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HARMONIC MOTION

- In this presentation, you will learn that
- The time period T of a harmonic oscillator is

INDEPENDENT of its amplitude. - This property is ideal for time-keeping.
- Although nothing is constant about this type of

motion, we can relate it directly to CIRCULAR

MOTION, which has a constant (angular frequency

in rad s-1). - Simple geometry will lead us to a useful

expression of displacement, amplitude and

frequency of oscillation.

Harmonic oscillators are all around us

They can destroy

They can cure

Start with a pendulum

A

- A

Displacement s

A

Displacement s

time

- A

Periodic time T

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Imagine a pendulum oscillating over an

old-fashioned record turntable

Looking from above

start

Turntable has moved to here

Pendulum bob moves to here

Looking from above

Although the pendulum bob is moving up and down

in a complex sinusoidal way, the arrow is moving

at a CONSTANT SPEED round the circle.

The moves up and down the vertical

line. The arrow goes round the circle

anticlockwise.

?

The phasor picture

Answer to ?

The clock arrow rotates at constant angular

frequency ? in rad s-1

Phase angle

Deg rad

0 0

45 ?/4

90 ?/2

135 3 ?/4

180 ?

225 5 ?/4

270 3 ?/2

315 7 ?/4

360 2 ?

s A sin ?

? ?t

Radius is the Amplitude A

? ? /t in rad s-1

? is the phase angle

Some basics .

- Clock arrow rotates 2? in time period T.
- Angle ? 2? (t/T)
- T 1/f where f frequency
- So Angle ? 2?ft
- And since ? ? /t, so ? 2 ?f.
- Displacement s A sin? A sin 2?ft

- The language of oscillators
- f 1/T in Hz
- 2?f in rad s-1
- Always work out ? first, if your oscillator is

harmonic! Its then easy to find your

displacement s - s A sin 2 ?ft s A sin ?t

Note that s A sin 2 ?ft when s 0 when t

0. If s A when t 0, there is a ?/2 phase

difference and s A cos 2 ?ft.

Try this example A childs swing oscillates at a

frequency of 0.5 Hz. Its amplitude at the start

of its swing is 2 m.

1. Whats its angular frequency? (Dont be put

off by it not moving in a circle!)

Answer ? 2?f 2 x 3.14 x 0.5 3.1 rad s-1

2. What will be the displacement of the swing

after 3.4 seconds? In which direction?

Answer s A sin ?t 2 x sin ? x 3.4 0.37 m

towards equilibrium.

3. What will be the displacement after 8 secs?

Answer since f 1/T, T 2 secs. Displacement

will be 0 m.

Two very useful equations (easy to remember)

Max velocity in any cycle ?A (where A is

Amplitude.)

Max acceleration in any cycle ?2A

Summary

All harmonic oscillators have these things in

common

- They are accelerated towards an equilibrium

position by a spring-like force. This always

pulls it back towards the equilibrium position.

(F ma) - At the equilibrium position, the velocity of the

oscillator is unchanged and at its maximum. So,

there is no acceleration here and no resultant

force! - Its time trace is sinusoidal, oscillating at the

NATURAL FREQUENCY fo of the oscillator.

What about the energy of the oscillator?

- It stores energy.
- The energy goes back and forth,being stored by a

sort of spring (P.E.) at the extremes, and then

carried by the motion (K.E.) as it passes through

the equilibrium position. - Resistive forces gradually drain the oscillator

of its energy. Its amplitude gradually decreases

until more energy is fed back into it to

compensate.