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Physics 321. Hour 11 Simple and Damped Harmonic Oscillators. Find U and T Equilibrium Equation of motion ω , T Energy plot . Mass on a Massless Spring. y. y. Assume Find U and T Equation of motion ω , T . x. A Shallow Frictionless Bowl. z. y. Assume Find U and T

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## Physics 321

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**Physics 321**Hour 11 Simple and Damped Harmonic Oscillators**Find U and T**Equilibrium Equation of motion ω, T Energy plot Mass on a Massless Spring y**y**Assume Find U and T Equation of motion ω, T x A Shallow Frictionless Bowl z**y**Assume Find U and T Newton’s Equations x Another Bowl z**Lissajous.nb**Example**The equation:**A little rearranging: Damped Oscillator A trial solution:**Underdamped:**• Critically damped: • Overdamped: Three Regimes**Solution:**or Underdamped**Solution:**Overdamped**Solution:**Critically Damped**DampOsc5_4.nb**Example**Physics 321**Hour 12 Driven Harmonic Oscillators**The equation:**Let The oscillator wants to oscillate at but the driver forces it to oscillate at . This leads to transient vs steady state behavior! Driven Oscillator**Driven_Osc.nb**Example**We assume a solution something like**But So we employ a trick… The driving force is the real part of Driven Oscillator**We assume a solution of the form**This gives: Driven Oscillator**Conclusion 1:**Driven Oscillator**Real parts:**Imaginary parts: Driven Oscillator**Conclusion 2:**And finally the steady state solution is: Driven Oscillator

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