Oscillation a periodic often sinusoidal variation example mass on a spring
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Why study oscillations? PowerPoint PPT Presentation


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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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Oscillation a periodic often sinusoidal variation example mass on a spring

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


Why study oscillations

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.


Why study oscillations

  • 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


Why study waves

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?


Connections between waves oscillations

1. Waves are made from interacting oscillators.

Connections between waves & Oscillations


Why study oscillations

2. Waves are produced by oscillators.


Why study oscillations

3. Waves cause oscillators to start oscillating.


Simple harmonic oscillation

Simple Harmonic oscillation

In one cycle, the argument of the cosine must change by 2

Angular frequency


Why study oscillations

Why is this mathematical behavior so universal?


Review of taylor series

Review of Taylor Series


Why study oscillations

At the minimum, U = 0 and

so

Applying to our system in stable equilibrium:


Hooke s law

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.)


Why study oscillations

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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