The Nonlinear Pendulum
|
|
- Constance Jenkins
- 6 years ago
- Views:
Transcription
1 The Nonlinear Pendulum - Pádraig Ó Conbhuí TP Monday 1. Abstract This experiment was performed to examine the effects that linearizing equations has on the accuracy of results and to find ways of accurately determining the outcome of equations without linearization, specifically in the case of the nonlinear pendulum. It was found that for a simple pendulum, linearized equations were accurate for small values of θ and for damped oscillators, linearized equations were accurate after long time periods. It was also shown that certain characteristics of a system can be lost with linearization. The chaotic nature of a damped, driven pendulum was also probed, with graphs drawn of the phase space of the oscillator as a control parameter was incremented. 2. Background and Theory A common method of solving equations is by linear approximation, often by use of a Taylor Series expansion. In the case of a simple pendulum, the equation of motion is: d 2 s dt = θ 2 Ld2 = g sin(θ) dt2 Where s is the displacement in the x-direction, θ is the angular displacement of the pendulum from equilibrium and L is the radius of the arc it traces out. By expanding sin(θ) about 0, it can be shown that sin(θ) = θ + θ3, so: 3! Using this, the equation can be rewritten: sin(θ) θ for θ 0 F gθ This method is good for finding solutions for the system when theta is small, but at higher values, the term in θ 3 starts growing rapidly. This method, however, is only good for finding solutions to specific examples of the general case. In general, θ would be much greater than 0. It remains to find a method of resolving the general case. While it may be impossible to find an analytical solution for the pendulums position in time, the motion of the pendulum can, itself, be simulated, and the results noted. This simulation can be done much quicker than real time and on almost any scale in which the equation is still applicable on a computer. A program can be written which makes use of an iterative solution of motion, of which there are several. Two of these methods shall be detailed here. 1
2 2.1 The Euler Method From the equations of motion, there is a second order equation, namely d 2 θ = f(θ, ω, t) dt2 Where f is some function of the force on the pendulum. This can be made into two first order equations: dθ(t) = ω(t) and dω(t) = f(θ, ω, t) dt dt These can then be expanded in a Taylor series to find their next timestep. These are found to be θ(t + t) = θ n+1 = θ n + ω n t + O( t 2 ) ω(t + t) = ω n+1 = ω n θ n t + O( t 2 ) And using the Trapezoid Rule, based on the integral being the area under a graph, an even better approximation can be derived. k 1a = ω n t k 1b = f(θ n, ω n, t) t k 2a = (ω n + k 1b ) t k 2b = f(θ n + k1a, ω n + k 1b, t + h) t θ n+1 = θ n + k 1a+k 2a 2 ω n+1 = ω n + k 1b+k 2b 2 Iterating over these functions θ n+2 and ω n+2 can be gotten and so forth for however long is desired. By listing the values obtained, a graph of the pendulum s motion can be drawn. This method, however, is only accurate to O( t 2 ). 2
3 2.2 The Runge-Kata Method The Runge-Kata Method is different from the Trapezoid Rule by way of integrating about the midpoint of a timestep instead of at the end, as is done in the latter. This makes it a much more accurate method, but it requires more equations to be solved and computed. It is computed in a similar fashion, but has a higher order of accuracy. The Algorithm outlined here is accurate to a 4 th order of the timestep and is as follows k 1a = ω n t k 2a = (ω n + k 1b /2) t k 3a = (ω n + k 2b /2) t k 4a = (ω n + k 3b ) t k 1b = f(θ, ω n, t) t k 2b = f(θ + k 1a 2, ω n + k 1b, t + t) t 2 2 k 3b = f(θ + k 2a 2, ω n + k 2b, t + t) t 2 2 k 4b = f(θ + k 3a, ω n + k 3b, t + h) t θ(t + t) = θ(t) + k 1a+2k 2a +2k 3a +k 4a 6 ω(t + t) = ω(t) + k 1b+2k 2b +2k 3b +k 4b 6 This greater accuracy is necessary for modelling damped driven oscillators. In the case given, the acceleration, f, of the pendulum was of the form f = β 2 sin(θ) kω + A cos(ω) 3
4 3. Method of Implementation Programs written in C were used to compute these iterative functions, comprising of a simple for loop which would first calculate the values of θ and ω at the current time, increase the time by t and then write the time, the position and the velocity of the pendulum to a document. The points written to the document were then graphed using gnuplot. Results were compared and contrasted and results were concluded on the accuracies of each method and the nature of the pendulums. For the sake of speed, system calls were made from the program to automate the running of gnuplot and any filesystem manipulations were done through bash scripting. The basic structure of the program was 1. Accept variables from the command line 2. If no variables are given, display quick manual message. 3. Begin iterative calculation. If linear option is included in the command line, do the linear calculation simultaneously. Write current variables to out.dat file on each iteration 4. Customize gnuplot input based on command line options, mostly just filename manipulation so the parameters used in a plot can be identified from the name. 5. Write gnuplot commands to out.plt file. Run a system call asking gnuplot to run out.plt. Files were written as png. 6. If command line option is used, edit newly created file names and create a directory for them, so they can be easily identified with eachother. 7. Run Eye Of Gnome (eog) through a system call asking it to open all the recently created files so they can be inspected. 8. Done. The first program written was to compare the linear approximation to actual nonlinear function discussed in section 2 of this paper using the Euler Method. The program simultaneously ran the simulation of the pendulum using both the linear and the nonlinear function. The second program was pretty much the exact same thing, but using the Runge-Kata algorithm instead. 4
5 4. Results and Analysis The following graphs were obtained by running the data gathered from the compiled programs through gnuplot. 4.1 Effects of Linearization Phase Spaces - First, an example of a typical phase space for a simple pendulum using the Euler Method and a small value of θ. 5
6 As can be seen, there is little difference in the overall shape of the phase spaces, they are both harmonic oscillators. However, for large θ, there is an interesting result. As can be seen, the nonlinear phase diagram is no longer closed. This is indicative of the pendulum spinning about its pivot. This is an important result and will be discussed in the next section. 6
7 Trajectories - Below are trajectories from several simulated pendulums using the Euler Method. 7
8 As can clearly be seen, the smaller the maximum displacement of θ is, ie the smaller θ is (not just it s starting position) the closer the linearized trajectory is to the actual trajectory. In the last graph here is further proof that this pendulum is spinning about it s axis. 8
9 The Damped Pendulum - 9
10 This is an assortment of different initial values for damped oscillators, simulated using the Runge-Kata method. Here, linearized equations were compared to the actual motion of the oscillator. In all cases, the linearized version approximated the nonlinear version quite well given enough time, even if just coming to a halt. 10
11 4.2 Nonlinear Systems In this section, the Runge-Kata method was used to simulate a nonlinear system. To show how it compares to the Euler Method, two simulations of the same system are below, the first using the Euler Method and the second Runge-Kata. Obviously these are not the same. Considering The Runge-Kata method is accurate O( t 4 ), and the graph looks more believable, it is likely more accurate. 11
12 Advantage is taken of the accuracy of this method to calculate the trajectory of an otherwise unpredictable system with reasonable accuracy. Below are graphs of the phase space of a system as its control parameter is tweaked. Figure 1. Figure 2. 12
13 Figure 3. Figure 4. 13
14 Figure 5. As can be seen, as the parameter A is increased, the system becomes more and more chaotic, until it reaches a totally chaotic state. 14
15 5. Conclusions and Discussion 5.1 Effects of Linearization Given that θ is small, linearization gives a very good approximation of the actual trajectory. In most cases, for a reasonable θ, the linearized trajectory more or less follows the actual path. Comparing the diagrams of section shows this quite clearly. In all cases, however, the linear approximation is accurate for some small amount of time. A similar effect is seen in section 4.1.3, where the linear approximation becomes accurate after some long amount of time. This is due to the constant retarding force in the governing equation which is proportional to the velocity. For a damped oscillator, eventually, linearization doesn t make a difference. However, upon linearization, certain attributes of the system may be completely lost. Take the example of the pendulum spinning about its axis from section In linearization, the pendulum is likened to a spring, meaning the greater the angular displacement, the greater the return force. However, in the actual governing equation, sin(θ) is used, which starts decreasing again past a certain threshold, meaning a pendulum need not necessarily oscillate. This is an example of a behaviour completely lost in linearization. For this reason it is important to be able to solve, or at least simulate nonlinear systems without linearization. 5.2 Nonlinearity and Chaos As was seen in section 4.2, eventually the damped, driven oscillator became chaotic. This is a direct result of its nonlinearity. The form of the acceleration was that of having a force field, a dampening force and a driving force act on the pendulum. Given a low driving amplitude, the pendulum remained rather well behaved. Upon increasing the drive amplitude, the periodicity of the pendulum began to change. In figure 1, the pendulum exhibits a rather normal oscillatory behaviour. In figure 2, while the shape is still closed, it is no longer one path, but seemingly two. It is, however, still periodic. In figure 3, it is no longer a harmonic oscillator, since it doesn t follow a closed path. In figure 4, a bifurcation is observed before becoming chaotic in figure 5. Nonlinearity and chaos, as subjects, are rather recent, due in no small part to the absolute necessity of a computer to process the given data and to perform the required iterative calculations. By modelling a system like this on a computer, it makes understanding real world applications of such systems much easier. 15
The Nonlinear Pendulum
The Nonlinear Pendulum Evan Sheridan 11367741 Feburary 18th 013 Abstract Both the non-linear linear pendulum are investigated compared using the pendulum.c program that utilizes the trapezoid method for
More informationTheoretical physics. Deterministic chaos in classical physics. Martin Scholtz
Theoretical physics Deterministic chaos in classical physics Martin Scholtz scholtzzz@gmail.com Fundamental physical theories and role of classical mechanics. Intuitive characteristics of chaos. Newton
More informationNonlinear Oscillators: Free Response
20 Nonlinear Oscillators: Free Response Tools Used in Lab 20 Pendulums To the Instructor: This lab is just an introduction to the nonlinear phase portraits, but the connection between phase portraits and
More informationMechanical Resonance and Chaos
Mechanical Resonance and Chaos You will use the apparatus in Figure 1 to investigate regimes of increasing complexity. Figure 1. The rotary pendulum (from DeSerio, www.phys.ufl.edu/courses/phy483l/group_iv/chaos/chaos.pdf).
More informationPhysics 235 Chapter 4. Chapter 4 Non-Linear Oscillations and Chaos
Chapter 4 Non-Linear Oscillations and Chaos Non-Linear Differential Equations Up to now we have considered differential equations with terms that are proportional to the acceleration, the velocity, and
More informationPH 120 Project # 2: Pendulum and chaos
PH 120 Project # 2: Pendulum and chaos Due: Friday, January 16, 2004 In PH109, you studied a simple pendulum, which is an effectively massless rod of length l that is fixed at one end with a small mass
More informationParametric Resonance and Elastic Pendulums
Parametric Resonance and Elastic Pendulums Ravitej Uppu Abstract In this I try to extend the theoretical conception of Elastic Pendulum that can be explained by the Driven Pendulums that I presented during
More informationSimple Harmonic Motion
Simple Harmonic Motion (FIZ 101E - Summer 2018) July 29, 2018 Contents 1 Introduction 2 2 The Spring-Mass System 2 3 The Energy in SHM 5 4 The Simple Pendulum 6 5 The Physical Pendulum 8 6 The Damped Oscillations
More information11/17/10. Chapter 14. Oscillations. Chapter 14. Oscillations Topics: Simple Harmonic Motion. Simple Harmonic Motion
11/17/10 Chapter 14. Oscillations This striking computergenerated image demonstrates an important type of motion: oscillatory motion. Examples of oscillatory motion include a car bouncing up and down,
More informationChapter 2 PARAMETRIC OSCILLATOR
CHAPTER- Chapter PARAMETRIC OSCILLATOR.1 Introduction A simple pendulum consists of a mass m suspended from a string of length L which is fixed at a pivot P. When simple pendulum is displaced to an initial
More informationMAS212 Assignment #2: The damped driven pendulum
MAS Assignment #: The damped driven pendulum Sam Dolan (January 8 Introduction In this assignment we study the motion of a rigid pendulum of length l and mass m, shown in Fig., using both analytical and
More informationLaboratory Instruction-Record Pages
Laboratory Instruction-Record Pages The Driven Pendulum Geology 200 - Evolutionary Systems James Madison University Lynn S. Fichter and Steven J. Baedke Brief History of Swinging Many things in this universe
More informationChapter 13 Lecture. Essential University Physics Richard Wolfson 2 nd Edition. Oscillatory Motion Pearson Education, Inc.
Chapter 13 Lecture Essential University Physics Richard Wolfson nd Edition Oscillatory Motion Slide 13-1 In this lecture you ll learn To describe the conditions under which oscillatory motion occurs To
More informationChaotic motion. Phys 420/580 Lecture 10
Chaotic motion Phys 420/580 Lecture 10 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t
More informationPhysics Mechanics. Lecture 32 Oscillations II
Physics 170 - Mechanics Lecture 32 Oscillations II Gravitational Potential Energy A plot of the gravitational potential energy U g looks like this: Energy Conservation Total mechanical energy of an object
More informationSimulation of Simple Pendulum
International Journal of Engineering Science Invention ISSN (Online): 2319 6734, ISSN (Print): 2319 6726 Volume 6 Issue 4 April 2017 PP. 33-38 Abdalftah Elbori 1, Ltfei Abdalsmd 2 1 (MODES Department Atilim
More informationPreClass Notes: Chapter 13, Sections
PreClass Notes: Chapter 13, Sections 13.3-13.7 From Essential University Physics 3 rd Edition by Richard Wolfson, Middlebury College 2016 by Pearson Education, Inc. Narration and extra little notes by
More informationChapter 14 Periodic Motion
Chapter 14 Periodic Motion 1 Describing Oscillation First, we want to describe the kinematical and dynamical quantities associated with Simple Harmonic Motion (SHM), for example, x, v x, a x, and F x.
More informationOscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. small angle approximation. Oscillatory solution
Oscillatory Motion Simple pendulum: linear Hooke s Law restoring force for small angular deviations d 2 θ dt 2 = g l θ small angle approximation θ l Oscillatory solution θ(t) =θ 0 sin(ωt + φ) F with characteristic
More informationComputational Physics (6810): Session 8
Computational Physics (6810): Session 8 Dick Furnstahl Nuclear Theory Group OSU Physics Department February 24, 2014 Differential equation solving Session 7 Preview Session 8 Stuff Solving differential
More informationOscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. Oscillatory solution
Oscillatory Motion Simple pendulum: linear Hooke s Law restoring force for small angular deviations d 2 θ dt 2 = g l θ θ l Oscillatory solution θ(t) =θ 0 sin(ωt + φ) F with characteristic angular frequency
More informationChapter 14. PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman. Lectures by Wayne Anderson
Chapter 14 Periodic Motion PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Exam 3 results Class Average - 57 (Approximate grade
More informationOscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is
Dr. Alain Brizard College Physics I (PY 10) Oscillations Textbook Reference: Chapter 14 sections 1-8. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring
More informationContents. Contents. Contents
Physics 121 for Majors Class 18 Linear Harmonic Last Class We saw how motion in a circle is mathematically similar to motion in a straight line. We learned that there is a centripetal acceleration (and
More informationOscillatory Motion. Solutions of Selected Problems
Chapter 15 Oscillatory Motion. Solutions of Selected Problems 15.1 Problem 15.18 (In the text book) A block-spring system oscillates with an amplitude of 3.50 cm. If the spring constant is 250 N/m and
More informationOscillatory Motion SHM
Chapter 15 Oscillatory Motion SHM Dr. Armen Kocharian Periodic Motion Periodic motion is motion of an object that regularly repeats The object returns to a given position after a fixed time interval A
More information2nd Year Computational Physics Week 4 (standard): Ordinary differential equations
2nd Year Computational Physics Week 4 (standard): Ordinary differential equations Last compiled September 28, 2017 1 Contents 1 Introduction 4 2 Prelab Questions 4 3 Simple Differential Equations 5 3.1
More informationChaotic motion. Phys 750 Lecture 9
Chaotic motion Phys 750 Lecture 9 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t =0to
More informationPhysics 1C. Lecture 12B
Physics 1C Lecture 12B SHM: Mathematical Model! Equations of motion for SHM:! Remember, simple harmonic motion is not uniformly accelerated motion SHM: Mathematical Model! The maximum values of velocity
More informationPHYSICS 210 SOLUTION OF THE NONLINEAR PENDULUM EQUATION USING FDAS
PHYSICS 210 SOLUTION OF THE NONLINEAR PENDULUM EQUATION USING FDAS 1. PHYSICAL & MATHEMATICAL FORMULATION O θ L r T m W 1 1.1 Derivation of the equation of motion O Consider idealized pendulum: Mass of
More informationClass 4: More Pendulum results
Class 4: More Pendulum results The pendulum is a mechanical problem with a long and interesting history, from Galileo s first ansatz that the period was independent of the amplitude based on watching priests
More informationLecture V : Oscillatory motion and spectral analysis
Lecture V : Oscillatory motion and spectral analysis I. IDEAL PENDULUM AND STABILITY ANALYSIS Let us remind ourselves of the equation of motion for the pendulum. Remembering that the external torque applied
More informationOSCILLATIONS ABOUT EQUILIBRIUM
OSCILLATIONS ABOUT EQUILIBRIUM Chapter 13 Units of Chapter 13 Periodic Motion Simple Harmonic Motion Connections between Uniform Circular Motion and Simple Harmonic Motion The Period of a Mass on a Spring
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 informationThe Harmonic Oscillator
The Harmonic Oscillator Math 4: Ordinary Differential Equations Chris Meyer May 3, 008 Introduction The harmonic oscillator is a common model used in physics because of the wide range of problems it can
More informationTransitioning to Chaos in a Simple Mechanical Oscillator
Transitioning to Chaos in a Simple Mechanical Oscillator Hwan Bae Physics Department, The College of Wooster, Wooster, Ohio 69, USA (Dated: May 9, 8) We vary the magnetic damping, driver frequency, and
More informationChapter 14. PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman. Lectures by Wayne Anderson
Chapter 14 Periodic Motion PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Goals for Chapter 14 To describe oscillations in
More informationImportant because SHM is a good model to describe vibrations of a guitar string, vibrations of atoms in molecules, etc.
Simple Harmonic Motion Oscillatory motion under a restoring force proportional to the amount of displacement from equilibrium A restoring force is a force that tries to move the system back to equilibrium
More informationChapter 14. Oscillations. Oscillations Introductory Terminology Simple Harmonic Motion:
Chapter 14 Oscillations Oscillations Introductory Terminology Simple Harmonic Motion: Kinematics Energy Examples of Simple Harmonic Oscillators Damped and Forced Oscillations. Resonance. Periodic Motion
More informationMidterm EXAM PHYS 401 (Spring 2012), 03/20/12
Midterm EXAM PHYS 401 (Spring 2012), 03/20/12 Name: Signature: Duration: 75 minutes Show all your work for full/partial credit! In taking this exam you confirm to adhere to the Aggie Honor Code: An Aggie
More informationChapter 15 - Oscillations
The pendulum of the mind oscillates between sense and nonsense, not between right and wrong. -Carl Gustav Jung David J. Starling Penn State Hazleton PHYS 211 Oscillatory motion is motion that is periodic
More informationA plane autonomous system is a pair of simultaneous first-order differential equations,
Chapter 11 Phase-Plane Techniques 11.1 Plane Autonomous Systems A plane autonomous system is a pair of simultaneous first-order differential equations, ẋ = f(x, y), ẏ = g(x, y). This system has an equilibrium
More informationCHAPTER 12 OSCILLATORY MOTION
CHAPTER 1 OSCILLATORY MOTION Before starting the discussion of the chapter s concepts it is worth to define some terms we will use frequently in this chapter: 1. The period of the motion, T, is the time
More informationPreLab 2 - Simple Harmonic Motion: Pendulum (adapted from PASCO- PS-2826 Manual)
Musical Acoustics Lab, C. Bertulani, 2012 PreLab 2 - Simple Harmonic Motion: Pendulum (adapted from PASCO- PS-2826 Manual) A body is said to be in a position of stable equilibrium if, after displacement
More informationUpdated 2013 (Mathematica Version) M1.1. Lab M1: The Simple Pendulum
Updated 2013 (Mathematica Version) M1.1 Introduction. Lab M1: The Simple Pendulum The simple pendulum is a favorite introductory exercise because Galileo's experiments on pendulums in the early 1600s are
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 informationComputers, Lies and the Fishing Season
1/47 Computers, Lies and the Fishing Season Liz Arnold May 21, 23 Introduction Computers, lies and the fishing season takes a look at computer software programs. As mathematicians, we depend on computers
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. Chapter 15 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc.
PHYSICS FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E Chapter 15 Lecture RANDALL D. KNIGHT Chapter 15 Oscillations IN THIS CHAPTER, you will learn about systems that oscillate in simple harmonic
More informationMAS212 Assignment #2: The damped driven pendulum
MAS Assignment #: The damped driven pendulum Dr. Sam Dolan (s.dolan@sheffield.ac.uk) Introduction: In this assignment you will use Python to investigate a non-linear differential equation which models
More informationChapter 14 Oscillations
Chapter 14 Oscillations Chapter Goal: To understand systems that oscillate with simple harmonic motion. Slide 14-2 Chapter 14 Preview Slide 14-3 Chapter 14 Preview Slide 14-4 Chapter 14 Preview Slide 14-5
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 informationExamining the Effect of Additional Driving Forces on a Mechanical Oscillator
Examining the Effect of Additional Driving Forces on a Mechanical Oscillator Nate Stone Physics Department, The College of Wooster, Wooster, Ohio 44691, USA (Dated: May 7, 2015) In this experiment, the
More informationPhysics 106b: Lecture 7 25 January, 2018
Physics 106b: Lecture 7 25 January, 2018 Hamiltonian Chaos: Introduction Integrable Systems We start with systems that do not exhibit chaos, but instead have simple periodic motion (like the SHO) with
More informationChapter 1. Harmonic Oscillator. 1.1 Energy Analysis
Chapter 1 Harmonic Oscillator Figure 1.1 illustrates the prototypical harmonic oscillator, the mass-spring system. A mass is attached to one end of a spring. The other end of the spring is attached to
More informationEntrainment Alex Bowie April 7, 2004
Entrainment Alex Bowie April 7, 2004 Abstract The driven Van der Pol oscillator displays entrainment, quasiperiodicity, and chaos. The characteristics of these different modes are discussed as well as
More informationSolution Set Five. 2 Problem #2: The Pendulum of Doom Equation of Motion Simple Pendulum Limit Visualization...
: Solution Set Five Northwestern University, Classical Mechanics Classical Mechanics, Third Ed.- Goldstein November 4, 2015 Contents 1 Problem #1: Coupled Mass Oscillator System. 2 1.1 Normal Modes.......................................
More information1 Pushing your Friend on a Swing
Massachusetts Institute of Technology MITES 017 Physics III Lecture 05: Driven Oscillations In these notes, we derive the properties of both an undamped and damped harmonic oscillator under the influence
More informationChasing Chaos With a Magnetic Pendulum
Chasing Chaos With a Magnetic Pendulum PHY 300 - Junior Phyics Laboratory Hassan Bukhari Roll no: 2012-10-0152 Department of Physcis LUMS SSE Saturday, October, 23, 2010 1 Introduction Chaos expresses
More informationPREMED COURSE, 14/08/2015 OSCILLATIONS
PREMED COURSE, 14/08/2015 OSCILLATIONS PERIODIC MOTIONS Mechanical Metronom Laser Optical Bunjee jumping Electrical Astronomical Pulsar Biological ECG AC 50 Hz Another biological exampe PERIODIC MOTIONS
More informationChapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx
Chapter 1 Lecture Notes Chapter 1 Oscillatory Motion Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx When the mass is released, the spring will pull
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 informationAssessment Schedule 2016 Physics: Demonstrate understanding of mechanical systems (91524)
NCEA Level 3 Physics (91524) 2016 page 1 of 6 Assessment Schedule 2016 Physics: Demonstrate understanding of mechanical systems (91524) Evidence Statement NØ N1 N2 A3 A4 M5 M6 E7 E8 No response; no relevant
More informationSlide 1 / 70. Simple Harmonic Motion
Slide 1 / 70 Simple Harmonic Motion Slide 2 / 70 SHM and Circular Motion There is a deep connection between Simple Harmonic Motion (SHM) and Uniform Circular Motion (UCM). Simple Harmonic Motion can be
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 informationThe Pendulum - Stating the Problem
The Pendulum - Stating the Problem The physics of the pendulum evokes a wide range of applications from circuits to ecology. We start with a simple pendulum consisting of a point mass on a negligibly light
More informationChapter 15 Oscillations
Chapter 15 Oscillations Summary Simple harmonic motion Hook s Law Energy F = kx Pendulums: Simple. Physical, Meter stick Simple Picture of an Oscillation x Frictionless surface F = -kx x SHM in vertical
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 informationHOMEWORK ANSWERS. Lesson 4.1: Simple Harmonic Motion
DEVIL PHYSICS HOMEWORK ANSWERS Tsokos, Chapter 3 Test Lesson 4.1: Simple Harmonic Motion 1. Objectives. By the end of this class you should be able to: a) Understand that in simple harmonic motion there
More information本教材僅供教學使用, 勿做其他用途, 以維護智慧財產權
本教材內容主要取自課本 Physics for Scientists and Engineers with Modern Physics 7th Edition. Jewett & Serway. 注意 本教材僅供教學使用, 勿做其他用途, 以維護智慧財產權 教材網址 : https://sites.google.com/site/ndhugp1 1 Chapter 15 Oscillatory Motion
More informationLecture V: The game-engine loop & Time Integration
Lecture V: The game-engine loop & Time Integration The Basic Game-Engine Loop Previous state: " #, %(#) ( #, )(#) Forces -(#) Integrate velocities and positions Resolve Interpenetrations Per-body change
More informationFaculty of Computers and Information. Basic Science Department
18--018 FCI 1 Faculty of Computers and Information Basic Science Department 017-018 Prof. Nabila.M.Hassan 18--018 FCI Aims of Course: The graduates have to know the nature of vibration wave motions with
More informationPhysics 132 3/31/17. March 31, 2017 Physics 132 Prof. E. F. Redish Theme Music: Benny Goodman. Swing, Swing, Swing. Cartoon: Bill Watterson
March 31, 2017 Physics 132 Prof. E. F. Redish Theme Music: Benny Goodman Swing, Swing, Swing Cartoon: Bill Watterson Calvin & Hobbes 1 Outline The makeup exam Recap: the math of the harmonic oscillator
More informationMechanics IV: Oscillations
Mechanics IV: Oscillations Chapter 4 of Morin covers oscillations, including damped and driven oscillators in detail. Also see chapter 10 of Kleppner and Kolenkow. For more on normal modes, see any book
More informationChapter 7 Hooke s Force law and Simple Harmonic Oscillations
Chapter 7 Hooke s Force law and Simple Harmonic Oscillations Hooke s Law An empirically derived relationship that approximately works for many materials over a limited range. Exactly true for a massless,
More informationIn-Class Problems 30-32: Moment of Inertia, Torque, and Pendulum: Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01 TEAL Fall Term 004 In-Class Problems 30-3: Moment of Inertia, Torque, and Pendulum: Solutions Problem 30 Moment of Inertia of a
More informationPHYS2330 Intermediate Mechanics Fall Final Exam Tuesday, 21 Dec 2010
Name: PHYS2330 Intermediate Mechanics Fall 2010 Final Exam Tuesday, 21 Dec 2010 This exam has two parts. Part I has 20 multiple choice questions, worth two points each. Part II consists of six relatively
More informationOscillations Simple Harmonic Motion
Oscillations Simple Harmonic Motion Lana Sheridan De Anza College Dec 1, 2017 Overview oscillations simple harmonic motion (SHM) spring systems energy in SHM pendula damped oscillations Oscillations and
More informationEDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1 - TRIGONOMETRICAL GRAPHS
EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1 - TRIGONOMETRICAL GRAPHS CONTENTS 3 Be able to understand how to manipulate trigonometric expressions and apply
More informationHarmonic Oscillator. Outline. Oscillatory Motion or Simple Harmonic Motion. Oscillatory Motion or Simple Harmonic Motion
Harmonic Oscillator Mass-Spring Oscillator Resonance The Pendulum Physics 109, Class Period 13 Experiment Number 11 in the Physics 121 Lab Manual (page 65) Outline Simple harmonic motion The vertical mass-spring
More informationLecture IV: Time Discretization
Lecture IV: Time Discretization Motivation Kinematics: continuous motion in continuous time Computer simulation: Discrete time steps t Discrete Space (mesh particles) Updating Position Force induces acceleration.
More information4. Complex Oscillations
4. Complex Oscillations The most common use of complex numbers in physics is for analyzing oscillations and waves. We will illustrate this with a simple but crucially important model, the damped harmonic
More informationPhysics 141, Lecture 7. Outline. Course Information. Course information: Homework set # 3 Exam # 1. Quiz. Continuation of the discussion of Chapter 4.
Physics 141, Lecture 7. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 07, Page 1 Outline. Course information: Homework set # 3 Exam # 1 Quiz. Continuation of the
More informationChapter 4. Oscillatory Motion. 4.1 The Important Stuff Simple Harmonic Motion
Chapter 4 Oscillatory Motion 4.1 The Important Stuff 4.1.1 Simple Harmonic Motion In this chapter we consider systems which have a motion which repeats itself in time, that is, it is periodic. In particular
More informationDamped Oscillations *
OpenStax-CNX module: m58365 1 Damped Oscillations * OpenStax This wor is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 By the end of this section, you will be
More informationIn the presence of viscous damping, a more generalized form of the Lagrange s equation of motion can be written as
2 MODELING Once the control target is identified, which includes the state variable to be controlled (ex. speed, position, temperature, flow rate, etc), and once the system drives are identified (ex. force,
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 informationAIMS Exercise Set # 1
AIMS Exercise Set #. Determine the form of the single precision floating point arithmetic used in the computers at AIMS. What is the largest number that can be accurately represented? What is the smallest
More information28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod)
28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) θ + ω 2 sin θ = 0. Indicate the stable equilibrium points as well as the unstable equilibrium points.
More informationSimple harmonic motion the motion of springs is a very important topic in physics.
Chapter 11 Potential and Kinetic Energy Together: Simple Harmonic Motion In This Chapter Using Hooke s law Working with simple harmonic motion Calculating simple harmonic motion velcoity Finding simple
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 informationInvestigating the Relationship Between Cavendish Temperature Fluctuation and Torsional Oscillation
Investigating the Relationship Between Cavendish Temperature Fluctuation and Torsional Oscillation John Grasel 4 March 2010 Abstract The Cavendish apparatus measures the gravitational attraction between
More informationMaps and differential equations
Maps and differential equations Marc R. Roussel November 8, 2005 Maps are algebraic rules for computing the next state of dynamical systems in discrete time. Differential equations and maps have a number
More informationMass on a Horizontal Spring
Course- B.Sc. Applied Physical Science (Computer Science) Year- IInd, Sem- IVth Subject Physics Paper- XIVth, Electromagnetic Theory Lecture No. 22, Simple Harmonic Motion Introduction Hello friends in
More informationChapter 13: Oscillatory Motions
Chapter 13: Oscillatory Motions Simple harmonic motion Spring and Hooe s law When a mass hanging from a spring and in equilibrium, the Newton s nd law says: Fy ma Fs Fg 0 Fs Fg This means the force due
More informationAssignments VIII and IX, PHYS 301 (Classical Mechanics) Spring 2014 Due 3/21/14 at start of class
Assignments VIII and IX, PHYS 301 (Classical Mechanics) Spring 2014 Due 3/21/14 at start of class Homeworks VIII and IX both center on Lagrangian mechanics and involve many of the same skills. Therefore,
More informationChap. 15: Simple Harmonic Motion
Chap. 15: Simple Harmonic Motion Announcements: CAPA is due next Tuesday and next Friday. Web page: http://www.colorado.edu/physics/phys1110/phys1110_sp12/ Examples of periodic motion vibrating guitar
More informationChapter 14 Oscillations. Copyright 2009 Pearson Education, Inc.
Chapter 14 Oscillations 14-1 Oscillations of a Spring 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
More informationHarmonic Oscillator. Mass-Spring Oscillator Resonance The Pendulum. Physics 109 Experiment Number 12
Harmonic Oscillator Mass-Spring Oscillator Resonance The Pendulum Physics 109 Experiment Number 12 Outline Simple harmonic motion The vertical mass-spring system Driven oscillations and resonance The pendulum
More informationNewton s laws. Chapter 1. Not: Quantum Mechanics / Relativistic Mechanics
PHYB54 Revision Chapter 1 Newton s laws Not: Quantum Mechanics / Relativistic Mechanics Isaac Newton 1642-1727 Classical mechanics breaks down if: 1) high speed, v ~ c 2) microscopic/elementary particles
More information