PH421: Oscillations; do not distribute 11/12/07 Summary so far: • Free, undamped, linear (harmonic) oscillator • Free, undamped, non-linear oscillator • Free, damped linear oscillator Starting today: • Driven, damped linear oscillator • Laboratory to investigate LRC circuit as example of driven, damped oscillator • Time and frequency representations • Fourier series Lecture 5/6 - damped oscllations

PH 421: Oscillations - do not distribute 11/12/2018 THE DRIVEN, DAMPED HARMONIC OSCILLATOR Reading: Main 5.1, 6.1 Taylor 5.5, 5.6 Lecture 6 - Driven oscillations

PH 421: Oscillations - do not distribute Natural motion of damped, driven harmonic oscillator 11/12/2018 x m k viscous medium F0coswt Note w and w0 are not the same thing! is driving frequency w0 is natural frequency Lecture 6 - Driven oscillations

PH 421: Oscillations - do not distribute Natural motion of damped, driven harmonic oscillator 11/12/2018 Apply Kirchoff’s laws L R C I Vocoswt http://www.sciencejoywagon.com/physicszone/lesson/otherpub/wfendt/accircuit.htm Lecture 6 - Driven oscillations

PH 421: Oscillations - do not distribute 11/12/2018 V0 real, constant, and known Let's assume this form for q(t) But now q0 is complex: This solution makes sure q(t) is oscillatory (and at the same frequency as Fext), but may not be in phase with the driving force. Task #1: Substitute this assumed form into the equation of motion, and find the values of |q0| and fq in terms of the known quantities. Note that these constants depend on driving frequency w (but not on t – that's why they're "constants"). How does the shape vary with w? Lecture 6 - Driven oscillations

PH 421: Oscillations - do not distribute 11/12/2018 Assume V0 real, and constant Task #2: In the lab, you'll actually measured I (current) or dq/dt. So let's look at that: Having found q(t), find I(t) and think about how the shape of the amplitude and phase of I change with frequency. Lecture 6 - Driven oscillations

PH 421: Oscillations - do not distribute 11/12/2018 Assume V0 real, and constant Task #1: Substitute this assumed form into the equation of motion, and find the values of |q0| and f in terms of the known quantities. Note that these constants depend on w (but not on t – that's why they're “constants”). How does the shape vary with w? Lecture 6 - Driven oscillations

PH 421: Oscillations - do not distribute 11/12/2018 “Resonance” Charge Amplitude |q0| w0 Driving Frequency------> Current Amplitude |I0| w0 Lecture 6 - Driven oscillations

PH 421: Oscillations - do not distribute 11/12/2018 Task #2: In the lab, you actually measured I (current) or dq/dt. So let's look at that: Having found q(t), find I(t) and think about how the shape of the amplitude and phase of I change with frequency. Lecture 6 - Driven oscillations

PH 421: Oscillations - do not distribute 11/12/2018 “Resonance” Current Amplitude |I0| Driving Frequency------> π/2 Current Phase -π/2 Lecture 6 - Driven oscillations

PH 421: Oscillations - do not distribute 11/12/2018 Charge Phase fq -π/2 -π w0 Driving Frequency------> π/2 Current Phase -π/2 w0 Lecture 6 - Driven oscillations