424: Oscillations & Waves

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

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

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

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

REPRESENTING SIMPLE HARMONIC MOTION PH421:Oscillations F09 6/8/2018 REPRESENTING SIMPLE HARMONIC MOTION simple not simple http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/shm.gif Please do not distribute

Simple Harmonic Motion PH421:Oscillations F09 6/8/2018 Simple Harmonic Motion y(t) Watch as time evolves Please do not distribute

determined by physical system amplitude phase angle PH421:Oscillations F09 6/8/2018 period angular freq (cyclic) freq determined by physical system amplitude phase angle determined by initial conditions A -A Please do not distribute

Position (cm) Velocity (cm/s) Acceleration (cm/s2) time (s) PH421:Oscillations F09 6/8/2018 Position (cm) Velocity (cm/s) Acceleration (cm/s2) time (s) Please do not distribute

x(t) is real-valued variable in all cases. PH421:Oscillations F09 6/8/2018 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 . Please do not distribute

Example: initial conditions PH421:Oscillations F09 6/8/2018 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 Please do not distribute

PH421:Oscillations F09 6/8/2018 Please do not distribute

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 Please do not distribute

PH421:Oscillations F09 6/8/2018 Please do not distribute

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)

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

Clicker Questions

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.

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.

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) 2 (d) 4

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.

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.

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.

At the moment when the mass is at the point P it has PH421:Oscillations F09 6/8/2018 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 Please do not distribute

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.

Optional Review of Complex Numbers

Complex numbers Imag |z| b f Real a Argand diagram PH421:Oscillations F09 6/8/2018 Complex numbers Real Imag b f |z| a Argand diagram Please do not distribute

Euler’s relation PH421:Oscillations F09 6/8/2018 Please do not distribute

Equate imaginary parts: PH421:Oscillations F09 6/8/2018 Consistency argument If these represent the same thing, then the assumed Euler relationship says: Equate real parts: Equate imaginary parts: Please do not distribute

Imag t = T0/4 t = 0, T0, 2T0 PHASOR Real t = t PH421:Oscillations F09 6/8/2018 Real Imag t = T0/4 t = 0, T0, 2T0 t = t PHASOR Please do not distribute

Adding complex numbers is easy in rectangular form PH421:Oscillations F09 6/8/2018 Adding complex numbers is easy in rectangular form Real Imag b a c d Please do not distribute

Multiplication and division of complex numbers is easy in polar form PH421:Oscillations F09 6/8/2018 Multiplication and division of complex numbers is easy in polar form Real Imag |z| q+f q f |w| Please do not distribute

The product of a complex number and its complex conjugate is REAL. PH421:Oscillations F09 6/8/2018 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” Please do not distribute

PH421:Oscillations F09 6/8/2018 And finally, rationalizing complex numbers, or: what to do when there's an i in the denominator? Please do not distribute