Chapter 14 Periodic Motion

Hooke’s Law

Potential Energy in a Spring See also section 7.3

Simple Harmonic Oscillator

Notations This is the simple harmonic oscillation equation. Very very important! ALL You want to write ALL oscillation equations in this form.

Other Examples Simple pendulum Tuning fork Skyscraper (Inverted Pendulum)

In general

Rewriting Formulae Equations

All equations looks the same ALL You want to write ALL oscillation equations in this form.

Solution

(Natural) Frequency, Period, etc…

Example

What it looks like

x,v & a

Using initial conditions

Example Given moment of inertia I and CM at l, find the angular frequency.

Example

Simple Pendulum

Example A lead ball is attached to a string 3m long. Find the natural period of the pendulum.

Energy of SHO

Conservation of Energy

Energy of a pendulum (reminder)

Energy of a pendulum

DampedOscillation

Damping Force FrFr

Newton Second Law

Oscillations with damping

Solving the Equation Try the solution:

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Under-damped Over-damped Critically damped Three Cases

Under-damped

Under-damped Movie

Three cases: 

Under-damped: Over-damped: Critically damped: Three Cases

Under-damped Over-damped Critically damped

Under-damped Critically damped Over-damped

under damped over damped critically damped system slows down fastest when critically damped Too much damping Is counter-productive!

Resonance Pushing a swing

Driven / Forced Oscillations

Amplitude A(ω d )

Resonance