N coupled oscillators
|
|
- Griffin O’Connor’
- 5 years ago
- Views:
Transcription
1 Waves 1 1
2 Waves 1 1. N coupled oscillators towards the continuous limit. Stretched string and the wave equation 3. The d Alembert solution 4. Sinusoidal waves, wave characteristics and notation
3 T 1 T N coupled oscillators Consider fleible elastic string to which are attached N particles of mass m, each a distance l apart. The string is fied at each end. Small transverse displacements are applied transverse oscillations p1 p p 1 p 1 l p N N 1 3
4 N coupled oscillators: special cases Let s first consider the special cases N=1 and N= 4
5 N coupled oscillators: general case Now let s tr and find solution for a general value N 5
6 N coupled oscillators: the solution T 1 T p1 p p 1 p 1 l p N N 1 Displacement for mass p when oscillating in mode n and angular frequenc: pn pn ( Cn sin cos( nt n) N 1 n n sin 0 ( N 1) 0 T / ml Although the value of n can go beond N, this just generates duplicate solutions, i.e. there are N normal modes in total. 6
7 N coupled oscillators: modes for N=5 Look at each mode for N=5, with snapshot taken at t=0 n=1 n= n=5 n=3 n=4 Note how the displacement of ever particle falls on a sine curve! 7
8 N coupled oscillators: N ver large Let s eplore the scenario where N is ver large, which starts to approimate case of a real, continuous, string 8
9 Sstem of springs and N masses: longitudinal oscillations k m m k k m k m k u p Let u p be displacement from equilibrium position of mass p 9
10 Stretched string Consider a segment of string of linear densit ρ stretched under tension T T T, small 10
11 Stretched string and wave equation Will show that the displacements on a stretched string obe t T which is the wave equation 1 t c T c with 11
12 Jean-Baptiste le Rond d Alembert Lived in Paris Mathematician and phsicist Also a music theorist and co-editor with Diderot of a famous encclopaedia 1
13 d Alembert solution of wave equation We will show how the wave equation can be solved to ield solutions of form: (, f ( c g( c Here f and g are an functions of (-c & (+c, determined b initial conditions. We will then interpret this solution. 13
14 Interpretation of D Alembert solution (, f ( c Focus on =0 and consider situations at t=0 and t=δt (0,0) ( 0, c t Wave moves to right with speed c 14
15 Interpretation of D Alembert solution (, g( c Focus on =0 and consider situations at t=0 and t=δt (0,0) ( 0, c t Wave moves to left with speed c 15
16 d Alembert s solution with boundar conditions Eample: rectangular wave of length a released from rest 1 (, U ( c U( c t 0 t a / c U() a a a a t a / c t 3a / c a a a a 16
17 Sinusoidal waves A ver common functional dependence for f and g... (, f ( c g( c...is sinusoidal. In this case it is usual to write: (, Acos( k Bcos( k with k and ω (and A and B) constants or Asin(k-ω... etc (choice doesn t matter, unless we are comparing one wave with another and then relative phases become importan speed of wave c / k 1/ T frequenc where ω is angular frequenc f / k / wavelength where k is the wave-number (or wave-vector if also used to indicate direction of wave) 17
18 Notation choices Sinusoidal solution (, Acos( k (writing here, for compactness, onl the forward-going solution) Using the relationships between k,ω, λ & c this can be epressed in man forms (, Acos[ k( c] (, Acos( t k) Also note that sometimes it is convenient to write Changes nothing (for cosine, triviall so, & practicall not even for sine function, as overall sign can be absorbed in constan & still describes forward-going wave. A ver frequent approach is to use comple notation (we alread made use of this when analsing normal modes, and ou will have seen it in circuit analsis) (, Re Aep[ i( k] or (, Im Aep[ i( k] if its important to pick out sine function. Note that often the Re or Im is implicit, and it gets omitted in discussion. 18
19 Phase differences Often important to specif phase shifts. Onl meaningful to do so when we are comparing one wave to another. 1(, Acos( k (, Acos( kt ) wave wave 1 In this eample wave leads wave 1 b π/, i.e. φ=-π/ Can be epressed with comple notation (, Re Aep[ i( kt )] Nicer still to subsume phase into amplitude (, Re Aep[ i( k] with π/ A A ep( i) k=0 ωt 19
One-Dimensional Wave Propagation (without distortion or attenuation)
Phsics 306: Waves Lecture 1 1//008 Phsics 306 Spring, 008 Waves and Optics Sllabus To get a good grade: Stud hard Come to class Email: satapal@phsics.gmu.edu Surve of waves One-Dimensional Wave Propagation
More informationChapter 13. F =!kx. Vibrations and Waves. ! = 2" f = 2" T. Hooke s Law Reviewed. Sinusoidal Oscillation Graphing x vs. t. Phases.
Chapter 13 Vibrations and Waves Hooke s Law Reviewed F =!k When is positive, F is negative ; When at equilibrium (=0, F = 0 ; When is negative, F is positive ; 1 2 Sinusoidal Oscillation Graphing vs. t
More informationChapter 16 Mechanical Waves
Chapter 6 Mechanical Waves A wave is a disturbance that travels, or propagates, without the transport of matter. Examples: sound/ultrasonic wave, EM waves, and earthquake wave. Mechanical waves, such as
More informationy y m y t 0 t > 3 t 0 x y t y m Harmonic waves Only pattern travels, not medium. Travelling wave f(x vt) is a wave travelling at v in +x dir n :
Waves and Sound for PHYS1169. Joe Wolfe, UNSW Waves are moving pattern of displacements. Ma transmit energ and signals. 1169 Sllabus Travelling waves, superposition and interference, velocit, reflection
More informationWave Phenomena Physics 15c
Wave Phenomena Phsics 15c Lecture 13 Multi-Dimensional Waves (H&L Chapter 7) Term Paper Topics! Have ou found a topic for the paper?! 2/3 of the class have, or have scheduled a meeting with me! If ou haven
More informationWave Phenomena Physics 15c
Wave Phenomena Phsics 15c Lecture 13 Multi-Dimensional Waves (H&L Chapter 7) Term Paper Topics! Have ou found a topic for the paper?! 2/3 of the class have, or have scheduled a meeting with me! If ou haven
More informationModule 8: Sinusoidal Waves Lecture 8: Sinusoidal Waves
Module 8: Sinusoidal Waves Lecture 8: Sinusoidal Waves We shift our attention to oscillations that propagate in space as time evolves. This is referred to as a wave. The sinusoidal wave a(,t) = A cos(ωt
More information25.2. Applications of PDEs. Introduction. Prerequisites. Learning Outcomes
Applications of PDEs 25.2 Introduction In this Section we discuss briefly some of the most important PDEs that arise in various branches of science and engineering. We shall see that some equations can
More informationChapter 9. Electromagnetic Waves
Chapter 9. Electromagnetic Waves 9.1 Waves in One Dimension 9.1.1 The Wave Equation What is a "wave?" Let's start with the simple case: fixed shape, constant speed: How would you represent such a string
More informationOscillatory Motion and Wave Motion
Oscillatory Motion and Wave Motion Oscillatory Motion Simple Harmonic Motion Wave Motion Waves Motion of an Object Attached to a Spring The Pendulum Transverse and Longitudinal Waves Sinusoidal Wave Function
More informationAdditional Topics in Differential Equations
6 Additional Topics in Differential Equations 6. Eact First-Order Equations 6. Second-Order Homogeneous Linear Equations 6.3 Second-Order Nonhomogeneous Linear Equations 6.4 Series Solutions of Differential
More informationSurvey of Wave Types and Characteristics
Seminar: Vibrations and Structure-Borne Sound in Civil Engineering Theor and Applications Surve of Wave Tpes and Characteristics Xiuu Gao April 1 st, 2006 Abstract Mechanical waves are waves which propagate
More information15. Eigenvalues, Eigenvectors
5 Eigenvalues, Eigenvectors Matri of a Linear Transformation Consider a linear ( transformation ) L : a b R 2 R 2 Suppose we know that L and L Then c d because of linearit, we can determine what L does
More informationStanding Waves If the same type of waves move through a common region and their frequencies, f, are the same then so are their wavelengths, λ.
Standing Waves I the same type o waves move through a common region and their requencies,, are the same then so are their wavelengths,. This ollows rom: v=. Since the waves move through a common region,
More informationWave Motion A wave is a self-propagating disturbance in a medium. Waves carry energy, momentum, information, but not matter.
wae-1 Wae Motion A wae is a self-propagating disturbance in a medium. Waes carr energ, momentum, information, but not matter. Eamples: Sound waes (pressure waes) in air (or in an gas or solid or liquid)
More informationChapter 15 Mechanical Waves
Chapter 15 Mechanical Waves 1 Types of Mechanical Waves This chapter and the next are about mechanical waves waves that travel within some material called a medium. Waves play an important role in how
More informationAdditional Topics in Differential Equations
0537_cop6.qd 0/8/08 :6 PM Page 3 6 Additional Topics in Differential Equations In Chapter 6, ou studied differential equations. In this chapter, ou will learn additional techniques for solving differential
More informationChapter 15. Mechanical Waves
Chapter 15 Mechanical Waves A wave is any disturbance from an equilibrium condition, which travels or propagates with time from one region of space to another. A harmonic wave is a periodic wave in which
More informationChapter 14: Wave Motion Tuesday April 7 th
Chapter 14: Wave Motion Tuesday April 7 th Wave superposition Spatial interference Temporal interference (beating) Standing waves and resonance Sources of musical sound Doppler effect Sonic boom Examples,
More informationOscillations. Oscillations and Simple Harmonic Motion
Oscillations AP Physics C Oscillations and Simple Harmonic Motion 1 Equilibrium and Oscillations A marble that is free to roll inside a spherical bowl has an equilibrium position at the bottom of the bowl
More informationChapter 16 Waves. Types of waves Mechanical waves. Electromagnetic waves. Matter waves
Chapter 16 Waves Types of waves Mechanical waves exist only within a material medium. e.g. water waves, sound waves, etc. Electromagnetic waves require no material medium to exist. e.g. light, radio, microwaves,
More informationEP225 Note No. 4 Wave Motion
EP5 Note No. 4 Wave Motion 4. Sinusoidal Waves, Wave Number Waves propagate in space in contrast to oscillations which are con ned in limited regions. In describing wave motion, spatial coordinates enter
More informationNo Lecture on Wed. But, there is a lecture on Thursday, at your normal recitation time, so please be sure to come!
Announcements Quiz 6 tomorrow Driscoll Auditorium Covers: Chapter 15 (lecture and homework, look at Questions, Checkpoint, and Summary) Chapter 16 (Lecture material covered, associated Checkpoints and
More informationFaculty of Computers and Information Fayoum University 2017/ 2018 Physics 2 (Waves)
Faculty of Computers and Information Fayoum University 2017/ 2018 Physics 2 (Waves) 3/10/2018 1 Using these definitions, we see that Example : A sinusoidal wave traveling in the positive x direction has
More informationIntroduction to Differential Equations. National Chiao Tung University Chun-Jen Tsai 9/14/2011
Introduction to Differential Equations National Chiao Tung Universit Chun-Jen Tsai 9/14/011 Differential Equations Definition: An equation containing the derivatives of one or more dependent variables,
More informationSecond-Order Linear Differential Equations C 2
C8 APPENDIX C Additional Topics in Differential Equations APPENDIX C. Second-Order Homogeneous Linear Equations Second-Order Linear Differential Equations Higher-Order Linear Differential Equations Application
More information1. Types of Waves. There are three main types of waves:
Chapter 16 WAVES I 1. Types of Waves There are three main types of waves: https://youtu.be/kvc7obkzq9u?t=3m49s 1. Mechanical waves: These are the most familiar waves. Examples include water waves, sound
More informationAP Physics 1 Waves and Simple Harmonic Motion Practice Test
AP Physics 1 Waves and Simple Harmonic Motion Practice Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) An object is attached to a vertical
More informationLecture 4 Notes: 06 / 30. Energy carried by a wave
Lecture 4 Notes: 06 / 30 Energy carried by a wave We want to find the total energy (kinetic and potential) in a sine wave on a string. A small segment of a string at a fixed point x 0 behaves as a harmonic
More informationSuperposition and Standing Waves
Physics 1051 Lecture 9 Superposition and Standing Waves Lecture 09 - Contents 14.5 Standing Waves in Air Columns 14.6 Beats: Interference in Time 14.7 Non-sinusoidal Waves Trivia Questions 1 How many wavelengths
More informationHÉNON HEILES HAMILTONIAN CHAOS IN 2 D
ABSTRACT HÉNON HEILES HAMILTONIAN CHAOS IN D MODELING CHAOS & COMPLEXITY 008 YOUVAL DAR PHYSICS DAR@PHYSICS.UCDAVIS.EDU Chaos in two degrees of freedom, demonstrated b using the Hénon Heiles Hamiltonian
More informationPhysics 1C. Lecture 12C
Physics 1C Lecture 12C Simple Pendulum The simple pendulum is another example of simple harmonic motion. Making a quick force diagram of the situation, we find:! The tension in the string cancels out with
More informationPeriodic Structures in FDTD
EE 5303 Electromagnetic Analsis Using Finite Difference Time Domain Lecture #19 Periodic Structures in FDTD Lecture 19 These notes ma contain coprighted material obtained under fair use rules. Distribution
More informationChapter 16: Oscillatory Motion and Waves. Simple Harmonic Motion (SHM)
Chapter 6: Oscillatory Motion and Waves Hooke s Law (revisited) F = - k x Tthe elastic potential energy of a stretched or compressed spring is PE elastic = kx / Spring-block Note: To consider the potential
More informationTime-Frequency Analysis: Fourier Transforms and Wavelets
Chapter 4 Time-Frequenc Analsis: Fourier Transforms and Wavelets 4. Basics of Fourier Series 4.. Introduction Joseph Fourier (768-83) who gave his name to Fourier series, was not the first to use Fourier
More informationFourier Analysis Fourier Series C H A P T E R 1 1
C H A P T E R Fourier Analysis 474 This chapter on Fourier analysis covers three broad areas: Fourier series in Secs...4, more general orthonormal series called Sturm iouville epansions in Secs..5 and.6
More informationspring mass equilibrium position +v max
Lecture 20 Oscillations (Chapter 11) Review of Simple Harmonic Motion Parameters Graphical Representation of SHM Review of mass-spring pendulum periods Let s review Simple Harmonic Motion. Recall we used
More information21.55 Worksheet 7 - preparation problems - question 1:
Dynamics 76. Worksheet 7 - preparation problems - question : A coupled oscillator with two masses m and positions x (t) and x (t) is described by the following equations of motion: ẍ x + 8x ẍ x +x A. Write
More information!T = 2# T = 2! " The velocity and acceleration of the object are found by taking the first and second derivative of the position:
A pendulum swinging back and forth or a mass oscillating on a spring are two examples of (SHM.) SHM occurs any time the position of an object as a function of time can be represented by a sine wave. We
More informationI. Degrees and Radians minutes equal 1 degree seconds equal 1 minute. 3. Also, 3600 seconds equal 1 degree. 3.
0//0 I. Degrees and Radians A. A degree is a unit of angular measure equal to /80 th of a straight angle. B. A degree is broken up into minutes and seconds (in the DMS degree minute second sstem) as follows:.
More informationTime-Frequency Analysis: Fourier Transforms and Wavelets
Chapter 4 Time-Frequenc Analsis: Fourier Transforms and Wavelets 4. Basics of Fourier Series 4.. Introduction Joseph Fourier (768-83) who gave his name to Fourier series, was not the first to use Fourier
More informationRaymond A. Serway Chris Vuille. Chapter Thirteen. Vibrations and Waves
Raymond A. Serway Chris Vuille Chapter Thirteen Vibrations and Waves Periodic Motion and Waves Periodic motion is one of the most important kinds of physical behavior Will include a closer look at Hooke
More informationWave Motion: v=λf [m/s=m 1/s] Example 1: A person on a pier observes a set of incoming waves that have a sinusoidal form with a distance of 1.
Wave Motion: v=λf [m/s=m 1/s] Example 1: A person on a pier observes a set of incoming waves that have a sinusoidal form with a distance of 1.6 m between the crests. If a wave laps against the pier every
More informationThe Force Table Introduction: Theory:
1 The Force Table Introduction: "The Force Table" is a simple tool for demonstrating Newton s First Law and the vector nature of forces. This tool is based on the principle of equilibrium. An object is
More informationCHAPTER 15 Wave Motion. 1. The speed of the wave is
CHAPTER 15 Wave Motion 1. The speed of the wave is v = fλ = λ/t = (9.0 m)/(4.0 s) = 2.3 m/s. 7. We find the tension from the speed of the wave: v = [F T /(m/l)] 1/2 ; (4.8 m)/(0.85 s) = {F T /[(0.40 kg)/(4.8
More informationEnergy stored in a mechanical wave
Waves 3 1. Energy stored in a mechanical wave. Wave equation revisited - separation of variables 3. Wave on string with fied ends 4. Waves at boundaries 5. mpedance 6. Other eamples: - Lossless transmission
More informationKINEMATIC RELATIONS IN DEFORMATION OF SOLIDS
Chapter 8 KINEMATIC RELATIONS IN DEFORMATION OF SOLIDS Figure 8.1: 195 196 CHAPTER 8. KINEMATIC RELATIONS IN DEFORMATION OF SOLIDS 8.1 Motivation In Chapter 3, the conservation of linear momentum for a
More information(Total 1 mark) IB Questionbank Physics 1
1. A transverse wave travels from left to right. The diagram below shows how, at a particular instant of time, the displacement of particles in the medium varies with position. Which arrow represents the
More informationFIRST YEAR MATHS FOR PHYSICS STUDENTS NORMAL MODES AND WAVES. Hilary Term Prof. G.G.Ross. Question Sheet 1: Normal Modes
FIRST YEAR MATHS FOR PHYSICS STUDENTS NORMAL MODES AND WAVES Hilary Term 008. Prof. G.G.Ross Question Sheet : Normal Modes [Questions marked with an asterisk (*) cover topics also covered by the unstarred
More informationChapter 14 (Oscillations) Key concept: Downloaded from
Chapter 14 (Oscillations) Multiple Choice Questions Single Correct Answer Type Q1. The displacement of a particle is represented by the equation. The motion of the particle is (a) simple harmonic with
More informationSeismic Waves Propagation in Complex Media
H4.SMR/1586-1 "7th Workshop on Three-Dimensional Modelling of Seismic Waves Generation and their Propagation" 5 October - 5 November 004 Seismic Waves Propagation in Comple Media Fabio ROMANELLI Dept.
More informationChapter 11 Vibrations and Waves
Chapter 11 Vibrations and Waves If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system
More informationPhysics 101 Lecture 18 Vibrations, SHM, Waves (II)
Physics 101 Lecture 18 Vibrations, SHM, Waves (II) Reminder: simple harmonic motion is the result if we have a restoring force that is linear with the displacement: F = -k x What would happen if you could
More informationClass Average = 71. Counts Scores
30 Class Average = 71 25 20 Counts 15 10 5 0 0 20 10 30 40 50 60 70 80 90 100 Scores Chapter 12 Mechanical Waves and Sound To describe mechanical waves. To study superposition, standing waves, and interference.
More information1 A mass on a spring undergoes SHM. The maximum displacement from the equilibrium is called?
Slide 1 / 20 1 mass on a spring undergoes SHM. The maximum displacement from the equilibrium is called? Period Frequency mplitude Wavelength Speed Slide 2 / 20 2 In a periodic process, the number of cycles
More information1 f. result from periodic disturbance same period (frequency) as source Longitudinal or Transverse Waves Characterized by
result from periodic disturbance same period (frequency) as source Longitudinal or Transverse Waves Characterized by amplitude (how far do the bits move from their equilibrium positions? Amplitude of MEDIUM)
More informationy R T However, the calculations are easier, when carried out using the polar set of co-ordinates ϕ,r. The relations between the co-ordinates are:
Curved beams. Introduction Curved beams also called arches were invented about ears ago. he purpose was to form such a structure that would transfer loads, mainl the dead weight, to the ground b the elements
More informationx y plane is the plane in which the stresses act, yy xy xy Figure 3.5.1: non-zero stress components acting in the x y plane
3.5 Plane Stress This section is concerned with a special two-dimensional state of stress called plane stress. It is important for two reasons: () it arises in real components (particularl in thin components
More informationChapter 11 Vibrations and Waves
Chapter 11 Vibrations and Waves 11-1 Simple Harmonic Motion If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic.
More informationLecture 17. Mechanical waves. Transverse waves. Sound waves. Standing Waves.
Lecture 17 Mechanical waves. Transverse waves. Sound waves. Standing Waves. What is a wave? A wave is a traveling disturbance that transports energy but not matter. Examples: Sound waves (air moves back
More informationPhysics 142 Mechanical Waves Page 1. Mechanical Waves
Physics 142 Mechanical Waves Page 1 Mechanical Waves This set of notes contains a review of wave motion in mechanics, emphasizing the mathematical formulation that will be used in our discussion of electromagnetic
More informationSection 3.7: Mechanical and Electrical Vibrations
Section 3.7: Mechanical and Electrical Vibrations Second order linear equations with constant coefficients serve as mathematical models for mechanical and electrical oscillations. For example, the motion
More informationOscillation the vibration of an object. Wave a transfer of energy without a transfer of matter
Oscillation the vibration of an object Wave a transfer of energy without a transfer of matter Equilibrium Position position of object at rest (mean position) Displacement (x) distance in a particular direction
More informationFixed Point Theorem and Sequences in One or Two Dimensions
Fied Point Theorem and Sequences in One or Two Dimensions Dr. Wei-Chi Yang Let us consider a recursive sequence of n+ = n + sin n and the initial value can be an real number. Then we would like to ask
More informationEquations. A body executing simple harmonic motion has maximum acceleration ) At the mean positions ) At the two extreme position 3) At any position 4) he question is irrelevant. A particle moves on the
More information3.1 Particles in Two-Dimensional Force Systems
3.1 Particles in Two-Dimensional Force Sstems + 3.1 Particles in Two-Dimensional Force Sstems Eample 1, page 1 of 1 1. Determine the tension in cables and. 30 90 lb 1 Free-bod diagram of connection F 2
More informationChapter 16 Waves in One Dimension
Chapter 16 Waves in One Dimension Slide 16-1 Reading Quiz 16.05 f = c Slide 16-2 Reading Quiz 16.06 Slide 16-3 Reading Quiz 16.07 Heavier portion looks like a fixed end, pulse is inverted on reflection.
More informationENGI 9420 Engineering Analysis Solutions to Additional Exercises
ENGI 940 Engineering Analsis Solutions to Additional Exercises 0 Fall [Partial differential equations; Chapter 8] The function ux (, ) satisfies u u u + = 0, subject to the x x u x,0 = u x, =. Classif
More informationTraveling Waves: Energy Transport
Traveling Waves: Energ Transport wave is a traveling disturbance that transports energ but not matter. Intensit: I P power rea Intensit I power per unit area (measured in Watts/m 2 ) Intensit is proportional
More informationLecture XXVI. Morris Swartz Dept. of Physics and Astronomy Johns Hopkins University November 5, 2003
Lecture XXVI Morris Swartz Dept. of Physics and Astronomy Johns Hopins University morris@jhu.edu November 5, 2003 Lecture XXVI: Oscillations Oscillations are periodic motions. There are many examples of
More informationOld Exams - Questions Ch-16
Old Exams - Questions Ch-16 T081 : Q1. The displacement of a string carrying a traveling sinusoidal wave is given by: y( x, t) = y sin( kx ω t + ϕ). At time t = 0 the point at x = 0 m has a displacement
More informationPhysics 1501 Lecture 28
Phsics 1501 Lecture 28 Phsics 1501: Lecture 28 Toda s Agenda Homework #10 (due Frida No. 11) Midterm 2: No. 16 Topics 1-D traeling waes Waes on a string Superposition Power Phsics 1501: Lecture 28, Pg
More informationChapter 16 Waves in One Dimension
Lecture Outline Chapter 16 Waves in One Dimension Slide 16-1 Chapter 16: Waves in One Dimension Chapter Goal: To study the kinematic and dynamics of wave motion, i.e., the transport of energy through a
More informationDouble Spring Harmonic Oscillator Lab
Dylan Humenik and Benjamin Daily Double Spring Harmonic Oscillator Lab Objectives: -Experimentally determine harmonic equations for a double spring system using various methods Part 1 Determining k of
More informationMathematical aspects of mechanical systems eigentones
Seminar: Vibrations and Structure-Borne Sound in Civil Engineering Theor and Applications Mathematical aspects of mechanical sstems eigentones Andre Kuzmin April st 6 Abstract Computational methods of
More informationExam 3 Review. Chapter 10: Elasticity and Oscillations A stress will deform a body and that body can be set into periodic oscillations.
Exam 3 Review Chapter 10: Elasticity and Oscillations stress will deform a body and that body can be set into periodic oscillations. Elastic Deformations of Solids Elastic objects return to their original
More informationSection 8.5 Parametric Equations
504 Chapter 8 Section 8.5 Parametric Equations Man shapes, even ones as simple as circles, cannot be represented as an equation where is a function of. Consider, for eample, the path a moon follows as
More informationBorn of the Wave Equation
Corso di Laurea in Fisica - UNITS ISTITUZIONI DI FISICA PER IL SISTEMA TERRA Born of the Wave Equation FABIO ROMANELLI Department of Mathematics & Geosciences Universit of Trieste romanel@units.it http://moodle.units.it/course/view.php?id=887
More informationPHY 113 C General Physics I 11 AM 12:15 PM TR Olin 101
PHY 113 C General Phsics I 11 AM 1:15 PM R Olin 101 Plan for Lectre 16: Chapter 16 Phsics of wave motion 1. Review of SHM. Eamples of wave motion 3. What determines the wave velocit 4. Properties of periodic
More informationMath 214 Spring problem set (a) Consider these two first order equations. (I) dy dx = x + 1 dy
Math 4 Spring 08 problem set. (a) Consider these two first order equations. (I) d d = + d (II) d = Below are four direction fields. Match the differential equations above to their direction fields. Provide
More informationTHE GENERAL ELASTICITY PROBLEM IN SOLIDS
Chapter 10 TH GNRAL LASTICITY PROBLM IN SOLIDS In Chapters 3-5 and 8-9, we have developed equilibrium, kinematic and constitutive equations for a general three-dimensional elastic deformable solid bod.
More informationGeneral Physics I Spring Oscillations
General Physics I Spring 2011 Oscillations 1 Oscillations A quantity is said to exhibit oscillations if it varies with time about an equilibrium or reference value in a repetitive fashion. Oscillations
More informationWaves Part 1: Travelling Waves
Waves Part 1: Travelling Waves Last modified: 15/05/2018 Links Contents Travelling Waves Harmonic Waves Wavelength Period & Frequency Summary Example 1 Example 2 Example 3 Example 4 Transverse & Longitudinal
More informationWaves Solutions to the Wave Equation Sine Waves Transverse Speed and Acceleration
Waves Solutions to the Wave Equation Sine Waves Transverse Speed and Acceleration Lana Sheridan De Anza College May 17, 2018 Last time pulse propagation the wave equation Overview solutions to the wave
More informationChapter 5 Oscillatory Motion
Chapter 5 Oscillatory Motion Simple Harmonic Motion An object moves with simple harmonic motion whenever its acceleration is proportional to its displacement from some equilibrium position and is oppositely
More informationChapter 16: Oscillations
Chapter 16: Oscillations Brent Royuk Phys-111 Concordia University Periodic Motion Periodic Motion is any motion that repeats itself. The Period (T) is the time it takes for one complete cycle of motion.
More information= C. on q 1 to the left. Using Coulomb s law, on q 2 to the right, and the charge q 2 exerts a force F 2 on 1 ( )
Phsics Solutions to Chapter 5 5.. Model: Use the charge model. Solve: (a) In the process of charging b rubbing, electrons are removed from one material and transferred to the other because the are relativel
More informationSolution to Problems for the 1-D Wave Equation
Solution to Problems for the -D Wave Equation 8. Linear Partial Differential Equations Matthew J. Hancock Fall 5 Problem (i) Suppose that an infinite string has an initial displacement +, u (, ) = f ()
More informationPhysics 41: Waves, Optics, Thermo
Physics 41: Waves, Optics, Thermo Particles & Waves Localized in Space: LOCAL Have Mass & Momentum No Superposition: Two particles cannot occupy the same space at the same time! Particles have energy.
More informationAP Physics 1. April 11, Simple Harmonic Motion. Table of Contents. Period. SHM and Circular Motion
AP Physics 1 2016-07-20 www.njctl.org Table of Contents Click on the topic to go to that section Period and Frequency SHM and UCM Spring Pendulum Simple Pendulum Sinusoidal Nature of SHM Period and Frequency
More information4.1 KINEMATICS OF SIMPLE HARMONIC MOTION 4.2 ENERGY CHANGES DURING SIMPLE HARMONIC MOTION 4.3 FORCED OSCILLATIONS AND RESONANCE Notes
4.1 KINEMATICS OF SIMPLE HARMONIC MOTION 4.2 ENERGY CHANGES DURING SIMPLE HARMONIC MOTION 4.3 FORCED OSCILLATIONS AND RESONANCE Notes I. DEFINING TERMS A. HOW ARE OSCILLATIONS RELATED TO WAVES? II. EQUATIONS
More informationTrigonometric Functions
Trigonometric Functions This section reviews radian measure and the basic trigonometric functions. C ' θ r s ' ngles ngles are measured in degrees or radians. The number of radians in the central angle
More informationPhysics 43 Chapter 41 Homework #11 Key
Physics 43 Chapter 4 Homework # Key π sin. A particle in an infinitely deep square well has a wave function given by ( ) for and zero otherwise. Determine the epectation value of. Determine the probability
More informationHonors Differential Equations
MIT OpenCourseWare http://ocw.mit.edu 18.034 Honors Differential Equations Spring 009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. LECTURE 7. MECHANICAL
More informationMechanics Departmental Exam Last updated November 2013
Mechanics Departmental Eam Last updated November 213 1. Two satellites are moving about each other in circular orbits under the influence of their mutual gravitational attractions. The satellites have
More informationPhysics 4A Lab: Simple Harmonic Motion
Name: Date: Lab Partner: Physics 4A Lab: Simple Harmonic Motion Objective: To investigate the simple harmonic motion associated with a mass hanging on a spring. To use hook s law and SHM graphs to calculate
More informationPhysics General Physics. Lecture 24 Oscillating Systems. Fall 2016 Semester Prof. Matthew Jones
Physics 22000 General Physics Lecture 24 Oscillating Systems Fall 2016 Semester Prof. Matthew Jones 1 2 Oscillating Motion We have studied linear motion objects moving in straight lines at either constant
More informationSolutions to Problem Set #11 Physics 151
Solutions to Problem Set #11 Physics 151 Problem 1 The Hamiltonian is p k H = + m Relevant Poisson brackets are H p [ H, ] = =, p m H [ p, H] = = k We can now write the formal solution for (t as 3 t t
More informationBorn simulation report
Born simulation report Name: The atoms in a solid are in constant thermally induced motion. In born we study the dynamics of a linear chain of atoms. We assume that the atomic arrangement that has minimum
More informationChapter 14: Periodic motion
Chapter 14: Periodic motion Describing oscillations Simple harmonic motion Energy of simple harmonic motion Applications of simple harmonic motion Simple pendulum & physical pendulum Damped oscillations
More information