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Why study oscillations?

Oscillation: A periodic (often sinusoidal) variation. Example: Mass on a spring. Why are oscillations so important?. They are everywhere, and are central to science and engineering. Examples from everyday life:. Why study oscillations?. Examples from science & engineering:

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Why study oscillations?

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  1. Oscillation: A periodic (often sinusoidal) variation. Example: Mass on a spring Why are oscillations so important? They are everywhere, and are central to science and engineering. Examples from everyday life: Why study oscillations? Examples from science & engineering: Thermal vibrations of atoms Vibrations of cantilevers in atomic force microscopes The quartz crystal that sets the clock rate of a computer The tuner circuit in a radio or TV Water molecules in a microwave oven Protons in Magnetic Resonance Imaging

  2. 2. Oscillations show some quite surprising behaviors, as we’ll explore. 3. The ways of thinking and mathematical techniques you will master in this course prepare you perfectly for later physics and engineering courses, especially quantum mechanics.

  3. Complex functions (functions of complex variables) • Differential equations • Orthogonal functions, including Fourier analysis • Hilbert space: a space in which the axes • correspond to orthogonal functions • Bra-ket notation (a convenient way of representing • vectors in Hilbert space) • Matrix math • Eigenvalue equations: Schrödinger’s equation

  4. They are everywhere, and are central to science and technology. Examples from everyday life: Examples from science & engineering: Seismic waves Gravity waves Matter waves (quantum mechanical waves) Why study waves?

  5. 1. Waves are made from interacting oscillators. Connections between waves & Oscillations

  6. 2. Waves are produced by oscillators.

  7. 3. Waves cause oscillators to start oscillating.

  8. Simple Harmonic oscillation In one cycle, the argument of the cosine must change by 2 Angular frequency

  9. Why is this mathematical behavior so universal?

  10. Review of Taylor Series

  11. At the minimum, U = 0 and so Applying to our system in stable equilibrium:

  12. Hooke’s Law For now, we assume that only conservative forces act Hooke’s Law So, virtually any system in stable equilibrium can be modeled with U = ½ kx2or with F = - kx. As we’ll see shortly, this explains why all these seemingly unrelated systems show oscillations that are exactly sinusoidal (for small amplitudes.)

  13. A couple of examples: Again, for now we ignore air resistance. If x is measured relative to equilibrium – see section 1.4

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