1. 424: Oscillations & Waves

2. PH424: Homework 30%; Laboratory reports 35%; Final 35%.
Feb & March 2017 ~
Mon
Tue
Wed
Thu
Fri 13
Simple Harmonic Motion
- 4 representations
14 - Free motion of an oscillator
-Free damped oscillations
15 -PS1a due
16 Lab & Discussion: the LCR circuit
17 Upload data -PS1b due Forced motion of a damped oscillator LCR circuit resonance
20 Forced motion of a damped oscillator -data workshop
o phase shifts o resonance
21 Forced motion & resonances
22 -PS2a due (Math Methods) 23
(Math Methods)
24 -PS2b due (Math Methods)
27 (Math Methods)
Formal LCR Lab Report Due (in class)
28 (Math Methods)
-The Fourier Series & Transform
1 -PS3a due
Intro to Wave Reflection & Transmission
Pre-Lab 2
Lab & Discussion: Coax Cable Lab Workshop
3 -PS3b due
Lab Data Workshop
Upload data 6
-Multiple Driving Frequencies 7
Fourier coefficients & transform
Fast Fourier Methods & Impulse Demo Lab
8 -PS4a due
Fast Fourier Transform 9
Wave Mechanics 10
-PS4b due
14  
15 -PS5a due 16
17 -PS5b due
Paradigms 424 Review I

3. Introducction to Formal Technical Writing
Two “formal” lab reports (35%) are required. Good technical writing is very similar to writing an essay with sub-headings. We want to hear a convincing scientific story, not a shopping list of everything you did.
Check out course web-site
We will have data & write-up workshops

4. Are oscillations ubiquitous or are they merely a paradigm?
Superposition of brain neuron activity

5. Oscillations  modulations in time Waves  modulations in space
for our purposes ….
Oscillations  modulations in time Waves  modulations in space
Superposition of brain neuron activity

6. REPRESENTING SIMPLE HARMONIC MOTION
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REPRESENTING SIMPLE HARMONIC MOTION
simple
not simple
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7. Simple Harmonic Motion
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Simple Harmonic Motion
y(t)
Watch as time evolves
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8. determined by physical system amplitude phase angle
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period
angular freq
(cyclic) freq
determined by physical system
amplitude
phase angle
determined by initial conditions A -A
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9. Position (cm) Velocity (cm/s) Acceleration (cm/s2) time (s)
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Position (cm)
Velocity (cm/s)
Acceleration (cm/s2)
time (s)
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10. x(t) is real-valued variable in all cases.
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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, f, Bp and Bq are REAL; C and D are COMPLEX.
x(t) is real-valued variable in all cases. A: B: C: D:
Engrave these on your soul - and know how to derive the relationships among A & f; Bp & Bq; C; and D .
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11. Example: initial conditions
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Example: initial conditions x m k
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
form B
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12. PH421:Oscillations F09
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13. PH421:Oscillations F09
Using complex numbers: initial conditions. Same example as before, but now use the "C" and "D" forms
6/8/2018 x m k
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
form D
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14. PH421:Oscillations F09
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15. Differential Equation Representation
our equation
is a second-order ODE
our basis can be
{cos(wt), sin(wt)}
But all linear combinations are solutions, i.e., Acos(wt) + Bsin(wt)

16. Consider all the equivalent solutions
what about Asin(wt+) ?
Asin(wt+)
=A{sin(wt) cos() + cos(wt) sin()}
=Acos() • sin(wt) + Asin()•cos(wt)
=A' sin(wt) + A' tan()•cos(wt)
= A' sin(wt) + B' cos(wt)
so Asin(wt+) gives all linear combinations of basis solutions
… spans the space

17. Clicker Questions

18. A particle executes simple harmonic motion.
When the velocity of the particle is a maximum which one of the following gives the correct values of potential energy and acceleration of the particle.
(a)potential energy is maximum and acceleration is maximum.
(b)potential energy is maximum and acceleration is zero.
(c)potential energy is minimum and acceleration is maximum.
(d)potential energy is minimum and acceleration is zero.

19. A particle executes simple harmonic motion.
When the velocity of the particle is a maximum which one of the following gives the correct values of potential energy and acceleration of the particle.
(a)potential energy is maximum and acceleration is maximum.
(b)potential energy is maximum and acceleration is zero.
(c)potential energy is minimum and acceleration is maximum.
(d)potential energy is minimum and acceleration is zero.
Answer (d). When velocity is maximum displacement is zero so potential energy and acceleration are both zero.

20. A mass vibrates on the end of the spring
A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The mass was changed by a factor of
1/4 (b) ½ (c) (d) 4

21. A mass vibrates on the end of the spring
A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The mass was changed by a factor of
1/4 (b) 1/2 (c) 2 (d) 4
Answer (a). Since the frequency has increased the mass must have decreased. Frequency is inversely proportional to the square root of mass, so to double frequency the mass must
change by a factor of 1/4.

22. A mass vibrates on the end of the spring
A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The maximum acceleration of the mass:
remains the same.
is halved.
is doubled.
is quadrupled.

23. A mass vibrates on the end of the spring
A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The maximum acceleration of the mass:
remains the same.
is halved.
is doubled.
is quadrupled.
Answer (d). Acceleration is proportional to frequency squared. If frequency is doubled than acceleration is quadrupled.

24. At the moment when the mass is at the point P it has
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A particle oscillates on the end of a spring and its position as a function of time is shown below.
At the moment when the mass is at the point P it has
(a) positive velocity and positive acceleration
(b) positive velocity and negative acceleration
(c) negative velocity and negative acceleration
(d) negative velocity and positive acceleration
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25. A particle oscillates on the end of a spring and its position as a function of time is shown below.
At the moment when the mass is at the point P it has
(a) positive velocity and positive acceleration
(b) positive velocity and negative acceleration
(c) negative velocity and negative acceleration
(d) negative velocity and positive acceleration
Answer (b). The slope is positive so velocity is positive. Since the slope is getting smaller with time the acceleration is negative.

26. Optional Review of Complex Numbers

27. Complex numbers Imag |z| b f Real a Argand diagram
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Complex numbers
Real
Imag b f
|z| a
Argand diagram
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28. Euler’s relation PH421:Oscillations F09 6/8/2018
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29. Equate imaginary parts:
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Consistency argument
If these represent the same thing, then the assumed Euler relationship says:
Equate real parts:
Equate imaginary parts:
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30. Imag t = T0/4 t = 0, T0, 2T0 PHASOR Real t = t PH421:Oscillations F09
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Real
Imag
t = T0/4
t = 0, T0, 2T0
t = t
PHASOR
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31. Adding complex numbers is easy in rectangular form
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Adding complex numbers is easy in rectangular form
Real
Imag b a c d
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32. Multiplication and division of complex numbers is easy in polar form
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Multiplication and division of complex numbers is easy in polar form
Real
Imag
|z|
q+f q f
|w|
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33. The product of a complex number and its complex conjugate is REAL.
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Another important idea is the COMPLEX CONJUGATE of a complex number. To form the c.c., change i -> -i
Real
Imag
|z| b f a
The product of a complex number and its complex conjugate is REAL.
We say “zz* equals mod z squared”
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34. PH421:Oscillations F09
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And finally, rationalizing complex numbers, or: what to do when there's an i in the denominator?
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