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Oscillations: the Essentials
3/22/07
Complex Numbers
Be familiar with the complex plane:
i
x
iy
re
q
+
=
, definitions:
mod
, phase of
.
z
z
r
z
=
=
=
To multiply complex numbers, multiply the mods,
add
the phases. Know the allimportant
formula:
cos
sin
,
i
e
i
q
=
+
and be able to interpret it as a point on the
unit circle
.
Some useful small x approximations:
( )
( )
1
1
2
1
,
ln 1
,
1
1
,
1
1
x
e
x
x
x
x
x
x
x

@ +
+
@
+
@ +
+
@ 
and for small angles
2
1
2
sin
tan
,
cos
1
.
@
@
@ 
Undamped Simple Harmonic Oscillator:
Be able to solve the equation
2
2
, or
,
d x
F
ma
m
kx
dt
=
= 
and write down the velocity and kinetic energy at any time. Be able to sketch a graph of the
potential energy
as a function of position, both for a horizontal and a vertical spring. Be able to
derive the angular frequency and the period. Know how to find the dependence of the period on
k
,
m
using dimensional arguments.
A Heavily Damped Oscillator:
You should know the equation of motion
2
2
d x
dx
m
kx
b
dt
dt
= 

and that a solution is
0
.
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This note was uploaded on 12/07/2011 for the course PHYSICS 152 taught by Professor Michaelfowler during the Fall '07 term at UVA.
 Fall '07
 MichaelFowler

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