DOCSLIB.ORG

Lagrangian & Hamiltonian Mechanics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lagrangian & Hamiltonian Mechanics Lagrangian & Hamiltonian mechanics Proof of the Beltrami Identity Newton’s Second Law for a single particle 1 2 2 2 Rather than starting from the traditional beginnings dL L L L L m x y z V(,,) x y z xx Chain Rule 2 of Conservation of Energy, Conservation of momentum ii dt ti xii x zˆ rx xˆ yy ˆ z zˆ or Newton’s Second Law, the Lagrangian (and related d L L Euler-Lagrange Equation Hamiltonian) formulations of dynamics offer a more d L L d L dx L dx L dt dt x x L x,, x t ii systematic solution mechanism for mechanical dt x x dt x dt x dt t dt ii d V V yˆ systems, especially those with multiple variables. mx mx dL L d L L dt xi x xxii ˆ xyzˆ ˆ ˆ 1 In Classical Mechanics, the Lagrangian L is defined as: dt t dt x x d V V x i ii my my dt y y LTV dL L d L i xi Product rule d V V dt ti dt xi mz mz Kinetic Potential dt zi z energy energy d L L Lx i mVr dti xi t d L L LL ˆ ˆ ˆ The Euler-Lagrange equation i.e. using rx x y y z z 0 Lx i constant yields the equations of motion Cartesian dt xii x txi i VVV basis vectors V xyzˆ ˆ ˆ x y z Leonhard Euler xi Position or angular coordinate 1707-1783 dx Conservation of Energy for a single particle i.e. we associate a force with a gradient x i Rate of change of position or of potential energy, which is entirely i angle with time t 1 2 dt L2 mv V(,,) x y z consistent with any conservative field such as gravity or electromagnetism. L1 m x2 y 2 z 2 V(,,) x y z The idea is that the Euer-Lagrange equation 2 extremizes* a quantity called the Action L Hamiltonians & total energy Beltrami Identity L xi k i x t2 i From the penultimate line of the proof S Lxx 1, 2 ,... xxxNN , 1 , 2 ,... xtdt , t of the Beltrami Identity: 1 LLL Joseph-Louis k L x y z x y z d L L Lagrange 1736-1813 In many problems the Lagrangian is independent Lxi 2 2 2 2 1 dti x t of time, so the Euler-Lagrange equation reduces k2 m x y z V xmx ymy zmz i to the Beltrami Identity. This is often the best starting 2 point for problem solving in Lagrangian mechanics. k 1 m x2 y 2 z 2 V Define the Hamiltonian 2 ‘Canonical 2 H p x L 1 2 2 2 ii monentum’ L E 2 m x y z V() E k i If 0 this means L t LHpi i.e. we can associate energy with the (negative) x L pxii i Lxconstant of the constant in the Beltrami Identity xpii i x i i dH L So we can associate the Hamiltonian with the total dt t energy. H is constant if L is Sir William Rowan time invariant. Hamilton 1805–1865 *i.e. yields either a maximum or minimum value Mathematics topic handout – Mechanics – Lagrangian and Hamiltonian dynamics Dr Andrew French. www.eclecticon.info PAGE 1 Example #1: dynamics of a Hookean spring From above: An ellipse defined in plane 2 polar coordinates ˆ L11 mx22 kx 22JJJ θ 22 rr rˆ x is the spring extension m mr2 m 2 r 3 rd d L L k is the spring constant 1 2 2 GM dA 2 r d dt x x rr r2 Semi- mx kx i.e. Hooke’s Law, SHM... minor 2 2 is the eccentricity r ba 1- J GM axis r 2 3 2 of the ellipse This is perhaps a bit contrived to start with the m r r 2b quadratic Elastic Potential Energy expression! b2 .. But at least this shows consistency with the 1 If orbits are elliptical (Kepler’s First Law) 2 a(1 ) Newtonian method. a aa1122 r cos 1 Example #2: Kepler’s Laws for a 1 cos r planet orbiting a massive star (i.e. Semi-major axis 2 2a no movement of star due to gravitational a 1 sin r2 sin r attraction to planet, only vice versa) 1 cos 2 a 1 2 dA122 1 J 1 2 2 2 2 2r 2 2 G1 a vrr rθˆ v r r Plane polar J sin dt m coordinates r 2 GMm ma 1 1 2 2 2 So equal areas are swept out in equal times, which is L2 m r r Note sign! r 2 GPE is negative JJJcos a 1 Kepler’s Second Law. so –V becomes d L L r 22 1 2 positive ma11 r ma mr dt r r Since equal areas are swept out in equal times, the orbital period is the area of the ellipse divided by the rate GMm 2 of area sweep. This proves Kepler’s Third Law. mr mr 2 2 r J22 GMa 1 J 1 GM Confirm Newton II, with 2 3 2 2 2 2 rr 2 m r r r m a1 r 22 2 correct formula for ab a 1 Ellipse area is r centripetal acceleration PP J2 GM J 2 J 2 dA 1 GM1 2 a ab d L L 2 m2 r 3 r 2 m 2 r 3 m2ar1 22 dt dt 2 234 d 2 22 Pa mr 0 Conservation of angular momentum J GMm a 1 GM dt mr2 J So elliptical orbits satisfy the laws of Kepler’s Third Law: The square of the orbital motion, and yield an expression for total period of a planet is directly proportional to the The last result is the first glimpse of the utility of angular momentum J, which is constant. cube of the semi-major axis of its orbit. the Lagrangian method. The conservation of angular momentum comes straight from the fact that the Lagrangian has no explicit dependence on angle, since KE doesn’t, and Newton’s Law of Gravitation acts purely radially. Mathematics topic handout – Mechanics – Lagrangian and Hamiltonian dynamics Dr Andrew French. www.eclecticon.info PAGE 2 d L L Applying the Euler-Lagrange equation Example #3: Moving plane* for mass m dt x22 x h x g d 2 mx2 m x 1 x 2 tan mg tan dt d 2 mx2 m x 1 x 2 tan mg tan d Hence: Mx12 mx 0 Mx mx 0 Which is a statement of conservation dt 12 M of linear momentum in the horizontal Consider a rectangular block of mass m xx21 direction i.e. where there is no m sliding frictionlessly from the top net external force (there is gravity, of a planar wedge of mass M. This also M 2 but this acts downwards) Mx11 mx1 tan mg tan slides frictionlessly on a horizontal surface. m 22 Assuming the block has not reached the Mx1 mx 1tan Mx 1 tan mg tan horizontal surface, after t seconds the horizontal x M1 tan22 m tan mg tan separation is: 1 2 M sin sin 1 sin x m mg 2 x x12 x 1 22 1 tan 2 , tan cos cos cos cos cos h x x tan vertical drop of block 12 2 x1 M msin mg sin cos The Lagrangian for the system is: LTV mg sin cosg sin cos x1 22 2 2 2 M 1 1 1 Mmsinm sin T2 Mx12 2 mx 2 mh Mgsin cos g sin cos Kinetic energy x 2 22m V mgh Mmsin 1M sin Gravitational potential energy L 1 Mx2 1 mx 2 1 m x x2 tan 2 mg x x tan i.e. constant acceleration motion for both 21 2 2 2 1 2 1 2 block and wedge. If system starts from rest: Limiting case when Mm g sin cos g sin cos x1 0 x2 gt sin cos d L L Applying the Euler-Lagrange equation xt1 2 xt2 2 M m 1 2 m sin 1 sin x 0 for mass M 1 x2 2 gt sin cos dt x11 x M g sin cos g sin cos xt 1 2 xt 1 2 d 2 1 2 M 2 2 2 m 2 Mx1 m x 1 x 2 tan mg tan m sin 1 M sin dt 2 .. which is consistent with no horizontal movement when the angle tends to 90o. Mx11 m x x2 tan mg tan i.e. in this case the block simply falls vertically! * Morin, Classical Mechanics problem 6.1 Mathematics topic handout – Mechanics – Lagrangian and Hamiltonian dynamics Dr Andrew French. www.eclecticon.info PAGE 3 Example #4: Hoop & Pulley* Now since the string is inextensible: sin sin eq sin sin cos cos sin g h R h h0 R eq eq m V MgR MgRcos mgh0 mgR sin M cos eq cos22 sin 1 m2 sin m 1 2 M M 2 cos 1 sin The Lagrangian for the system is therefore: Positive root O m 1 22 sin MM Mm in range of germane h LTV m 1 to this system 1 22 sin MM (M m )( M m ) L2 m M R MgRcos MgR mgh0 mgR Hence: Applying the Euler-Lagrange equation: Mass M is fixed to a circular hoop g m M sin d L L of negligible mass, which can freely rotate R m M about fixed origin O.
Load more

Recommended publications
  • Ch 11 Vibrations and Waves Simple Harmonic Motion Simple Harmonic Motion
    Ch 11 Vibrations and Waves Simple Harmonic Motion Simple Harmonic Motion A vibration (oscillation) back & forth taking the same amount of time for each cycle is periodic. Each vibration has an equilibrium position from which it is somehow disturbed by a given energy source. The disturbance produces a displacement from equilibrium. This is followed by a restoring force. Vibrations transfer energy. Recall Hooke’s Law The restoring force of a spring is proportional to the displacement, x. F = -kx. K is the proportionality constant and we choose the equilibrium position of x = 0. The minus sign reminds us the restoring force is always opposite the displacement, x. F is not constant but varies with position. Acceleration of the mass is not constant therefore. http://www.youtube.com/watch?v=eeYRkW8V7Vg&feature=pl ayer_embedded Key Terms Displacement- distance from equilibrium Amplitude- maximum displacement Cycle- one complete to and fro motion Period (T)- Time for one complete cycle (s) Frequency (f)- number of cycles per second (Hz) * period and frequency are inversely related: T = 1/f f = 1/T Energy in SHOs (Simple Harmonic Oscillators) In stretching or compressing a spring, work is required and potential energy is stored. Elastic PE is given by: PE = ½ kx2 Total mechanical energy E of the mass-spring system = sum of KE + PE E = ½ mv2 + ½ kx2 Here v is velocity of the mass at x position from equilibrium. E remains constant w/o friction. Energy Transformations As a mass oscillates on a spring, the energy changes from PE to KE while the total E remains constant.
    [Show full text]
  • Euler Equation and Geodesics R
    Euler Equation and Geodesics R. Herman February 2, 2018 Introduction Newton formulated the laws of motion in his 1687 volumes, col- lectively called the Philosophiae Naturalis Principia Mathematica, or simply the Principia. However, Newton’s development was geometrical and is not how we see classical dynamics presented when we first learn mechanics. The laws of mechanics are what are now considered analytical mechanics, in which classical dynamics is presented in a more elegant way. It is based upon variational principles, whose foundations began with the work of Eu- ler and Lagrange and have been refined by other now-famous figures in the eighteenth and nineteenth centuries. Euler coined the term the calculus of variations in 1756, though it is also called variational calculus. The goal is to find minima or maxima of func- tions of the form f : M ! R, where M can be a set of numbers, functions, paths, curves, surfaces, etc. Interest in extrema problems in classical mechan- ics began near the end of the seventeenth century with Newton and Leibniz. In the Principia, Newton was interested in the least resistance of a surface of revolution as it moves through a fluid. Seeking extrema at the time was not new, as the Egyptians knew that the shortest path between two points is a straight line and that a circle encloses the largest area for a given perimeter. Heron, an Alexandrian scholar, deter- mined that light travels along the shortest path. This problem was later taken up by Willibrord Snellius (1580–1626) after whom Snell’s law of refraction is named.
    [Show full text]
  • Chapter 1 Chapter 2 Chapter 3
    Notes CHAPTER 1 1. Herbert Westren Turnbull, The Great Mathematicians in The World of Mathematics. James R. Newrnan, ed. New York: Sirnon & Schuster, 1956. 2. Will Durant, The Story of Philosophy. New York: Sirnon & Schuster, 1961, p. 41. 3. lbid., p. 44. 4. G. E. L. Owen, "Aristotle," Dictionary of Scientific Biography. New York: Char1es Scribner's Sons, Vol. 1, 1970, p. 250. 5. Durant, op. cit., p. 44. 6. Owen, op. cit., p. 251. 7. Durant, op. cit., p. 53. CHAPTER 2 1. Williarn H. Stahl, '' Aristarchus of Samos,'' Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 1, 1970, p. 246. 2. Jbid., p. 247. 3. G. J. Toorner, "Ptolerny," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 11, 1975, p. 187. CHAPTER 3 1. Stephen F. Mason, A History of the Sciences. New York: Abelard-Schurnan Ltd., 1962, p. 127. 2. Edward Rosen, "Nicolaus Copernicus," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 3, 1971, pp. 401-402. 3. Mason, op. cit., p. 128. 4. Rosen, op. cit., p. 403. 391 392 NOTES 5. David Pingree, "Tycho Brahe," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 2, 1970, p. 401. 6. lbid.. p. 402. 7. Jbid., pp. 402-403. 8. lbid., p. 413. 9. Owen Gingerich, "Johannes Kepler," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 7, 1970, p. 289. 10. lbid.• p. 290. 11. Mason, op. cit., p. 135. 12. Jbid .. p. 136. 13. Gingerich, op. cit., p. 305. CHAPTER 4 1.
    [Show full text]
  • Oscillations
    CHAPTER FOURTEEN OSCILLATIONS 14.1 INTRODUCTION In our daily life we come across various kinds of motions. You have already learnt about some of them, e.g., rectilinear 14.1 Introduction motion and motion of a projectile. Both these motions are 14.2 Periodic and oscillatory non-repetitive. We have also learnt about uniform circular motions motion and orbital motion of planets in the solar system. In 14.3 Simple harmonic motion these cases, the motion is repeated after a certain interval of 14.4 Simple harmonic motion time, that is, it is periodic. In your childhood, you must have and uniform circular enjoyed rocking in a cradle or swinging on a swing. Both motion these motions are repetitive in nature but different from the 14.5 Velocity and acceleration periodic motion of a planet. Here, the object moves to and fro in simple harmonic motion about a mean position. The pendulum of a wall clock executes 14.6 Force law for simple a similar motion. Examples of such periodic to and fro harmonic motion motion abound: a boat tossing up and down in a river, the 14.7 Energy in simple harmonic piston in a steam engine going back and forth, etc. Such a motion motion is termed as oscillatory motion. In this chapter we 14.8 Some systems executing study this motion. simple harmonic motion The study of oscillatory motion is basic to physics; its 14.9 Damped simple harmonic motion concepts are required for the understanding of many physical 14.10 Forced oscillations and phenomena. In musical instruments, like the sitar, the guitar resonance or the violin, we come across vibrating strings that produce pleasing sounds.
    [Show full text]
  • Exact Solution for the Nonlinear Pendulum (Solu¸C˜Aoexata Do Pˆendulon˜Aolinear)
    Revista Brasileira de Ensino de F¶³sica, v. 29, n. 4, p. 645-648, (2007) www.sb¯sica.org.br Notas e Discuss~oes Exact solution for the nonlinear pendulum (Solu»c~aoexata do p^endulon~aolinear) A. Bel¶endez1, C. Pascual, D.I. M¶endez,T. Bel¶endezand C. Neipp Departamento de F¶³sica, Ingenier¶³ade Sistemas y Teor¶³ade la Se~nal,Universidad de Alicante, Alicante, Spain Recebido em 30/7/2007; Aceito em 28/8/2007 This paper deals with the nonlinear oscillation of a simple pendulum and presents not only the exact formula for the period but also the exact expression of the angular displacement as a function of the time, the amplitude of oscillations and the angular frequency for small oscillations. This angular displacement is written in terms of the Jacobi elliptic function sn(u;m) using the following initial conditions: the initial angular displacement is di®erent from zero while the initial angular velocity is zero. The angular displacements are plotted using Mathematica, an available symbolic computer program that allows us to plot easily the function obtained. As we will see, even for amplitudes as high as 0.75¼ (135±) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency ! in terms of the complete elliptic integral of the ¯rst kind. We can conclude that for amplitudes lower than 135o the periodic motion exhibited by a simple pendulum is practically harmonic but its oscillations are not isochronous (the period is a function of the initial amplitude).
    [Show full text]
  • Lecture I, Aug25, 2014 Newton, Lagrange and Hamilton's Equations of Classical Mechanics
    Lecture I, Aug25, 2014 Newton, Lagrange and Hamilton’s Equations of Classical Mechanics Introduction What this course is about... BOOK Goldstein is a classic Text Book Herbert Goldstein ( June 26, 1922 January 12, 2005) PH D MIT in 1943; Then at Harvard and Columbia first edition of CM book was published in 1950 ( 399 pages, each page is about 3/4 in area compared to new edition) Third edition appeared in 2002 ... But it is an old Text book What we will do different ?? Start with Newton’s Lagrange and Hamilton’s equation one after the other Small Oscillations: Marion, why is it important ??? we start right in the beginning talking about small oscillations, simple limit Phase Space Plots ( Marian, page 159 ) , Touch nonlinear physics and Chaos, Symmetries Order in which Chapters are covered is posted on the course web page Classical Encore 1-4pm, Sept 29, Oct 27, Dec 1 Last week of the Month: Three Body Problem.. ( chaos ), Solitons, may be General Relativity We may not cover scattering and Rigid body dynamics.. the topics that you have covered in Phys303 and I think there is less to gain there.... —————————————————————————— Newton’s Equation, Lagrange and Hamilton’s Equations Beauty Contest: Write Three equations and See which one are the prettiest?? (Simplicity, Mathematical Beauty...) Same Equations disguised in three different forms (I)Lagrange and Hamilton’s equations are scalar equations unlike Newton’s equation.. (II) To apply Newtons’ equation, Forces of constraints are needed to describe constrained motion (III) Symmetries are best described in the Lagrangian formulation (IV)For rectangular coordinates, Newtons’ s Eqn are the easiest.
    [Show full text]
  • HOOKE's LAW and Sihlple HARMONIC MOTION by DR
    Hooke’s Law by Dr. James E. Parks Department of Physics and Astronomy 401 Nielsen Physics Building The University of Tennessee Knoxville, Tennessee 37996-1200 Copyright June, 2000 by James Edgar Parks* *All rights are reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the author. Objective The objectives of this experiment are: (1) to study simple harmonic motion, (2) to learn the requirements for simple harmonic motion, (3) to learn Hooke's Law, (4) to verify Hooke's Law for a simple spring, (5) to measure the force constant of a spiral spring, (6) to learn the definitions of period and frequency and the relationships between them, (7) to learn the definition of amplitude, (8) to learn the relationship between the period, mass, and force constant of a vibrating mass on a spring undergoing simple harmonic motion, (9) to determine the period of a vibrating spring with different masses, and (10) to compare the measured periods of vibration with those calculated from theory. Theory The motion of a body that oscillates back and forth is defined as simple harmonic motion if there exists a restoring force F that is opposite and directly proportional to the distance x that the body is displaced from its equilibrium position. This relationship between the restoring force F and the displacement x may be written as F=-kx (1) where k is a constant of proportionality. The minus sign indicates that the force is oppositely directed to the displacement and is always towards the equilibrium position.
    [Show full text]
  • Simple Harmonic Motion
    Simple Harmonic Motion Ø http://www.youtube.com/watch?v=yVkdfJ9PkRQ Ø What it shows: Fifteen uncoupled simple pendulums of increasing lengths dance together to produce visual traveling waves, standing waves, beating, and random motion. Ø How it works: The period of one complete cycle of the dance is 60 seconds. The length of the longest pendulum has been adjusted so that it executes 51 oscillations in this 60 second period. The length of each successively shorter pendulum is carefully adjusted so that it executes one additional oscillation in this period. Thus, the 15th pendulum (shortest) undergoes 65 oscillations. When all 15 pendulums are started together, they quickly fall out of sync—their relative phases continuously change because of their different periods of oscillation. However, after 60 seconds they will all have executed an integral number of oscillations and be back in sync again at that instant, ready to repeat the dance. The Ideal Spring & Simple Harmonic Motion Ø Hooke’s Law: The force (F) applied directly to a spring moves or causes a displacement (x) in the spring directly proportional to the amount of force. Ø F ∝ x; ∴ F ∝ x and F ∝ x Ø Hooke’s Law Equation: F = -k·x Ø Ideal springs follow this relationship. Ø F = Fs = restoring force = the force applied by the spring. Units: N. Ø k = the spring constant = “stiffness” of spring (unique for each spring… depends on material and depends on number of coils) Units: N/m. Ø x = displacement of spring from unstrained length. Units: m. Ø F = -k·x; this is negative b/c restoring force always points in a direction opposite to the displacement of the spring.
    [Show full text]
  • (PH003) Classical Mechanics the Inverted Pendulum
    CALIFORNIA INSTITUTE OF TECHNOLOGY PHYSICS MATHEMATICS AND ASTRONOMY DIVISION Freshman Physics Laboratory (PH003) Classical Mechanics The Inverted Pendulum Kenneth G Libbrecht, Virginio de Oliveira Sannibale, 2010 (Revision October 2012) Chapter 3 The Inverted Pendulum 3.1 Introduction The purpose of this lab is to explore the dynamics of the harmonic me- chanical oscillator. To make things a bit more interesting, we will model and study the motion of an inverted pendulum (IP), which is a special type of tunable mechanical oscillator. As we will see below, the IP contains two restoring forces, one positive and one negative. By adjusting the relative strengths of these two forces, we can change the oscillation frequency of the pendulum over a wide range. As usual (see section3.2), we will first make a mathematical model of the IP, and then you will characterize the system by measuring various parameters in the model. Finally, you will observe the motion of the pen- dulum and see if it agrees with the model to within experimental uncer- tainties. The IP is a fairly simple mechanical device, so you should be able to analyze and characterize the system almost completely. At the same time, the inverted pendulum exhibits some interesting dynamics, and it demon- strates several important principles in physics. Waves and oscillators are everywhere in physics and engineering, and one of the best ways to un- derstand oscillatory phenomenon is to carefully analyze a relatively sim- ple system like the inverted pendulum. 27 DRAFT 28 CHAPTER 3. THE INVERTED PENDULUM 3.2 Modeling the Inverted Pendulum (IP) 3.2.1 The Simple Harmonic Oscillator We begin our discussion with the most basic harmonic oscillator – a mass connected to an ideal spring.
    [Show full text]
  • Table of Contents Variable Forces and Differential Equations
    Copyright © 2011 Casa Software Ltd. www.casaxps.com Table of Contents Variable Forces and Differential Equations ........................................................................................... 2 Differential Equations ........................................................................................................................ 3 Second Order Linear Differential Equations with Constant Coefficients ...................................... 6 Reduction of Differential Equations to Standard Forms by Substitution .................................... 18 Simple Harmonic Motion..................................................................................................................... 21 The Simple Pendulum ...................................................................................................................... 23 Solving Problems using Simple Harmonic Motion ...................................................................... 24 Circular Motion .................................................................................................................................... 28 Mathematical Background .............................................................................................................. 28 Polar Coordinate System ............................................................................................................. 28 Polar Coordinates and Motion .................................................................................................... 31 Examples of Circular
    [Show full text]
  • The Dynamics of the Elastic Pendulum
    MATH MODELING MIDTERM REPORT MATH 485, Instructor: Dr. Ildar Gabitov, Mentor: Joseph Gibney THE DYNAMICS OF THE ELASTIC PENDULUM A group project proposal with preliminary research by Corey Zammit, Nirantha Balagopal, Zijun Li, Shenghao Xia, and Qisong Xiao 1. An Introduction to the System Considered The system of the elastic pendulum consists of a spring, connected to a pivot, suspending a mass. This spring can have many different properties among these are stiffness which can be considered a constant in some practical cases, so the spring has a linear reaction force when extended and compressed. Typically a spring that one would find in real-life applications is either an extension spring or a compression spring (and this will be discussed in greater detail in section 5) however for simplified models and preliminary research we find it practical to consider the case where this spring behaves in a Hookian manor in both extension and compression. This assumption, however, does beg this question among others: Does the spring bend as shown in Figure 1 when compressed ever, and how would this effect the behavior of the system? One way to answer this question is to acknowledge that assumptions such as the ones that follow need to be made to analyze this system, and in any case, behaviors such as these are not easy to analyze and would involve making many more assumptions that could be equally unsatisfying. We will be considering two regimes of this system in our preliminary research with the same assumptions and will pose questions for further research with suggestions for different assumptions.
    [Show full text]
  • Chapter 15 Oscillations and Waves
    Chapter 15 Oscillations and Waves Oscillations and Waves • Simple Harmonic Motion • Energy in SHM • Some Oscillating Systems • Damped Oscillations • Driven Oscillations • Resonance MFMcGraw-PHY 2425 Chap 15Ha-Oscillations-Revised 10/13/2012 2 Simple Harmonic Motion Simple harmonic motion (SHM) occurs when the restoring force (the force directed toward a stable equilibrium point ) is proportional to the displacement from equilibrium. MFMcGraw-PHY 2425 Chap 15Ha-Oscillations-Revised 10/13/2012 3 Characteristics of SHM • Repetitive motion through a central equilibrium point. • Symmetry of maximum displacement. • Period of each cycle is constant. • Force causing the motion is directed toward the equilibrium point (minus sign). • F directly proportional to the displacement from equilibrium. Acceleration = - ω2 x Displacement MFMcGraw-PHY 2425 Chap 15Ha-Oscillations-Revised 10/13/2012 4 A Simple Harmonic Oscillator (SHO) Frictionless surface The restoring force is F = −kx. MFMcGraw-PHY 2425 Chap 15Ha-Oscillations-Revised 10/13/2012 5 Two Springs with Different Amplitudes Frictionless surface MFMcGraw-PHY 2425 Chap 15Ha-Oscillations-Revised 10/13/2012 6 SHO Period is Independent of the Amplitude MFMcGraw-PHY 2425 Chap 15Ha-Oscillations-Revised 10/13/2012 7 The Period and the Angular Frequency 2π T = . The period of oscillation is ω where ω is the angular frequency of k the oscillations, k is the spring ω = m constant and m is the mass of the block. MFMcGraw-PHY 2425 Chap 15Ha-Oscillations-Revised 10/13/2012 8 Simple Harmonic Motion At the equilibrium point x = 0 so, a = 0 also. When the stretch is a maximum, a will be a maximum too.
    [Show full text]