π = βπ½ ± π½ 2 β π02 Three Regimes β’ Underdamped: π½ < π0 β’ Critically damped: π½ = π0 β’ Overdamped: π½ > π0 Underdamped π½ < π0 , π = βπ½ ± π½ 2 β π02 π = βπ½ ± π π02 β π½ 2 Solution: π₯ π‘ = π΄π or βπ½π‘ +π π02 βπ½ 2 π‘ π + π΅π βπ½π‘ βπ π02 βπ½ 2 π‘ π π₯ π‘ = π΄π βπ½π‘ sin( π02 β π½ 2 π‘) + π΅π βπ½π‘ cos( π02 β π½ 2 π‘) Note: Ο = π02 β π½ 2 Overdamped π = βπ½ ± π½ 2 β π02 π½ > π0 , Solution: π₯ π‘ = π΄π βπ½π‘ + π½2 βπ02 π‘ π + π΅π βπ½π‘ β π½2 βπ02 π‘ π Critically Damped π½ = π0 , π = βπ½ ± π½ 2 β π02 π = βπ½ Solution: π₯ π‘ = π΄π βπ½π‘ + π΅π‘π βπ½π‘ Example DampOsc5_4.nb