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HARMONIC MOTION

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HARMONIC MOTION Phase angle Some basics . Phase angle Some basics . In this presentation, you will learn that: The time period T of a harmonic ... – PowerPoint PPT presentation

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Title: HARMONIC MOTION


1
HARMONIC MOTION
2
  • 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.

3
Harmonic oscillators are all around us
4
They can destroy
5
They can cure
6
Start with a pendulum
A
- A
Displacement s
7
A
Displacement s
time
- A
Periodic time T
8
(No Transcript)
9
Imagine a pendulum oscillating over an
old-fashioned record turntable
Looking from above
start
Turntable has moved to here
Pendulum bob moves to here
10
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.
11
The moves up and down the vertical
line. The arrow goes round the circle
anticlockwise.
?
The phasor picture
12
Answer to ?
The clock arrow rotates at constant angular
frequency ? in rad s-1
13
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
14
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

15
  • 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.
16
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.
17
Two very useful equations (easy to remember)
Max velocity in any cycle ?A (where A is
Amplitude.)
Max acceleration in any cycle ?2A
18
Summary
19
All harmonic oscillators have these things in
common
  1. They are accelerated towards an equilibrium
    position by a spring-like force. This always
    pulls it back towards the equilibrium position.
    (F ma)
  2. 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!
  3. Its time trace is sinusoidal, oscillating at the
    NATURAL FREQUENCY fo of the oscillator.

20
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.
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