Exact Solution for the Nonlinear Pendulum (Solu¸C˜Aoexata Do Pˆendulon˜Aolinear)
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Chapter 7. the Principle of Least Action
Chapter 7. The Principle of Least Action 7.1 Force Methods vs. Energy Methods We have so far studied two distinct ways of analyzing physics problems: force methods, basically consisting of the application of Newton’s Laws, and energy methods, consisting of the application of the principle of conservation of energy (the conservations of linear and angular momenta can also be considered as part of this). Both have their advantages and disadvantages. More precisely, energy methods often involve scalar quantities (e.g., work, kinetic energy, and potential energy) and are thus easier to handle than forces, which are vectorial in nature. However, forces tell us more. The simple example of a particle subjected to the earth’s gravitation field will clearly illustrate this. We know that when a particle moves from position y0 to y in the earth’s gravitational field, the conservation of energy tells us that ΔK + ΔUgrav = 0 (7.1) 1 2 2 m v − v0 + mg(y − y0 ) = 0, 2 ( ) which implies that the final speed of the particle, of mass m , is 2 v = v0 + 2g(y0 − y). (7.2) Although we know the final speed v of the particle, given its initial speed v0 and the initial and final positions y0 and y , we do not know how its position and velocity (and its speed) evolve with time. On the other hand, if we apply Newton’s Second Law, then we write −mgey = maey dv (7.3) = m e . dt y This equation is easily manipulated to yield t v = dv ∫t0 t = − gdτ + v0 (7.4) ∫t0 = −g(t − t0 ) + v0 , - 138 - where v0 is the initial velocity (it appears in equation (7.4) as a constant of integration). -
Dynamics of the Elastic Pendulum Qisong Xiao; Shenghao Xia ; Corey Zammit; Nirantha Balagopal; Zijun Li Agenda
Dynamics of the Elastic Pendulum Qisong Xiao; Shenghao Xia ; Corey Zammit; Nirantha Balagopal; Zijun Li Agenda • Introduction to the elastic pendulum problem • Derivations of the equations of motion • Real-life examples of an elastic pendulum • Trivial cases & equilibrium states • MATLAB models The Elastic Problem (Simple Harmonic Motion) 푑2푥 푑2푥 푘 • 퐹 = 푚 = −푘푥 = − 푥 푛푒푡 푑푡2 푑푡2 푚 • Solve this differential equation to find 푥 푡 = 푐1 cos 휔푡 + 푐2 sin 휔푡 = 퐴푐표푠(휔푡 − 휑) • With velocity and acceleration 푣 푡 = −퐴휔 sin 휔푡 + 휑 푎 푡 = −퐴휔2cos(휔푡 + 휑) • Total energy of the system 퐸 = 퐾 푡 + 푈 푡 1 1 1 = 푚푣푡2 + 푘푥2 = 푘퐴2 2 2 2 The Pendulum Problem (with some assumptions) • With position vector of point mass 푥 = 푙 푠푖푛휃푖 − 푐표푠휃푗 , define 푟 such that 푥 = 푙푟 and 휃 = 푐표푠휃푖 + 푠푖푛휃푗 • Find the first and second derivatives of the position vector: 푑푥 푑휃 = 푙 휃 푑푡 푑푡 2 푑2푥 푑2휃 푑휃 = 푙 휃 − 푙 푟 푑푡2 푑푡2 푑푡 • From Newton’s Law, (neglecting frictional force) 푑2푥 푚 = 퐹 + 퐹 푑푡2 푔 푡 The Pendulum Problem (with some assumptions) Defining force of gravity as 퐹푔 = −푚푔푗 = 푚푔푐표푠휃푟 − 푚푔푠푖푛휃휃 and tension of the string as 퐹푡 = −푇푟 : 2 푑휃 −푚푙 = 푚푔푐표푠휃 − 푇 푑푡 푑2휃 푚푙 = −푚푔푠푖푛휃 푑푡2 Define 휔0 = 푔/푙 to find the solution: 푑2휃 푔 = − 푠푖푛휃 = −휔2푠푖푛휃 푑푡2 푙 0 Derivation of Equations of Motion • m = pendulum mass • mspring = spring mass • l = unstreatched spring length • k = spring constant • g = acceleration due to gravity • Ft = pre-tension of spring 푚푔−퐹 • r = static spring stretch, 푟 = 푡 s 푠 푘 • rd = dynamic spring stretch • r = total spring stretch 푟푠 + 푟푑 Derivation of Equations of Motion -
Forced Mechanical Oscillations
169 Carl von Ossietzky Universität Oldenburg – Faculty V - Institute of Physics Module Introductory laboratory course physics – Part I Forced mechanical oscillations Keywords: HOOKE's law, harmonic oscillation, harmonic oscillator, eigenfrequency, damped harmonic oscillator, resonance, amplitude resonance, energy resonance, resonance curves References: /1/ DEMTRÖDER, W.: „Experimentalphysik 1 – Mechanik und Wärme“, Springer-Verlag, Berlin among others. /2/ TIPLER, P.A.: „Physik“, Spektrum Akademischer Verlag, Heidelberg among others. /3/ ALONSO, M., FINN, E. J.: „Fundamental University Physics, Vol. 1: Mechanics“, Addison-Wesley Publishing Company, Reading (Mass.) among others. 1 Introduction It is the object of this experiment to study the properties of a „harmonic oscillator“ in a simple mechanical model. Such harmonic oscillators will be encountered in different fields of physics again and again, for example in electrodynamics (see experiment on electromagnetic resonant circuit) and atomic physics. Therefore it is very important to understand this experiment, especially the importance of the amplitude resonance and phase curves. 2 Theory 2.1 Undamped harmonic oscillator Let us observe a set-up according to Fig. 1, where a sphere of mass mK is vertically suspended (x-direc- tion) on a spring. Let us neglect the effects of friction for the moment. When the sphere is at rest, there is an equilibrium between the force of gravity, which points downwards, and the dragging resilience which points upwards; the centre of the sphere is then in the position x = 0. A deflection of the sphere from its equilibrium position by x causes a proportional dragging force FR opposite to x: (1) FxR ∝− The proportionality constant (elastic or spring constant or directional quantity) is denoted D, and Eq. -
Measuring Earth's Gravitational Constant with a Pendulum
Measuring Earth’s Gravitational Constant with a Pendulum Philippe Lewalle, Tony Dimino PHY 141 Lab TA, Fall 2014, Prof. Frank Wolfs University of Rochester November 30, 2014 Abstract In this lab we aim to calculate Earth’s gravitational constant by measuring the period of a pendulum. We obtain a value of 9.79 ± 0.02m/s2. This measurement agrees with the accepted value of g = 9.81m/s2 to within the precision limits of our procedure. Limitations of the techniques and assumptions used to calculate these values are discussed. The pedagogical context of this example report for PHY 141 is also discussed in the final remarks. 1 Theory The gravitational acceleration g near the surface of the Earth is known to be approximately constant, disregarding small effects due to geological variations and altitude shifts. We aim to measure the value of that acceleration in our lab, by observing the motion of a pendulum, whose motion depends both on g and the length L of the pendulum. It is a well known result that a pendulum consisting of a point mass and attached to a massless rod of length L obeys the relationship shown in eq. (1), d2θ g = − sin θ, (1) dt2 L where θ is the angle from the vertical, as showng in Fig. 1. For small displacements (ie small θ), we make a small angle approximation such that sin θ ≈ θ, which yields eq. (2). d2θ g = − θ (2) dt2 L Eq. (2) admits sinusoidal solutions with an angular frequency ω. We relate this to the period T of oscillation, to obtain an expression for g in terms of the period and length of the pendulum, shown in eq. -
The Damped Harmonic Oscillator
THE DAMPED HARMONIC OSCILLATOR Reading: Main 3.1, 3.2, 3.3 Taylor 5.4 Giancoli 14.7, 14.8 Free, undamped oscillators – other examples k m L No friction I C k m q 1 x m!x! = !kx q!! = ! q LC ! ! r; r L = θ Common notation for all g !! 2 T ! " # ! !!! + " ! = 0 m L 0 mg k friction m 1 LI! + q + RI = 0 x C 1 Lq!!+ q + Rq! = 0 C m!x! = !kx ! bx! ! r L = cm θ Common notation for all g !! ! 2 T ! " # ! # b'! !!! + 2"!! +# ! = 0 m L 0 mg Natural motion of damped harmonic oscillator Force = mx˙˙ restoring force + resistive force = mx˙˙ ! !kx ! k Need a model for this. m Try restoring force proportional to velocity k m x !bx! How do we choose a model? Physically reasonable, mathematically tractable … Validation comes IF it describes the experimental system accurately Natural motion of damped harmonic oscillator Force = mx˙˙ restoring force + resistive force = mx˙˙ !kx ! bx! = m!x! ! Divide by coefficient of d2x/dt2 ! and rearrange: x 2 x 2 x 0 !!+ ! ! + " 0 = inverse time β and ω0 (rate or frequency) are generic to any oscillating system This is the notation of TM; Main uses γ = 2β. Natural motion of damped harmonic oscillator 2 x˙˙ + 2"x˙ +#0 x = 0 Try x(t) = Ce pt C, p are unknown constants ! x˙ (t) = px(t), x˙˙ (t) = p2 x(t) p2 2 p 2 x(t) 0 Substitute: ( + ! + " 0 ) = ! 2 2 Now p is known (and p = !" ± " ! # 0 there are 2 p values) p t p t x(t) = Ce + + C'e " Must be sure to make x real! ! Natural motion of damped HO Can identify 3 cases " < #0 underdamped ! " > #0 overdamped ! " = #0 critically damped time ---> ! underdamped " < #0 # 2 !1 = ! 0 1" 2 ! 0 ! time ---> 2 2 p = !" ± " ! # 0 = !" ± i#1 x(t) = Ce"#t+i$1t +C*e"#t"i$1t Keep x(t) real "#t x(t) = Ae [cos($1t +%)] complex <-> amp/phase System oscillates at "frequency" ω1 (very close to ω0) ! - but in fact there is not only one single frequency associated with the motion as we will see. -
VIBRATIONAL SPECTROSCOPY • the Vibrational Energy V(R) Can Be Calculated Using the (Classical) Model of the Harmonic Oscillator
VIBRATIONAL SPECTROSCOPY • The vibrational energy V(r) can be calculated using the (classical) model of the harmonic oscillator: • Using this potential energy function in the Schrödinger equation, the vibrational frequency can be calculated: The vibrational frequency is increasing with: • increasing force constant f = increasing bond strength • decreasing atomic mass • Example: f cc > f c=c > f c-c Vibrational spectra (I): Harmonic oscillator model • Infrared radiation in the range from 10,000 – 100 cm –1 is absorbed and converted by an organic molecule into energy of molecular vibration –> this absorption is quantized: A simple harmonic oscillator is a mechanical system consisting of a point mass connected to a massless spring. The mass is under action of a restoring force proportional to the displacement of particle from its equilibrium position and the force constant f (also k in followings) of the spring. MOLECULES I: Vibrational We model the vibrational motion as a harmonic oscillator, two masses attached by a spring. nu and vee! Solving the Schrödinger equation for the 1 v h(v 2 ) harmonic oscillator you find the following quantized energy levels: v 0,1,2,... The energy levels The level are non-degenerate, that is gv=1 for all values of v. The energy levels are equally spaced by hn. The energy of the lowest state is NOT zero. This is called the zero-point energy. 1 R h Re 0 2 Vibrational spectra (III): Rotation-vibration transitions The vibrational spectra appear as bands rather than lines. When vibrational spectra of gaseous diatomic molecules are observed under high-resolution conditions, each band can be found to contain a large number of closely spaced components— band spectra. -
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. -
Harmonic Oscillator with Time-Dependent Effective-Mass And
Harmonic oscillator with time-dependent effective-mass and frequency with a possible application to 'chirped tidal' gravitational waves forces affecting interferometric detectors Yacob Ben-Aryeh Physics Department, Technion-Israel Institute of Technology, Haifa,32000,Israel e-mail: [email protected] ; Fax: 972-4-8295755 Abstract The general theory of time-dependent frequency and time-dependent mass ('effective mass') is described. The general theory for time-dependent harmonic-oscillator is applied in the present research for studying certain quantum effects in the interferometers for detecting gravitational waves. When an astronomical binary system approaches its point of coalescence the gravitational wave intensity and frequency are increasing and this can lead to strong deviations from the simple description of harmonic oscillations for the interferometric masses on which the mirrors are placed. It is shown that under such conditions the harmonic oscillations of these masses can be described by mechanical harmonic-oscillators with time- dependent frequency and effective-mass. In the present theoretical model the effective- mass is decreasing with time describing pumping phenomena in which the oscillator amplitude is increasing with time. The quantization of this system is analyzed by the use of the adiabatic approximation. It is found that the increase of the gravitational wave intensity, within the adiabatic approximation, leads to squeezing phenomena where the quantum noise in one quadrature is increased and in the other quadrature it is decreased. PACS numbers: 04.80.Nn, 03.65.Bz, 42.50.Dv. Keywords: Gravitational waves, harmonic-oscillator with time-dependent effective- mass 1 1.Introduction The problem of harmonic-oscillator with time-dependent mass has been related to a quantum damped oscillator [1-7]. -
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. -
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. -
22.51 Course Notes, Chapter 9: Harmonic Oscillator
9. Harmonic Oscillator 9.1 Harmonic Oscillator 9.1.1 Classical harmonic oscillator and h.o. model 9.1.2 Oscillator Hamiltonian: Position and momentum operators 9.1.3 Position representation 9.1.4 Heisenberg picture 9.1.5 Schr¨odinger picture 9.2 Uncertainty relationships 9.3 Coherent States 9.3.1 Expansion in terms of number states 9.3.2 Non-Orthogonality 9.3.3 Uncertainty relationships 9.3.4 X-representation 9.4 Phonons 9.4.1 Harmonic oscillator model for a crystal 9.4.2 Phonons as normal modes of the lattice vibration 9.4.3 Thermal energy density and Specific Heat 9.1 Harmonic Oscillator We have considered up to this moment only systems with a finite number of energy levels; we are now going to consider a system with an infinite number of energy levels: the quantum harmonic oscillator (h.o.). The quantum h.o. is a model that describes systems with a characteristic energy spectrum, given by a ladder of evenly spaced energy levels. The energy difference between two consecutive levels is ∆E. The number of levels is infinite, but there must exist a minimum energy, since the energy must always be positive. Given this spectrum, we expect the Hamiltonian will have the form 1 n = n + ~ω n , H | i 2 | i where each level in the ladder is identified by a number n. The name of the model is due to the analogy with characteristics of classical h.o., which we will review first. 9.1.1 Classical harmonic oscillator and h.o. -
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.