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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