1 Are oscillations ubiquitous or are they merely a paradigm? Superposition of brain neuron activity

LRC Resonance Lab Average 74/100 (note BlackBoard Score 5 pts lower, pre-lab) (scores 85-95: all elements excellent scores <69  I was not convinced you found a physical to explain your data) NOT about finding a mathematical model for LRC circuit Lab is about finding a physical model to explain your observations of (i) current enhancement (ii) phase shift ; Why do these occur? What’s happening the circuit? What is the origin of resonance? Explain why your results are physically interesting and useful

WEEK 1 SUMMARY

Energy approach to equation of motion: i.e. find trajectory x(t) if U(x) is known Special case of U(x)=(½)kx 2 found period T (indep of A), found x(t) = A cos(  t+  ) Found 3 other forms of A cos(  t+  ) Learned to apply initial conditions to determine A, , and also the arbitrary parameters in other 3 forms

Energy approach to equation of motion: Harder case of U(  )= MgL cm (1-cos  ) found -> HO for small theta found T numerically measured T for pendulum Learned to argue qualitatively about time to move certain distances, comparing T for diff U Did NOT learn equation of motion  (t)

Complex numbers: rectangular and polar form and Argand diag. complex conjugate Euler relation solving one complex equation is actually solving 2 simultaneous equations

WEEK 2 SUMMARY

x m m k k Free, undamped oscillators No friction  mg m T L C I q Common notation for all

Force approach to equation of motion of FREE, UNDAMPED HARMONIC OSCILLATOR: i.e. find trajectory  (t) if F(  ) is known Special case of F(  )=-sin(  ) -> small angle approx: F(  )=-  =>2 nd order DE, Found sinusoidal motion Applied initial conditions as before.

 mg m T Common notation for all x m k friction Free, damped oscillators

Force approach to equation of motion of FREE, DAMPED OSCILLATOR Add damping force to eqn of motion Found decaying sinusoid

FREE, DAMPED OSCILLATOR Damping time  measures number of oscillations in decay time apply initial conditions, energy decay

WEEK 3 SUMMARY

L R C I V o cos  t 14 DRIVEN, DAMPED OSCILLATOR

Charge Amplitude |q 0 | Charge Phase  q Driving Frequency  > "Resonance" 0 -π -π/2   CHARGE

Current Amplitude |I 0 | Current Phase  I “Resonance” 0 π/2 -π/2   Driving Frequency  > CURRENT

Admittance Amplitude |Y 0 | Addmittance Phase  I “Resonance” 0 π/2 -π/2   Driving Frequency  > ADMITTANCE NOT time dependent, but IS freq dependent.

L R C I V o cos  t DRIVEN, DAMPED OSCILLATOR can also rewrite diff eq in terms of I and solve directly (same result of course)

FOURIER SERIES – periodic functions are sums of sines and cosines of integer multiples of a fundamental frequency. These “basis functions are orthonormal

ODD functions f(t) =  f(  t). Their Fourier representation must also be in terms of odd functions, namely sines. Suppose we have an odd periodic function f(t) like our sawtooth wave and you have to find its Fourier series Then the unknown coefficients can be evaluated this way the function the harmonic Integrate over the period of the fundamental normalize properly Here’s the coefficient of the sin(  n t) term! Plot it on your spectrum!

frequency B the function the harmonic Integrate over the period of the fundamental normalize properly coefficient of the sin(  n t) term! Time domain Frequency domain Fundamantal freq = 2π/T

frequency B DRIVING AN OCILLATOR WITH A PERIODIC FORCING FUNCTION THAT IS NOT A PURE SINE Forcing function Oscillator response Important to know where the fundamental freq of the forcing function lies in relation to the oscillator max response freq!

DRIVING AN OCILLATOR WITH A PERIODIC FORCING FUNCTION THAT IS NOT A PURE SINE

time frequency ? This was harmonic response expt - you know what black box (LRC) does to a single freq Observe what (LRC) black box does to an impulse function FT - you know this Are these connected by FT?? They’d better be - you find out! Black box Z(  ) Black box Z(  )

Series LRC Circuit Which admittance plot |Y(  )| corresponds to which free decay plot I(t)? (All plots scaled to 1 at maximum value) (A) i 1, ii 2 (B) i 2, ii 1 (i) (ii)(2) (1)

Series LRC Circuit If the admittance |Y(  )| and the phase  I response of a series LCR circuit are as given on the left below, then which oscilloscope trace on the right corresponds to a circuit driven below resonance? Red is drive voltage, blue is current, represented by V R. (A) (B) (C) (D)

 Suppose that, if you apply a (red) sinusoidal voltage across a series LRC circuit, you measure the (blue) voltage response across the resistor.  Now, if you now apply a (red) square-wave voltage with the same period to the same circuit, and you measure the (blue) voltage response across the resistor, will you get this ….?  Or this?  Or this? Or something else?

 Suppose that, if you apply a (red) sinusoidal voltage across a series LRC circuit, you measure the (blue) voltage response across the resistor.  Now, if you now apply a (red) square-wave voltage at the same frequency the same circuit, and you measure the (blue) voltage response across the resistor, will you get this ….  Or this?  Or this? Or something else?

The last note of the piece Mozart chose for his piano recital was a sustained middle C (256 Hz). While he listened for the last note to fade away to a ghost of its initial amplitude, he had time calculate the quality factor Q of that oscillation. If the energy died away to 1% of its initial value after 15 seconds, approximately what was Q? None of the above