# Physics 235 Chapter 5. Chapter 5 Gravitation

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus on specifying the popeties of the foce and its associated foce field, using the vecto notions we have intoduced in the pevious Chaptes. The Gavitational Foce The gavitational foce between two point-like paticles is popotional to the poduct of thei masses, and invesely popotional to the squae of the distance between them. The foce is always attactive, and diected along the line connection the two paticles. F = G mm ˆ The constant G is the gavitational constant, whose value is x Nm 2 /kg 2. This elation is known as Newton s law of univesal gavitation. The pinciple of supeposition allows us to calculate the foce on mass m due to multiple othe point-like masses M 1, M 2, : F = Gm n i=1 M i i 2 ˆ i whee i is the distance between mass M i and mass m. The pevious elation is coect only if both masses ae point-like objects. If one of the masses is a continuous, we must eplace the sum with an integal: F = Gm ( ) ˆ ' ' 2 ρ ' dv' The Gavitational Field The gavitational field geneated by the gavitational foce is defined in much the same way as we defined the electostatic field geneated by the electostatic foce: g = F m = G ( ) ˆ ' ' 2 ρ ' dv' - 1 -

2 The units of the gavitational field ae N/kg = m/s 2. The Gavitational Potential The gavitational potential is defined as the scala function Φ whose gadient is equal to the opposite of the gavitational field: g = Φ One way to detemine if we can find such a scala function is to calculate the cul of the gavitational field: g = Φ = ˆx ŷ ẑ x Φ x y Φ y z Φ z = 0 In the case of the gavitational field we find g = GM ˆ = GM 1 sinθ φ 1 ˆθ 1 θ 1 ˆφ = 0 and we conclude that thee is a scala function that can geneate the gavitational field. Since the gavitational field depends just on, we expect that the gavitational potential is also just a function of. The following scala function can geneate the gavitational field: Φ = G M This is the gavitational potential due to a point mass M. If we have a continuous mass distibution, the gavitational potential will be equal to Φ = G In this equation we have assumed that the constant of integation is equal to 0 (o that the gavitational potential is 0 at infinity). ( ) ' ρ ' dv' - 2 -

3 One of the easons that the gavitational potential is intoduced is that even fo extended mass distibutions, it is in geneal easie to calculate the gavitational potential (which is a scala) instead of the gavitational potential enegy. Once we have detemined the gavitational potential, we can detemine the gavitational foce by calculating the gadient of the gavitational potential. The Gavitational Potential Enegy The gavitational potential can be used to detemine the gavitational potential enegy U of a paticle of mass m. Recall fom Physics 121 o Physics 141 that the change in the potential enegy of an object, when it moves fom one position to anothe position, is the opposite of the wok done by the foces acting on the object. Conside an object that moves in the gavitational field of a point mass M located at the oigin of a coodinate system. The wok done on the object of unit mass by the gavitational foce is equal to dw = g d = ( Φ) d = i Φ dx i = dφ x i If we move the object of unit mass fom infinity to a specific position the wok done is equal to W ( ) = dφ = Φ( ) The wok done to move an object of mass m to this position is thus equal to W ( ) = mφ( ) The potential enegy of the object at this position is thus equal to U ( ) = mφ( ) Infomation about the foce on the object can be obtained by taking the gadient of the potential enegy: F( ) = U ( ) = G mm ˆ = G mm ˆ which is of couse equal to the gavitational foce we used as ou stating point

4 isualization of the Gavitational Potential We can visualize the gavitational potential in a numbe of diffeent ways. The most common ways ae the contou plot, showing equipotential sufaces, and 3D plots showing the gavitational potential as function of the two-dimensional position (x, y). A simple pogam that can be used to geneate such plots is gavitationalpotential which can be found in the Mathematica folde unde Computing Tools on ou website: (* Make a 3D plot of the gavitational potential due to two point masses of mass 1 and mass 4, located at (-2, -2) and (2, 2), espectively. *) Plot3D[(1/Sqt[(x + 2)^2 + (y + 2)^2]) + (4/Sqt[(x - 2)^2 + (y - 2)^2]), {x, -5, 5}, {y, -5, 5}, PlotRange -> {0, 7.5}, PlotPoints -> 50, Ticks -> {Automatic, Automatic, Automatic}] (* Make a 2D contou plot of the gavitational potential due to two point masses of mass 1 and mass 4, located at (-2, -2) and (2, 2), espectively. *) ContouPlot[(1/Sqt[(x + 2)^2 + (y + 2)^2]) + (4/ Sqt[(x - 2)^2 + (y - 2)^2]), {x, -5, 5}, {y, -5, 5}]; The plots that ae geneated using this pogam ae shown in Figue 1. Figue 1. Diffeent ways to visualize the gavitational potential due to two point masses of mass 1 and mass 3, located at (-2,-2) and (2,2), espectively

5 The Shell Theoem When we calculate the gavitational foce o the gavitational potential geneated by a mass distibution, we can always use the most geneal expession fo these quantities in tems of the volume integal ove the mass distibution. Howeve, if the mass distibution has spheical symmety, we can use the shell theoem to calculate the gavitation foce and the gavitational potential. The shell theoem states that: The gavitational potential at any point outside a spheically symmetic mass distibution is independent of the size of the distibution, and we can conside all of its mass to be concentated at the cente of the mass distibution. The gavitational potential is zeo at any point inside a spheically symmetic mass distibution. Using the shell theoem it is easy to calculate the gavitational potential and the gavitational foce due to a spheical shell, which is shown in Figue 2. Figue 2. The gavitational potential and the gavitational foce due to a spheical shell. The shell theoem can be used to make impotant pedictions about obital motion of planets aound the cental sta and sola systems aound the cente of a galaxy. The obsevation of the otational motion of the planets aound the sun let to the fist detemination of its mass. Since the sun is much moe massive than any of the planets in the sola system, the motion of the - 5 -

6 planets could be descibed in tems of the gavitational foce due to just the mass of the sun (see Figue 3. Figue 3. The obital velocity of the planets in ou sola system as function of thei distance fom the sun. The theoetical dependence, which depends on the mass of the sun, is shown by the solid cuve, and does an excellent job descibing the tend in the data. Figue 4. Measued obital velocity of stas as function of distance fom the cente of the galaxy. The otational cuve can only be explained if we assume that thee is halo of dak matte in univese, distibuted thoughout the galaxy. The sola systems in most galaxies cay out an obit aound the cente of the galaxy. Since it is assumed that a massive black hole is located in the cente of most galaxies, we expect to see a tend in the obital velocity vesus distance, simila to the tend seen in ou sola system (see - 6 -

7 Figue 3). In eality, we see a distibution that deceases at a much smalle ate as function of the distance fom the cente of the galaxy (see Figue 4). This implies that thee is moe mass in the galaxy than was assumed, but also that this exta mass is NOT located in the cente of the galaxy, but thoughout the galaxy. This exta matte, that we can not see diectly, is called dak matte and you teache is looking fo it in a mine in England. The Poisson Equation In Electicity and Magnetism we geatly benefited fom the Gauss law, which elated the electic flux though a closed suface with the total chage enclosed by that suface. Given the fact that the natue of the gavitational foce (its 1/ dependence) is simila to the natue of the electic foce, we expect that simila laws apply to the gavitational foce. Based on the appoach we took in Electicity and Magnetism, we define the gavitational flux due a point mass m in the following manne: Φ gav = S ( ˆn g)da whee S is an abitay suface suounding the mass m (see Figue 5). Using the definition of the gavitational field, we can ewite this equation as Φ gav = Gm S cosθ da π 2π cosθ = Gm sinθdθdφ = 2πGm cosθ sinθdθ = 4πGm θ=0 φ=0 π θ=0 Figue 5. Suface used to calculate gavitational flux associated with a point mass m

8 When we have a mass distibution inside the suface S we need to eplace m by a volume integal ove the mass distibution: Φ gav = 4πG ρdv The left-hand side of this equation can be ewitten using Gauss s divegence theoem: We thus conclude that fo any volume and thus Φ gav = ( ˆn g)da = ( g)dv S ( g)dv + 4πG ρdv = {( g) + 4πGρ}dv = 0 ( g) = 4πGρ This equation can also be expessed in tems of the gavitational potential and becomes o This equation is known as Poisson s equation. ( Φ) = 2 Φ = 4πGρ 2 Φ = 4πGρ The Tides We all know that the tides ae caused by the motion of the moon aound the eath, but most of us have a moe difficult time to explain why we have two high tides a day. Ou simple pictue would suggest that you have high tide on that side of the eath closest to the moon, but this would only explain one high tide a day. The calculation of the effect of the moon on the wate on eath is complicated by the fact that the eath is not a good inetial system, so we have to assume that we can find a good inetial fame in which we can descibe the foces acting on the wate (see Figue 6). Conside the foces on a volume of wate of mass m, located on the suface of the eath (see Figue 6), due to the moon and the eath: - 8 -

9 F m = m m = GmM E ˆ GmM m R 2 ˆR Figue 6. Geomety used to detemine the foces on a volume of wate of mass m, located on the suface of the eath. The foce exeted by the moon on the cente of the eath is equal to F E = M E E = GM m M E D 2 ˆD When we view the motion of the wate on eath, we view its motion with espect to the eath. We have used a coect inetial fame to detemine the acceleation of the individual components of ou system, and we can now tansfom to a efeence fame in which the eath is at est and centeed aound the oigin (note: this is a non-inetial efeence fame). In this efeence fame, the acceleation of mass m is equal to = m E = GM E ˆ GM m R 2 ˆR + GM m D 2 GM ˆD = E ˆR ˆ GM m R 2 ˆD D 2-9 -

10 The fist tem on the ight-hand side is just the gavitational foce on mass m due to the eath. It will be the same anywhee on the suface of the eath and is not esponsible fo the tides. The second pat of the ight-hand side is equal to the acceleation associated with the tidal foce. It is elated to the diffeence between the gavitational pull of the moon on the cente of the eath and on the suface of the eath. Let us conside what this equation tells us about the magnitude and the diection of the tidal foce at 4 diffeent positions on the suface of the eath (see Figue 6). Point a: At point a, the unit vectos associated with R and D ae pointing in the same diection. Since R > D, the (1/R 2 ) tems will be smalle than the (1/D 2 ) tem, and the tidal acceleation will be diected towads the ight: 1 T = GM m D 1 2 R 2 ˆD 1 = GM m D 1 2 D + ( ) 2 ˆD = GM m 1 1 D 2 1+ D 2 ˆD = 2GM m D 3 ˆD Point b: At point b, the unit vecto associated with R and D ae pointing in the same diection. But, since R < D, the (1/R 2 ) tems will be lage than the (1/D 2 ) tem, and the tidal acceleation will be diected towads the left but with the same magnitude as the acceleation at point a: 1 T = GM m D 1 2 R 2 ˆD 1 = GM m D 1 2 D ( ) 2 ˆD = GM m 1 1 D 2 1 D 2 ˆD = 2GM m D 3 ˆD Point c: The x components of the vectos associated with R and D ae simila, and the x components of the acceleation will cancel. The vecto associated with D will have no y component, and the acceleation will be due to the y component associated with R (which points towads the cente of the eath): ˆR T = GM m R 2 ˆD = GM m D 2 ˆR y R 2 = GM m 1 D 2 D ŷ = GM m D 3 ŷ Point d: The calculation fo the foce on point c is simila to the calculation of the foce on point a. The foce will be diected towads the cente of the eath and will be equal to T = GM m D 3 ŷ

11 We thus conclude that the acceleation of the wate on the suface of the eath is diected away fom the suface at two diffeent locations. It will thus be high tide at these two locations and since the moon otates aound the eath in one day, each location will see a high tide twice a day (one time when the moon that location is facing the moon, and one time when the moon is located on the opposite side of the eath fom that location). A summay of the tidal foces on the suface of the eath is shown in Figue 7. Figue 7. Tidal foces at vaious places on the suface of the eath

### Chapter 2. Electrostatics

Chapte. Electostatics.. The Electostatic Field To calculate the foce exeted by some electic chages,,, 3,... (the souce chages) on anothe chage Q (the test chage) we can use the pinciple of supeposition.

### The Role of Gravity in Orbital Motion

! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State

### PHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013

PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0

### Voltage ( = Electric Potential )

V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage

### Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

### A r. (Can you see that this just gives the formula we had above?)

24-1 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down - you can pedict (o contol) motion

### Analytical Proof of Newton's Force Laws

Analytical Poof of Newton s Foce Laws Page 1 1 Intouction Analytical Poof of Newton's Foce Laws Many stuents intuitively assume that Newton's inetial an gavitational foce laws, F = ma an Mm F = G, ae tue

### CHAPTER 5 GRAVITATIONAL FIELD AND POTENTIAL

CHATER 5 GRAVITATIONAL FIELD AND OTENTIAL 5. Intoduction. This chapte deals with the calculation of gavitational fields and potentials in the vicinity of vaious shapes and sizes of massive bodies. The

### Lesson 7 Gauss s Law and Electric Fields

Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual

### Forces & Magnetic Dipoles. r r τ = μ B r

Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent

### UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

### 2. Orbital dynamics and tides

2. Obital dynamics and tides 2.1 The two-body poblem This efes to the mutual gavitational inteaction of two bodies. An exact mathematical solution is possible and staightfowad. In the case that one body

### Structure and evolution of circumstellar disks during the early phase of accretion from a parent cloud

Cente fo Tubulence Reseach Annual Reseach Biefs 2001 209 Stuctue and evolution of cicumstella disks duing the ealy phase of accetion fom a paent cloud By Olusola C. Idowu 1. Motivation and Backgound The

### VISCOSITY OF BIO-DIESEL FUELS

VISCOSITY OF BIO-DIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use

### Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

### Chapter 4: Fluid Kinematics

Oveview Fluid kinematics deals with the motion of fluids without consideing the foces and moments which ceate the motion. Items discussed in this Chapte. Mateial deivative and its elationship to Lagangian

### 4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

### 10. Collisions. Before During After

10. Collisions Use conseation of momentum and enegy and the cente of mass to undestand collisions between two objects. Duing a collision, two o moe objects exet a foce on one anothe fo a shot time: -F(t)

### Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.

### Multiple choice questions [70 points]

Multiple choice questions [70 points] Answe all of the following questions. Read each question caefull. Fill the coect bubble on ou scanton sheet. Each question has exactl one coect answe. All questions

### Gravitational Mechanics of the Mars-Phobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning

Gavitational Mechanics of the Mas-Phobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This

### Excitation energies for molecules by Time-Dependent. based on Effective Exact Exchange Kohn-Sham potential

Excitation enegies fo molecules by Time-Dependent Density-Functional Theoy based on Effective Exact Exchange Kohn-Sham potential Fabio Della Sala National Nanotechnology Laboatoies Lecce Italy A. Göling

### PAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII - SPETO - 1995. pod patronatem. Summary

PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8 - TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC

Chapte 3 Is Gavitation A Results Of Asymmetic Coulomb Chage Inteactions? Jounal of Undegaduate Reseach èjurè Univesity of Utah è1992è, Vol. 3, No. 1, pp. 56í61. Jeæey F. Gold Depatment of Physics, Depatment

### Deflection of Electrons by Electric and Magnetic Fields

Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An

### CHAPTER 9 THE TWO BODY PROBLEM IN TWO DIMENSIONS

9. Intoduction CHAPTER 9 THE TWO BODY PROBLEM IN TWO DIMENSIONS In this chapte we show how Keple s laws can be deived fom Newton s laws of motion and gavitation, and consevation of angula momentum, and

### The Effect of Modified Gravity on Solar System Scales

The Effect of Modified Gavity on Sola System Scales Dane Pittock Physics Depatment Case Westen Reseve Univesity Cleveland, Ohio 44106 USA May 3, 013 Abstact Duing my senio poject, I have exploed the effects

### Skills Needed for Success in Calculus 1

Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

### Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing

M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow

### Phys 2101 Gabriela González. cos. sin. sin

1 Phys 101 Gabiela González a m t t ma ma m m T α φ ω φ sin cos α τ α φ τ sin m m α τ I We know all of that aleady!! 3 The figue shows the massive shield doo at a neuton test facility at Lawence Livemoe

### Relativistic Quantum Mechanics

Chapte Relativistic Quantum Mechanics In this Chapte we will addess the issue that the laws of physics must be fomulated in a fom which is Loentz invaiant, i.e., the desciption should not allow one to

### SELF-INDUCTANCE AND INDUCTORS

MISN-0-144 SELF-INDUCTANCE AND INDUCTORS SELF-INDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction.............................................. 1 A 2. Self-Inductance L.........................................

### Problem Set # 9 Solutions

Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease

### MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION

MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe- and subsolution method with

### est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.

9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,

### STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

### The Supply of Loanable Funds: A Comment on the Misconception and Its Implications

JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently Fields-Hat

### Complex Envelope Vectorization for the solution of mid-high frequency acoustic problems. A. Sestieri

Complex Envelope Vectoization fo the solution of mid-high fequency acoustic poblems A. Sestiei Depatment of Mechanical and Aeospace Engineeing Univesity of Rome la Sapienza Pesentation layout - Low fequency

### AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM

AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,

### Strength Analysis and Optimization Design about the key parts of the Robot

Intenational Jounal of Reseach in Engineeing and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Pint): 2320-9356 www.ijes.og Volume 3 Issue 3 ǁ Mach 2015 ǁ PP.25-29 Stength Analysis and Optimization Design

### INVESTIGATION OF FLOW INSIDE AN AXIAL-FLOW PUMP OF GV IMP TYPE

1 INVESTIGATION OF FLOW INSIDE AN AXIAL-FLOW PUMP OF GV IMP TYPE ANATOLIY A. YEVTUSHENKO 1, ALEXEY N. KOCHEVSKY 1, NATALYA A. FEDOTOVA 1, ALEXANDER Y. SCHELYAEV 2, VLADIMIR N. KONSHIN 2 1 Depatment of

### ON THE (Q, R) POLICY IN PRODUCTION-INVENTORY SYSTEMS

ON THE R POLICY IN PRODUCTION-INVENTORY SYSTEMS Saifallah Benjaafa and Joon-Seok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poduction-inventoy

### 2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

### Define What Type of Trader Are you?

Define What Type of Tade Ae you? Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 1 Disclaime and Risk Wanings Tading any financial maket involves isk. The content of this

### Risk Sensitive Portfolio Management With Cox-Ingersoll-Ross Interest Rates: the HJB Equation

Risk Sensitive Potfolio Management With Cox-Ingesoll-Ross Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,

### Impulse and Linear Momentum 5

Implse and Linea Momentm 5 How does jet poplsion wok? How can yo mease the speed of a bllet? Wold a meteoite collision significantly change Eath s obit? In pevios chaptes we discoveed that the pshing inteaction

### Construction of semi-dynamic model of subduction zone with given plate kinematics in 3D sphere

Eath Planets Space, 62, 665 673, 2010 Constuction of semi-dynamic model of subduction zone with given plate kinematics in 3D sphee M. Moishige 1, S. Honda 1, and P. J. Tackley 2 1 Eathquake Reseach Institute,

### www.sakshieducation.com

Viscosity. The popety of viscosity in gas is due to ) Cohesive foces between the moecues ) Coisions between the moecues ) Not having a definite voume ) Not having a definite size. When tempeatue is inceased

### Continuous Compounding and Annualization

Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

DYNAMIS AND STRUTURAL LOADING IN WIND TURBINES M. Ragheb 12/30/2008 INTRODUTION The loading egimes to which wind tubines ae subject to ae extemely complex equiing special attention in thei design, opeation

### Efficient Redundancy Techniques for Latency Reduction in Cloud Systems

Efficient Redundancy Techniques fo Latency Reduction in Cloud Systems 1 Gaui Joshi, Emina Soljanin, and Gegoy Wonell Abstact In cloud computing systems, assigning a task to multiple seves and waiting fo

### Characterization of particles and particle systems

W. Pabst / E. Gegoová Chaacteization of paticles and paticle systems ICT Pague 007 Tyto studijní mateiály vznikly v ámci pojektu FRVŠ 674 / 007 F1 / b Tvoba předmětu Chaakteizace částic a částicových soustav.

### Channel selection in e-commerce age: A strategic analysis of co-op advertising models

Jounal of Industial Engineeing and Management JIEM, 013 6(1):89-103 Online ISSN: 013-0953 Pint ISSN: 013-843 http://dx.doi.og/10.396/jiem.664 Channel selection in e-commece age: A stategic analysis of

### Questions for Review. By buying bonds This period you save s, next period you get s(1+r)

MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the two-peiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume

### Chapter 2 Modelling of Fluid Flow and Heat Transfer in Rotating-Disk Systems

Chapte 2 Modelling of Fluid Flow and Heat Tansfe in Rotating-Disk Systems 2.1 Diffeential and Integal Equations 2.1.1 Diffeential Navie Stokes and Enegy Equations We will conside hee stationay axisymmetic

### Comparing Availability of Various Rack Power Redundancy Configurations

Compaing Availability of Vaious Rack Powe Redundancy Configuations White Pape 48 Revision by Victo Avela > Executive summay Tansfe switches and dual-path powe distibution to IT equipment ae used to enhance

### NBER WORKING PAPER SERIES FISCAL ZONING AND SALES TAXES: DO HIGHER SALES TAXES LEAD TO MORE RETAILING AND LESS MANUFACTURING?

NBER WORKING PAPER SERIES FISCAL ZONING AND SALES TAXES: DO HIGHER SALES TAXES LEAD TO MORE RETAILING AND LESS MANUFACTURING? Daia Bunes David Neumak Michelle J. White Woking Pape 16932 http://www.nbe.og/papes/w16932

### INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS

INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in

### Data Center Demand Response: Avoiding the Coincident Peak via Workload Shifting and Local Generation

(213) 1 28 Data Cente Demand Response: Avoiding the Coincident Peak via Wokload Shifting and Local Geneation Zhenhua Liu 1, Adam Wieman 1, Yuan Chen 2, Benjamin Razon 1, Niangjun Chen 1 1 Califonia Institute

### How do investments in heat pumps affect household energy consumption?

Discussion Papes Statistics Noway Reseach depatment No. 737 Apil 203 Bente Halvosen and Bodil Meethe Lasen How do investments in heat pumps affect household enegy consumption? Discussion Papes No. 737,

### Comparing Availability of Various Rack Power Redundancy Configurations

Compaing Availability of Vaious Rack Powe Redundancy Configuations By Victo Avela White Pape #48 Executive Summay Tansfe switches and dual-path powe distibution to IT equipment ae used to enhance the availability

### Explicit, analytical solution of scaling quantum graphs. Abstract

Explicit, analytical solution of scaling quantum gaphs Yu. Dabaghian and R. Blümel Depatment of Physics, Wesleyan Univesity, Middletown, CT 06459-0155, USA E-mail: ydabaghian@wesleyan.edu (Januay 6, 2003)

### Concept and Experiences on using a Wiki-based System for Software-related Seminar Papers

Concept and Expeiences on using a Wiki-based System fo Softwae-elated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wth-aachen.de,

### Chapter 2 Coulomb s Law

Chapte Coulomb s Law.1 lectic Chage...-3. Coulomb's Law...-3 Animation.1: Van de Gaaff Geneato...-4.3 Pinciple of Supeposition...-5 xample.1: Thee Chages...-5.4 lectic Field...-7 Animation.: lectic Field

### Valuation of Floating Rate Bonds 1

Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY PRINCIPLES OF CARDIAC ELECTROPHYSIOLOGY

Havad-MIT Division of Health Sciences and Technology HST.542J: Quantitative Physiology: Ogan Tanspot Systems Instucto: Roge Mak MASSACHUSETTS INSTITUTE OF TECHNOLOGY Pof. Roge G. Mak, 2004 Depatments of

### STABILITY ANALYSIS IN MILLING BASED ON OPERATIONAL MODAL DATA 1. INTRODUCTION

Jounal of Machine Engineeing, Vol. 11, No. 4, 211 Batosz POWALKA 1 Macin CHODZKO 1 Kzysztof JEMIELNIAK 2 milling, chatte, opeational modal analysis STABILITY ANALYSIS IN MILLING BASED ON OPERATIONAL MODAL

### Instituto Superior Técnico Av. Rovisco Pais, 1 1049-001 Lisboa E-mail: virginia.infante@ist.utl.pt

FATIGUE LIFE TIME PREDICTIO OF POAF EPSILO TB-30 AIRCRAFT - PART I: IMPLEMETATIO OF DIFERET CYCLE COUTIG METHODS TO PREDICT THE ACCUMULATED DAMAGE B. A. S. Seano 1, V. I. M.. Infante 2, B. S. D. Maado

### Tracking/Fusion and Deghosting with Doppler Frequency from Two Passive Acoustic Sensors

Tacking/Fusion and Deghosting with Dopple Fequency fom Two Passive Acoustic Sensos Rong Yang, Gee Wah Ng DSO National Laboatoies 2 Science Pak Dive Singapoe 11823 Emails: yong@dso.og.sg, ngeewah@dso.og.sg

### Software Engineering and Development

I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining

### Chris J. Skinner The probability of identification: applying ideas from forensic statistics to disclosure risk assessment

Chis J. Skinne The pobability of identification: applying ideas fom foensic statistics to disclosue isk assessment Aticle (Accepted vesion) (Refeeed) Oiginal citation: Skinne, Chis J. (2007) The pobability

### CONCEPTUAL FRAMEWORK FOR DEVELOPING AND VERIFICATION OF ATTRIBUTION MODELS. ARITHMETIC ATTRIBUTION MODELS

CONCEPUAL FAMEOK FO DEVELOPING AND VEIFICAION OF AIBUION MODELS. AIHMEIC AIBUION MODELS Yui K. Shestopaloff, is Diecto of eseach & Deelopment at SegmentSoft Inc. He is a Docto of Sciences and has a Ph.D.

### A Capacitated Commodity Trading Model with Market Power

A Capacitated Commodity Tading Model with Maket Powe Victo Matínez-de-Albéniz Josep Maia Vendell Simón IESE Business School, Univesity of Navaa, Av. Peason 1, 08034 Bacelona, Spain VAlbeniz@iese.edu JMVendell@iese.edu

### Intertemporal Macroeconomics

Intetempoal Macoeconomics Genot Doppelhofe* May 2009 Fothcoming in J. McCombie and N. Allington (eds.), Cambidge Essays in Applied Economics, Cambidge UP This chapte eviews models of intetempoal choice

### Modeling and Verifying a Price Model for Congestion Control in Computer Networks Using PROMELA/SPIN

Modeling and Veifying a Pice Model fo Congestion Contol in Compute Netwoks Using PROMELA/SPIN Clement Yuen and Wei Tjioe Depatment of Compute Science Univesity of Toonto 1 King s College Road, Toonto,

### Supplementary Material for EpiDiff

Supplementay Mateial fo EpiDiff Supplementay Text S1. Pocessing of aw chomatin modification data In ode to obtain the chomatin modification levels in each of the egions submitted by the use QDCMR module

### An Epidemic Model of Mobile Phone Virus

An Epidemic Model of Mobile Phone Vius Hui Zheng, Dong Li, Zhuo Gao 3 Netwok Reseach Cente, Tsinghua Univesity, P. R. China zh@tsinghua.edu.cn School of Compute Science and Technology, Huazhong Univesity

### METHODOLOGICAL APPROACH TO STRATEGIC PERFORMANCE OPTIMIZATION

ETHODOOGICA APPOACH TO STATEGIC PEFOANCE OPTIIZATION ao Hell * Stjepan Vidačić ** Željo Gaača *** eceived: 4. 07. 2009 Peliminay communication Accepted: 5. 0. 2009 UDC 65.02.4 This pape pesents a matix

### Electricity transmission network optimization model of supply and demand the case in Taiwan electricity transmission system

Electicity tansmission netwok optimization model of supply and demand the case in Taiwan electicity tansmission system Miao-Sheng Chen a Chien-Liang Wang b,c, Sheng-Chuan Wang d,e a Taichung Banch Gaduate

### Electric Potential. otherwise to move the object from initial point i to final point f

PHY2061 Enched Physcs 2 Lectue Notes Electc Potental Electc Potental Dsclame: These lectue notes ae not meant to eplace the couse textbook. The content may be ncomplete. Some topcs may be unclea. These

### Strategic Betting for Competitive Agents

Stategic Betting fo Competitive Agents Liad Wagman Depatment of Economics Duke Univesity Duham, NC, USA liad.wagman@duke.edu Vincent Conitze Depts. of Compute Science and Economics Duke Univesity Duham,

### Ilona V. Tregub, ScD., Professor

Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation

### An Infrastructure Cost Evaluation of Single- and Multi-Access Networks with Heterogeneous Traffic Density

An Infastuctue Cost Evaluation of Single- and Multi-Access Netwoks with Heteogeneous Taffic Density Andes Fuuskä and Magnus Almgen Wieless Access Netwoks Eicsson Reseach Kista, Sweden [andes.fuuska, magnus.almgen]@eicsson.com

### Things to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.

Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to

NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA THESIS CONTOUR TRACKING CONTROL FOR THE REMUS AUTONOMOUS UNDERWATER VEHICLE by Alan Robet Van Reet June 2005 Thesis Adviso: Anthony Healey Appoved fo public

### Load Balancing in Processor Sharing Systems

Load Balancing in ocesso Shaing Systems Eitan Altman INRIA Sophia Antipolis 2004, oute des Lucioles 06902 Sophia Antipolis, Fance altman@sophia.inia.f Utzi Ayesta LAAS-CNRS Univesité de Toulouse 7, Avenue

### Load Balancing in Processor Sharing Systems

Load Balancing in ocesso Shaing Systems Eitan Altman INRIA Sophia Antipolis 2004, oute des Lucioles 06902 Sophia Antipolis, Fance altman@sophia.inia.f Utzi Ayesta LAAS-CNRS Univesité de Toulouse 7, Avenue

### Summary: Vectors. This theorem is used to find any points (or position vectors) on a given line (direction vector). Two ways RT can be applied:

Summ: Vectos ) Rtio Theoem (RT) This theoem is used to find n points (o position vectos) on given line (diection vecto). Two ws RT cn e pplied: Cse : If the point lies BETWEEN two known position vectos

### UNIVERSIDAD DE CANTABRIA TESIS DOCTORAL

UNIVERSIDAD DE CANABRIA Depatamento de Ingenieía de Comunicaciones ESIS DOCORAL Cyogenic echnology in the Micowave Engineeing: Application to MIC and MMIC Vey Low Noise Amplifie Design Juan Luis Cano de

### THE DISTRIBUTED LOCATION RESOLUTION PROBLEM AND ITS EFFICIENT SOLUTION

IADIS Intenational Confeence Applied Computing 2006 THE DISTRIBUTED LOCATION RESOLUTION PROBLEM AND ITS EFFICIENT SOLUTION Jög Roth Univesity of Hagen 58084 Hagen, Gemany Joeg.Roth@Fenuni-hagen.de ABSTRACT

### Avoided emissions kgco 2eq /m 2 Reflecting surfaces 130

ANALYSIS OF GLOBAL WARMING MITIGATION BY WHITE REFLECTING SURFACES Fedeico Rossi, Andea Nicolini Univesity of Peugia, CIRIAF Via G.Duanti 67 0615 Peugia, Italy T: +9-075-585846; F: +9-075-5848470; E: fossi@unipg.it

### The Incidence of Social Security Taxes in Economies with Partial. Compliance: Evidence from the SS Reform in Mexico

The Incidence of Social Secuity Taxes in Economies ith Patial Compliance: Evidence fom the SS Refom in Mexico Gecia M. Maufo Abstact Looking at impovements in social secuity benefits in Mexico, this pape

### Numerical Simulation of Coal Boiler at Electric Thermal Plants Using Computational Fluid Dynamics

10th Intenational Symposium on ocess Systems Engineeing - SE2009 Rita Maia de Bito Alves, Claudio Augusto Olle do Nascimento and Evaisto Chalbaud Biscaia J. (Editos) 2009 Elsevie B.V. All ights eseved.

### 3 Molecules in Electric and Magnetic Fields

Chapte, page Molecules in Electic and Magnetic Fields. Basic Equations fom Electodynamics The basis of the desciption of the behaviou of molecules in electic and magnetic fields ae the mateial equations

### Hip Hop solutions of the 2N Body problem

Hip Hop solutions of the N Boy poblem Esthe Baabés baabes@ima.ug.es Depatament Infomàtica i Matemàtica Aplicaa, Univesitat e Giona. Josep Maia Cos cos@eupm.upc.es Depatament e Matemàtica Aplicaa III, Univesitat

### The impact of migration on the provision. of UK public services (SRG.10.039.4) Final Report. December 2011

The impact of migation on the povision of UK public sevices (SRG.10.039.4) Final Repot Decembe 2011 The obustness The obustness of the analysis of the is analysis the esponsibility is the esponsibility

### A Web Application for Geothermal Borefield Design

Poceedings Wold Geothemal Congess 205 Melboune, Austalia, 9-25 Apil 205 A Web Application fo Geothemal Boefield Design Davide Rolando,2, José Acuna and Maco Fossa 2 KT Royal Institute of Technology, Binellvägen

### YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH

nd INTERNATIONAL TEXTILE, CLOTHING & ESIGN CONFERENCE Magic Wold of Textiles Octobe 03 d to 06 th 004, UBROVNIK, CROATIA YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH Jana VOBOROVA; Ashish GARG; Bohuslav