Assignment 6. Using the result for the Lagrangian for a double pendulum in Problem 1.22, we get
|
|
- Dwight Ray
- 5 years ago
- Views:
Transcription
1 Assignment 6 Goldstein 6.4 Obtain the normal modes of vibration for the double pendulum shown in Figure.4, assuming equal lengths, but not equal masses. Show that when the lower mass is small compared to the upper one, the two resonant frequencies are almost equal. If the pendula are set in motion by pulling the upper mass slighlty away from the vertical and then releasing it, show that subsequent motion is such that at regular intervals one pendulum is at rest while the other has its maximum amplitude. This is the familiar phenomenon of beats. Using the result for the Lagrangian for a double pendulum in Problem., we get L = (m + m ) l θ + m l θ m l θ θ cos(θ + θ ) + (m + m ) gl cos θ + m gl cos θ where we have set the lengths to l and kept the masses distinct. Note too that we have followed Goldstein s convention in the figure and taken θ as a counterclockwise measured angle and θ as a clockwise measured angle. As we are working within the assumption of small oscillations, we can make further simplifications to the Lagrangian. Working to second order in the potential, we can let cos θ θ / and similarly for cos θ. However, we only work to zeroth order in the kinetic energy and so cos(θ + θ ). While this seems contradictory, it is the basic assumption of small oscillations. The Lagrangian becomes L = (m + m ) l θ + m l θ m l θ θ (m + m ) gl θ m gl θ where we have dropped some constant terms from the potential energy piece as they do not contribute to the equations of motion. If we define θ η = θ the Lagrangian is L = η T T η ηt V η where the kinetic and potential energy matrices are T = l m + m m m m m + m V = gl m With these in hand, we can calculate the characteristic frequences from the secular equation: = det ( V ω T ) (m = + m )(gl l ω ) l m ω l m ω m (gl l ω ) = l 4 m (ω ω ) ω 4 m where, for convenience, we have defined ω = g/l and = m + m. The eigenfrequencies are ω = ω ± m m
2 The normal modes come from solving V η ± = ω±t η ± for η ±. equation we get the relevant equations Putting the eigenvalues into this matrix which, for the eigenfrequencies, reduce to and provide the (unnormalized) eigenvectors ( ω ω ±) θ + m ω ± θ = m ω ± θ + m ( ω ω ±) θ = m η + = m θ = ± θ η = m Note that if m << m, then we can write the frequencies as ω ± ω ( ± ɛ) where ɛ = m /m and the eigenfrequencies are very nearly the same. The normal modes are just the combination of these eigenvectors in the form ζ = A η where A is the matrix comprised of the above eigenvectors m m A = The time dependance is included in a simple way since the normal modes are just solutions of the harmonic oscillator equation for each eigenmode with the corresponding eigenfrequency: Putting all of this together, we get ζ ± + ω ± ζ ± = ζ ± (t) = C ± e iω±t where the constants C ± are complex and the normal modes are ζ ± (t) = θ (t) ± m θ (t) = C ± e iω± t where we know to take the real part. The initial conditions to consider are θ () = θ together with the vanishing of θ, θ and θ at t =. This yields the conditions θ = Re (C ± ) We thus have θ (t) ± = Re ( iω ± C ± ) Combining these so as to get θ and θ alone, we have θ (t) = θ θ (t) = θ m θ (t) = θ cos(ω ± t) cos(ω + t) + cos(ω t) cos(ω + t) cos(ω t) m
3 In order to analyze beat phenomena, it is useful to consider two different frequencies, the average and difference of the eigenfrequencies: In terms of these, we have and we get ω = (ω + + ω ) ω = (ω + ω ) ω ± = ω ± ω θ (t) = θ cos( ω t) cos( ωt) θ (t) = θ sin( ω t) sin( ωt). m In this form we see the overall oscillation of both pendula with frequency ω > ω together with a modulating oscillation in the amplitude of frequency ω. This modulating amplitude is 9 out of phase between the two pendula. Said another way, when the modulated amplitude for θ is a maximum (say ωt n = nπ), the amplitude of θ is zero and vice versa. This is the phenomena of beats. 3
4 Goldstein 6.8 The equilibrium configuration of a molecule is represented by three atoms of equal mass at the vertices of a 45 right triangle connected by springs of equal force constant. Obtain the secular determinant for the modes of vibration in the plane and show by rearrangement of the columns that the secular equation has a triple root ω =. Reduce the determinant to one of third rank and obtain the nonvanishing frequencies of free vibration. Perhaps the hardest part of this problem is setting things up in a clear way. We begin by orienting an x-y coordinate system so that the atom at the right angle of the 45 triangle is at the origin and the hypotenuse is in the first quadrant. With this orientation, the equilibrium positions (no motion) of the particles are (, ), (, a) and (a, ). For clarity, we will further label our particles clockwise from the origin so that particle in equilibrium is at (, ), particle is at (, a) and particle 3 is at (a, ). The absolute coordinate locations of these particles are then (x, y ), (x, y ) and (x 3, y 3 ). Each of these coordinates is, of course, a function of time. We also give the displacements of these particles from their equilibrium positions, namely (η, ξ ) = (x, y ) (η, ξ ) = (x a, y ) (η 3, ξ 3 ) = (x 3, y 3 a) It is these displacements that we will use as our generalized coordinates. The kinetic energy is obvious. Indeed, the kinetic energy matrix will be the identity matrix multiplied by the mass, m. The potential energy is a bit harder to come by. The potential energy in the spring between particles and is given by displacement of the spring away from its equilibrium length. Said another way, we want the difference between the length of the spring at a moment of time, t, and the equilibrium length a. This is (x x ) + (y y ) a Thus the potential energy associated with that spring is V = k (η + a η ) + (ξ ξ ) a The other pieces of the potential energy will look similar, namely V 3 = k (η3 η ) + (ξ 3 + a ξ ) a V 3 = k (η3 η a) + (ξ 3 + a ξ ) a The total potential energy is the sum of these three pieces. It is a mess, to say the least. It is also not a very convenient form for the methods we want to use. But we need to realize that we are considering small oscillations and we must expand the potential about the equilibrium positions. The practical way of saying and doing this is to note that the equilibrium lengths a are much larger than either types of displacements, η, ξ. So we expand the square roots in powers of the (small) displacements: V = k = k a + a (η η ) + a (η η ) + a (ξ ξ ) a ( a + ) a (η η ) + a k (η η ) where we have thrown away all terms inside the original square brackets second order and higher in the displacements. Similar analyses with the other potential terms yields V 3 k (ξ 3 ξ ) V 3 k 4 (η 3 η ) (ξ 3 ξ ). 4
5 The total potential energy is thus V = k (η η ) + k (ξ 3 ξ ) + k (η 3 η ) + k (ξ 3 ξ ) k (η 3 η )(ξ 3 ξ ) The potential energy matrix for η, ξ, η, ξ, η 3, ξ 3 is thus and the characteristic equation becomes V ω T = k 3/ / / / V = k / / / / / / / / / / / 3/ mω /k mω /k 3/ mω /k / / / / / mω /k / / / / / mω /k / / / / 3/ mω /k The best way to evaluate this is to use some computer algebra program. Doing so yields V ω T = k 5 ( mω ) 3( mω k )( mω k )( mω 3k ) which is a sixth order polynomial in ω. One of the solutions is a third order zero (ω = ). These zero eigenfrequencies correspond to the three possible rigid body motions (modes) of the triangular molecule, two independent translations in the x-y plane and a single rotation in the plane (about an axis perpendicular to the plane a z axis if you will). The remaining frequencies are the frequencies of free vibration, namely ω = k m, k m, 3k m 5
6 Goldstein 6. A uniform bar of length l and mass m is suspended by two equal springs of equilibrium length b and force constant k, as shown in a diagram in Goldstein. Find the normal modes of small oscillation in the plane. Referring to the figure, we will place a coordinate system in the following way. In the equilibrium configuration when the system is not moving, we place the origin of our coordinate system at the center of the bar. The right end of the bar is at (x, y) = (l/, ) while the left end of the bar is at (x, y) = ( l/, ). The upper right support for the right spring is at a coordinate location of (x, y) = (l/ + b sin θ, b cos θ ) while the left support has a coordinatization of (x, y) = ( l/ b sin θ, b cos θ ). The equilibrium length of both springs under the influence of gravity is b. It is important to note that this already takes into account the potential energy due to gravity. Because we will be considering small displacements (oscillations) away from equilibrium, any linear terms in the potential will already have been discarded. In other words, we are working at energies slightly above the minimum of the potential. There are three degrees of freedom in the problem. They include the two dimensional motion of the center of mass of the bar. These displacements we will call x and y, displacements from the origin of coordinates. There is also a rotational degree of freedom associated with the bar s motion about its center of mass. We will identify this with the angle φ and measure it from the horizontal (equilibrium) position and take it to be positive in the counterclockwise direction. Our kinetic energy becomes T = m (ẋ + ẏ ) + I φ where the moment of inertia of our rod about the center of mass is I = ml /. Because of how φ appears later, we will write the kinetic energy as T = m ẋ + ẏ ) + (l φ/) 3 Now we consider the potential terms. The potential energy in the right spring we will take to be related to the difference in the lengths of the stretched and unstretched springs, namely V = k { (x l + cos φ) ( l + b sin θ ) (y l + + sin φ) b cos θ b k { } (x ) ( b sin θ + y + lφ b cos θ ) b k x sin θ + ( y + lφ) cos θ where in the second line we have assumed small oscillations in the form that cos φ and sin φ φ. To get the third line, we have simply expanded and kept only terms from the square root that are linear in our small displacements, x, y and φ. Expanding the other potential energy we get something similar V = k { (x l cos φ) ( l b sin θ ) (y l } + sin φ) b cos θ b k x sin θ ( y lφ) cos θ } Combining these we get for the total potential (again, we drop the linear terms as we are working only with the small oscillations) V k ( lφ ) ( lφ ) x sin θ + y cos θ + cos θ + 4x sin θ cos θ 6
7 Constructing the kinetic and potential matrices, we have T = m m m/3 V = k cos θ sin θ sin θ cos θ sin θ cos θ cos θ The characteristic equation, det(v ω T) =, becomes = ( k cos θ mω ) ( k sin θ mω )( k cos θ 3 mω) 4k sin θ cos θ This has solutions for the eigenvalues of ω = ω = k m cos θ ω 3 = k m ( + cos θ ) The corresponding (unnormalized) eigenvectors can now be found as η = cos θ sin θ η = η 3 = sin θ 3 cos θ Note that the zero mode corresponds to a pure rigid body motion. It amounts to a horizontal motion of the center of mass combined with a rotation of the bar about the center of mass. No motion takes place in the vertical direction. The normal mode associated with η is a purely vertical oscillation of the center of mass. The third normal mode is the highest frequency mode and produces oscillations which combine a horizontal and rotational aspect. 7
8 Goldstein 6. Two particles move in one dimension at the junction of three springs, as shown in the figure in Goldstein. The springs all have unstretched lengths equal to a, and the force constants and masses are shown. Find the eigenfrequencies and normal modes of the system. After the other problems this is perhaps the easiest of the lot. For two particles moving in one dimension, we have only two degrees of freedom. The distance between the walls is 3a. The absolute coordinate location of the first particle is x and the second is x as measured from an origin at the left wall. The corresponding displacements from eqilibrium are η = x a The kinetic energy of the system is and the potential energy is η = x a T = m ( η + η ) V = k ( x a ) 3k ( + (x a) (x a) ) k ( + x a ) = k η + 3(η η ) + η Using the matrix approach, we have the kinetic and potential matrices given by m T = m 4k 3k V = 3k 4k The characteristic equation for this system thus becomes which gives us the eigenfrequencies of det ( V ω T ) = ( k mω )( 7k mω ) = ω = k m, 7k m We now want the normal modes of the system, i.e. the characteristic motions for η and η. To this end, we must solve V η = ω T η for η T = (η η ). In the case that ω + = k/m, we find the relevant equation to be η = η and our unnormalized eigenvector is η ω+ =. In the case that ω = 7k/m, we find the relevant equation to be η = η with an unnormalized eigenvetor of η ω+ =. Both, of course, can be normalized with an extra factor of /. The modes, then, correspond to symmetric motion in phase for the ω + mode and antisymmetric motion out of phase for the ω mode. 8
In which we count degrees of freedom and find the normal modes of a mess o masses and springs, which is a lovely model of a solid.
Coupled Oscillators Wednesday, 26 October 2011 In which we count degrees of freedom and find the normal modes of a mess o masses and springs, which is a lovely model of a solid. Physics 111 θ 1 l k θ 2
More informationCoupled Oscillators Monday, 29 October 2012 normal mode Figure 1:
Coupled Oscillators Monday, 29 October 2012 In which we count degrees of freedom and find the normal modes of a mess o masses and springs, which is a lovely model of a solid. Physics 111 θ 1 l m k θ 2
More informationOscillations and Waves
Oscillations and Waves Somnath Bharadwaj and S. Pratik Khastgir Department of Physics and Meteorology IIT Kharagpur Module : Oscillations Lecture : Oscillations Oscillations are ubiquitous. It would be
More informationSeminar 6: COUPLED HARMONIC OSCILLATORS
Seminar 6: COUPLED HARMONIC OSCILLATORS 1. Lagrangian Equations of Motion Let consider a system consisting of two harmonic oscillators that are coupled together. As a model, we will use two particles attached
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 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 informationMath 1302, Week 8: Oscillations
Math 302, Week 8: Oscillations T y eq Y y = y eq + Y mg Figure : Simple harmonic motion. At equilibrium the string is of total length y eq. During the motion we let Y be the extension beyond equilibrium,
More informationL = 1 2 a(q) q2 V (q).
Physics 3550, Fall 2011 Motion near equilibrium - Small Oscillations Relevant Sections in Text: 5.1 5.6 Motion near equilibrium 1 degree of freedom One of the most important situations in physics is motion
More informationLecture 9: Eigenvalues and Eigenvectors in Classical Mechanics (See Section 3.12 in Boas)
Lecture 9: Eigenvalues and Eigenvectors in Classical Mechanics (See Section 3 in Boas) As suggested in Lecture 8 the formalism of eigenvalues/eigenvectors has many applications in physics, especially in
More informationTOPIC E: OSCILLATIONS SPRING 2019
TOPIC E: OSCILLATIONS SPRING 2019 1. Introduction 1.1 Overview 1.2 Degrees of freedom 1.3 Simple harmonic motion 2. Undamped free oscillation 2.1 Generalised mass-spring system: simple harmonic motion
More informationLab 1: Damped, Driven Harmonic Oscillator
1 Introduction Lab 1: Damped, Driven Harmonic Oscillator The purpose of this experiment is to study the resonant properties of a driven, damped harmonic oscillator. This type of motion is characteristic
More informationLab 1: damped, driven harmonic oscillator
Lab 1: damped, driven harmonic oscillator 1 Introduction The purpose of this experiment is to study the resonant properties of a driven, damped harmonic oscillator. This type of motion is characteristic
More informationSimple Harmonic Motion Practice Problems PSI AP Physics B
Simple Harmonic Motion Practice Problems PSI AP Physics B Name Multiple Choice 1. A block with a mass M is attached to a spring with a spring constant k. The block undergoes SHM. Where is the block located
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 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 informationLecture 18. In other words, if you double the stress, you double the resulting strain.
Lecture 18 Stress and Strain and Springs Simple Harmonic Motion Cutnell+Johnson: 10.1-10.4,10.7-10.8 Stress and Strain and Springs So far we ve dealt with rigid objects. A rigid object doesn t change shape
More information! 4 4! o! +! h 4 o=0! ±= ± p i And back-substituting into the linear equations gave us the ratios of the amplitudes of oscillation:.»» = A p e i! +t»»
Topic 6: Coupled Oscillators and Normal Modes Reading assignment: Hand and Finch Chapter 9 We are going to be considering the general case of a system with N degrees of freedome close to one of its stable
More informationM2A2 Problem Sheet 3 - Hamiltonian Mechanics
MA Problem Sheet 3 - Hamiltonian Mechanics. The particle in a cone. A particle slides under gravity, inside a smooth circular cone with a vertical axis, z = k x + y. Write down its Lagrangian in a) Cartesian,
More informationIntroductory Physics. Week 2015/05/29
2015/05/29 Part I Summary of week 6 Summary of week 6 We studied the motion of a projectile under uniform gravity, and constrained rectilinear motion, introducing the concept of constraint force. Then
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 informationFigure 5.16 Compound pendulum: (a) At rest in equilibrium, (b) General position with coordinate θ, Freebody
Lecture 27. THE COMPOUND PENDULUM Figure 5.16 Compound pendulum: (a) At rest in equilibrium, (b) General position with coordinate θ, Freebody diagram The term compound is used to distinguish the present
More informationOscillations. PHYS 101 Previous Exam Problems CHAPTER. Simple harmonic motion Mass-spring system Energy in SHM Pendulums
PHYS 101 Previous Exam Problems CHAPTER 15 Oscillations Simple harmonic motion Mass-spring system Energy in SHM Pendulums 1. The displacement of a particle oscillating along the x axis is given as a function
More informationRigid bodies - general theory
Rigid bodies - general theory Kinetic Energy: based on FW-26 Consider a system on N particles with all their relative separations fixed: it has 3 translational and 3 rotational degrees of freedom. Motion
More informationDamped harmonic motion
Damped harmonic motion March 3, 016 Harmonic motion is studied in the presence of a damping force proportional to the velocity. The complex method is introduced, and the different cases of under-damping,
More informationAP Pd 3 Rotational Dynamics.notebook. May 08, 2014
1 Rotational Dynamics Why do objects spin? Objects can travel in different ways: Translation all points on the body travel in parallel paths Rotation all points on the body move around a fixed point An
More informationProblem Set Number 01, MIT (Winter-Spring 2018)
Problem Set Number 01, 18.377 MIT (Winter-Spring 2018) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) February 28, 2018 Due Thursday, March 8, 2018. Turn it in (by 3PM) at the Math.
More informationKeble College - Hilary 2014 CP3&4: Mathematical methods I&II Tutorial 5 - Waves and normal modes II
Tomi Johnson 1 Keble College - Hilary 2014 CP3&4: Mathematical methods I&II Tutorial 5 - Waves and normal modes II Prepare full solutions to the problems with a self assessment of your progress on a cover
More informationMECHANICS LAB AM 317 EXP 8 FREE VIBRATION OF COUPLED PENDULUMS
MECHANICS LAB AM 37 EXP 8 FREE VIBRATIN F CUPLED PENDULUMS I. BJECTIVES I. To observe the normal modes of oscillation of a two degree-of-freedom system. I. To determine the natural frequencies and mode
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 informationUnit 7: Oscillations
Text: Chapter 15 Unit 7: Oscillations NAME: Problems (p. 405-412) #1: 1, 7, 13, 17, 24, 26, 28, 32, 35 (simple harmonic motion, springs) #2: 45, 46, 49, 51, 75 (pendulums) Vocabulary: simple harmonic motion,
More informationRigid Body Dynamics, SG2150 Solutions to Exam,
KTH Mechanics 011 10 Calculational problems Rigid Body Dynamics, SG150 Solutions to Eam, 011 10 Problem 1: A slender homogeneous rod of mass m and length a can rotate in a vertical plane about a fied smooth
More informationSmall oscillations and normal modes
Chapter 4 Small oscillations and normal modes 4.1 Linear oscillations Discuss a generalization of the harmonic oscillator problem: oscillations of a system of several degrees of freedom near the position
More informationClassical Mechanics Comprehensive Exam Solution
Classical Mechanics Comprehensive Exam Solution January 31, 011, 1:00 pm 5:pm Solve the following six problems. In the following problems, e x, e y, and e z are unit vectors in the x, y, and z directions,
More information1.1 To observe the normal modes of oscillation of a two degree of freedom system.
I. BJECTIVES. To observe the normal modes of oscillation of a two degree of freedom system.. To determine the natural frequencies and mode shapes of the system from solution of the Eigenvalue problem..3
More informationStructural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.
Outline of Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear
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 informationHB Coupled Pendulums Lab Coupled Pendulums
HB 04-19-00 Coupled Pendulums Lab 1 1 Coupled Pendulums Equipment Rotary Motion sensors mounted on a horizontal rod, vertical rods to hold horizontal rod, bench clamps to hold the vertical rods, rod clamps
More informationP321(b), Assignement 1
P31(b), Assignement 1 1 Exercise 3.1 (Fetter and Walecka) a) The problem is that of a point mass rotating along a circle of radius a, rotating with a constant angular velocity Ω. Generally, 3 coordinates
More informationM04M.1 Particles on a Line
Part I Mechanics M04M.1 Particles on a Line M04M.1 Particles on a Line Two elastic spherical particles with masses m and M (m M) are constrained to move along a straight line with an elastically reflecting
More informationNORMAL MODES, WAVE MOTION AND THE WAVE EQUATION. Professor G.G.Ross. Oxford University Hilary Term 2009
NORMAL MODES, WAVE MOTION AND THE WAVE EQUATION Professor G.G.Ross Oxford University Hilary Term 009 This course of twelve lectures covers material for the paper CP4: Differential Equations, Waves and
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 informationL = 1 2 a(q) q2 V (q).
Physics 3550 Motion near equilibrium - Small Oscillations Relevant Sections in Text: 5.1 5.6, 11.1 11.3 Motion near equilibrium 1 degree of freedom One of the most important situations in physics is motion
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 informationImaginary. Axis. Real. Axis
Name ME6 Final. I certify that I upheld the Stanford Honor code during this exam Monday December 2, 25 3:3-6:3 p.m. ffl Print your name and sign the honor code statement ffl You may use your course notes,
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 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 informationSeminar 8. HAMILTON S EQUATIONS. p = L q = m q q = p m, (2) The Hamiltonian (3) creates Hamilton s equations as follows: = p ṗ = H = kq (5)
Problem 31. Derive Hamilton s equations for a one-dimensional harmonic oscillator. Seminar 8. HAMILTON S EQUATIONS Solution: The Lagrangian L = T V = 1 m q 1 kq (1) yields and hence the Hamiltonian is
More informationDifferential Equations and Linear Algebra Exercises. Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS
Differential Equations and Linear Algebra Exercises Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS CHAPTER 1 Linear second order ODEs Exercises 1.1. (*) 1 The following differential
More informationSimple Harmonic Motion Practice Problems PSI AP Physics 1
Simple Harmonic Motion Practice Problems PSI AP Physics 1 Name Multiple Choice Questions 1. A block with a mass M is attached to a spring with a spring constant k. The block undergoes SHM. Where is the
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 informationReflections and Rotations in R 3
Reflections and Rotations in R 3 P. J. Ryan May 29, 21 Rotations as Compositions of Reflections Recall that the reflection in the hyperplane H through the origin in R n is given by f(x) = x 2 ξ, x ξ (1)
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 2001/2051 The Compound Pendulum Experiment 4 and Helical Springs
PY001/051 Compound Pendulum and Helical Springs Experiment 4 Physics 001/051 The Compound Pendulum Experiment 4 and Helical Springs Prelab 1 Read the following background/setup and ensure you are familiar
More informationEngineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Introduction to vibration
Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Introduction to vibration Module 15 Lecture 38 Vibration of Rigid Bodies Part-1 Today,
More information4 A mass-spring oscillating system undergoes SHM with a period T. What is the period of the system if the amplitude is doubled?
Slide 1 / 52 1 A block with a mass M is attached to a spring with a spring constant k. The block undergoes SHM. Where is the block located when its velocity is a maximum in magnitude? A 0 B + or - A C
More informationIntroduction. Physics E-1a Expt 5: The Sweet Spot Fall 2006
Physics E-1a Expt 5: The Sweet Spot all 2006 The center of percussion () is the place on a bat where it may be struck without causing a reaction at the point of support. When a ball is hit at the the contact
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 information02 Coupled Oscillators
Utah State University DigitalCommons@USU Foundations of Wave Phenomena Physics, Department of --4 Coupled Oscillators Charles G. Torre Department of Physics, Utah State University, Charles.Torre@usu.edu
More informationStatic Equilibrium, Gravitation, Periodic Motion
This test covers static equilibrium, universal gravitation, and simple harmonic motion, with some problems requiring a knowledge of basic calculus. Part I. Multiple Choice 1. 60 A B 10 kg A mass of 10
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 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 informationPhys 7221 Homework # 8
Phys 71 Homework # 8 Gabriela González November 15, 6 Derivation 5-6: Torque free symmetric top In a torque free, symmetric top, with I x = I y = I, the angular velocity vector ω in body coordinates with
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 information7 Pendulum. Part II: More complicated situations
MATH 35, by T. Lakoba, University of Vermont 60 7 Pendulum. Part II: More complicated situations In this Lecture, we will pursue two main goals. First, we will take a glimpse at a method of Classical Mechanics
More informationTheory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati
Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 2 Simpul Rotors Lecture - 2 Jeffcott Rotor Model In the
More informationVibrations Qualifying Exam Study Material
Vibrations Qualifying Exam Study Material The candidate is expected to have a thorough understanding of engineering vibrations topics. These topics are listed below for clarification. Not all instructors
More informationRotational motion problems
Rotational motion problems. (Massive pulley) Masses m and m 2 are connected by a string that runs over a pulley of radius R and moment of inertia I. Find the acceleration of the two masses, as well as
More informationWhy Is CO 2 a Greenhouse Gas?
Why Is CO 2 a Greenhouse Gas? The Earth is warming and the cause is the increase in greenhouse gases like carbon dioxide (CO 2 ) in the atmosphere. Carbon dioxide is a linear, triatomic molecule with a
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 informationChapter 9 Notes. x cm =
Chapter 9 Notes Chapter 8 begins the discussion of rigid bodies, a system of particles with fixed relative positions. Previously we have dealt with translation of a particle: if a rigid body does not rotate
More informationA Level. A Level Physics. Oscillations (Answers) AQA, Edexcel. Name: Total Marks: /30
Visit http://www.mathsmadeeasy.co.uk/ for more fantastic resources. AQA, Edexcel A Level A Level Physics Oscillations (Answers) Name: Total Marks: /30 Maths Made Easy Complete Tuition Ltd 2017 1. The graph
More informationPhysics 161 Lecture 17 Simple Harmonic Motion. October 30, 2018
Physics 161 Lecture 17 Simple Harmonic Motion October 30, 2018 1 Lecture 17: learning objectives Review from lecture 16 - Second law of thermodynamics. - In pv cycle process: ΔU = 0, Q add = W by gass
More informationIf the symmetry axes of a uniform symmetric body coincide with the coordinate axes, the products of inertia (Ixy etc.
Prof. O. B. Wright, Autumn 007 Mechanics Lecture 9 More on rigid bodies, coupled vibrations Principal axes of the inertia tensor If the symmetry axes of a uniform symmetric body coincide with the coordinate
More informationCorso di Laurea in LOGOPEDIA FISICA ACUSTICA MOTO OSCILLATORIO
Corso di Laurea in LOGOPEDIA FISICA ACUSTICA MOTO OSCILLATORIO Fabio Romanelli Department of Mathematics & Geosciences University of Trieste Email: romanel@units.it What is an Oscillation? Oscillation
More informationNonlinear Dynamic Systems Homework 1
Nonlinear Dynamic Systems Homework 1 1. A particle of mass m is constrained to travel along the path shown in Figure 1, which is described by the following function yx = 5x + 1x 4, 1 where x is defined
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 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 informationLECTURE 3 ENERGY AND PENDULUM MOTION. Instructor: Kazumi Tolich
LECTURE 3 ENERGY AND PENDULUM MOTION Instructor: Kazumi Tolich Lecture 3 2 14.4: Energy in simple harmonic motion Finding the frequency for simple harmonic motion 14.5: Pendulum motion Physical pendulum
More informationFinal Exam December 11, 2017
Final Exam Instructions: You have 120 minutes to complete this exam. This is a closed-book, closed-notes exam. You are NOT allowed to use a calculator with communication capabilities during the exam. Usage
More informationSOLUTIONS, PROBLEM SET 11
SOLUTIONS, PROBLEM SET 11 1 In this problem we investigate the Lagrangian formulation of dynamics in a rotating frame. Consider a frame of reference which we will consider to be inertial. Suppose that
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 15+ Revisit Oscillations and Simple Harmonic Motion
Chapter 15+ Revisit Oscillations and Simple Harmonic Motion Revisit: Oscillations Simple harmonic motion To-Do: Pendulum oscillations Derive the parallel axis theorem for moments of inertia and apply it
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 informationSpecial Classical Physical Systems
Chapter 6 Special Classical Physical Systems 6.1 Introduction In order to understand the ideas of modern physics, it is essential to understand the operations of some special classical systems. Not only
More informationThe distance of the object from the equilibrium position is m.
Answers, Even-Numbered Problems, Chapter..4.6.8.0..4.6.8 (a) A = 0.0 m (b).60 s (c) 0.65 Hz Whenever the object is released from rest, its initial displacement equals the amplitude of its SHM. (a) so 0.065
More informationt = g = 10 m/s 2 = 2 s T = 2π g
Annotated Answers to the 1984 AP Physics C Mechanics Multiple Choice 1. D. Torque is the rotational analogue of force; F net = ma corresponds to τ net = Iα. 2. C. The horizontal speed does not affect the
More informationMITOCW R11. Double Pendulum System
MITOCW R11. Double Pendulum System The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for
More informationLecture 38: Equations of Rigid-Body Motion
Lecture 38: Equations of Rigid-Body Motion It s going to be easiest to find the equations of motion for the object in the body frame i.e., the frame where the axes are principal axes In general, we can
More informationConstrained motion and generalized coordinates
Constrained motion and generalized coordinates based on FW-13 Often, the motion of particles is restricted by constraints, and we want to: work only with independent degrees of freedom (coordinates) k
More informationRotational Kinetic Energy
Lecture 17, Chapter 10: Rotational Energy and Angular Momentum 1 Rotational Kinetic Energy Consider a rigid body rotating with an angular velocity ω about an axis. Clearly every point in the rigid body
More informationLecture 5: Harmonic oscillator, Morse Oscillator, 1D Rigid Rotor
Lecture 5: Harmonic oscillator, Morse Oscillator, 1D Rigid Rotor It turns out that the boundary condition of the wavefunction going to zero at infinity is sufficient to quantize the value of energy that
More informationEngineering Science OUTCOME 2 - TUTORIAL 3 FREE VIBRATIONS
Unit 2: Unit code: QCF Level: 4 Credit value: 5 Engineering Science L/60/404 OUTCOME 2 - TUTORIAL 3 FREE VIBRATIONS UNIT CONTENT OUTCOME 2 Be able to determine the behavioural characteristics of elements
More informationUnforced Oscillations
Unforced Oscillations Simple Harmonic Motion Hooke s Law Newton s Second Law Method of Force Competition Visualization of Harmonic Motion Phase-Amplitude Conversion The Simple Pendulum and The Linearized
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 informationCharged objects in Conducting Fluids
Charged objects in Conducting Fluids Net charge in a sphere of radius λ D is approximately zero. λ D 2 = ε 0κ k B T c 0 e 2 Z 2 k B T k c = 1 / 4πε 0 c 0 Z e κ Thermal energy (Joules) Coulomb constant
More informationAP Physics. Harmonic Motion. Multiple Choice. Test E
AP Physics Harmonic Motion Multiple Choice Test E A 0.10-Kg block is attached to a spring, initially unstretched, of force constant k = 40 N m as shown below. The block is released from rest at t = 0 sec.
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 informationColumbia University Department of Physics QUALIFYING EXAMINATION
Columbia University Department of Physics QUALIFYING EXAMINATION Monday, January 11, 2016 1:00PM to 3:00PM Classical Physics Section 1. Classical Mechanics Two hours are permitted for the completion of
More informationNoninertial Reference Frames
Chapter 12 Non Reference Frames 12.1 Accelerated Coordinate Systems A reference frame which is fixed with respect to a rotating rigid is not. The parade example of this is an observer fixed on the surface
More informationWaves & Oscillations
Physics 42200 Waves & Oscillations Lecture 21 Review Spring 2013 Semester Matthew Jones Midterm Exam: Date: Wednesday, March 6 th Time: 8:00 10:00 pm Room: PHYS 203 Material: French, chapters 1-8 Review
More information