|
|
- Jeffrey Gold
- 8 years ago
- Views:
Transcription
1 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 of Mathematics Univesity of Utah Abstact Many attempts have been made to equate gavitational foces with manifestations of othe phenomena. In these emaks we exploe the consequences of fomulating gavitational foces as asymmetic Coulomb chage inteactions. This is contay to some established theoies, fo the model pedicts diæeential acceleations dependent on the elemental composition of the test mass. The pedicted diæeentials of acceleation of vaious elemental masses ae compaed to those diæeentials that have been obtained expeimentally. Although the model tuns out to fail, the constuction of this model is a useful intellectual and pedagogical execise. 1
2 CHAPTER 3. ASYMMETRIC COULOMB CHARGE INTERACTIONS 2 Intoduction The similaities in the expessions fo Newtonian gavitation and Coulomb electostatic inteactions, ç ~Fg =,G m1m2 ^ ~F 2 c = k q1q2 ^ 2 ae appaent, paticulaly the invese-squae natue of both foces and the binay poduct of the masses and chages. Based on this similaity, and the physicists' goal to unify all fundamental foces of natue, it is easonable to conjectue that gavitation is some manifestation of electostatic phenomena. With these consideations in mind, we constuct a simple gavity model; howeve, the model fails. The model pedicts diæeential acceleations dependent on the elemental composition of the test mass. This is in keeping with vaious Fifth-Foce èhypechageè hypotheses and e-evaluations of the Eíotvíos expeiments which postulate the existence of non-newtonian and bayon-dependent foces ove shot anges. Such hypotheses ae pesently contovesial, at best. T.M. Niebaue et al. have shown that the diæeential acceleation æg, of two diæeent mateials, is less than 5 pats in of the gavitational acceleation g = cmès 2. Expeiments veifying the Equivalence Pinciple have been pefomed by Dicke et al., demonstating that gavitational mass and inetial mass ae equivalent to within 1 pat in These expeiments ule out Fifth- Foce hypotheses as well as ou simple gavity model. Ou model will show that the calculated diæeential acceleation æg calc, fo the same mateials, is on the ode of 8.37 cmès 2, in clea contadiction to the limits established by the expeiments of Niebaue et al. The oiginal motivation fo this fomulation of gavitation was to calculate the appoximate change æk, of the electostatic constant k = 1è4çæ 0 = 8:9876 æ10 9 Nm 2 èc 2, in an asymmetic Coulomb chage inteaction, necessay to account fo gavitational foces. By this we mean that foces between like chages ae expessed as ~F 1;2 = k q 1q 2 ^ ; è3.1è 2 while foces between unlike chages ae expessed as ~F 1;2 =èk + ækè q 1q 2 2 ^ ; è3.2è with æk é 0. We will show that æk=k is on the ode of 10,37. The idea of expessing gavity as a Coulomb inteaction is not new; it had been exploed by Wilhelm Webe è1804í1891è of Gíottingen and Fiedich Zíollne è1834í1882è of Leipzig. Because Webe and Zíollne pefomed thei ëgedanken" expeiment befoe the discovey of atomic paticles, they wee unable to foesee the consequences of combining chages in a densely packed nucleus. In these emaks we takeinto account the mass defects due to the binding enegies of nuclei.
3 CHAPTER 3. ASYMMETRIC COULOMB CHARGE INTERACTIONS 3 Asymmetic Coulomb Chage Inteactions Suppose two spheical masses, m 1 and m 2, ae sepaated by a distance connecting thei centes of mass. Each massm i is composed of p i positive chages and e i negative chages. Let F ++ denote the foce between the positive chages of mass m 1 and the positive chages of mass m 2 ; then the thee emaining foces F +,, F,+, and F,, ae labelled appopiately to epesent thetype of inteaction between the chages of m 1 and m 2. We expess the positive fundamental chage as q and negative fundamental chage as,q in ode to maintain the pope oientation of these cental foces ë1ë. The foces on one of the masses ae 8 ~F ++ =,k q2 p 1p 2 ^ é ~F,, 2 =,k q2 e 1e 2 ^ 2 ~F +, = èk + ækè é: q2 p 1e 2 è3.3è ^ 2 ~F,+ = èk + ækè q2 e 1p 2 ^ : 2 If we combine these foces ë2ë, ~F tot =,k q2 èp 1 p 2 + e 1 e 2 è 2 o, upon eaanging the tems, we ænd that ~F tot =,k q2 èp 1, e 1 èèp 2, e 2 è 2 ^ +èk + ækè q2 èp 1 e 2 + e 1 p 2 è 2 ^ ; è3.4è ^ + æk q2 èp 1 e 2 + e 1 p 2 è 2 ^ : è3.5è An examination of the æst tem of equation è3.5è eveals that it is a efomulation of the Coulomb foce equation. Note that this tem vanishes if eithe mass is electically neutal. The last tem, æk q2 èp 1 e 2 + e 1 p 2 è 2 ^ ; è3.6è does not change sign, egadless of the p i and e i ; like gavity, itis always attactive. Next we wish to ænd what the p i and e i ae fo each mass. This is simpliæed if we assume that m 1 and m 2 ae electically neutal masses. This implies that e 1 = p 1 and e 2 = p 2, and that ~F =2æk q2 p 1 p 2 2 ^ : è3.7è When chages ae combined, a vey small faction of the constituent paticle masses is lost to the binding enegy; this is efeed to as a mass defect. We
4 CHAPTER 3. ASYMMETRIC COULOMB CHARGE INTERACTIONS 4 denote the actual mass of an atom by m i, and its mass defect by B i. elationships of these quantities to the constituent paticle masses ae m 1 + B 1 = Z 1 èm p + m e è+n 1 m n m 2 + B 2 = Z 2 èm p + m e è+n 2 m n : The è3.8è Hee Z i epesents the numbe of potons, and n i epesents the numbe of neutons contained in the electically neutal mass m i ;thequantities m p, m e, and m n epesent the masses of potons, electons, and neutons, espectively. It is cucial to ou model that each neuton is composed of one positive and one negative chage as indicated in the disintegation: n! p + e, + ç e,. This implies that p 1 = Z 1 + n 1 è3.9è p 2 = Z 2 + n 2 : The numbe of neutons can be expessed as a function of the numbe of potons ë3ë, that is, n i = ç i Z i, whee ç i =èa i, Z i è=z i. Substituting ç 1 Z 1 fo n 1 and ç 2 Z 2 fo n 2 in equations è3.9è, we ænd p 1 = Z 1 è1 + ç 1 è p 2 = Z 2 è1 + ç 2 è : è3.10è By eplacing n i with ç i Z i in equations è3.8è, we have m 1 + B 1 = Z 1 èm p + m e + ç 1 m n è m 2 + B 2 = Z 2 èm p + m e + ç 2 m n è : è3.11è Solving fo Z i in equations è3.11è, and substituting these, in tun, into equations è3.10è, we have p 1 = èm1+b1èè1+ç1è èm p+m e+ç 1m nè è3.12è p 2 = èm2+b2èè1+ç2è èm : p+m e+ç 2m nè Finally, equation è3.7è becomes ~F = èm p + m e è 2 ç è1 + ç1 èè1 + ç 2 èè1 + ç 1 èè1 + ç 2 è è1 + ç 1 æèè1 + ç 2 æè whee ç i = Bi m i, ç i = Ni,Zi Z i,andæ= mn m p+m e =1: ç m1 m 2 2 ^ ; è3.13è If we assume the binding enegies ae negligible, that is, ç i =0,and that æ ç 1, then equation è3.13è becomes ~F = èm p + m e è 2 m 1 m 2 2 ^ : è3.14è
5 CHAPTER 3. ASYMMETRIC COULOMB CHARGE INTERACTIONS 5 This allows us to equate the constant, G =6:672 æ 10,11 Nm 2 èkg 2,toænd èm p+m eè 2, with the gavitational constant, æk = G æ èm p + m e è 2,27 Nm2 =3:639 æ 10 : è3.15è 2q 2 C 2 Dividing this value by k, we conclude that ækèk ç 10,37. Diæeential Acceleations Unlike Newtonian o Einsteinian gavitation, ou model pedicts small diæeences in the acceleations of diæeent test masses. These elative diæeences in g ae dependent on the mass defects B i, and the neutonèpoton atios, ç i. The foce exeted by the eath, M eath, on a test mass m 1,is ~F = èm p + m e è 2 ç è1 + çeath èè1 + ç 1 èè1 + ç eath èè1 + ç 1 è ç Meath m 1 2 ^ : è1 + ç eath æè è1 + ç 1 æè è3.16è If anothe test mass, m 2, of diæeing composition, is acceleating unde the same conditions, the atio of the acceleations of masses m 1 and m 2 is j~a 1 j j~a 2 j = ç çç 1+ç1 1+ç1 1+ç 2 1+ç 2 çç ç 1+ç2 æ 1+ç 1 æ : è3.17è In the feefall expeiment ofniebaueet al., the test masses epesented extemes in elemental composition. Onetestmasswas 40.0 gams of coppe è 63 29Cuè and the othe was gams of depleted Uanium è Uè. The following table fo ç i and ç i is deived fom these data ë4ë: Table I. Coppe è 63 29Cuè Uanium è 238 test mass èmè 40.0 gm gm atomic mass èm a è amu amu ç ç Uè Substituting the values of Table èiè into equation è3.17è, we ænd j~a 1 j j~a 2 j =1:00854 ; è3.18è so that æg calc =g = Next we constuct the following table fo the elements aluminum èalè and gold èauè used in the expeiments of Dicke et al.
6 CHAPTER 3. ASYMMETRIC COULOMB CHARGE INTERACTIONS 6 Table II. Aluminum è 27 13Alè Gold è Auè atomic mass èm a è amu amu ç ç The diæeential acceleation of aluminum and gold pedicted by ou model is æg calc èg = Both values of æg calc, namely cmès 2 and cmès 2, fa exceed the limit æg é4:9033 æ 10,7 cmès 2 expeimentally found by Niebaue et al. Conclusion The model pedicts diæeential acceleations that cannot be econciled with those obtained by expeiment. This does not peclude some elation o uniæcation of gavitation and electomagnetism, but it does show that the simple model of Coulomb asymmety is not viable. Acknowledgements I thank Pofesso Richad H. Pice fo the immense assistance povided in the pepaation of this manuscipt. Thanks ae also extended to Vince Fedeick fo his assistance in ænding vaious bibliogaphic souces. Special thanks ae extended to Pofesso Don H. Tucke fo the suppot I have eceived fo this and many othe pojects. Notes 1. The magnitudes of the fundamental positive and negative chages ae equivalent to within 1 pat in Note that we have not taken into account any Coulombic shielding that would occu. 3. Typical values of ç =èa, Zè=Z ç 1í1:5, whee A is the bayon numbe and Z is the atomic èo potonè numbe. 4. 1amu èatomic mass unitè = 1:661 æ 10,24 gams.
7 CHAPTER 3. ASYMMETRIC COULOMB CHARGE INTERACTIONS 7 Bibliogaphy Emsley, J. è1989è The Elements, èpp. 12í13, pp. 54í55, pp. 78í79, pp. 202í203è. Claendon Pess: Oxfod. Eíotvíos, R. V., Pekça, D., and Fekete, E. è1922è Ann. Phys. èleipzigè 68, 11. Fischbach, E., Sudasky, D., Szafe, A., Talmadge, C., and Aonson, S. è1986è Phys. Rev. Lett. 56, 3. King, J. G. è1960è Phys. Rev. Lett. 5, 562. Kittel, C., Knight, W., Rudeman, M., Helmholz, A., and Moye, B. è1973è Mechanics, Bekeley Physics Couse, Volume 1, Second Edition, èpp. 398í408è. McGaw-Hill Book Company, Inc.:New Yok. Livesey, D. L. è1966è Atomic and Nuclea Physics, èpp è. Blaisdell Publishing Company: Massachusetts. Niebaue, T., McHugh, M., and Falle, J. è1987è Phys. Rev. Lett. 59, 6. Petley, B.W.è1988èThe Fundamental Physical Constants and the Fontie of Measuement, èpp. 282í287è. Adam Hilga: Bistol. Pucell, E. M. è1985è Electicity and Magnetism, Bekeley Physics Couse, Volume 2, Second Edition, èpp. 5í6è. McGaw-Hill Book Company, Inc.: New Yok. Roll, P. G., Kotkov, R., and Dicke, R. H. è1964è Ann. Phys. èn.y.è 26, 442. Stubbs, C., Adelbege, E., Raab, F., Gundlach, J., Heckel, B., McMuy, K., Swanson, H., and Watanabe, R. è1987è Phys. Rev. Lett. 58, 11. Thiebege, P. è1987è Phys. Rev. Lett. 58, 11. Wheele, J. A. è1990è A Jouney into Gavity and Spacetime, èpp. 24í27è. Scientiæc Ameican Libay: New Yok. Whittake, E. è1960è A Histoy of the Theoies of Aethe & Electicity, Vol. II, èpp. 144í196è. Hape & Bothes: New Yok.
FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.
Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing
More information(a) The centripetal acceleration of a point on the equator of the Earth is given by v2. The velocity of the earth can be found by taking the ratio of
Homewok VI Ch. 7 - Poblems 15, 19, 22, 25, 35, 43, 51. Poblem 15 (a) The centipetal acceleation of a point on the equato of the Eath is given by v2. The velocity of the eath can be found by taking the
More informationThe 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
More informationPHYSICS 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
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
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
More informationThe force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges
The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee
More informationEpisode 401: Newton s law of universal gravitation
Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce
More informationDeflection 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
More informationCharges, Coulomb s Law, and Electric Fields
Q&E -1 Chages, Coulomb s Law, and Electic ields Some expeimental facts: Expeimental fact 1: Electic chage comes in two types, which we call (+) and ( ). An atom consists of a heavy (+) chaged nucleus suounded
More informationMagnetic Field and Magnetic Forces. Young and Freedman Chapter 27
Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field
More informationContinuous 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
More informationPAN 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
More informationChapter 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,
More information1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2
Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the
More informationConcept 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,
More informationest 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,
More informationFinancing Terms in the EOQ Model
Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad
More informationSTUDENT 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
More informationVoltage ( = 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
More informationGravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2
F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,
More informationINITIAL 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
More informationLab #7: Energy Conservation
Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 1-4 Intoduction: Pehaps one of the most unusual
More informationComparing 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
More informationPY1052 Problem Set 8 Autumn 2004 Solutions
PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what
More informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation
More informationF G r. Don't confuse G with g: "Big G" and "little g" are totally different things.
G-1 Gavity Newton's Univesal Law of Gavitation (fist stated by Newton): any two masses m 1 and m exet an attactive gavitational foce on each othe accoding to m m G 1 This applies to all masses, not just
More informationON 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
More informationChapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom
Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in
More informationChapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6
Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe
More informationVISCOSITY 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
More informationComparing 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
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationExplicit, 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)
More informationVoltage ( = Electric Potential )
V-1 Voltage ( = Electic 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 is
More informationStructure 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
More informationNUCLEAR MAGNETIC RESONANCE
19 Jul 04 NMR.1 NUCLEAR MAGNETIC RESONANCE In this expeiment the phenomenon of nuclea magnetic esonance will be used as the basis fo a method to accuately measue magnetic field stength, and to study magnetic
More informationMechanics 1: Motion in a Central Force Field
Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.
More informationSELF-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.........................................
More informationChapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.
Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium-39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming
More informationGravitation. AP Physics C
Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What
More informationExam 3: Equation Summary
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P
More informationGravitational 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
More information7 Circular Motion. 7-1 Centripetal Acceleration and Force. Period, Frequency, and Speed. Vocabulary
7 Cicula Motion 7-1 Centipetal Acceleation and Foce Peiod, Fequency, and Speed Vocabulay Vocabulay Peiod: he time it takes fo one full otation o evolution of an object. Fequency: he numbe of otations o
More informationAN 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,
More informationAn Introduction to Omega
An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom
More informationDisplacement, Velocity And Acceleration
Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,
More informationVector 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
More informationExperiment 6: Centripetal Force
Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee
More informationThe 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
More informationChannel 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
More information883 Brochure A5 GENE ss vernis.indd 1-2
ess x a eu / u e a. p o.eu c e / :/ http EURAXESS Reseaches in Motion is the gateway to attactive eseach caees in Euope and to a pool of wold-class eseach talent. By suppoting the mobility of eseaches,
More informationResearch on Risk Assessment of the Transformer Based on Life Cycle Cost
ntenational Jounal of Smat Gid and lean Enegy eseach on isk Assessment of the Tansfome Based on Life ycle ost Hui Zhou a, Guowei Wu a, Weiwei Pan a, Yunhe Hou b, hong Wang b * a Zhejiang Electic Powe opoation,
More informationAn Efficient Group Key Agreement Protocol for Ad hoc Networks
An Efficient Goup Key Ageement Potocol fo Ad hoc Netwoks Daniel Augot, Raghav haska, Valéie Issany and Daniele Sacchetti INRIA Rocquencout 78153 Le Chesnay Fance {Daniel.Augot, Raghav.haska, Valéie.Issany,
More informationCarter-Penrose diagrams and black holes
Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example
More informationGravity. A. Law of Gravity. Gravity. Physics: Mechanics. A. The Law of Gravity. Dr. Bill Pezzaglia. B. Gravitational Field. C.
Physics: Mechanics 1 Gavity D. Bill Pezzaglia A. The Law of Gavity Gavity B. Gavitational Field C. Tides Updated: 01Jul09 A. Law of Gavity 3 1a. Invese Squae Law 4 1. Invese Squae Law. Newton s 4 th law
More informationMULTIPLE 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
More informationAnalytical 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
More informationTrading Volume and Serial Correlation in Stock Returns in Pakistan. Abstract
Tading Volume and Seial Coelation in Stock Retuns in Pakistan Khalid Mustafa Assistant Pofesso Depatment of Economics, Univesity of Kaachi e-mail: khalidku@yahoo.com and Mohammed Nishat Pofesso and Chaiman,
More informationChapter 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.
More informationMultiband Microstrip Patch Antenna for Microwave Applications
IOSR Jounal of Electonics and Communication Engineeing (IOSR-JECE) ISSN: 2278-2834, ISBN: 2278-8735. Volume 3, Issue 5 (Sep. - Oct. 2012), PP 43-48 Multiband Micostip Patch Antenna fo Micowave Applications
More informationEpdf Sulf petroleum, Eflecti and Eeflecti
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
More information2. 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
More informationLesson 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
More informationRelativistic 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
More informationOn Some Functions Involving the lcm and gcd of Integer Tuples
SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 6, 2 (2014), 91-100. On Some Functions Involving the lcm and gcd of Intege Tuples O. Bagdasa Abstact:
More informationQuestions & 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
More informationModal Characteristics study of CEM-1 Single-Layer Printed Circuit Board Using Experimental Modal Analysis
Available online at www.sciencediect.com Pocedia Engineeing 41 (2012 ) 1360 1366 Intenational Symposium on Robotics and Intelligent Sensos 2012 (IRIS 2012) Modal Chaacteistics study of CEM-1 Single-Laye
More informationThe LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero.
Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the
More informationExperiment MF Magnetic Force
Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuent-caying conducto is basic to evey electic moto -- tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating
More informationGauss Law. Physics 231 Lecture 2-1
Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More informationInstituto 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
More informationThe Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = -W/q 0 1V [Volt] =1 Nm/C
Geneal Physics - PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit
More informationTORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION
MISN-0-34 TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION shaft TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION by Kiby Mogan, Chalotte, Michigan 1. Intoduction..............................................
More informationDYNAMICS AND STRUCTURAL LOADING IN WIND TURBINES
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
More informationNontrivial lower bounds for the least common multiple of some finite sequences of integers
J. Numbe Theoy, 15 (007), p. 393-411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to
More information12.1. FÖRSTER RESONANCE ENERGY TRANSFER
ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 1-1 1.1. FÖRSTER RESONNCE ENERGY TRNSFER Föste esonance enegy tansfe (FR) efes to the nonadiative tansfe of an electonic excitation fom a dono molecule to
More informationCONCEPTUAL 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.
More informationThings 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
More informationMining Relatedness Graphs for Data Integration
Mining Relatedness Gaphs fo Data Integation Jeemy T. Engle (jtengle@indiana.edu) Ying Feng (yingfeng@indiana.edu) Robet L. Goldstone (goldsto@indiana.edu) Indiana Univesity Bloomington, IN. 47405 USA Abstact
More informationUNIT 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: + = - - -
More informationThe Gravity Field of the Earth - Part 1 (Copyright 2002, David T. Sandwell)
1 The Gavity Field of the Eath - Pat 1 (Copyight 00, David T. Sandwell) This chapte coves physical geodesy - the shape of the Eath and its gavity field. This is just electostatic theoy applied to the Eath.
More informationPhysics HSC Course Stage 6. Space. Part 1: Earth s gravitational field
Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe
More informationLoyalty Rewards and Gift Card Programs: Basic Actuarial Estimation Techniques
Loyalty Rewads and Gift Cad Pogams: Basic Actuaial Estimation Techniques Tim A. Gault, ACAS, MAAA, Len Llaguno, FCAS, MAAA and Matin Ménad, FCAS, MAAA Abstact In this pape we establish an actuaial famewok
More informationFigure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!
1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the
More informationCHAPTER 10 Aggregate Demand I
CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income
More informationSTABILITY 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
More information4a 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
More informationCHAPTER 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
More informationSkills 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
More informationMETHODOLOGICAL 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
More information9:6.4 Sample Questions/Requests for Managing Underwriter Candidates
9:6.4 INITIAL PUBLIC OFFERINGS 9:6.4 Sample Questions/Requests fo Managing Undewite Candidates Recent IPO Expeience Please povide a list of all completed o withdawn IPOs in which you fim has paticipated
More informationThe transport performance evaluation system building of logistics enterprises
Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194-114 Online ISSN: 213-953 Pint ISSN: 213-8423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics
More informationA 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
More informationConverting knowledge Into Practice
Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading
More informationDatabase Management Systems
Contents Database Management Systems (COP 5725) D. Makus Schneide Depatment of Compute & Infomation Science & Engineeing (CISE) Database Systems Reseach & Development Cente Couse Syllabus 1 Sping 2012
More informationUnveiling the MPLS Structure on Internet Topology
Unveiling the MPLS Stuctue on Intenet Topology Gabiel Davila Revelo, Mauicio Andeson Ricci, Benoit Donnet, José Ignacio Alvaez-Hamelin INTECIN, Facultad de Ingenieía, Univesidad de Buenos Aies Agentina
More informationModule Availability at Regent s School of Drama, Film and Media Autumn 2016 and Spring 2017 *subject to change*
Availability at Regent s School of Dama, Film and Media Autumn 2016 and Sping 2017 *subject to change* 1. Choose you modules caefully You must discuss the module options available with you academic adviso/
More informationThe 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
More informationPromised Lead-Time Contracts Under Asymmetric Information
OPERATIONS RESEARCH Vol. 56, No. 4, July August 28, pp. 898 915 issn 3-364X eissn 1526-5463 8 564 898 infoms doi 1.1287/ope.18.514 28 INFORMS Pomised Lead-Time Contacts Unde Asymmetic Infomation Holly
More informationAn application of stochastic programming in solving capacity allocation and migration planning problem under uncertainty
An application of stochastic pogamming in solving capacity allocation and migation planning poblem unde uncetainty Yin-Yann Chen * and Hsiao-Yao Fan Depatment of Industial Management, National Fomosa Univesity,
More information