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PRIOR READING:Main 1.1 & 1.2Taylor 5.2
REPRESENTING SIMPLE HARMONIC MOTION
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/shm.gif
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y(t)
Simple Harmonic Motion
Watch as time evolves
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x(t) =Acos ω0t+φ( )
-1
1
-0.25 0.25 0.75 1.25 1.75
time
position
-A
A
T =2πω0
=1f
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x0
t =−φω0
amplitude phase angle
determined by initial conditions
period
angular freq (cyclic) freq
determined by physical system3
Position (cm)
Velocity (cm/s)
Acceleration (cm/s2)
-40
-20
0
20
40
a
0 1 2t
-6
-4
-2
0
2
4
6
v
0 1 2t
-1
-0.5
0
0.5
1
x
0 1 2t
time (s)
x(t) =Acos ω0t+φ( )
&x(t) =Aω0 cos ω0t+φ+ π2( )
&&x(t) =Aω02 cos ω0t+φ+π( )
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x(t) = Acos ω0t +φ( )
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x(t) = Bp cosω0t + Bq sinω0t
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x(t) = C exp iω0t( ) +C * exp −iω0t( )
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x(t) = Re Dexp iω0t( )[ ]
These representations of the position of a simple harmonic oscillator as a function of time are all equivalent - there are 2 arbitrary constants in each. Note that A, , Bp and Bq are REAL; C and D are COMPLEX. x(t) is real-valued variable in all cases.
Engrave these on your soul - and know how to derive the relationships among A & ; Bp & Bq; C; and D .
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x(t) = Acos ω0t +φ( )
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x(t) = Bp cosω0t + Bq sinω0t
m = 0.01 kg; k = 36 Nm-1. At t = 0, m is displaced 50mm to the right and is moving to the right at 1.7 ms-1. Express the motion in form A (even groups)form B (odd groups)
x
m
mk
k
Example: initial conditions
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x(t) =57.5cos 60s−1⎡⎣ ⎤⎦t−0.516( )mm
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x(t) = Bp cosω0t + Bq sinω0t€
x(t) = Acos ω0t +φ( )
x(t) =50mmcos 60s−1⎡⎣ ⎤⎦t( ) + 28.3mmsin 60s−1⎡⎣ ⎤⎦t( )
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Bp = Acosφ
Bq = −Asinφ
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A = Bp2 + Bq
2
tanφ = −Bq
Bp 7
Real
Imag
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z = z e iφ
a
b
|z|
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z = a2 + b2
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tanφ =b
a
Complex numbers
Argand diagram
i = −1
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z =a+ ib
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a + ib = z cosφ + i z sinφ
Consistency argument
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z = a + ib
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z = z e iφ
If these represent the same thing, then the assumed Euler relationship says:
Equate real parts:
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a = z cosφ
Equate imaginary parts:
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b = z sinφ
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z = a2 + b2
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tanφ =b
a
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exp iφ( ) = cosφ + isinφ
Euler’s relation
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exp iω0t( ) = cos ω0t( ) + isin ω0t( )10
Real
Imag
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x(t) = Re Aeiφeiω0t[ ] = Re Aei ω0t+φ( )
[ ]
t = 0, T0, 2T0
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φ
t = T0/4
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π2
+φ
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ω0t +φ
t = t
PHASOR
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Adding complex numbers is easy in rectangular form
z =a+ ib w =c+ id
z +w= a+ c[ ] + i b+d[ ]
Real
Imag
ab
d
c
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Multiplication and division of complex numbers is easy in polar form
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z = z e iφ w =weiθ
zw = z wei φ+θ[ ]
Real
Imag
|z|
|w|
θθ
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z =a+ ib z* =a−ib
z + z* = a+ a[ ] + i b−b[ ] =2a
Real
Imag
ab
Another important idea is the COMPLEX CONJUGATE of a complex number. To form the c.c., change i -> -i
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z = z e iφ z*= z e−iφ
zz*= z eiφ z e−iφ = z 2
The product of a complex number and its complex conjugate is REAL.We say “zz* equals mod z squared”
|z|
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And finally, rationalizing complex numbers, or: what to do when there’s an i in the denominator?
z =a+ ibc+ id
z=a+ ibc+ id
×c−idc−id
z =ac+bd+ i bc−ad( )
c2 +d2
=ac+bdc2 +d2
Re z( )1 24 34
+ ibc−ad( )c2 +d2
Im z( )1 24 34
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x(t) = C exp iω0t( ) +C *exp −iω0t( )
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x(t) = Re Dexp iω0t( )[ ]
m = 0.01 kg; k = 36 Nm-1. At t = 0, m is displaced 50mm to the right and is moving to the right at 1.7 ms-1. Express the motion in form C (even groups), form D (odd groups)
x
m
mk
k
Using complex numbers: initial conditions. Same example as before, but now use the “C” and “D” forms
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x(t) = 25e−i0.516e i60s−1t + 25e+i0.516e−i60s−1t( ) mm
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Acosφ = Bp = 2Re C[ ] = Re D[ ]
Asinφ = −Bq = 2Im C[ ] = Im D[ ]
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D = 2C = A
tanφ =Im D[ ]Re D[ ]
=Im C[ ]Re C[ ]
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x(t) = C exp iω0t( ) +C * exp −iω0t( )
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x(t) = Re Dexp iω0t( )[ ]
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x(t) = Re 50e−i0.516( )e
i60s−1t[ ] mm
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