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… constant forces  integrate EOM  parabolic trajectories.

Real Oscillators. … constant forces  integrate EOM  parabolic trajectories. … linear restoring force  guess EOM solution  SHM. … nonlinear restoring forces  ?. nonlinear spring?. linear spring. F. F. x. x. 0. 0. The spring of air :. use Ideal Gas Law: PV=NRT.

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… constant forces  integrate EOM  parabolic trajectories.

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  1. Real Oscillators … constant forces  integrate EOM  parabolic trajectories. … linear restoring force  guess EOM solution  SHM … nonlinear restoring forces  ? nonlinear spring? linear spring F F x x

  2. 0 0 The spring of air : use Ideal Gas Law: PV=NRT Patm chamber volume: V=Ax WTF! (whoa there, fella) m EOM A +x Stable Equilibrium at xeq = NRT / (mg + APatm) P, V 0

  3. Taylor Series Expansions: Turns a function into a polynomial near x = a Example:

  4. Expand around x = -3: 2nd order 0th order 1st order

  5. Expand around x = 2: 0th order 1st order 2nd order

  6. Expand NRT/x around xeq: Is it safe to linearize it? Better check a unitless ratio. How about: (Yes, excellent choice Dr. Hafner!)

  7. .. Displacement 5% of xeq: 0 .05 .0025 …. Perhaps you would prefer…. SHM with

  8. -x Simple Pendulum: Stable Equilibrium: Length: L Mass: m Q Displace by Q: mg cosQ T mg cosQ sinQ mg cosQ EOM: mg Expand it! mg

  9. Derivatives:

  10. Now express as a unitless ratio of the dependent variable and some parameter of the system: Displacement 5% of length: 0 .05 0 .0000625 … SHM with

  11. The world is not linear. However, one can use a Taylor expansion to linearize an EOM by assuming only small perturbations around a point of stable equilibrium (which may not be the origin).

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