Free SHM

1
Superposition
EOM
Free SHM
Solutions
Superposition the sum of solutions to an EOM is
also a solution . . . . . . if the EOM is
linear.
cos(w1tf1) terms
cos(w2tf2) terms
2
Is the linear combination useful?
initial displacement xo , vo 0
initial velocity vo , xo 0
3
initial displacement xo and velocity vo
Solve each for A, equate, solve for f
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.
Plug back into top equation to get A
Solution
4
The trig is getting complicated, lets try
something else
Imagine SHM in 1D
as a projection of 2D motion.
x
5
The trig is getting complicated, lets try
something else
Imagine SHM in 1D
as a projection of 2D motion.
x
imaginary y-axis
real x-axis
6
Describe the position
z
r
OR
geometrically
algebraically
j -gt rotate 90 degrees
j2b -gt go along original axis in opposite
direction
an imaginary number!
7
complex number
z
r
real
imaginary
complex plane
8
expansion of exp(jt)!
Eulers Formula - imaginary exponentials
oscillate !
9
Eulers Identity
..we cannot understand it, and we dont know
what it means, but we have proved it, and
therefore we know it must be the
truth. -Benjamin Peirce
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
this amazing jewel the most remarkable formula
in mathematics. -Feynman
10
Physical Graffiti
11
Describe SHM in the complex plane
Does it work?
Yes, if
A and f take any value
Keepin it real !
Dont forget!!!
12
Same A and w, but different f
13
x1 and x2 in phase
x1 and x2 out of phase
x
x
x1
x1
x2
x2
t
t
14
Same A and f, different w
Stick with trig
Then
yields beats.
15
Systems with linear EOMs obey the Principle of
Superposition solutions to the EOM can be
summed to make more complicated solutions.
Re
Im