A new Lagrangian of the simple harmonic oscillator 1 revisited
|
|
- Beryl Dawson
- 5 years ago
- Views:
Transcription
1 A new Lagrangian of the siple haronic oscillator 1 revisited Faisal Ain Yassein Abdelohssin Sudan Institute for Natural Sciences, P.O.BOX 3045, Khartou, Sudan Abstract A better and syetric new Lagrangian functional of the siple haronic oscillator has been proposed. The derived equation of otion is eactly the sae as that derived fro the first variation s Lagrangian functional. The equation of otion is derived fro Euler-Lagrange equation by perforing partial derivatives on the Lagrangian functional of the second variation of the calculus of variations. Although the better new Hailtonian functional is off that derived fro the first variation by a factor of two, it syetric than in the previous paper. PACS nubers: b, 0.30.Hq, 0.30.X 3. Keywords: General physics, Haronic oscillator, Ordinary differential Equations, Analytic echanics, Euler-Lagrange equation. Introduction The siple haronic oscillator odel is very iportant in physics (Classical and Quantu). Haronic oscillators occur widely in nature and are eploited in any anade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves. Discussion (1) First variation of the Calculus of Variation It is known that the Euler-Lagrange equation resulting fro applying the first variation of the Calculus of Variations of a Lagrangian functional L( t, q( t), q of a single independent variable q( t ), its first derivative q ( t) of following action I[ q( t)] L( t, q( t), q when varied with respect to the arguents of integrand and the variation are set to zero, i.e. 1 vixra: , A new Lagrangian of the Siple Haronic Oscillator, Faisal Ain Yassein Abdelohssin, Category: Classical Physics. f.a.y.abdelohssin@gail.co 3 (1)
2 is given by 0 I[ q( t)] L( t, q( t), q [ L( t, q( t), q ] d d t q q t q q q d 0 q q Provided that the variation q vanishes at the end points of the integration and the Lagrangian function doesn t depend eplicitly on tie (i.e. 0 ). t Defining the generalized oentu p as p q Then, the Euler-Lagrange equation ay be written as p q Defining the generalized force F as F q Then, the Euler-Lagrange equation has the sae atheatical for as Newton s second law of otion: F p (i) The Lagrangian functional of siple haronic oscillator The Lagrangian functional of siple haronic oscillator in one diension is written as: 1 1 L k The first ter is the potential energy and the second ter is kinetic energy of the siple haronic oscillator. The equation of otion of the siple haronic oscillator is derived fro the Euler-Lagrange equation: d 0 To give k 0 ()
3 This is the sae as the equation of otion of the siple haronic oscillator resulted fro application of Newton's second law to a ass attached to spring of spring constant k and displaced to a position fro equilibriu position. Solving this differential equation, we find that the otion is described by the function ( t) cos( t ), 0 0 k where 0 ( t t0) and 0. T (ii) The first Hailtonian functional of siple haronic oscillator The Hailtonian functional H H( q, p) is derived fro the first Lagrangain with the use of the Legendre transfo; H pq L and defining p as the generalized oentu. Calculating the right hand q side in the equation defining the Hailtonian, we get 1 1 H k p () Second Variations of the Calculus of Variations It is known that the Euler-Lagrange equation resulting fro applying the second variations of the Calculus of Variations of a Lagrangian functional L( t, q( t), q ( t), q of a single independent variable q( t ), its first and second derivatives q ( t), q ( t) of following action I[ q( t)] L( t, q( t), q ( t), q When varied with respect to the arguents of integrand and the variation are set to zero, i.e. 0 I[ q( t)] L( t, q( t), q ( t), q [ L( t, q( t), q ( t), q ] d d d d t q q q q t q q q q q q is given by q q q L d L d L 0 Provided that the variations q and q vanish at the end points of the integration. (3)
4 The Better Model (1) The new Lagrangian functional of the siple haronic oscillator In the previous paper the new Lagrangian was given by 1 L k The better new Lagrangian functional of the siple haronic oscillator in one diension ay now be written as L k The first ter (the potential energy) is ade twice as the one in the previous paper. The equation of otion is derived fro Euler-Lagrange equation by perforing the partial derivatives on the Lagrangian functional L( ( t), ( t), : d d 0 With the ters calculated as follows k ; 0;. The equation of otion is k 0 Or, k 0 Dividing both sides by, we get the standard equation of otion of the SHO. k 0 () The new Hailtonian functional of the siple haronic oscillator The Hailtonian functional H H( q( t), q ( t), p, ) of the siple haronic q oscillator in the second variation can obtained for the Euler-Lagrange equation of the second variation as follows: First, define the generalized oentu in the second variation as d p Then, the Euler-Lagrange equation ay be written as (4)
5 d d 0 d d [ ] L d [ p ] p This yield p This has the sae atheatical for as of the Euler-Lagrange equation of the first variation and the Newton s second law of otion. The corresponding Legendre transforation in the second variations is written as: H pq q L q Substituting the corresponding variables of the siple haronic oscillator d p d 0 ( ) i.e. p in the Legendre transforation above to obtain the Hailtonian of the SHO as: H pq q L q p p( ) ( ) ( k ) p p( ) k p k which is twice the Hailtonian obtained by the ethod of the first variation. Conclusion: The second variation of the ethod of calculus of variation is rich in its applicability than the first variation. Although there was no kinetic energy ter (first derivative) in the new Lagrangian functional of the siple haronic (5)
6 oscillator we obtained the sae equation of otion siilar to those derived fro the first variation and fro the Newton s second law of otion. The Hailtonian function is off by a factor of two of the one derived fro the first variation. The second variation of the calculus of variations is proising in constructing Lagrangian of dynaical syste which were difficult to construct by following the first variation. It is possible to construct the long sought for: The Lagrangian of the daped haronic oscillator using the second variation of the calculus of variations. References [1] Feynan R, Leighton R, and Sands M. The Feynan Lectures on Physics. 3 Volues, ISBN (006) [] Goldstein. H, Poole. C, Safko. J, Addison Wesley. Classical Mechanics, Third edition, July, (000) [3] Serway, Rayond A., Jewett, John W. (003). Physics for Scientists and Engineers. Brooks / Cole. ISBN [4] Tipler, Paul (1998). Physics for Scientists and Engineers: Vol. 1 (4th Edition). W. H. Freean. ISBN [5] Wylie, C. R. (1975). Advanced Engineering Matheatics (4th edition). McGraw-Hill. ISBN [6] Hayek, Sabih I. (15 Apr 003). "Mechanical Vibration and Daping". Encyclopedia of Applied Physics. WILEY-VCH Verlag GbH & Co KGaA. ISBN doi:10.100/ eap31. [7] Hazewinkel, Michiel, ed. (001) [1994], "Oscillator, haronic", Encyclopedia of Matheatics, Springer Science+Business Media B.V. / Kluwer Acadeic Publishers, ISBN [8] Cornelius Lanczos, The Variational Principles of Mechanics, University of Toronto Press, Toronto, 1970 edition. [9] F. A. Y. Abdelohssin, Equation of Motion of a Particle in a Potential Proportional to Square of Second Derivative of Position W.r.t Tie in Its Lagrangian, vixra: subitted on :34:18, Category: Classical Physics. [10] F. A. Y. Abdelohssin, A haronic oscillator in a potential energy Proportional to the square of the second Derivative of the coordinate with Respect to tie, vixra: subitted on :15:04, Category: Classical Physics [11] F. A. Y. Abdelohssin, Euler-Lagrange Equations of the Einstein-Hilbert Action, ViXra: subitted on :49:6, Category: Relativity and Cosology. [1] F. A. Y. Abdelohssin, A new Lagrangian of the Siple Haronic Oscillator, vixra: subitted on :4:55, Category: Classical Physics. (6)
A new Lagrangian of the simple harmonic oscillator
A new Lagrangian of the simple harmonic oscillator Faisal Amin Yassein Abdelmohssin 1 Sudan Institute for Natural Sciences, P.O.BOX 3045, Khartoum, Sudan Abstract A new Lagrangian functional of the simple
More informationOscillations: Review (Chapter 12)
Oscillations: Review (Chapter 1) Oscillations: otions that are periodic in tie (i.e. repetitive) o Swinging object (pendulu) o Vibrating object (spring, guitar string, etc.) o Part of ediu (i.e. string,
More informationLecture #8-3 Oscillations, Simple Harmonic Motion
Lecture #8-3 Oscillations Siple Haronic Motion So far we have considered two basic types of otion: translation and rotation. But these are not the only two types of otion we can observe in every day life.
More informationThe Lagrangian Method vs. other methods (COMPARATIVE EXAMPLE)
The Lagrangian ethod vs. other ethods () This aterial written by Jozef HANC, jozef.hanc@tuke.sk Technical University, Kosice, Slovakia For Edwin Taylor s website http://www.eftaylor.co/ 6 January 003 The
More informationPhysics 221B: Solution to HW # 6. 1) Born-Oppenheimer for Coupled Harmonic Oscillators
Physics B: Solution to HW # 6 ) Born-Oppenheier for Coupled Haronic Oscillators This proble is eant to convince you of the validity of the Born-Oppenheier BO) Approxiation through a toy odel of coupled
More informationSimple Harmonic Motion
Siple Haronic Motion Physics Enhanceent Prograe for Gifted Students The Hong Kong Acadey for Gifted Education and Departent of Physics, HKBU Departent of Physics Siple haronic otion In echanical physics,
More informationHamilton-Jacobi Approach for Power-Law Potentials
Brazilian Journal of Physics, vol. 36, no. 4A, Deceber, 26 1257 Hailton-Jacobi Approach for Power-Law Potentials R. C. Santos 1, J. Santos 1, J. A. S. Lia 2 1 Departaento de Física, UFRN, 5972-97, Natal,
More informationChapter 1: Basics of Vibrations for Simple Mechanical Systems
Chapter 1: Basics of Vibrations for Siple Mechanical Systes Introduction: The fundaentals of Sound and Vibrations are part of the broader field of echanics, with strong connections to classical echanics,
More informationLecture 8 Symmetries, conserved quantities, and the labeling of states Angular Momentum
Lecture 8 Syetries, conserved quantities, and the labeling of states Angular Moentu Today s Progra: 1. Syetries and conserved quantities labeling of states. hrenfest Theore the greatest theore of all ties
More informationForce and dynamics with a spring, analytic approach
Force and dynaics with a spring, analytic approach It ay strie you as strange that the first force we will discuss will be that of a spring. It is not one of the four Universal forces and we don t use
More informationQuantum Ground States as Equilibrium Particle Vacuum Interaction States
Quantu Ground States as Euilibriu article Vacuu Interaction States Harold E uthoff Abstract A rearkable feature of atoic ground states is that they are observed to be radiationless in nature despite (fro
More informationGolden ratio in a coupled-oscillator problem
IOP PUBLISHING Eur. J. Phys. 28 (2007) 897 902 EUROPEAN JOURNAL OF PHYSICS doi:10.1088/0143-0807/28/5/013 Golden ratio in a coupled-oscillator proble Crystal M Mooran and John Eric Goff School of Sciences,
More informationPearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world
Pearson Education Liited Edinburgh Gate Harlow Esse CM0 JE England and Associated Copanies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk Pearson Education Liited 04 All rights
More informationProblem Set 14: Oscillations AP Physics C Supplementary Problems
Proble Set 14: Oscillations AP Physics C Suppleentary Probles 1 An oscillator consists of a bloc of ass 050 g connected to a spring When set into oscillation with aplitude 35 c, it is observed to repeat
More informationPh 20.3 Numerical Solution of Ordinary Differential Equations
Ph 20.3 Nuerical Solution of Ordinary Differential Equations Due: Week 5 -v20170314- This Assignent So far, your assignents have tried to failiarize you with the hardware and software in the Physics Coputing
More informationCHECKLIST. r r. Newton s Second Law. natural frequency ω o (rad.s -1 ) (Eq ) a03/p1/waves/waves doc 9:19 AM 29/03/05 1
PHYS12 Physics 1 FUNDAMENTALS Module 3 OSCILLATIONS & WAVES Text Physics by Hecht Chapter 1 OSCILLATIONS Sections: 1.5 1.6 Exaples: 1.6 1.7 1.8 1.9 CHECKLIST Haronic otion, periodic otion, siple haronic
More informationJOURNAL OF PHYSICAL AND CHEMICAL SCIENCES
JOURNAL OF PHYSIAL AND HEMIAL SIENES Journal hoepage: http://scienceq.org/journals/jps.php Review Open Access A Review of Siple Haronic Motion for Mass Spring Syste and Its Analogy to the Oscillations
More informationA GENERAL FORM FOR THE ELECTRIC FIELD LINES EQUATION CONCERNING AN AXIALLY SYMMETRIC CONTINUOUS CHARGE DISTRIBUTION
A GENEAL FOM FO THE ELECTIC FIELD LINES EQUATION CONCENING AN AXIALLY SYMMETIC CONTINUOUS CHAGE DISTIBUTION BY MUGU B. ăuţ Abstract..By using an unexpected approach it results a general for for the electric
More informationSimple and Compound Harmonic Motion
Siple Copound Haronic Motion Prelab: visit this site: http://en.wiipedia.org/wii/noral_odes Purpose To deterine the noral ode frequencies of two systes:. a single ass - two springs syste (Figure );. two
More informationPhysics 2107 Oscillations using Springs Experiment 2
PY07 Oscillations using Springs Experient Physics 07 Oscillations using Springs Experient Prelab Read the following bacground/setup and ensure you are failiar with the concepts and theory required for
More information1B If the stick is pivoted about point P a distance h = 10 cm from the center of mass, the period of oscillation is equal to (in seconds)
05/07/03 HYSICS 3 Exa #1 Use g 10 /s in your calculations. NAME Feynan lease write your nae also on the back side of this exa 1. 1A A unifor thin stick of ass M 0. Kg and length 60 c is pivoted at one
More information1 Graded problems. PHY 5246: Theoretical Dynamics, Fall November 23 rd, 2015 Assignment # 12, Solutions. Problem 1
PHY 546: Theoretical Dynaics, Fall 05 Noveber 3 rd, 05 Assignent #, Solutions Graded probles Proble.a) Given the -diensional syste we want to show that is a constant of the otion. Indeed,.b) dd dt Now
More informationCourse Information. Physics 1C Waves, optics and modern physics. Grades. Class Schedule. Clickers. Homework
Course Inforation Physics 1C Waves, optics and odern physics Instructor: Melvin Oaura eail: oaura@physics.ucsd.edu Course Syllabus on the web page http://physics.ucsd.edu/ students/courses/fall2009/physics1c
More informationm A 1 m mgd k m v ( C) AP Physics Multiple Choice Practice Oscillations
P Physics Multiple Choice Practice Oscillations. ass, attached to a horizontal assless spring with spring constant, is set into siple haronic otion. Its axiu displaceent fro its equilibriu position is.
More informationPhysics 41 HW Set 1 Chapter 15 Serway 7 th Edition
Physics HW Set Chapter 5 Serway 7 th Edition Conceptual Questions:, 3, 5,, 6, 9 Q53 You can take φ = π, or equally well, φ = π At t= 0, the particle is at its turning point on the negative side of equilibriu,
More informationSIMPLE HARMONIC MOTION: NEWTON S LAW
SIMPLE HARMONIC MOTION: NEWTON S LAW siple not siple PRIOR READING: Main 1.1, 2.1 Taylor 5.1, 5.2 http://www.yoops.org/twocw/it/nr/rdonlyres/physics/8-012fall-2005/7cce46ac-405d-4652-a724-64f831e70388/0/chp_physi_pndul.jpg
More informationIII. Quantization of electromagnetic field
III. Quantization of electroagnetic field Using the fraework presented in the previous chapter, this chapter describes lightwave in ters of quantu echanics. First, how to write a physical quantity operator
More informationTUTORIAL 1 SIMPLE HARMONIC MOTION. Instructor: Kazumi Tolich
TUTORIAL 1 SIMPLE HARMONIC MOTION Instructor: Kazui Tolich About tutorials 2 Tutorials are conceptual exercises that should be worked on in groups. Each slide will consist of a series of questions that
More informationMore Oscillations! (Today: Harmonic Oscillators)
More Oscillations! (oday: Haronic Oscillators) Movie assignent reinder! Final due HURSDAY April 20 Subit through ecapus Different rubric; reeber to chec it even if you got 00% on your draft: http://sarahspolaor.faculty.wvu.edu/hoe/physics-0
More information13 Harmonic oscillator revisited: Dirac s approach and introduction to Second Quantization
3 Haronic oscillator revisited: Dirac s approach and introduction to Second Quantization. Dirac cae up with a ore elegant way to solve the haronic oscillator proble. We will now study this approach. The
More informationFour-vector, Dirac spinor representation and Lorentz Transformations
Available online at www.pelagiaresearchlibrary.co Advances in Applied Science Research, 2012, 3 (2):749-756 Four-vector, Dirac spinor representation and Lorentz Transforations S. B. Khasare 1, J. N. Rateke
More informationIn the session you will be divided into groups and perform four separate experiments:
Mechanics Lab (Civil Engineers) Nae (please print): Tutor (please print): Lab group: Date of lab: Experients In the session you will be divided into groups and perfor four separate experients: (1) air-track
More informationSimple Harmonic Motion of Spring
Nae P Physics Date iple Haronic Motion and prings Hooean pring W x U ( x iple Haronic Motion of pring. What are the two criteria for siple haronic otion? - Only restoring forces cause siple haronic otion.
More informationDefinition of Work, The basics
Physics 07 Lecture 16 Lecture 16 Chapter 11 (Work) v Eploy conservative and non-conservative forces v Relate force to potential energy v Use the concept of power (i.e., energy per tie) Chapter 1 v Define
More informationThe path integral approach in the frame work of causal interpretation
Annales de la Fondation Louis de Broglie, Volue 28 no 1, 2003 1 The path integral approach in the frae work of causal interpretation M. Abolhasani 1,2 and M. Golshani 1,2 1 Institute for Studies in Theoretical
More informationDispersion. February 12, 2014
Dispersion February 1, 014 In aterials, the dielectric constant and pereability are actually frequency dependent. This does not affect our results for single frequency odes, but when we have a superposition
More informationPeriodic Motion is everywhere
Lecture 19 Goals: Chapter 14 Interrelate the physics and atheatics of oscillations. Draw and interpret oscillatory graphs. Learn the concepts of phase and phase constant. Understand and use energy conservation
More informationPHYS 1443 Section 003 Lecture #21 Wednesday, Nov. 19, 2003 Dr. Mystery Lecturer
PHYS 443 Section 003 Lecture # Wednesday, Nov. 9, 003 Dr. Mystery Lecturer. Fluid Dyanics : Flow rate and Continuity Equation. Bernoulli s Equation 3. Siple Haronic Motion 4. Siple Bloc-Spring Syste 5.
More informationNUMERICAL MODELLING OF THE TYRE/ROAD CONTACT
NUMERICAL MODELLING OF THE TYRE/ROAD CONTACT PACS REFERENCE: 43.5.LJ Krister Larsson Departent of Applied Acoustics Chalers University of Technology SE-412 96 Sweden Tel: +46 ()31 772 22 Fax: +46 ()31
More informationNote that an that the liit li! k+? k li P!;! h (k)? ((k? )) li! i i+? i + U( i ) is just a Rieann su representation of the continuous integral h h j +
G5.65: Statistical Mechanics Notes for Lecture 5 I. THE FUNCTIONAL INTEGRAL REPRESENTATION OF THE PATH INTEGRAL A. The continuous liit In taking the liit P!, it will prove useful to ene a paraeter h P
More informationwhich is the moment of inertia mm -- the center of mass is given by: m11 r m2r 2
Chapter 6: The Rigid Rotator * Energy Levels of the Rigid Rotator - this is the odel for icrowave/rotational spectroscopy - a rotating diatoic is odeled as a rigid rotator -- we have two atos with asses
More informationDetermination of Active and Reactive Power in Multi-Phase Systems through Analytical Signals Associated Current and Voltage Signals
56 ACA ELECROEHNICA Deterination of Active and Reactive Power in ulti-phase Systes through Analytical Signals Associated Current and Voltage Signals Gheorghe ODORAN, Oana UNEAN and Anca BUZURA Suary -
More informationdt dt THE AIR TRACK (II)
THE AIR TRACK (II) References: [] The Air Track (I) - First Year Physics Laoratory Manual (PHY38Y and PHYY) [] Berkeley Physics Laoratory, nd edition, McGraw-Hill Book Copany [3] E. Hecht: Physics: Calculus,
More information5.2. Example: Landau levels and quantum Hall effect
68 Phs460.nb i ħ (-i ħ -q A') -q φ' ψ' = + V(r) ψ' (5.49) t i.e., using the new gauge, the Schrodinger equation takes eactl the sae for (i.e. the phsics law reains the sae). 5.. Eaple: Lau levels quantu
More informationTOWARDS THE GEOMETRIC REDUCTION OF CONTROLLED THREE-DIMENSIONAL BIPEDAL ROBOTIC WALKERS 1
TOWARDS THE GEOMETRIC REDUCTION OF CONTROLLED THREE-DIMENSIONAL BIPEDAL ROBOTIC WALKERS 1 Aaron D. Aes, 2 Robert D. Gregg, Eric D.B. Wendel and Shankar Sastry Departent of Electrical Engineering and Coputer
More informationClassical Mechanics Small Oscillations
Classical Mechanics Sall Oscillations Dipan Kuar Ghosh UM-DAE Centre for Excellence in Basic Sciences, Kalina Mubai 400098 Septeber 4, 06 Introduction When a conservative syste is displaced slightly fro
More informationACTIVE VIBRATION CONTROL FOR STRUCTURE HAVING NON- LINEAR BEHAVIOR UNDER EARTHQUAKE EXCITATION
International onference on Earthquae Engineering and Disaster itigation, Jaarta, April 14-15, 8 ATIVE VIBRATION ONTROL FOR TRUTURE HAVING NON- LINEAR BEHAVIOR UNDER EARTHQUAE EXITATION Herlien D. etio
More informationOn the Diffusion Coefficient: The Einstein Relation and Beyond 3
Stoch. Models, Vol. 19, No. 3, 2003, (383-405) Research Report No. 424, 2001, Dept. Theoret. Statist. Aarhus On the Diffusion Coefficient: The Einstein Relation and Beyond 3 GORAN PESKIR 33 We present
More informationSimple Harmonic Motion
Reading: Chapter 15 Siple Haronic Motion Siple Haronic Motion Frequency f Period T T 1. f Siple haronic otion x ( t) x cos( t ). Aplitude x Phase Angular frequency Since the otion returns to its initial
More informationFinal Exam Classical Mechanics
Final Ea Classical Mechanics. Consider the otion in one diension of a article subjected to otential V= (where =constant). Use action-angle variables to find the eriod of the otion as a function of energ.
More informationTHE NEW EXTREMAL CRITERION OF STABILITY
UDC 517.9+531.3 T. G. Stryzhak THE NEW EXTREMAL CRITERION OF STABILITY Introduction It would be rather difficult to list all the publications about the Pendulu. Throughout history, oscillations of pendulu
More informationPH 221-1D Spring Oscillations. Lectures Chapter 15 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition)
PH 1-1D Spring 013 Oscillations Lectures 35-37 Chapter 15 (Halliday/Resnick/Walker, Fundaentals of Physics 9 th edition) 1 Chapter 15 Oscillations In this chapter we will cover the following topics: Displaceent,
More informationXI PHYSICS M. AFFAN KHAN LECTURER PHYSICS, AKHSS, K. https://promotephysics.wordpress.com
XI PHYSICS M. AFFAN KHAN LECTURER PHYSICS, AKHSS, K affan_414@live.co https://prootephysics.wordpress.co [MOTION] CHAPTER NO. 3 In this chapter we are going to discuss otion in one diension in which we
More informationPage 1. Physics 131: Lecture 22. Today s Agenda. SHM and Circles. Position
Physics 3: ecture Today s genda Siple haronic otion Deinition Period and requency Position, velocity, and acceleration Period o a ass on a spring Vertical spring Energy and siple haronic otion Energy o
More information(a) Why cannot the Carnot cycle be applied in the real world? Because it would have to run infinitely slowly, which is not useful.
PHSX 446 FINAL EXAM Spring 25 First, soe basic knowledge questions You need not show work here; just give the answer More than one answer ight apply Don t waste tie transcribing answers; just write on
More informationOn the summations involving Wigner rotation matrix elements
Journal of Matheatical Cheistry 24 (1998 123 132 123 On the suations involving Wigner rotation atrix eleents Shan-Tao Lai a, Pancracio Palting b, Ying-Nan Chiu b and Harris J. Silverstone c a Vitreous
More informationMechanics Physics 151
Mechanics Physics 5 Lecture Oscillations (Chapter 6) What We Did Last Tie Analyzed the otion of a heavy top Reduced into -diensional proble of θ Qualitative behavior Precession + nutation Initial condition
More informationQuestion 1. [14 Marks]
6 Question 1. [14 Marks] R r T! A string is attached to the dru (radius r) of a spool (radius R) as shown in side and end views here. (A spool is device for storing string, thread etc.) A tension T is
More informationSHM stuff the story continues
SHM stuff the story continues Siple haronic Motion && + ω solution A cos t ( ω + α ) Siple haronic Motion + viscous daping b & + ω & + Viscous daping force A e b t Viscous daped aplitude Viscous daped
More informationSome consequences of a Universal Tension arising from Dark Energy for structures from Atomic Nuclei to Galaxy Clusters
unning Head: Universal Tension fro DE Article Type: Original esearch Soe consequences of a Universal Tension arising fro Dark Energy for structures fro Atoic Nuclei to Galaxy Clusters C Sivara Indian Institute
More informationWork, Energy and Momentum
Work, Energy and Moentu Work: When a body oves a distance d along straight line, while acted on by a constant force of agnitude F in the sae direction as the otion, the work done by the force is tered
More informationOSCILLATIONS AND WAVES
OSCILLATIONS AND WAVES OSCILLATION IS AN EXAMPLE OF PERIODIC MOTION No stories this tie, we are going to get straight to the topic. We say that an event is Periodic in nature when it repeats itself in
More informationOn Lotka-Volterra Evolution Law
Advanced Studies in Biology, Vol. 3, 0, no. 4, 6 67 On Lota-Volterra Evolution Law Farruh Muhaedov Faculty of Science, International Islaic University Malaysia P.O. Box, 4, 570, Kuantan, Pahang, Malaysia
More information8.1 Force Laws Hooke s Law
8.1 Force Laws There are forces that don't change appreciably fro one instant to another, which we refer to as constant in tie, and forces that don't change appreciably fro one point to another, which
More informationUnit 14 Harmonic Motion. Your Comments
Today s Concepts: Periodic Motion Siple - Mass on spring Daped Forced Resonance Siple - Pendulu Unit 1, Slide 1 Your Coents Please go through the three equations for siple haronic otion and phase angle
More informationRationality Problems of the Principles of Equivalence and General Relativity
Rationality Probles of the Principles of Equivalence and General Relativity Mei Xiaochun (Departent of Physics, Fuzhou University, E-ail: xc1@163.co Tel:86-591-8761414) (N.7-B, South Building, Zhongfu
More information= T. Oscillations and Waves. Example of an Oscillating System IB 12 IB 12
Oscillation: the vibration of an object Oscillations and Waves Eaple of an Oscillating Syste A ass oscillates on a horizontal spring without friction as shown below. At each position, analyze its displaceent,
More informationClassical systems in equilibrium
35 Classical systes in equilibriu Ideal gas Distinguishable particles Here we assue that every particle can be labeled by an index i... and distinguished fro any other particle by its label if not by any
More information27 Oscillations: Introduction, Mass on a Spring
Chapter 7 Oscillations: Introduction, Mass on a Spring 7 Oscillations: Introduction, Mass on a Spring If a siple haronic oscillation proble does not involve the tie, you should probably be using conservation
More informationChapter 11: Vibration Isolation of the Source [Part I]
Chapter : Vibration Isolation of the Source [Part I] Eaple 3.4 Consider the achine arrangeent illustrated in figure 3.. An electric otor is elastically ounted, by way of identical isolators, to a - thick
More informationQ5 We know that a mass at the end of a spring when displaced will perform simple m harmonic oscillations with a period given by T = 2!
Chapter 4.1 Q1 n oscillation is any otion in which the displaceent of a particle fro a fixed point keeps changing direction and there is a periodicity in the otion i.e. the otion repeats in soe way. In
More informationP235 Midterm Examination Prof. Cline
P235 Mier Exaination Prof. Cline THIS IS A CLOSED BOOK EXAMINATION. Do all parts of all four questions. Show all steps to get full credit. 7:00-10.00p, 30 October 2009 1:(20pts) Consider a rocket fired
More informationPhysics 2210 Fall smartphysics 20 Conservation of Angular Momentum 21 Simple Harmonic Motion 11/23/2015
Physics 2210 Fall 2015 sartphysics 20 Conservation of Angular Moentu 21 Siple Haronic Motion 11/23/2015 Exa 4: sartphysics units 14-20 Midter Exa 2: Day: Fri Dec. 04, 2015 Tie: regular class tie Section
More informationNumerical Investigations of Hydraulic Transients in Pipelines having Centrifugal Pumps
International Journal of Applied Engineering Research ISSN 0973-4562 Volue 13, Nuber 8 (2018) pp. 5999-6003 Research India Publications. http://www.ripublication.co Nuerical Investigations of Hydraulic
More informationROTATIONAL MOTION FROM TRANSLATIONAL MOTION
ROTATIONAL MOTION FROM TRANSLATIONAL MOTION Velocity Acceleration 1-D otion 3-D otion Linear oentu TO We have shown that, the translational otion of a acroscopic object is equivalent to the translational
More informationVibrations: Two Degrees of Freedom Systems - Wilberforce Pendulum and Bode Plots
.003J/.053J Dynaics and Control, Spring 007 Professor Peacock 5/4/007 Lecture 3 Vibrations: Two Degrees of Freedo Systes - Wilberforce Pendulu and Bode Plots Wilberforce Pendulu Figure : Wilberforce Pendulu.
More information2009 Academic Challenge
009 Acadeic Challenge PHYSICS TEST - REGIONAL This Test Consists of 5 Questions Physics Test Production Tea Len Stor, Eastern Illinois University Author/Tea Leader Doug Brandt, Eastern Illinois University
More information3. Period Law: Simplified proof for circular orbits Equate gravitational and centripetal forces
Physics 106 Lecture 10 Kepler s Laws and Planetary Motion-continued SJ 7 th ed.: Chap 1., 1.6 Kepler s laws of planetary otion Orbit Law Area Law Period Law Satellite and planetary orbits Orbits, potential,
More informationTutorial Exercises: Incorporating constraints
Tutorial Exercises: Incorporating constraints 1. A siple pendulu of length l ass is suspended fro a pivot of ass M that is free to slide on a frictionless wire frae in the shape of a parabola y = ax. The
More informationi ij j ( ) sin cos x y z x x x interchangeably.)
Tensor Operators Michael Fowler,2/3/12 Introduction: Cartesian Vectors and Tensors Physics is full of vectors: x, L, S and so on Classically, a (three-diensional) vector is defined by its properties under
More informationTHE ROCKET EXPERIMENT 1. «Homogenous» gravitational field
THE OCKET EXPEIENT. «Hoogenous» gravitational field Let s assue, fig., that we have a body of ass Μ and radius. fig. As it is known, the gravitational field of ass Μ (both in ters of geoetry and dynaics)
More informationOn the approximation of Feynman-Kac path integrals
On the approxiation of Feynan-Kac path integrals Stephen D. Bond, Brian B. Laird, and Benedict J. Leikuhler University of California, San Diego, Departents of Matheatics and Cheistry, La Jolla, CA 993,
More informationPHYSICS 110A : CLASSICAL MECHANICS MIDTERM EXAM #2
PHYSICS 110A : CLASSICAL MECHANICS MIDTERM EXAM #2 [1] Two blocks connected by a spring of spring constant k are free to slide frictionlessly along a horizontal surface, as shown in Fig. 1. The unstretched
More informationUSEFUL HINTS FOR SOLVING PHYSICS OLYMPIAD PROBLEMS. By: Ian Blokland, Augustana Campus, University of Alberta
1 USEFUL HINTS FOR SOLVING PHYSICS OLYMPIAD PROBLEMS By: Ian Bloland, Augustana Capus, University of Alberta For: Physics Olypiad Weeend, April 6, 008, UofA Introduction: Physicists often attept to solve
More informationChapter 2: Introduction to Damping in Free and Forced Vibrations
Chapter 2: Introduction to Daping in Free and Forced Vibrations This chapter ainly deals with the effect of daping in two conditions like free and forced excitation of echanical systes. Daping plays an
More information9 HOOKE S LAW AND SIMPLE HARMONIC MOTION
Experient 9 HOOKE S LAW AND SIMPLE HARMONIC MOTION Objectives 1. Verify Hoo s law,. Measure the force constant of a spring, and 3. Measure the period of oscillation of a spring-ass syste and copare it
More informationPart IA Paper 1: Mechanical Engineering MECHANICAL VIBRATIONS Examples paper 3
ENGINEERING Part IA Paper 1: Mechanical Engineering MECHANICAL VIBRATIONS Exaples paper 3 IRST YEAR Straightforward questions are ared with a Tripos standard questions are ared *. Systes with two or ore
More informationSupporting Information
Supporting Inforation Nash et al. 10.1073/pnas.1507413112 Equation of Motion If a gyroscope is spinning with a large constant angular frequency, ω, around its principal axis, ^l, then its dynaics are deterined
More informationSOLUTIONS for Homework #3
SOLUTIONS for Hoework #3 1. In the potential of given for there is no unboun states. Boun states have positive energies E n labele by an integer n. For each energy level E, two syetrically locate classical
More informationSimple Schemes of Multi anchored Flexible Walls Dynamic Behavior
6 th International Conference on Earthquake Geotechnical Engineering -4 Noveber 05 Christchurch, New Zealand Siple Schees of Multi anchored Flexible Walls Dynaic Behavior A. D. Garini ABSTRACT Siple schees
More informationOn the characterization of non-linear diffusion equations. An application in soil mechanics
On the characterization of non-linear diffusion equations. An application in soil echanics GARCÍA-ROS, G., ALHAMA, I., CÁNOVAS, M *. Civil Engineering Departent Universidad Politécnica de Cartagena Paseo
More informationOn the Bell- Kochen -Specker paradox
On the Bell- Kochen -Specker paradox Koji Nagata and Tadao Nakaura Departent of Physics, Korea Advanced Institute of Science and Technology, Daejeon, Korea E-ail: ko_i_na@yahoo.co.jp Departent of Inforation
More informationPH 221-2A Fall Waves - I. Lectures Chapter 16 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition)
PH 1-A Fall 014 Waves - I Lectures 4-5 Chapter 16 (Halliday/Resnick/Walker, Fundaentals of Physics 9 th edition) 1 Chapter 16 Waves I In this chapter we will start the discussion on wave phenoena. We will
More informationObject Oriented Programming for Partial Differential Equations
Procedia Coputer Science Volue 51, 2015, Pages 1013 1022 ICCS 2015 International Conference On Coputational Science Object Oriented Prograing for Partial Differential Equations E. Alberdi Celaya 1 and
More informationDRAFT. Memo. Contents. To whom it may concern SVN: Jan Mooiman +31 (0) nl
Meo To To who it ay concern Date Reference Nuber of pages 219-1-16 SVN: 5744 22 Fro Direct line E-ail Jan Mooian +31 )88 335 8568 jan.ooian@deltares nl +31 6 4691 4571 Subject PID controller ass-spring-daper
More informationAnalysis of Impulsive Natural Phenomena through Finite Difference Methods A MATLAB Computational Project-Based Learning
Analysis of Ipulsive Natural Phenoena through Finite Difference Methods A MATLAB Coputational Project-Based Learning Nicholas Kuia, Christopher Chariah, Mechatronics Engineering, Vaughn College of Aeronautics
More informationBasic concept of dynamics 3 (Dynamics of a rigid body)
Vehicle Dynaics (Lecture 3-3) Basic concept of dynaics 3 (Dynaics of a rigid body) Oct. 1, 2015 김성수 Vehicle Dynaics Model q How to describe vehicle otion? Need Reference fraes and Coordinate systes 2 Equations
More informationACCUMULATION OF FLUID FLOW ENERGY BY VIBRATIONS EXCITATION IN SYSTEM WITH TWO DEGREE OF FREEDOM
ENGINEERING FOR RURAL DEVELOPMENT Jelgava, 9.-.5.8. ACCUMULATION OF FLUID FLOW ENERGY BY VIBRATION EXCITATION IN YTEM WITH TWO DEGREE OF FREEDOM Maris Eiduks, Janis Viba, Lauris tals Riga Technical University,
More informationIn this chapter we will start the discussion on wave phenomena. We will study the following topics:
Chapter 16 Waves I In this chapter we will start the discussion on wave phenoena. We will study the following topics: Types of waves Aplitude, phase, frequency, period, propagation speed of a wave Mechanical
More informationKinematics and dynamics, a computational approach
Kineatics and dynaics, a coputational approach We begin the discussion of nuerical approaches to echanics with the definition for the velocity r r ( t t) r ( t) v( t) li li or r( t t) r( t) v( t) t for
More information