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