xmkx Free SHM

1111 cos tAtx

Superposition

Superposition: the sum of solutions to an EOM is also a solution . . . . . . if the EOM is linear.

EOM:

Solutions:

2222 cos tAtx

x and its derivatives appear only to first power.

…or a linear combination: c1x1 + c2x2

Is the linear combination useful?

initial displacement, begin at restxo , vo = 0

0111 cos0 xAx

0sin0 111 oAx

txtx oo cos1

101 ,0 Ax

initial velocity, begin at originvo , xo = 0

0cos0 222 Ax

oo vAx 222 sin0

22 cos

tv

tx oo

o

o

ovA

2

22

initial displacement xo and velocity vo

oo vAx 333 sin0

0333 cos0 xAx

Solve each for A3, equate, solve for :f

oo

o

xv

1

3 tan

Plug back into top equation to get A3:

ooo

o

xvx

A

13 tancos

Solution:

oooo

ooo

o xvtxv

xtx

1

13 tancostancos

-2

-1

0

1

2

0 5 10 15 20

X

Time

-2

-1

0

1

2

0 5 10 15 20X

Time

-2

-1

0

1

2

0 5 10 15 20

X

Time

-2

-1

0

1

2

0 5 10 15 20

X

Time

Superposition: The motion resulting from two simultaneous initial conditions is equal to the sum of the motions resulting from each initial condition. . . . . . if the EOM is linear.

x1

x2

x1+x2

x3

+x

Imagine SHM in 1D…

…as a projection of 2D motion.

The trig is getting complicated, let’s try something else…

Imagine SHM in 1D…

…as a projection of 2D motion.+x

“real” x-axis

“imaginary” y-axis

The trig is getting complicated, let’s try something else…

“real” x-axis

“imaginary” y-axis

Describe the position…

j -> “rotate 90 degrees”

…algebraically…geometrically OR

jbiar ˆˆ jbaz

j2b -> “go along original axis in opposite direction”

1j

12 j

… an imaginary number!

rz

jbaz

“real” “imaginary”

“complex number”

“complex plane”

“real”x-axis

“imaginary”y-axis

rz

“Euler’s Formula” - imaginary exponentials oscillate !

!5!4!3!2

1)sin(cos5432 t

jtt

jt

jttjt

!5!4!3!2

15432 jtjtjtjt

jt

…expansion of exp(jt)!

jtetjt )sin()cos(

Circular motion in complex plane:

01 jeEuler’s Identity

“Greatest equation ever.”-Physics World

“The most famous equation in all mathematics.”-Constance Reid

“if this formula is not immediately apparent to a student being told it, the student would never be a first-class mathematician.”

-Gauss

“..we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth.”

-Benjamin Peirce

“this amazing jewel … the most remarkable formula in mathematics.”-Feynman

Physical Graffiti

Describe SHM in the complex plane…

Keepin’ it real !

tjAetz )(

tAtztx cosRe)(

Does it work?

Yes, if:

A and f take any value

tjtj Aedtd

mAek 2

2

tjtj emAkAe 2

2mk

Don’t forget!!!

Same A and w, but different f:

)(1

1 tjAez)(

22 tjAez

)(21

21 tjtj eeAz

2121

jjtj eeAez

121 121

jtj eAez

1

1

j

j

ee

021

x1 and x2 in phase

1221

tjAez

21

x1 and x2 out of phase

021 z

jtj eAez 11

21

x

t

x

t

x1

x2

x1

x2

121 cos2 tAx

021 x

Same A and f, different w:

ttAx 2121 coscos

22 coscos2coscos

Then:

ttAx

2cos

2cos2 2121

21

yields beats.21

Stick with trig…

Systems with linear EOMs obey the Principle of Superposition: solutions to the EOM can be summed to make more complicated solutions.

Re

Im