Physics 106 Lecture 12. Oscillations II. Recap: SHM using phasors (uniform circular motion) music structural and mechanical engineering waves
|
|
- Bruno Crawford
- 7 years ago
- Views:
Transcription
1 Physics 6 Lctur Oscillations II SJ 7 th Ed.: Chap 5.4, Rad only 5.6 & 5.7 Rcap: SHM using phasors (unifor circular otion) Physical pndulu xapl apd haronic oscillations Forcd oscillations and rsonanc. Rsonanc xapls and discussion usic structural and chanical nginring wavs Sapl probls Oscillations suary chart apd Oscillations nglct gravity Non-consrvativ forcs ay b prsnt Friction is a coon nonconsrvativ forc No longr an idal syst (such as thos dalt with so far) Th chanical nrgy of th syst diinishs in ti, otion is said to b dapd Th otion of th syst can b dcaying oscillations if th daping is wa. If daping is strong, otion ay di away without oscillating. Still no driving forc, onc syst has bn startd
2 Add aping: E ch not constant, oscillations not sipl nglct gravity Spring oscillator as bfor, but with dissipativ forc F dap such as th syst in th figur, with van oving in fluid. F dap viscous drag forc, proportional to vlocity F dap = bv Prvious forc quation gts a nw, daping forc tr F nt d d = = () b d d nw tr b d + = Solution for apd oscillator quation d nw tr b d + = Solution: odifid oscillations = x xponntially dcaying nvlop cos( ω' altrd frquncy ω can b ral or iaginary ω' b 4 ω = : natural frquncy ω' ω ( b/ ) Rcovr undapd solution for b
3 apd physical systs can b of thr typs Solution: dapd oscillations = x cos( ω' ω' b 4 Undrdapd: d d sall b< b <, for which ω is positiv. 4 Critically dapd: b= b ω for which ω' 4 Ovrdapd: b > ω for which ω' 4 x x Math Rviw: cos( ix) = cosh( x) = ( + ) / x x sin( ix) = sinh( x) = ( ) / is iaginary cos( ix + y) = cos( ix)cos( y) sin( ix)sin( y) Typs of aping, cont (Lin to Activ Fig.) a) an undrdapd oscillator b) a critically dapd oscillator c) an ovrdapd oscillator For critically dapd and ovrdapd oscillators thr is no priodic otion and th angular frquncy ω has a diffrnt aning 3
4 Waly dapd oscillator : b << ω 4 b ' 4 ω ω xt () = x cos( ω t+ ϕ) X = x - () (t) x slow dcay of aplitud nvlop cos( ωt + φ) sall fractional chang in aplitud during on coplt cycl Waly dapd oscillator : b ' 4 x (t) x - b << ω 4 ω ω xt () = x cos( ω t+ ϕ) Aplitud : X = A slow dcay of aplitud nvlop sall fractional chang in aplitud during on coplt cycl cos( ωt + φ) Vlocity with wa daping: find drivativ d v(t) = v axiu vlocity sin( ω't + φ) v = ωx xponntially dcaying nvlop altrd frquncy ~ ω 4
5 Mchanical nrgy dcays xponntially in an waly dapd oscillator (sall b) Ech = K(t) + U(t) = v (t) + x (t) () = cos( ω + ϕ) xt x t Vlocity with wa daping: find drivativ d v(t) = v axiu vlocity sin( ω't + φ) v = ωx xponntially dcaying nvlop altrd frquncy ~ ω b x cos( ωt+ ϕ) tr is ngligibl, bcaus b is sall.. Mchanical nrgy dcays xponntially in an waly dapd oscillator (sall b) Ech = K(t) + U(t) = v (t) + Substitut prvious solutions: E = x ch = cos( ω' ω x / As always: cos (x) + sin (x) = Also: ω E ch + (t) = x v(t) x ω x sin ( ω' x / (t) cos ( ω' / sin( ω' Initial chanical nrgy xponntial dcay at twic th rat of aplitud dcay 5
6 apd physical systs can b of thr typs Solution: dapd oscillations = x xponntially dcaying nvlop cos( ω' altrd frquncy ω can b ral or iaginary ω' b 4 b << ω for which ω' ω 4 Th rstoring forc is larg copard to th daping forc. Th syst oscillats with dcaying aplitud Undrdapd: Critically dapd: b ω for which ω' 4 Th rstoring forc and daping forc ar coparabl in ffct. Th syst can not oscillat; th aplitud dis away xponntially Ovrdapd: b 4 > ω for which ω' is iaginary Th daping forc is uch strongr than th rstoring forc. Th aplitud dis away as a odifid xponntial Not: Cos( ix ) = Cosh( x ) Forcd (rivn) Oscillations and Rsonanc An xtrnal driving forc starts oscillations in a stationary syst Th aplitud rains constant (or grows) if th nrgy input pr cycl xactly quals (or xcds) th nrgy loss fro daping Evntually, E driving = E lost and a stady-stat condition is rachd Oscillations thn continu with constant aplitud Oscillations ar at th driving frquncy ω F (t) = F cos( ω t + φ') Oscillating driving forc applid to a dapd d oscillator F (t) 6
7 Equation for Forcd (rivn) Oscillations ω = natural frquncy ω = ω = driving frquncy of xtrnal forc Extrnal driving forc function: F (t) = F cos( ω t + φ') dx t Fnt = F ( t) -b - = () d x() t F (t) Solution for Forcd (rivn) Oscillations dx t Fnt = F ( t) -b - = F (t) = F cos( ω () d x() t t + φ') Solution (stady stat solution): = Acos( ωt + φ) whr A = ( ω F ω ) / bω + ( ) F (t) Th syst always oscillats at th driving frquncy ω in stady-stat Th aplitud A dpnds on how clos ω is to natural frquncy ω rsonanc ω = 7
8 Aplitud of th drivn oscillations: Th largst aplitud oscillations occur at or nar RESONANCE (ω ~ ω ) As daping bcos war rsonanc sharpns & aplitud at rsonanc incrass. A = ( ω F ω ) / bω + ( rsonanc ) Rsonanc At rsonanc, th applid forc is in phas with th vlocity and th powr Fov transfrrd to th oscillator is a axiu. Th aplitud of rsonant oscillations can bco norous whn th daping is wa, storing norous aounts of nrgy Applications: buildings drivn by arthquas bridgs undr wind load all inds of radio dvics, icrowav othr nurous applications 8
9 Forcd rsonant torsional oscillations du to wind - Tacoa Narrows Bridg Roadway collaps - Tacoa Narrows Bridg 9
10 Twisting bridg at rsonanc frquncy Braing glass with voic
New Basis Functions. Section 8. Complex Fourier Series
Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ral-valud Fourir sris is xplaind and formula ar givn for convrting
More informationPhysics 211: Lab Oscillations. Simple Harmonic Motion.
Physics 11: Lab Oscillations. Siple Haronic Motion. Reading Assignent: Chapter 15 Introduction: As we learned in class, physical systes will undergo an oscillatory otion, when displaced fro a stable equilibriu.
More informationVan der Waals Forces Between Atoms
Van dr Waals Forcs twn tos Michal Fowlr /8/7 Introduction Th prfct gas quation of stat PV = NkT is anifstly incapabl of dscribing actual gass at low tpraturs, sinc thy undrgo a discontinuous chang of volu
More informationAnswer, Key Homework 7 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Hoework 7 David McIntyre 453 Mar 5, 004 This print-out should have 4 questions. Multiple-choice questions ay continue on the next colun or page find all choices before aking your selection.
More informationFinancial Mathematics
Financial Mathatics A ractical Guid for Actuaris and othr Businss rofssionals B Chris Ruckan, FSA & Jo Francis, FSA, CFA ublishd b B rofssional Education Solutions to practic qustions Chaptr 7 Solution
More informationQuestion 3: How do you find the relative extrema of a function?
ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating
More information2.2.C Analogy between electronic excitations in an atom and the mechanical motion of a forced harmonic oscillator"
..C Analgy btwn lctrnic xcitatins in an atm and th mchanical mtin f a frcd harmnic scillatr" Hw t chs th valu f th crrspnding spring cnstant k? Rsnant Absrptin Mchanical rsnanc W idntify th mchanical rsnanc
More informationSpring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations
Spring Simple Harmonic Oscillator Simple Harmonic Oscillations and Resonance We have an object attached to a spring. The object is on a horizontal frictionless surface. We move the object so the spring
More informationWork, Energy, Conservation of Energy
This test covers Work, echanical energy, kinetic energy, potential energy (gravitational and elastic), Hooke s Law, Conservation of Energy, heat energy, conservative and non-conservative forces, with soe
More informationChapter 14 Oscillations
Chapter 4 Oscillations Conceptual Probles 3 n object attached to a spring exhibits siple haronic otion with an aplitude o 4. c. When the object is. c ro the equilibriu position, what percentage o its total
More informationThe Virtual Spring Mass System
The Virtual Spring Mass Syste J. S. Freudenberg EECS 6 Ebedded Control Systes Huan Coputer Interaction A force feedbac syste, such as the haptic heel used in the EECS 6 lab, is capable of exhibiting a
More informationApplications of Second-Order Differential Equations
Applications of Second-Order Differential Equations Second-order linear differential equations have a variety of applications in science and engineering. In this section we explore two of them: the vibration
More informationSecond Order Linear Differential Equations
CHAPTER 2 Second Order Linear Differential Equations 2.. Homogeneous Equations A differential equation is a relation involving variables x y y y. A solution is a function f x such that the substitution
More information5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:
.4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This
More informationSimple Harmonic Motion(SHM) Period and Frequency. Period and Frequency. Cosines and Sines
Simple Harmonic Motion(SHM) Vibration (oscillation) Equilibrium position position of the natural length of a spring Amplitude maximum displacement Period and Frequency Period (T) Time for one complete
More informationME 612 Metal Forming and Theory of Plasticity. 6. Strain
Mtal Forming and Thory of Plasticity -mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.
More informationLecture 3: Diffusion: Fick s first law
Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th
More information2. The acceleration of a simple harmonic oscillator is zero whenever the oscillating object is at the equilibrium position.
CHAPTER : Vibrations and Waes Answers to Questions The acceleration o a siple haronic oscillator is zero wheneer the oscillating object is at the equilibriu position 5 The iu speed is gien by = A k Various
More informationAP Calculus AB 2008 Scoring Guidelines
AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos mission is to connct studnts to collg succss and opportunity.
More informationCHAPTER 4c. ROOTS OF EQUATIONS
CHAPTER c. ROOTS OF EQUATIONS A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring by Dr. Ibrahim A. Aakka Spring 00 ENCE 03 - Computation Mthod in Civil Enginring II Dpartmnt o Civil
More informationTraffic Flow Analysis (2)
Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. Gang-Ln Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,
More informationSolution: F = kx is Hooke s law for a mass and spring system. Angular frequency of this system is: k m therefore, k
Physics 1C Midterm 1 Summer Session II, 2011 Solutions 1. If F = kx, then k m is (a) A (b) ω (c) ω 2 (d) Aω (e) A 2 ω Solution: F = kx is Hooke s law for a mass and spring system. Angular frequency of
More informationLesson 44: Acceleration, Velocity, and Period in SHM
Lesson 44: Acceleration, Velocity, and Period in SHM Since there is a restoring force acting on objects in SHM it akes sense that the object will accelerate. In Physics 20 you are only required to explain
More informationPhysics 1120: Simple Harmonic Motion Solutions
Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured
More informationCUTTING METHODS AND CARTESIAN ROBOTS KESME YÖNTEMLERİ VE KARTEZYEN ROBOTLAR
ournal of Naval Scinc and Enginring 2009, Vol. 5, No.2, pp. 35-42 CUTTING METHODS AND CARTESIAN ROBOTS Asst. Prof. Ugur SIMSIR, Lt.Cdr. Turkish Naval Acady Mchanical Enginring Dpartnt Tuzla, Istanbul,Turkiy
More informationChapter 14 Oscillations
Chapter 4 Oscillations Conceptual Probles rue or false: (a) For a siple haronic oscillator, the period is proportional to the square of the aplitude. (b) For a siple haronic oscillator, the frequency does
More informationLong run: Law of one price Purchasing Power Parity. Short run: Market for foreign exchange Factors affecting the market for foreign exchange
Lctur 6: Th Forign xchang Markt xchang Rats in th long run CON 34 Mony and Banking Profssor Yamin Ahmad xchang Rats in th Short Run Intrst Parity Big Concpts Long run: Law of on pric Purchasing Powr Parity
More informationA Note on Approximating. the Normal Distribution Function
Applid Mathmatical Scincs, Vol, 00, no 9, 45-49 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and
More informationCIRCUITS AND ELECTRONICS. Basic Circuit Analysis Method (KVL and KCL method)
6. CIRCUITS AND ELECTRONICS Basic Circuit Analysis Mthod (KVL and KCL mthod) Cit as: Anant Agarwal and Jffry Lang, cours matrials for 6. Circuits and Elctronics, Spring 7. MIT 6. Fall Lctur Rviw Lumpd
More information[ ] These are the motor parameters that are needed: Motor voltage constant. J total (lb-in-sec^2)
MEASURING MOOR PARAMEERS Fil: Motor paramtrs hs ar th motor paramtrs that ar ndd: Motor voltag constant (volts-sc/rad Motor torqu constant (lb-in/amp Motor rsistanc R a (ohms Motor inductanc L a (Hnris
More informationVersion 001 test 1 review tubman (IBII201516) 1
Version 001 test 1 review tuban (IBII01516) 1 This print-out should have 44 questions. Multiple-choice questions ay continue on the next colun or page find all choices before answering. Crossbow Experient
More informationHOMEWORK FOR UNIT 5-1: FORCE AND MOTION
Nam Dat Partnrs HOMEWORK FOR UNIT 51: FORCE AND MOTION 1. You ar givn tn idntial springs. Dsrib how you would dvlop a sal of for (i., a mans of produing rpatabl fors of a varity of sizs) using ths springs.
More informationHOOKE S LAW AND SIMPLE HARMONIC MOTION
HOOKE S LAW AND SIMPLE HARMONIC MOTION Alexander Sapozhnikov, Brooklyn College CUNY, New York, alexs@brooklyn.cuny.edu Objectives Study Hooke s Law and measure the spring constant. Study Simple Harmonic
More informationExercise 4 INVESTIGATION OF THE ONE-DEGREE-OF-FREEDOM SYSTEM
Eercise 4 IVESTIGATIO OF THE OE-DEGREE-OF-FREEDOM SYSTEM 1. Ai of the eercise Identification of paraeters of the euation describing a one-degree-of- freedo (1 DOF) atheatical odel of the real vibrating
More informationFundamentals: NATURE OF HEAT, TEMPERATURE, AND ENERGY
Fundamntals: NATURE OF HEAT, TEMPERATURE, AND ENERGY DEFINITIONS: Quantum Mchanics study of individual intractions within atoms and molculs of particl associatd with occupid quantum stat of a singl particl
More informationOscillations. Vern Lindberg. June 10, 2010
Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1
More informationPhysics 231 Lecture 15
Physics 31 ecture 15 Main points of today s lecture: Simple harmonic motion Mass and Spring Pendulum Circular motion T 1/f; f 1/ T; ω πf for mass and spring ω x Acos( ωt) v ωasin( ωt) x ax ω Acos( ωt)
More informationLecture L9 - Linear Impulse and Momentum. Collisions
J. Peraire, S. Widnall 16.07 Dynaics Fall 009 Version.0 Lecture L9 - Linear Ipulse and Moentu. Collisions In this lecture, we will consider the equations that result fro integrating Newton s second law,
More informationForeign Exchange Markets and Exchange Rates
Microconomics Topic 1: Explain why xchang rats indicat th pric of intrnational currncis and how xchang rats ar dtrmind by supply and dmand for currncis in intrnational markts. Rfrnc: Grgory Mankiw s Principls
More informationExperimental and Theoretical Modeling of Moving Coil Meter
Experiental and Theoretical Modeling of Moving Coil Meter Prof. R.G. Longoria Updated Suer 010 Syste: Moving Coil Meter FRONT VIEW Electrical circuit odel Mechanical odel Meter oveent REAR VIEW needle
More informationby John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia
Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs
More informationNonlinear Control Design of Shunt Flexible AC Transmission System Devices for Damping Power System Oscillation
Journal o Coputer Science 7 (6): 854-858, ISSN 549-3636 Science Publications Nonlinear Control Design o Shunt Flexible AC Transission Syste Devices or Daping Power Syste Oscillation Prechanon Kukratug
More informationphysics 1/12/2016 Chapter 20 Lecture Chapter 20 Traveling Waves
Chapter 20 Lecture physics FOR SCIENTISTS AND ENGINEERS a strategic approach THIRD EDITION randall d. knight Chapter 20 Traveling Waves Chapter Goal: To learn the basic properties of traveling waves. Slide
More informationENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D225 TUTORIAL 1 LINEAR AND ANGULAR DISPLACEMENT, VELOCITY AND ACCELERATION
ENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D225 TUTORIAL 1 LINEAR AND ANGULAR DISPLACEMENT, VELOCITY AND ACCELERATION This tutorial covers pre-requisite material and should be skipped if you are
More informationNotice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case.
HW1 Possible Solutions Notice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case. Tipler 14.P.003 An object attached to a spring has simple
More informationLecture L26-3D Rigid Body Dynamics: The Inertia Tensor
J. Peraire, S. Widnall 16.07 Dynaics Fall 008 Lecture L6-3D Rigid Body Dynaics: The Inertia Tensor Version.1 In this lecture, we will derive an expression for the angular oentu of a 3D rigid body. We shall
More information8. Spring design. Introduction. Helical Compression springs. Fig 8.1 Common Types of Springs. Fig 8.1 Common Types of Springs
Objectives 8. Spring design Identify, describe, and understand principles of several types of springs including helical copression springs, helical extension springs,, torsion tubes, and leaf spring systes.
More informationEstablishing Wireless Conference Calls Under Delay Constraints
Establishing Wirlss Confrnc Calls Undr Dlay Constraints Aotz Bar-Noy aotz@sci.brooklyn.cuny.du Grzgorz Malwicz grg@cs.ua.du Novbr 17, 2003 Abstract A prvailing fatur of obil tlphony systs is that th cll
More informationDifferentiation of vectors
Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where
More informationMath 267 - Practice exam 2 - solutions
C Roettger, Fall 13 Math 267 - Practice exam 2 - solutions Problem 1 A solution of 10% perchlorate in water flows at a rate of 8 L/min into a tank holding 200L pure water. The solution is kept well stirred
More informationLecture 20: Emitter Follower and Differential Amplifiers
Whits, EE 3 Lctur 0 Pag of 8 Lctur 0: Emittr Followr and Diffrntial Amplifirs Th nxt two amplifir circuits w will discuss ar ry important to lctrical nginring in gnral, and to th NorCal 40A spcifically.
More informationHomework 8. problems: 10.40, 10.73, 11.55, 12.43
Hoework 8 probles: 0.0, 0.7,.55,. Proble 0.0 A block of ass kg an a block of ass 6 kg are connecte by a assless strint over a pulley in the shape of a soli isk having raius R0.5 an ass M0 kg. These blocks
More informationIncomplete 2-Port Vector Network Analyzer Calibration Methods
Incomplt -Port Vctor Ntwork nalyzr Calibration Mthods. Hnz, N. Tmpon, G. Monastrios, H. ilva 4 RF Mtrology Laboratory Instituto Nacional d Tcnología Industrial (INTI) Bunos irs, rgntina ahnz@inti.gov.ar
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More information2.2 Magic with complex exponentials
2.2. MAGIC WITH COMPLEX EXPONENTIALS 97 2.2 Magic with complex exponentials We don t really know what aspects of complex variables you learned about in high school, so the goal here is to start more or
More informationCE 3500 Fluid Mechanics / Fall 2014 / City College of New York
1 Drag Coefficient The force ( F ) of the wind blowing against a building is given by F=C D ρu 2 A/2, where U is the wind speed, ρ is density of the air, A the cross-sectional area of the building, and
More informationInternational Association of Scientific Innovation and Research (IASIR) (An Association Unifying the Sciences, Engineering, and Applied Research)
Intrnational Association of Scintific Innovation and Rsarch (IASIR) (An Association Unifing th Scincs, Enginring, and Applid Rsarch) ISSN (Print): 79-000 ISSN (Onlin): 79-009 Intrnational Journal of Enginring,
More informationSection 7.4: Exponential Growth and Decay
1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 1-17 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart
More informationPhysics 41 HW Set 1 Chapter 15
Physics 4 HW Set Chapter 5 Serway 8 th OC:, 4, 7 CQ: 4, 8 P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59, 67, 74 OC CQ P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59,
More informationAC Circuits Three-Phase Circuits
AC Circuits Thr-Phs Circuits Contnts Wht is Thr-Phs Circuit? Blnc Thr-Phs oltgs Blnc Thr-Phs Connction Powr in Blncd Systm Unblncd Thr-Phs Systms Aliction Rsidntil Wiring Sinusoidl voltg sourcs A siml
More informationSimple Harmonic Motion Experiment. 1 f
Simple Harmonic Motion Experiment In this experiment, a motion sensor is used to measure the position of an oscillating mass as a function of time. The frequency of oscillations will be obtained by measuring
More information19.7. Applications of Differential Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Applications of Differential Equations 19.7 Introduction Blocks 19.2 to 19.6 have introduced several techniques for solving commonly-occurring firstorder and second-order ordinary differential equations.
More information226 Chapter 15: OSCILLATIONS
Chapter 15: OSCILLATIONS 1. In simple harmonic motion, the restoring force must be proportional to the: A. amplitude B. frequency C. velocity D. displacement E. displacement squared 2. An oscillatory motion
More informationFactorials! Stirling s formula
Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical
More informationDesign, Manufacturing & Analysis of Differential Crown Gear and Pinion for MFWD Axle
IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE) ISSN(e) : 2278-1684, ISSN(p) : 220 4X, PP : 59-66 www.iosrjournals.org Design, Manufacturing & Analysis of Differential Crown Gear and Pinion
More informationProjections - 3D Viewing. Overview Lecture 4. Projection - 3D viewing. Projections. Projections Parallel Perspective
Ovrviw Lctur 4 Projctions - 3D Viwing Projctions Paralll Prspctiv 3D Viw Volum 3D Viwing Transformation Camra Modl - Assignmnt 2 OFF fils 3D mor compl than 2D On mor dimnsion Displa dvic still 2D Analog
More informationCHARACTERISTICS OF ELECTROMAGNETIC WAVE PROPAGATION THROUGH A MAGNETISED PLASMA SLAB WITH LINEARLY VARYING ELECTRON DEN- SITY
Progrss In Elctromagntics Rsarch B, Vol. 21, 385 398, 2010 CHARACTERISTICS OF ELECTROMAGNETIC WAVE PROPAGATION THROUGH A MAGNETISED PLASMA SLAB WITH LINEARLY VARYING ELECTRON DEN- SITY Ç. S. Gürl Dpartmnt
More informationA Gas Law And Absolute Zero Lab 11
HB 04-06-05 A Gas Law And Absolute Zero Lab 11 1 A Gas Law And Absolute Zero Lab 11 Equipent safety goggles, SWS, gas bulb with pressure gauge, 10 C to +110 C theroeter, 100 C to +50 C theroeter. Caution
More informationPHYSICS 151 Notes for Online Lecture #11
PHYSICS 151 ote for Online Lecture #11 A free-bod diagra i a wa to repreent all of the force that act on a bod. A free-bod diagra ake olving ewton econd law for a given ituation eaier, becaue ou re odeling
More informationI. INTRODUCTION. Figure 1, The Input Display II. DESIGN PROCEDURE
Ballast Dsign Softwar Ptr Grn, Snior ighting Systms Enginr, Intrnational Rctifir, ighting Group, 101S Spulvda Boulvard, El Sgundo, CA, 9045-438 as prsntd at PCIM Europ 0 Abstract: W hav dvlopd a Windows
More informationPHYSICS 151 Notes for Online Lecture 2.2
PHYSICS 151 otes for Online Lecture. A free-bod diagra is a wa to represent all of the forces that act on a bod. A free-bod diagra akes solving ewton s second law for a given situation easier, because
More informationC B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N
Three boxes are connected by massless strings and are resting on a frictionless table. Each box has a mass of 15 kg, and the tension T 1 in the right string is accelerating the boxes to the right at a
More informationhttp://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force
ctivation nrgis http://www.wwnorton.com/chmistry/tutorials/ch14.htm (back to collision thory...) Potntial and Kintic nrgy during a collision + + ngativly chargd lctron cloud Rpulsiv Forc ngativly chargd
More information2.016 Hydrodynamics Prof. A.H. Techet Fall 2005
.016 Hydrodynamics Reading #7.016 Hydrodynamics Prof. A.H. Techet Fall 005 Free Surface Water Waves I. Problem setu 1. Free surface water wave roblem. In order to determine an exact equation for the roblem
More informationCircuits with Transistors
ircuits with Transistors ontnts 1 Transistors 1 2 Amplifirs 2 2.1 h paramtrs.................................... 3 3 Bipolar Junction Transistor (BJT) 3 3.1 BJT as a switch...................................
More informationLet s examine the response of the circuit shown on Figure 1. The form of the source voltage Vs is shown on Figure 2. R. Figure 1.
Examples of Transient and RL Circuits. The Series RLC Circuit Impulse response of Circuit. Let s examine the response of the circuit shown on Figure 1. The form of the source voltage Vs is shown on Figure.
More informationHW 2. Q v. kt Step 1: Calculate N using one of two equivalent methods. Problem 4.2. a. To Find:
HW 2 Proble 4.2 a. To Find: Nuber of vacancies per cubic eter at a given teperature. b. Given: T 850 degrees C 1123 K Q v 1.08 ev/ato Density of Fe ( ρ ) 7.65 g/cc Fe toic weight of iron ( c. ssuptions:
More informationEffect of Design Parameter on the Performance of Lithium Ion Battery
Aadil Ahmad, Mohd. Parvz / Intrnational Journal of Enginring Rarch and Application Vol. 3, Iu 4, Jul-Aug 2013, pp.1196-1201 Effct of Dign Paramtr on th Prformanc of Lithium Ion Battry Aadil Ahmad 1, Mohd.
More informationColumbia University Department of Physics QUALIFYING EXAMINATION
Columbia University Department of Physics QUALIFYING EXAMINATION Monday, January 13, 2014 1:00PM to 3:00PM Classical Physics Section 1. Classical Mechanics Two hours are permitted for the completion of
More informationAnalyzing the Economic Efficiency of ebaylike Online Reputation Reporting Mechanisms
A rsarch and ducation initiativ at th MIT Sloan School of Managmnt Analyzing th Economic Efficincy of Baylik Onlin Rputation Rporting Mchanisms Papr Chrysanthos Dllarocas July For mor information, plas
More informationCAFA DIVERSITY JURISDICTION
Cla Action 101: CAFA Divrity Juridiction at a Glanc By Kathryn Honckr Jun 20, 2013 In thi dition of Cla Action 101, w giv a viual guid to th Cla Action Fairn Act (CAFA), 28 U.S.C. 1332(d)(2), to hlp you
More informationChapter 12 Driven RLC Circuits
hapter Driven ircuits. A Sources... -. A ircuits with a Source and One ircuit Element... -3.. Purely esistive oad... -3.. Purely Inductive oad... -6..3 Purely apacitive oad... -8.3 The Series ircuit...
More information7. Beats. sin( + λ) + sin( λ) = 2 cos(λ) sin( )
34 7. Beats 7.1. What beats are. Musicians tune their instruments using beats. Beats occur when two very nearby pitches are sounded simultaneously. We ll make a mathematical study of this effect, using
More informationMidterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m
Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of
More informationGATE 2009 Electrical Engineering Question Paper
GATE 9 Elctrical Enginring Qustion Papr Q. No. Carry On Mark Each. Th prssur coil of a dynamomtr typ wattmtr is highly inductiv highly rsistiv purly rsistiv purly inductiv. Th masurmnt systm shown in th
More informationPREDICTION OF MILKLINE FILL AND TRANSITION FROM STRATIFIED TO SLUG FLOW
PREDICTION OF MILKLINE FILL AND TRANSITION FROM STRATIFIED TO SLUG FLOW ABSTRACT: by Douglas J. Reineann, Ph.D. Assistant Professor of Agricultural Engineering and Graee A. Mein, Ph.D. Visiting Professor
More informationChapter 31. Current and Resistance. What quantity is represented by the symbol J?
Chapter 31. Current and Resistance Lights, sound systes, icrowave ovens, and coputers are all connected by wires to a battery or an electrical outlet. How and why does electric current flow through a wire?
More informationIntegration ALGEBRAIC FRACTIONS. Graham S McDonald and Silvia C Dalla
Integration ALGEBRAIC FRACTIONS Graham S McDonald and Silvia C Dalla A self-contained Tutorial Module for practising the integration of algebraic fractions Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk
More informationVibrations of a Free-Free Beam
Vibrations of a Free-Free Beam he bending vibrations of a beam are described by the following equation: y EI x y t 4 2 + ρ A 4 2 (1) y x L E, I, ρ, A are respectively the Young Modulus, second moment of
More informationOn a class of Hill s equations having explicit solutions. E-mail: m.bartuccelli@surrey.ac.uk, j.wright@surrey.ac.uk, gentile@mat.uniroma3.
On a class of Hill s equations having explicit solutions Michele Bartuccelli 1 Guido Gentile 2 James A. Wright 1 1 Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK. 2 Dipartimento
More informationLABORATORY 10 TIME AVERAGES, RMS VALUES AND THE BRIDGE RECTIFIER. Bridge Rectifier
LABORATORY 10 TIME AVERAGES, RMS VALUES AND THE BRIDGE RECTIFIER Full-wave Rectification: Bridge Rectifier For many electronic circuits, DC supply voltages are required but only AC voltages are available.
More informationMathematical Model for Glucose-Insulin Regulatory System of Diabetes Mellitus
Advances in Applied Matheatical Biosciences. ISSN 8-998 Volue, Nuber (0), pp. 9- International Research Publication House http://www.irphouse.co Matheatical Model for Glucose-Insulin Regulatory Syste of
More informationCPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions
CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags:
More informationInterference and Diffraction
Chapter 14 nterference and Diffraction 14.1 Superposition of Waves... 14-14. Young s Double-Slit Experiment... 14-4 Example 14.1: Double-Slit Experiment... 14-7 14.3 ntensity Distribution... 14-8 Example
More informationLAWS OF MOTION PROBLEM AND THEIR SOLUTION
http://www.rpauryascienceblog.co/ LWS OF OIO PROBLE D HEIR SOLUIO. What is the axiu value of the force F such that the F block shown in the arrangeent, does not ove? 60 = =3kg 3. particle of ass 3 kg oves
More informationSymplectic structures and Hamiltonians of a mechanical system
INVESTIGACIÓN REVISTA MEXICANA DE FÍSICA 49 (5) 445 449 OCTUBRE 2003 Syplectic structures and Hailtonians of a echanical syste G.F. Torres del Castillo Departaento de Física Mateática, Instituto de Ciencias,
More informationFaraday s Law of Induction
Chapter 10 Faraday s Law of Induction 10.1 Faraday s Law of Induction...10-10.1.1 Magnetic Flux...10-3 10.1. Lenz s Law...10-5 10. Motional EMF...10-7 10.3 Induced Electric Field...10-10 10.4 Generators...10-1
More informationPhysics 102: Lecture 13 RLC circuits & Resonance
Physics 102: Lecture 13 L circuits & esonance L Physics 102: Lecture 13, Slide 1 I = I max sin(2pft) V = I max sin(2pft) V in phase with I eview: A ircuit L V = I max X sin(2pft p/2) V lags I IE I V t
More information