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Why study oscillations?

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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:

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. 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.

- 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

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?

1. Waves are made from interacting oscillators.

Connections between waves & Oscillations

2. Waves are produced by oscillators.

3. Waves cause oscillators to start oscillating.

Simple Harmonic oscillation

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

Angular frequency

Why is this mathematical behavior so universal?

At the minimum, U = 0 and

so

Applying to our system in stable equilibrium:

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.)

A couple of examples:

Again, for now we ignore air resistance.

If x is measured relative

to equilibrium –

see section 1.4