Chapter 14Periodic 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!

You want to write ALL oscillation equations in this form.

Other Examples

Simplependulum

Tuning fork

Skyscraper(Inverted Pendulum)

In general

Rewriting Formulae

Equations

All equations looks the same

You want to write ALL oscillation equations in this form.

Solution

(Natural) Frequency, Period, etc…

Example

What it looks like

Math Reminder

x,v & a

Using initial conditions

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

Example

Simple Pendulum

Simple Pendulum

ExampleA 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

Fr

Newton Second Law

Oscillations with damping

Solving the Equation

Try the solution:

skip

Skip

Under-damped

Over-damped

Critically damped

Three Cases

Under-damped

Under-damped Movie

Three cases:

1.

2.

3.

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 dampingIs counter-productive!

ResonancePushing a swing

Driven / Forced Oscillations

Driven / Forced Oscillations

Amplitude A(ωd)

Resonance

Resonance