THE SOLAR SYSTEM. We begin with an inertial system and locate the planet and the sun with respect to it. Then. F m. Then

Size: px
Start display at page:

Download "THE SOLAR SYSTEM. We begin with an inertial system and locate the planet and the sun with respect to it. Then. F m. Then"

Transcription

1 THE SOLAR SYSTEM We now want to apply what we have learned to the olar ytem. Hitorially thi wa the great teting ground for mehani and provided ome of it greatet triumph, uh a the diovery of the outer planet. We will ue a implified model of the ytem baed on the fat that the ma of the un i muh greater than that of any of the planet or in fat of all of them ombined. We will alo neglet the effet of the planet on eah other, although it i preiely thi whih led to the diovery of the outer planet. Hene we will uppoe that the only objet in the univere are the un and the planet other objet we are onidering. The firt tep i to hooe a atifatory oordinate ytem. It mut be inertial if we want to avoid the ue of the fititiou fore we have diued earlier. Ideally we would hooe a oordinate ytem either entered on the earth (Pre Coperniu) or on the un (Pot Coperniu). At firt glane neither would be atifatory beaue of Newton third law. Eah exert a fore on the other and thu neither i unaelerated. However thi problem an be overome, a we will now ee. We begin with an inertial ytem and loate the planet and the un with repet to it. Then Rp R r Ap A a F F m A m A a p p p p p F F Fp ma A m p Then Fp Fp mp a m

2 mp Fp 1 mpa m and mm p Fp a a m m p where μ (the redued ma) = m m p /(m +m p). Thu we an hooe the enter of the un a the origin of our oordinate ytem if we merely replae the ma of the planet by it redued ma. In the ae of the olar ytem all mae are mall ompared to that of the un, and hene the orretion i uually negligible. However if we were to onider the moon going around the earth thi would not be the ae. The phyial reaon for thi effet i eay to undertand. An oberver on either objet ee both her aeleration and the aeleration of the other objet toward her, a diued in la. We now have a very imple fore diagram for planet. There i only one fore the gravitational attration of the un. Sine it i direted through the axi it produe no torque: RFp 0 Then beaue of Newton eond law written in term of angular quantitie: We mut have the angular momentum ontant. But ine angular momentum i a vetor it diretion mut be ontant. Sine: dr dt J rp rmv and both r and v are in the plane of the orbit we ee that J i perpendiular to the plane of the orbit. Then ine that diretion i fixed we get our firt onluion the planet mut move in a plane. We will find the orbit exatly in a moment, but firt we onider a peial ae in whih the orbit i irular. It turn out that mot of the planet move in nearly irular orbit o thi i a good tarting point. In thi ae the problem i very imple. The aeleration toward the enter of the irle mut be provided by the gravitational attration:

3 mp vp FG m p R R FM Rv p But vp R P where P i the period of the planet the time to go one around the un. Then or 3 4 R P 3 R p 4 (the ame for all objet in the olar ytem). Thi reult i Kepler third law, and provide a mean of meauring the ma of the un. We now turn to the general problem where the orbit are not irular. Beaue the orbit may repeat it will be muh more onvenient to ue polar rather than Carteian oordinate (note that we are in a plane o pherial oordinate are not needed). We now need to alulate the aeleration in polar oordinate. We do thi a before. To get ˆ dr / dt onider the keth r rrˆ dr dr drˆ v rˆ r dt dt dt

4 Then drˆ d ˆ dt dt Hene dr d v rˆ r ˆ dt dt Of oure thi i obviou. dr/dt i jut the omponent of veloity along the radiu d r r dt i jut the tangential veloity along the irumferene. Next dv d r dr drˆ dr d d d dˆ a rˆ ˆ r ˆ r dt dt dt dt dt dt dt dt dt Hene we need d ˆ /dt. We find thi from the keth Thu dˆ d ˆr dt dt

5 Then d r d ˆ d dr d a rˆ r r dt dt dt dt dt Again mot of thee term are obviou. d r/dt i jut the aeleration along r. i jut the familiar entripetal aeleration. d v v r r r dt r r d d r r r dt dt i jut the tangential aeleration along the urve. The lat term dr d dt dt i new beaue we have not previouly onidered aeleration in both the radial and tangential diretion at the ame time. Sine we know the fore we an now write Newton eond law in our hoen oordinate ytem. m F r p rˆ d r dr dt dt r (1) d dr d r 0 dt dt dt () Thi look a bit intimidating until we ue our knowledge of the phyi of the ituation. We know that angular momentum i ontant. We have already ued the fat that the diretion i ontant to ee that the motion i in a plane. We now ue the fat that the magnitude i ontant to olve ().

6 Hene dr d ˆ d J mr v mrrˆ rˆ r mr zˆ dt dt dt d r ontant dt Thu d d dr d d d dr d r 0 r r r r dt dt dt dt dt dt dt dt Thu d r ont A dt i the olution of (). We now ubtitute thi reult in (1) to get: A r dr dt r r (3) To olve (3) it i ueful to make a hange of variable. Let: 1 r u Then dr dt 1 du u dt dr du 1 du 3 dt dt u u dt and du 1 d u 3 Au dt u u dt 3 u (4)

7 Sine we are intereted in the hape of the orbit we really want u(θ) rather than u(t). We therefore make another hange of variable. We have: We now ue the hain rule to get: Subtituting thi in (4) give: or u(t) = u(θ(t)) du du d du A du Au dt d dt d r d d u d du d u du Au Au Au Au Au d d dt d d du 1 d u 1du 3 Au Au Au A u 3 u u d u d ud du 3 Au Au u d d u u d A But thi i an equation we know how to olve. We do it a the um of the general olution of the homogeneou equation plu one olution of the whole equation. The olution of the homogeneou equation i obviouly: uh Bo where B and φ are arbitrary ontant. A olution of the whole equation i: Thu u p A u uk up Bo A

8 To fix the arbitrary ontant we hooe θ to be zero at the point of loet approah. At that point we alo know the angular momentum i: But Hene Chooe φ = 0. Then rmv J A rv m r r u Bo rv Bo rv r Br o 1 rv 1 r Br 1 rv r rv 1 r r rv B 1 r r r rv 1 o 1 o rv rv 1 where 1 rv rv 1 rv

9 But thi i jut the equation of a oni etion with eentriity ε. There are now everal ae to onider depending on the value of ε. We begin by aking whether or not the planet will return, i.e. i it in a loed orbit or doe it ultimately leave the olar ytem. To anwer thi we need only look at the energy. Thi i given at the point of loet approah by: m 1 1 E mv m v r r If the planet i to reah infinite ditane it will have to have a veloity greater than or equal to zero at infinite ditane. Hene E mut be greater than or equal to zero. 1 1 r v v 0 v 1 r r If thi ondition i not met the planet will return and alo we an predit how far from the un it will get. To do o we note that both the angular momentum and the energy are onerved. Thu: or rv r v m m 1 1 v vm r r 1 r v p m m E m r r E 1 rm rm rv 0 m p r m E r v E m p mp 1/ Sine E/m p < 0 we an ue either ign. One give r, the other r m (the maximum ditane).

10 The ae to be onidered are then: Cae 1: < 1 Cae : ε = 1 Cae 3: ε = 0 Cae 4: ε > 1 CASE 3 Thi i the implet ituation. We then have: r v r r v rv or r 4 P Thi i jut the ae of irular orbit diued above. CASE Thi i a bit more ompliated, but not muh. In thi ae we have: rv rv 11 1 Thi i the ituation in whih the planet would jut reah infinite ditane with zero veloity. We an readily find the equation of the orbit a follow. Swithing to Carteian oordinate we have rv r 1 o

11 rv 1/ x y x 1 1/ x y 1/ r v rv rv rv x y x x y x xx But thi i jut the equation of a parabola. y r r x y x r r CASE 1 Thi i the ae for whih the objet will return, and hene the one that applie for planet in the olar ytem. Thi i more ompliated to analyze, o ome patiene i required. Now we have: rv rv r r 1/ r x y 1o x 1 1/ x y 1/ rv x y x r rv rv rv x y r x r r x x rv rv x 1 r xy r

12 We now omplete the quare on the x term x rv r y rv r x rv rv rv y r r r x rv r rv 1 r Sine < 1, thi an equation of the form x y 1 a b whih i the equation of an ellipe entered at x = -α with emi-major axi a and emi-minor axi b. Note that But rv rv rv rv rv rv r r 1 r 1 r 1 1 rv rv rv rv 1 1

13 r o r rm E 1 v m p r i the enter of the ellipe. Then rv rv o 1 1 v rv v r r r 1 r r r Thu we have the equation of an ellipe with origin at the enter of the ellipe. CASE 4 Now all that hange i the ign of (1 ε ). Thi hange the equation to: x y 1 a b whih i the equation of a hyperbola. We now turn to the quetion of the motion a a funtion of time. To do thi we note that: d A r d dt dt r A 3 r A d t d A 1 o o o 3 1 tan A in 1 1 tan 3/ 1/ o 1 1o 1 1 For example we an find the period of the orbit of a planet a:

14 P 3 1 tan 3 A 1 A tan 3/ 1/ 3/ But 1/ r r v A A a 1 1 a 1 3 3/ P A 1 a A a 3/ 1/ 4 3 P a Thi i Kepler third law. In other word the planet have the ame period a irular orbit of radiu equal to the emi-major axi would have. It i now a trivial problem to find the period of any planet if we know it ditane and veloity relative to the un at loet approah.

Inverse Kinematics 1 1/21/2018

Inverse Kinematics 1 1/21/2018 Invere Kinemati 1 Invere Kinemati 2 given the poe of the end effetor, find the joint variable that produe the end effetor poe for a -joint robot, given find 1 o R T 3 2 1,,,,, q q q q q q RPP + Spherial

More information

Where Standard Physics Runs into Infinite Challenges, Atomism Predicts Exact Limits

Where Standard Physics Runs into Infinite Challenges, Atomism Predicts Exact Limits Where Standard Phyi Run into Infinite Challenge, Atomim Predit Exat Limit Epen Gaarder Haug Norwegian Univerity of Life Siene Deember, 07 Abtrat Where tandard phyi run into infinite hallenge, atomim predit

More information

Einstein's Energy Formula Must Be Revised

Einstein's Energy Formula Must Be Revised Eintein' Energy Formula Mut Be Reied Le Van Cuong uong_le_an@yahoo.om Information from a iene journal how that the dilation of time in Eintein peial relatie theory wa proen by the experiment of ientit

More information

, an inverse square law.

, an inverse square law. Uniform irular motion Speed onstant, but eloity hanging. and a / t point to enter. s r θ > θ s/r t / r, also θ in small limit > t/r > a / r, entripetal aeleration Sine a points to enter of irle, F m a

More information

DISCHARGE MEASUREMENT IN TRAPEZOIDAL LINED CANALS UTILIZING HORIZONTAL AND VERTICAL TRANSITIONS

DISCHARGE MEASUREMENT IN TRAPEZOIDAL LINED CANALS UTILIZING HORIZONTAL AND VERTICAL TRANSITIONS Ninth International Water Tehnology Conferene, IWTC9 005, Sharm El-Sheikh, Egypt 63 DISCHARGE MEASUREMENT IN TRAPEZOIDAL LINED CANALS UTILIZING HORIZONTAL AND VERTICAL TRANSITIONS Haan Ibrahim Mohamed

More information

Chapter 6 Control Systems Design by Root-Locus Method. Lag-Lead Compensation. Lag lead Compensation Techniques Based on the Root-Locus Approach.

Chapter 6 Control Systems Design by Root-Locus Method. Lag-Lead Compensation. Lag lead Compensation Techniques Based on the Root-Locus Approach. hapter 6 ontrol Sytem Deign by Root-Lou Method Lag-Lead ompenation Lag lead ompenation ehnique Baed on the Root-Lou Approah. γ β K, ( γ >, β > ) In deigning lag lead ompenator, we onider two ae where γ

More information

@(; t) p(;,b t) +; t), (; t)) (( whih lat line follow from denition partial derivative. in relation quoted in leture. Th derive wave equation for ound

@(; t) p(;,b t) +; t), (; t)) (( whih lat line follow from denition partial derivative. in relation quoted in leture. Th derive wave equation for ound 24 Spring 99 Problem Set 5 Optional Problem Phy February 23, 999 Handout Derivation Wave Equation for Sound. one-dimenional wave equation for ound. Make ame ort Derive implifying aumption made in deriving

More information

To determine the biasing conditions needed to obtain a specific gain each stage must be considered.

To determine the biasing conditions needed to obtain a specific gain each stage must be considered. PHYSIS 56 Experiment 9: ommon Emitter Amplifier A. Introdution A ommon-emitter oltage amplifier will be tudied in thi experiment. You will inetigate the fator that ontrol the midfrequeny gain and the low-and

More information

UVa Course on Physics of Particle Accelerators

UVa Course on Physics of Particle Accelerators UVa Coure on Phyi of Partile Aelerator B. Norum Univerity of Virginia G. A. Krafft Jefferon Lab 3/7/6 Leture x dx d () () Peudoharmoni Solution = give β β β () ( o µ + α in µ ) β () () β x dx ( + α() α

More information

Chapter 4. Simulations. 4.1 Introduction

Chapter 4. Simulations. 4.1 Introduction Chapter 4 Simulation 4.1 Introdution In the previou hapter, a methodology ha been developed that will be ued to perform the ontrol needed for atuator haraterization. A tudy uing thi methodology allowed

More information

Thermochemistry and Calorimetry

Thermochemistry and Calorimetry WHY? ACTIVITY 06-1 Thermohemitry and Calorimetry Chemial reation releae or tore energy, uually in the form of thermal energy. Thermal energy i the kineti energy of motion of the atom and moleule ompriing

More information

Practice Exam 2 Solutions

Practice Exam 2 Solutions MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department o Physis Physis 801T Fall Term 004 Problem 1: stati equilibrium Pratie Exam Solutions You are able to hold out your arm in an outstrethed horizontal position

More information

Laplace Transformation

Laplace Transformation Univerity of Technology Electromechanical Department Energy Branch Advance Mathematic Laplace Tranformation nd Cla Lecture 6 Page of 7 Laplace Tranformation Definition Suppoe that f(t) i a piecewie continuou

More information

The Laws of Electromagnetism Maxwell s Equations Displacement Current Electromagnetic Radiation

The Laws of Electromagnetism Maxwell s Equations Displacement Current Electromagnetic Radiation The letromagneti petrum The Law of letromagnetim Maxwell quation Diplaement Current letromagneti Radiation Maxwell quation of letromagnetim in Vauum (no harge, no mae) lane letromagneti Wave d d z y (x,

More information

Energy-Work Connection Integration Scheme for Nonholonomic Hamiltonian Systems

Energy-Work Connection Integration Scheme for Nonholonomic Hamiltonian Systems Commun. Theor. Phy. Beiing China 50 2008 pp. 1041 1046 Chinee Phyial Soiety Vol. 50 No. 5 November 15 2008 Energy-Wor Connetion Integration Sheme for Nonholonomi Hamiltonian Sytem WANG Xian-Jun 1 and FU

More information

ES 247 Fracture Mechanics Zhigang Suo. Applications of Fracture Mechanics

ES 247 Fracture Mechanics   Zhigang Suo. Applications of Fracture Mechanics Appliation of Frature Mehani Many appliation of frature mehani are baed on the equation σ a Γ = β. E Young modulu i uually known. Of the other four quantitie, if three are known, the equation predit the

More information

COMM 602: Digital Signal Processing. Lecture 8. Digital Filter Design

COMM 602: Digital Signal Processing. Lecture 8. Digital Filter Design COMM 60: Digital Signal Proeing Leture 8 Digital Filter Deign Remember: Filter Type Filter Band Pratial Filter peifiation Pratial Filter peifiation H ellipti H Pratial Filter peifiation p p IIR Filter

More information

Moment of Inertia of an Equilateral Triangle with Pivot at one Vertex

Moment of Inertia of an Equilateral Triangle with Pivot at one Vertex oment of nertia of an Equilateral Triangle with Pivot at one Vertex There are two wa (at leat) to derive the expreion f an equilateral triangle that i rotated about one vertex, and ll how ou both here.

More information

Final Comprehensive Exam Physical Mechanics Friday December 15, Total 100 Points Time to complete the test: 120 minutes

Final Comprehensive Exam Physical Mechanics Friday December 15, Total 100 Points Time to complete the test: 120 minutes Final Comprehenive Exam Phyical Mechanic Friday December 15, 000 Total 100 Point Time to complete the tet: 10 minute Pleae Read the Quetion Carefully and Be Sure to Anwer All Part! In cae that you have

More information

Bogoliubov Transformation in Classical Mechanics

Bogoliubov Transformation in Classical Mechanics Bogoliubov Tranformation in Claical Mechanic Canonical Tranformation Suppoe we have a et of complex canonical variable, {a j }, and would like to conider another et of variable, {b }, b b ({a j }). How

More information

Wave Phenomena Physics 15c

Wave Phenomena Physics 15c Wave Phenomena Phyi 15 Leture 18 EM Wave in Matter (H&L Setion 9.7) What We Did Lat Time! Reviewed refletion and refration! Total internal refletion i more ubtle than it look! Imaginary wave extend a few

More information

Sound Propagation through Circular Ducts with Spiral Element Inside

Sound Propagation through Circular Ducts with Spiral Element Inside Exerpt from the Proeeding of the COMSOL Conferene 8 Hannover Sound Propagation through Cirular Dut with Spiral Element Inide Wojieh Łapka* Diviion of Vibroaouti and Sytem Biodynami, Intitute of Applied

More information

q-expansions of vector-valued modular forms of negative weight

q-expansions of vector-valued modular forms of negative weight Ramanujan J 2012 27:1 13 DOI 101007/11139-011-9299-9 q-expanion of vetor-valued modular form of negative weight Joe Gimenez Wiam Raji Reeived: 19 July 2010 / Aepted: 2 February 2011 / Publihed online:

More information

V V The circumflex (^) tells us this is a unit vector

V V The circumflex (^) tells us this is a unit vector Vector 1 Vector have Direction and Magnitude Mike ailey mjb@c.oregontate.edu Magnitude: V V V V x y z vector.pptx Vector Can lo e Defined a the oitional Difference etween Two oint 3 Unit Vector have a

More information

Physics 2. Angular Momentum. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Physics 2. Angular Momentum. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Phyic Angular Momentum For Campu earning Angular Momentum Thi i the rotational equivalent of linear momentum. t quantifie the momentum of a rotating object, or ytem of object. To get the angular momentum,

More information

5.2.6 COMPARISON OF QUALITY CONTROL AND VERIFICATION TESTS

5.2.6 COMPARISON OF QUALITY CONTROL AND VERIFICATION TESTS 5..6 COMPARISON OF QUALITY CONTROL AND VERIFICATION TESTS Thi proedure i arried out to ompare two different et of multiple tet reult for finding the ame parameter. Typial example would be omparing ontrator

More information

Physics 741 Graduate Quantum Mechanics 1 Solutions to Final Exam, Fall 2014

Physics 741 Graduate Quantum Mechanics 1 Solutions to Final Exam, Fall 2014 Phyic 7 Graduate Quantum Mechanic Solution to inal Eam all 0 Each quetion i worth 5 point with point for each part marked eparately Some poibly ueful formula appear at the end of the tet In four dimenion

More information

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS Laplace Tranform Paul Dawkin Table of Content Preface... Laplace Tranform... Introduction... The Definition... 5 Laplace Tranform... 9 Invere Laplace Tranform... Step Function...4

More information

DIFFERENTIAL EQUATIONS Laplace Transforms. Paul Dawkins

DIFFERENTIAL EQUATIONS Laplace Transforms. Paul Dawkins DIFFERENTIAL EQUATIONS Laplace Tranform Paul Dawkin Table of Content Preface... Laplace Tranform... Introduction... The Definition... 5 Laplace Tranform... 9 Invere Laplace Tranform... Step Function...

More information

Pulsed Magnet Crimping

Pulsed Magnet Crimping Puled Magnet Crimping Fred Niell 4/5/00 1 Magnetic Crimping Magnetoforming i a metal fabrication technique that ha been in ue for everal decade. A large capacitor bank i ued to tore energy that i ued to

More information

Period #8: Axial Load/Deformation in Indeterminate Members

Period #8: Axial Load/Deformation in Indeterminate Members ENGR:75 Meh. Def. odie Period #8: ial oad/deformation in Indeterminate Member. Review We are onidering aial member in tenion or ompreion in the linear, elati regime of behavior. Thu the magnitude of aial

More information

Mechanics Physics 151

Mechanics Physics 151 Mechanic Phyic 151 Lecture 7 Scattering Problem (Chapter 3) What We Did Lat Time Dicued Central Force Problem l Problem i reduced to one equation mr = + f () r 3 mr Analyzed qualitative behavior Unbounded,

More information

Correction for Simple System Example and Notes on Laplace Transforms / Deviation Variables ECHE 550 Fall 2002

Correction for Simple System Example and Notes on Laplace Transforms / Deviation Variables ECHE 550 Fall 2002 Correction for Simple Sytem Example and Note on Laplace Tranform / Deviation Variable ECHE 55 Fall 22 Conider a tank draining from an initial height of h o at time t =. With no flow into the tank (F in

More information

Analysis of Feedback Control Systems

Analysis of Feedback Control Systems Colorado Shool of Mine CHEN403 Feedbak Control Sytem Analyi of Feedbak Control Sytem ntrodution to Feedbak Control Sytem 1 Cloed oo Reone 3 Breaking Aart the Problem to Calulate the Overall Tranfer Funtion

More information

Physics 6A. Angular Momentum. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Physics 6A. Angular Momentum. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Phyic 6A Angular Momentum For Campu earning Angular Momentum Thi i the rotational equivalent of linear momentum. t quantifie the momentum of a rotating object, or ytem of object. f we imply tranlate the

More information

TMA4125 Matematikk 4N Spring 2016

TMA4125 Matematikk 4N Spring 2016 Norwegian Univerity of Science and Technology Department of Mathematical Science TMA45 Matematikk 4N Spring 6 Solution to problem et 6 In general, unle ele i noted, if f i a function, then F = L(f denote

More information

Chapter 26 Lecture Notes

Chapter 26 Lecture Notes Chapter 26 Leture Notes Physis 2424 - Strauss Formulas: t = t0 1 v L = L0 1 v m = m0 1 v E = m 0 2 + KE = m 2 KE = m 2 -m 0 2 mv 0 p= mv = 1 v E 2 = p 2 2 + m 2 0 4 v + u u = 2 1 + vu There were two revolutions

More information

Lecture 17: Analytic Functions and Integrals (See Chapter 14 in Boas)

Lecture 17: Analytic Functions and Integrals (See Chapter 14 in Boas) Lecture 7: Analytic Function and Integral (See Chapter 4 in Boa) Thi i a good point to take a brief detour and expand on our previou dicuion of complex variable and complex function of complex variable.

More information

Factor Analysis with Poisson Output

Factor Analysis with Poisson Output Factor Analyi with Poion Output Gopal Santhanam Byron Yu Krihna V. Shenoy, Department of Electrical Engineering, Neurocience Program Stanford Univerity Stanford, CA 94305, USA {gopal,byronyu,henoy}@tanford.edu

More information

into a discrete time function. Recall that the table of Laplace/z-transforms is constructed by (i) selecting to get

into a discrete time function. Recall that the table of Laplace/z-transforms is constructed by (i) selecting to get Lecture 25 Introduction to Some Matlab c2d Code in Relation to Sampled Sytem here are many way to convert a continuou time function, { h( t) ; t [0, )} into a dicrete time function { h ( k) ; k {0,,, }}

More information

Heat transfer and absorption of SO 2 of wet flue gas in a tube cooled

Heat transfer and absorption of SO 2 of wet flue gas in a tube cooled Heat tranfer and aborption of SO of wet flue ga in a tube ooled L. Jia Department of Power Engineering, Shool of Mehanial, Eletroni and Control Engineering, Beijing Jiaotong Univerity, Beijing 00044, China

More information

EE 333 Electricity and Magnetism, Fall 2009 Homework #11 solution

EE 333 Electricity and Magnetism, Fall 2009 Homework #11 solution EE 333 Eetriity and Magnetim, Fa 009 Homework #11 oution 4.4. At the interfae between two magneti materia hown in Fig P4.4, a urfae urrent denity J S = 0.1 ŷ i fowing. The magneti fied intenity H in region

More information

online learning Unit Workbook 4 RLC Transients

online learning Unit Workbook 4 RLC Transients online learning Pearon BTC Higher National in lectrical and lectronic ngineering (QCF) Unit 5: lectrical & lectronic Principle Unit Workbook 4 in a erie of 4 for thi unit Learning Outcome: RLC Tranient

More information

Effect of Anisotropy Asymmetry on the Switching. Behavior of Exchange Biased Bilayers

Effect of Anisotropy Asymmetry on the Switching. Behavior of Exchange Biased Bilayers Applied athematial Siene, Vol. 5, 0, no. 44, 95-06 Effet of Aniotropy Aymmetry on the Swithing Behavior of Exhange Biaed Bilayer Congxiao Liu a, atthew E. Edward b, in Sun and J. C. Wang b a Department

More information

Econ 455 Answers - Problem Set 4. where. ch ch ch ch ch ch ( ) ( ) us us ch ch us ch. (world price). Combining the above two equations implies: 40P

Econ 455 Answers - Problem Set 4. where. ch ch ch ch ch ch ( ) ( ) us us ch ch us ch. (world price). Combining the above two equations implies: 40P Fall 011 Eon 455 Harvey Lapan Eon 455 Anwer - roblem et 4 1. Conider the ae of two large ountrie: U: emand = 300 4 upply = 6 where h China: emand = 300 10 ; upply = 0 h where (a) Find autarky prie: U:

More information

Green s function for the wave equation

Green s function for the wave equation Green s funtion for the wave equation Non-relativisti ase January 2019 1 The wave equations In the Lorentz Gauge, the wave equations for the potentials are (Notes 1 eqns 43 and 44): 1 2 A 2 2 2 A = µ 0

More information

Deepak Rajput

Deepak Rajput General quetion about eletron and hole: A 1a) What ditinguihe an eletron from a hole? An) An eletron i a fundamental partile wherea hole i jut a onept. Eletron arry negative harge wherea hole are onidered

More information

arxiv:gr-qc/ v2 6 Feb 2004

arxiv:gr-qc/ v2 6 Feb 2004 Hubble Red Shift and the Anomalous Aeleration of Pioneer 0 and arxiv:gr-q/0402024v2 6 Feb 2004 Kostadin Trenčevski Faulty of Natural Sienes and Mathematis, P.O.Box 62, 000 Skopje, Maedonia Abstrat It this

More information

A Queueing Model for Call Blending in Call Centers

A Queueing Model for Call Blending in Call Centers 434 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 8, AUGUST 2003 A Queueing Model for Call Blending in Call Center Sandjai Bhulai and Ger Koole Abtrat Call enter that apply all blending obtain high-produtivity

More information

ELECTROMAGNETIC NORMAL MODES AND DISPERSION FORCES.

ELECTROMAGNETIC NORMAL MODES AND DISPERSION FORCES. ELECTROMAGNETIC NORMAL MODES AND DISPERSION FORCES. All systems with interation of some type have normal modes. One may desribe them as solutions in absene of soures; they are exitations of the system

More information

231 Outline Solutions Tutorial Sheet 7, 8 and January Which of the following vector fields are conservative?

231 Outline Solutions Tutorial Sheet 7, 8 and January Which of the following vector fields are conservative? 231 Outline olutions Tutorial heet 7, 8 and 9. 12 Problem heet 7 18 January 28 1. Whih of the following vetor fields are onservative? (a) F = yz sin x i + z osx j + y os x k. (b) F = 1 2 y i 1 2 x j. ()

More information

Halliday/Resnick/Walker 7e Chapter 6

Halliday/Resnick/Walker 7e Chapter 6 HRW 7e Chapter 6 Page of Halliday/Renick/Walker 7e Chapter 6 3. We do not conider the poibility that the bureau might tip, and treat thi a a purely horizontal motion problem (with the peron puh F in the

More information

Singular perturbation theory

Singular perturbation theory Singular perturbation theory Marc R. Rouel June 21, 2004 1 Introduction When we apply the teady-tate approximation (SSA) in chemical kinetic, we typically argue that ome of the intermediate are highly

More information

The elementary spike produced by a pure e + e pair-electromagnetic pulse from a Black Hole: The PEM Pulse

The elementary spike produced by a pure e + e pair-electromagnetic pulse from a Black Hole: The PEM Pulse Atronomy & Atrophyi manuript no. (will be inerted by hand later) The elementary pike produed by a pure e + e pair-eletromagneti pule from a Blak Hole: The PEM Pule Carlo Luiano Biano 1, Remo Ruffini 1,

More information

Critical Percolation Probabilities for the Next-Nearest-Neighboring Site Problems on Sierpinski Carpets

Critical Percolation Probabilities for the Next-Nearest-Neighboring Site Problems on Sierpinski Carpets Critial Perolation Probabilitie for the Next-Nearet-Neighboring Site Problem on Sierpinki Carpet H. B. Nie, B. M. Yu Department of Phyi, Huazhong Univerity of Siene and Tehnology, Wuhan 430074, China K.

More information

Notes on Phase Space Fall 2007, Physics 233B, Hitoshi Murayama

Notes on Phase Space Fall 2007, Physics 233B, Hitoshi Murayama Note on Phae Space Fall 007, Phyic 33B, Hitohi Murayama Two-Body Phae Space The two-body phae i the bai of computing higher body phae pace. We compute it in the ret frame of the two-body ytem, P p + p

More information

MATEMATIK Datum: Tid: eftermiddag. A.Heintz Telefonvakt: Anders Martinsson Tel.:

MATEMATIK Datum: Tid: eftermiddag. A.Heintz Telefonvakt: Anders Martinsson Tel.: MATEMATIK Datum: 20-08-25 Tid: eftermiddag GU, Chalmer Hjälpmedel: inga A.Heintz Telefonvakt: Ander Martinon Tel.: 073-07926. Löningar till tenta i ODE och matematik modellering, MMG5, MVE6. Define what

More information

The elementary spike produced by a pure e + e pair-electromagnetic pulse from a Black Hole: The PEM Pulse

The elementary spike produced by a pure e + e pair-electromagnetic pulse from a Black Hole: The PEM Pulse A&A 368, 377 39 (21) DOI: 1.151/4-6361:2556 ESO 21 Atronomy & Atrophyi The elementary pike produed by a pure e + e pair-eletromagneti pule from a Blak Hole: The PEM Pule C. L. Biano 1, R. Ruffini 1, and

More information

The Prime Number Theorem

The Prime Number Theorem he Prime Number heorem Yuuf Chebao he main purpoe of thee note i to preent a fairly readable verion of a proof of the Prime Number heorem (PN, epanded from Setion 7-8 of Davenport tet [3]. We intend to

More information

Tutorial 2 Euler Lagrange ( ) ( ) In one sentence: d dx

Tutorial 2 Euler Lagrange ( ) ( ) In one sentence: d dx Tutoril 2 Euler Lgrnge In one entene: d Fy = F d Importnt ft: ) The olution of EL eqution i lled eterml. 2) Minmum / Mimum of the "Mot Simple prolem" i lo n eterml. 3) It i eier to olve EL nd hek if we

More information

Chapter 4. The Laplace Transform Method

Chapter 4. The Laplace Transform Method Chapter 4. The Laplace Tranform Method The Laplace Tranform i a tranformation, meaning that it change a function into a new function. Actually, it i a linear tranformation, becaue it convert a linear combination

More information

Given the following circuit with unknown initial capacitor voltage v(0): X(s) Immediately, we know that the transfer function H(s) is

Given the following circuit with unknown initial capacitor voltage v(0): X(s) Immediately, we know that the transfer function H(s) is EE 4G Note: Chapter 6 Intructor: Cheung More about ZSR and ZIR. Finding unknown initial condition: Given the following circuit with unknown initial capacitor voltage v0: F v0/ / Input xt 0Ω Output yt -

More information

PHYS 110B - HW #6 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased

PHYS 110B - HW #6 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased PHYS B - HW #6 Spring 4, Solution by David Pace Any referenced equation are from Griffith Problem tatement are paraphraed. Problem. from Griffith Show that the following, A µo ɛ o A V + A ρ ɛ o Eq..4 A

More information

Dr G. I. Ogilvie Lent Term 2005

Dr G. I. Ogilvie Lent Term 2005 Aretion Diss Mathematial Tripos, Part III Dr G. I. Ogilvie Lent Term 2005 1.4. Visous evolution of an aretion dis 1.4.1. Introdution The evolution of an aretion dis is regulated by two onservation laws:

More information

INDIVIDUAL OVERTOPPING EVENTS AT DIKES

INDIVIDUAL OVERTOPPING EVENTS AT DIKES INDIVIDUAL OVEOPPING EVENS A DIKES Gij Boman 1, Jentje van der Meer 2, Gij offman 3, olger Shüttrumpf 4 and enk Jan Verhagen 5 eently, formulae have been derived for maximum flow depth and veloitie on

More information

Design Manual to EC2. LinkStudPSR. Version 3.1 BS EN : Specialists in Punching Shear Reinforcement.

Design Manual to EC2. LinkStudPSR. Version 3.1 BS EN : Specialists in Punching Shear Reinforcement. LinkStudPSR Speialit in Punhing Shear Reinforement Deign Manual to EC BS EN 199-1-1:004 Verion 3.1 January 018 LinkStud PSR Limited /o Brook Forging Ltd Doulton Road Cradley Heath Wet Midland B64 5QJ Tel:

More information

On the Stationary Convection of Thermohaline Problems of Veronis and Stern Types

On the Stationary Convection of Thermohaline Problems of Veronis and Stern Types Applied Mathemati, 00,, 00-05 doi:0.36/am.00.505 Publihed Online November 00 (http://www.sip.org/journal/am) On the Stationary Convetion of Thermohaline Problem of Veroni and Stern Type Abtrat Joginder

More information

p. (The electron is a point particle with radius r = 0.)

p. (The electron is a point particle with radius r = 0.) - pin ½ Recall that in the H-atom olution, we howed that the fact that the wavefunction Ψ(r) i ingle-valued require that the angular momentum quantum nbr be integer: l = 0,,.. However, operator algebra

More information

Nonlinear Dynamics of Single Bunch Instability in Accelerators *

Nonlinear Dynamics of Single Bunch Instability in Accelerators * SLAC-PUB-7377 Deember 996 Nonlinear Dynami of Single Bunh Intability in Aelerator * G. V. Stupakov Stanford Linear Aelerator Center Stanford Univerity, Stanford, CA 9439 B.N. Breizman and M.S. Pekker Intitute

More information

Astr 5465 Mar. 29, 2018 Galactic Dynamics I: Disks

Astr 5465 Mar. 29, 2018 Galactic Dynamics I: Disks Galati Dynamis Overview Astr 5465 Mar. 29, 2018 Subjet is omplex but we will hit the highlights Our goal is to develop an appreiation of the subjet whih we an use to interpret observational data See Binney

More information

Lecture 15 - Current. A Puzzle... Advanced Section: Image Charge for Spheres. Image Charge for a Grounded Spherical Shell

Lecture 15 - Current. A Puzzle... Advanced Section: Image Charge for Spheres. Image Charge for a Grounded Spherical Shell Lecture 15 - Current Puzzle... Suppoe an infinite grounded conducting plane lie at z = 0. charge q i located at a height h above the conducting plane. Show in three different way that the potential below

More information

Bernoulli s equation may be developed as a special form of the momentum or energy equation.

Bernoulli s equation may be developed as a special form of the momentum or energy equation. BERNOULLI S EQUATION Bernoulli equation may be developed a a pecial form of the momentum or energy equation. Here, we will develop it a pecial cae of momentum equation. Conider a teady incompreible flow

More information

Name Solutions to Test 1 September 23, 2016

Name Solutions to Test 1 September 23, 2016 Name Solutions to Test 1 September 3, 016 This test onsists of three parts. Please note that in parts II and III, you an skip one question of those offered. Possibly useful formulas: F qequb x xvt E Evpx

More information

PHYSICS 211 MIDTERM II 12 May 2004

PHYSICS 211 MIDTERM II 12 May 2004 PHYSIS IDTER II ay 004 Exa i cloed boo, cloed note. Ue only your forula heet. Write all wor and anwer in exa boolet. The bac of page will not be graded unle you o requet on the front of the page. Show

More information

Hylleraas wavefunction for He. dv 2. ,! r 2. )dv 1. in the trial function. A simple trial function that does include r 12. is ) f (r 12.

Hylleraas wavefunction for He. dv 2. ,! r 2. )dv 1. in the trial function. A simple trial function that does include r 12. is ) f (r 12. Hylleraa wavefunction for He The reaon why the Hartree method cannot reproduce the exact olution i due to the inability of the Hartree wave-function to account for electron correlation. We know that the

More information

Strauss PDEs 2e: Section Exercise 3 Page 1 of 13. u tt c 2 u xx = cos x. ( 2 t c 2 2 x)u = cos x. v = ( t c x )u

Strauss PDEs 2e: Section Exercise 3 Page 1 of 13. u tt c 2 u xx = cos x. ( 2 t c 2 2 x)u = cos x. v = ( t c x )u Strauss PDEs e: Setion 3.4 - Exerise 3 Page 1 of 13 Exerise 3 Solve u tt = u xx + os x, u(x, ) = sin x, u t (x, ) = 1 + x. Solution Solution by Operator Fatorization Bring u xx to the other side. Write

More information

ME 375 EXAM #1 Tuesday February 21, 2006

ME 375 EXAM #1 Tuesday February 21, 2006 ME 375 EXAM #1 Tueday February 1, 006 Diviion Adam 11:30 / Savran :30 (circle one) Name Intruction (1) Thi i a cloed book examination, but you are allowed one 8.5x11 crib heet. () You have one hour to

More information

Solving Radical Equations

Solving Radical Equations 10. Solving Radical Equation Eential Quetion How can you olve an equation that contain quare root? Analyzing a Free-Falling Object MODELING WITH MATHEMATICS To be proficient in math, you need to routinely

More information

Green s function for the wave equation

Green s function for the wave equation Green s funtion for the wave equation Non relativisti ase 1 The wave equations In the Lorentz Gauge, the wave equations for the potentials in Lorentz Gauge Gaussian units are: r 2 A 1 2 A 2 t = 4π 2 j

More information

Circular Motion Problem Solving

Circular Motion Problem Solving iula Motion Poblem Soling Aeleation o a hange in eloity i aued by a net foe: Newton nd Law An objet aeleate when eithe the magnitude o the dietion of the eloity hange We aw in the lat unit that an objet

More information

Reading assignment: In this chapter we will cover Sections Definition and the Laplace transform of simple functions

Reading assignment: In this chapter we will cover Sections Definition and the Laplace transform of simple functions Chapter 4 Laplace Tranform 4 Introduction Reading aignment: In thi chapter we will cover Section 4 45 4 Definition and the Laplace tranform of imple function Given f, a function of time, with value f(t

More information

Practice Problems - Week #7 Laplace - Step Functions, DE Solutions Solutions

Practice Problems - Week #7 Laplace - Step Functions, DE Solutions Solutions For Quetion -6, rewrite the piecewie function uing tep function, ketch their graph, and find F () = Lf(t). 0 0 < t < 2. f(t) = (t 2 4) 2 < t In tep-function form, f(t) = u 2 (t 2 4) The graph i the olid

More information

Math Skills. Scientific Notation. Uncertainty in Measurements. Appendix A5 SKILLS HANDBOOK

Math Skills. Scientific Notation. Uncertainty in Measurements. Appendix A5 SKILLS HANDBOOK ppendix 5 Scientific Notation It i difficult to work with very large or very mall number when they are written in common decimal notation. Uually it i poible to accommodate uch number by changing the SI

More information

The machines in the exercise work as follows:

The machines in the exercise work as follows: Tik-79.148 Spring 2001 Introduction to Theoretical Computer Science Tutorial 9 Solution to Demontration Exercie 4. Contructing a complex Turing machine can be very laboriou. With the help of machine chema

More information

Nonlinear Single-Particle Dynamics in High Energy Accelerators

Nonlinear Single-Particle Dynamics in High Energy Accelerators Nonlinear Single-Particle Dynamic in High Energy Accelerator Part 6: Canonical Perturbation Theory Nonlinear Single-Particle Dynamic in High Energy Accelerator Thi coure conit of eight lecture: 1. Introduction

More information

13.Prandtl-Meyer Expansion Flow

13.Prandtl-Meyer Expansion Flow 3.Prandtl-eyer Expansion Flow This hapter will treat flow over a expansive orner, i.e., one that turns the flow outward. But before we onsider expansion flow, we will return to onsider the details of the

More information

Advanced Computational Fluid Dynamics AA215A Lecture 4

Advanced Computational Fluid Dynamics AA215A Lecture 4 Advaned Computational Fluid Dynamis AA5A Leture 4 Antony Jameson Winter Quarter,, Stanford, CA Abstrat Leture 4 overs analysis of the equations of gas dynamis Contents Analysis of the equations of gas

More information

Universal Gravitation

Universal Gravitation Universal Gravitation Newton s Law of Universal Gravitation Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely

More information

V = 4 3 πr3. d dt V = d ( 4 dv dt. = 4 3 π d dt r3 dv π 3r2 dv. dt = 4πr 2 dr

V = 4 3 πr3. d dt V = d ( 4 dv dt. = 4 3 π d dt r3 dv π 3r2 dv. dt = 4πr 2 dr 0.1 Related Rate In many phyical ituation we have a relationhip between multiple quantitie, and we know the rate at which one of the quantitie i changing. Oftentime we can ue thi relationhip a a convenient

More information

Reading assignment: In this chapter we will cover Sections Definition and the Laplace transform of simple functions

Reading assignment: In this chapter we will cover Sections Definition and the Laplace transform of simple functions Chapter 4 Laplace Tranform 4 Introduction Reading aignment: In thi chapter we will cover Section 4 45 4 Definition and the Laplace tranform of imple function Given f, a function of time, with value f(t

More information

The gravitational phenomena without the curved spacetime

The gravitational phenomena without the curved spacetime The gravitational phenomena without the urved spaetime Mirosław J. Kubiak Abstrat: In this paper was presented a desription of the gravitational phenomena in the new medium, different than the urved spaetime,

More information

Constant Force: Projectile Motion

Constant Force: Projectile Motion Contant Force: Projectile Motion Abtract In thi lab, you will launch an object with a pecific initial velocity (magnitude and direction) and determine the angle at which the range i a maximum. Other tak,

More information

Impulse. calculate the impulse given to an object calculate the change in momentum as the result of an impulse

Impulse. calculate the impulse given to an object calculate the change in momentum as the result of an impulse Add Important Impule Page: 386 Note/Cue Here NGSS Standard: N/A Impule MA Curriculum Framework (2006): 2.5 AP Phyic 1 Learning Objective: 3.D.2.1, 3.D.2.2, 3.D.2.3, 3.D.2.4, 4.B.2.1, 4.B.2.2 Knowledge/Undertanding

More information

Relativity in Classical Physics

Relativity in Classical Physics Relativity in Classial Physis Main Points Introdution Galilean (Newtonian) Relativity Relativity & Eletromagnetism Mihelson-Morley Experiment Introdution The theory of relativity deals with the study of

More information

Solving Differential Equations by the Laplace Transform and by Numerical Methods

Solving Differential Equations by the Laplace Transform and by Numerical Methods 36CH_PHCalter_TechMath_95099 3//007 :8 PM Page Solving Differential Equation by the Laplace Tranform and by Numerical Method OBJECTIVES When you have completed thi chapter, you hould be able to: Find the

More information

Control Systems Analysis and Design by the Root-Locus Method

Control Systems Analysis and Design by the Root-Locus Method 6 Control Sytem Analyi and Deign by the Root-Locu Method 6 1 INTRODUCTION The baic characteritic of the tranient repone of a cloed-loop ytem i cloely related to the location of the cloed-loop pole. If

More information

The Hand of God, Building the Universe and Multiverse

The Hand of God, Building the Universe and Multiverse 1.0 Abtract What i the mathematical bai for the contruction of the univere? Thi paper intend to how a tart of how the univere i contructed. It alo anwer the quetion, did the hand of God build the univere?

More information

Sample Problems. Lecture Notes Related Rates page 1

Sample Problems. Lecture Notes Related Rates page 1 Lecture Note Related Rate page 1 Sample Problem 1. A city i of a circular hape. The area of the city i growing at a contant rate of mi y year). How fat i the radiu growing when it i exactly 15 mi? (quare

More information

Linear Momentum. calculate the momentum of an object solve problems involving the conservation of momentum. Labs, Activities & Demonstrations:

Linear Momentum. calculate the momentum of an object solve problems involving the conservation of momentum. Labs, Activities & Demonstrations: Add Important Linear Momentum Page: 369 Note/Cue Here NGSS Standard: HS-PS2-2 Linear Momentum MA Curriculum Framework (2006): 2.5 AP Phyic 1 Learning Objective: 3.D.1.1, 3.D.2.1, 3.D.2.2, 3.D.2.3, 3.D.2.4,

More information

V V V V. Vectors. Mike Bailey. Vectors have Direction and Magnitude. Magnitude: x y z. Computer Graphics.

V V V V. Vectors. Mike Bailey. Vectors have Direction and Magnitude. Magnitude: x y z. Computer Graphics. 1 Vector Mike Bailey mjb@c.oregontate.edu vector.pptx Vector have Direction and Magnitude Magnitude: V V V V x y z 1 Vector Can lo Be Defined a the Poitional Difference Between Two Point 3 ( x, y, z )

More information