Gravity: Newtonian, post-newtonian, Relativistic
|
|
- Robert Bradley
- 5 years ago
- Views:
Transcription
1 Gavity: Newtonian, post-newtonian, Cous de l IP Institut d stophysique Pais, 19 Septembe 24 Octobe 2016 Cliff Relativistic Pofesso Cliffod Will Distinguished Pofesso of Physics Univesity of Floida Checheu ssocié Institut d stophysique de Pais Bueau 118E cmw@phys.ufl.edu
2 Outline of the Lectues 1. Newtonian gavity (2 lectues) Foundations, Isolated gavitating bodies, Obital dynamics PW, Chaptes 1, 3 2. Post-Minkowskian theoy (0.5 lectue) Fomulation, Implementation PW, Chaptes Post-Newtonian theoy: Nea-zone physics (2 lectues) Implementation, PN celestial mechanics, astomety & timekeeping PW, Chaptes Post-Newtonian theoy: Fa-zone physics (1.5 lectues) Gavitational adiation, Radiation Reaction PW, Chaptes Textbook: Gavity: Newtonian, post-newtonian, Geneal Relativistic, by Eic Poisson and Cliffod Will (Cambidge U Pess, 2014) Lectue slides (in pdf) and selected PW chaptes available at
3 Foundations of Newtonian Gavity Newton s 2nd law and the law of gavitation: m I a = F The pinciple of equivalence: a = m G GM m I 3 If F = Gm G M/ 3 m G = m I (1 + ) Then, compaing the acceleation of two diffeent bodies o mateials a = a 1 a 2 = ( 1 2 ) GM 3
4 The Weak Equivalence Pinciple (WEP) 400 CE Ioannes Philiponus: let fall fom the same height two weights of which one is many times as heavy as the othe. the diffeence in time is a vey small one 1553 Giambattista Benedetti poposed equality 1586 Simon Stevin expeiments Galileo Galilei Leaning Towe of Pisa? Newton pendulum expeiments 1889, 1908 Baon R. von Eötvös tosion balance expeiments (10-9 ) UW (Eöt-Wash) tom intefeometes matte waves vs macoscopic object Bodies fall in a gavitational field with an acceleation that is independent of mass, composition o intenal stuctue
5 Tests of the Weak Equivalence Pinciple 10-8 Eötvös Matte waves 10-9 Renne Fee-fall Pinceton Boulde Fifth-foce seaches Eöt-Wash POLLO (LLR) Micoscope (2015) Futue: STEP, GG, STE-QUEST Moscow LLR Eöt-Wash a 1 -a 2 (a 1 +a 2 )/ YER OF EXPERIMENT
6 Newtonian equations of Hydodynamics Witing Equation of motion Field equation Genealize to multiple souces (sum ove M s) and ma = mu, U = GM/, dv dt = U + ( v) =0, p, 2 U = 4 G, d + v, p = p(, T,... ) Fomal solution of Poisson s field equation: Wite U(t, x) =G G(x, x 0 ) (t, x 0 )d 3 x 0, Eule equation of motion Continuity equation Poisson field equation Total o Lagangian deivative Equation of state Geen function 2 G(x, x 0 )= 4 (x x 0 ) ) G(x, x 0 )=1/ x x 0 U(t, x) =G (t, x 0 ) x x 0 d3 x 0
7 Rules of the oad Consequences of the continuity equation: fo any f(x,t): d (t, x)f(t, x) d 3 x + d 3 f ( v) d 3 I + v f d 3 x f v = df dt d3 x. (t, x 0 )f(t, x, x 0 ) d 3 x 0 + v0 0 f d 3 x 0, (t, x 0 )f(t, x, x 0 ) d 3 x 0 = 0 + v f + v0 0 f d 3 x 0 = 0 df dt d3 x 0
8 Global consevation laws M := (t, x) d 3 x = constant P := (t, x)v d 3 x = constant d( V)+pdV =0 v = V 1 dv/dt E := T (t)+ (t)+e int (t) = constant J := x v d 3 x = constant T (t) := 1 2 (t) := E int (t) := v 2 d 3 x 1 2 G d 3 x 0 x x 0 d3 x 0 d 3 x, R(t) := 1 M (t, x)x d 3 x = P M (t t 0)+R 0 d dt vd 3 x = ( U = G =0 p) d 3 x 0 x x 0 x x 0 3 d3 xd 3 x 0 I pnd 2 S
9 Spheical and nealy spheical bodies Spheical symmety @ = 4 G = Gm(t, ) 2 m(t, ) := 0 4 (t, 0 ) 02 d 0 U(t, ) = Gm(t, ) +4 G R (t, 0 ) 0 d 0. Outside the body U = GM/
10 Spheical and nealy spheical bodies Non-spheical bodies: the extenal field x 0 < x Taylo expansion: 1 x x 0 = 1 1X = `=0 1 x j x0j x k ( 1)` 1 x L `! Then the Newtonian potential outside the body becomes 1X ( 1)` 1 U ext (t, x) =G I hli, `! `=0 I hli (t) := (t, x 0 )x 0hLi d 3 x 0 x L := x i x j...(l L j...(l times) h...i := symmetic tacefee poduct
11 Symmetic tacefee (STF) tensos hijk...i Symmetic on all indices, and ij hijk...i =0 Example: gadients of j 1 = n j jk 1 = 3n j n k jk 3, jkn 1 = h15n j n k n n 3 n j kn + n k jn + n n jk L 1 hli 1 =( 1)`(2` 1)!! n hli `+1 Geneal fomula fo n <L> : n hli = [`/2] X p (2` 2p 1)!! h ( 1) 2P n +sym(q)i L 2P (2` 1)!! p=0 q := `!/[(` 2p)!(2p)!!]
12 Symmetic tacefee (STF) tensos Link between n <L> and spheical hamonics e hli n hli = n hli := `! (2` 1)!! P`(e n) 4 `! (2` + 1)!! `X m= ` Y hli `m Y`m(, ) Y hzi 10 = 3 Y hxxi 20 = 4, 5 16, Yhxi 3 11 = 8, Yhyyi 20 = 5 16, 3 Yhyi 11 = i 8, Yhzzi 20 =2 5 16, veage of n <L> ove a sphee: 8 hhn L ii := 1 I 1 < L/2 n L (`+1)!! +sym[(` 1)!!] ` =even d = 4 : 0 ` =odd
13 Spheical and nealy spheical bodies Example: axially symmetic body I hli = m RÀ(J`) e hli J` := 4 2` +1 1 MR` (t, x)`y `0(, e ) d 3 x U ext (t, x) = GM " 1 1X `=2 J` R ` P`(cos ) # Note that: J 2 = C MR 2 C
14 Measuing the Eath s multipole moments Gavity Recovey nd Climate Expeiment (GRCE) Eath: j (5) X = X 10, j (1) X = 9.6 X 10-7, j (4) X = 5.4 X 10-7, j 360,0 = 3.2 (5) X mgal 10-6 g
15 Motion of extended fluid bodies Main assumptions: Bodies small compaed to typical sepaation (R << ) isolated -- no mass flow T int ~ (R 3 /Gm) 1/2 << T ob ~ ( 3 /Gm) 1/2 -- quasi equilibium adiabatic esponse to tidal defomations -- nealy spheical Extenal poblem: detemine motions of bodies as functions (o functionals) of intenal paametes Intenal poblem: given motions, detemine evolution of intenal paametes Solve the two poblems self-consistently o iteatively Example: Eath-Moon system -- obital motion aises tides, tidally defomed fields affect motions
16 Motion of extended fluid bodies Basic definitions m := (t, x) d 3 x (t) := 1 (t, x)x d 3 x m dm /dt =0 v (t) := d dt a (t) := dv dt = 1 m = 1 m Is the cente of mass unique? pue convenience, should not wande outside the body not physically measuable almost impossible to define in GR m a = G G 0 x x 0 x x 0 3 d3 xd 3 x X 0 x x 0 B6= B x x 0 3 d3 x 0 5 d 3 x v d 3 x dv dt d3 x Define: x := (t)+ x x 0 := B (t)+ x 0 B := B
17 N-body point mass tems Motion of extended fluid bodies a j = G X B6= + 1X `=2 + 1 m ( m B 2 B n j B 1 h ( 1)`I hli B `! 1X 1X `=2 `0=2 ( 1)`0 `!`0! Moments of othe bodies + m i B I hli jl m B I hli i 1 IhL0 jll 0 B ) Effect of body s own moments Two-body system with only body 2 having non-zeo I <L> := 1 2, := R := (m m 2 2 )/m Moment-moment inteaction tems m := m 1 + m 2 µ := m 1 m 2 /m a j = Gm 2 nj + Gm 1X `=2 ( 1)` `! I hli 2 m jl 1
Gravity: Newtonian, post-newtonian, Relativistic
Gavity: Newtonian, post-newtonian, Relativistic A Mini-couse within PH 7608 Cliff Pofesso Cliffod Will Distinguished Pofesso of Physics Univesity of Floida Checheu Associé Institut d Astophysique de Pais
More informationIntroduction to GR: Newtonian Gravity
Introduction to GR: Newtonian Gravity General Relativity @99 DPG Physics School Physikzentrum Bad Honnef, 14 19 Sept 2014 Clifford Will Distinguished Professor of Physics University of Florida Chercheur
More informationOSCILLATIONS AND GRAVITATION
1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,
More informationMechanics Physics 151
Mechanics Physics 151 Lectue 5 Cental Foce Poblem (Chapte 3) What We Did Last Time Intoduced Hamilton s Pinciple Action integal is stationay fo the actual path Deived Lagange s Equations Used calculus
More informationExtra notes for circular motion: Circular motion : v keeps changing, maybe both speed and
Exta notes fo cicula motion: Cicula motion : v keeps changing, maybe both speed and diection ae changing. At least v diection is changing. Hence a 0. Acceleation NEEDED to stay on cicula obit: a cp v /,
More informationChapter 13 Gravitation
Chapte 13 Gavitation In this chapte we will exploe the following topics: -Newton s law of gavitation, which descibes the attactive foce between two point masses and its application to extended objects
More informationGravitation. AP/Honors Physics 1 Mr. Velazquez
Gavitation AP/Honos Physics 1 M. Velazquez Newton s Law of Gavitation Newton was the fist to make the connection between objects falling on Eath and the motion of the planets To illustate this connection
More informationDiffusion and Transport. 10. Friction and the Langevin Equation. Langevin Equation. f d. f ext. f () t f () t. Then Newton s second law is ma f f f t.
Diffusion and Tanspot 10. Fiction and the Langevin Equation Now let s elate the phenomena of ownian motion and diffusion to the concept of fiction, i.e., the esistance to movement that the paticle in the
More informationd 2 x 0a d d =0. Relative to an arbitrary (accelerating frame) specified by x a = x a (x 0b ), the latter becomes: d 2 x a d 2 + a dx b dx c
Chapte 6 Geneal Relativity 6.1 Towads the Einstein equations Thee ae seveal ways of motivating the Einstein equations. The most natual is pehaps though consideations involving the Equivalence Pinciple.
More informationDEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS LSN 10-: MOTION IN A GRAVITATIONAL FIELD Questions Fom Reading Activity? Gavity Waves? Essential Idea: Simila appoaches can be taken in analyzing electical
More informationPhysics 111. Ch 12: Gravity. Newton s Universal Gravity. R - hat. the equation. = Gm 1 m 2. F g 2 1. ˆr 2 1. Gravity G =
ics Announcements day, embe 9, 004 Ch 1: Gavity Univesal Law Potential Enegy Keple s Laws Ch 15: Fluids density hydostatic equilibium Pascal s Pinciple This week s lab will be anothe physics wokshop -
More informationChapter 5 Force and Motion
Chapte 5 Foce and Motion In Chaptes 2 and 4 we have studied kinematics, i.e., we descibed the motion of objects using paametes such as the position vecto, velocity, and acceleation without any insights
More informationPractice. Understanding Concepts. Answers J 2. (a) J (b) 2% m/s. Gravitation and Celestial Mechanics 287
Pactice Undestanding Concepts 1. Detemine the gavitational potential enegy of the Eath Moon system, given that the aveage distance between thei centes is 3.84 10 5 km, and the mass of the Moon is 0.0123
More informationDEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS TSOKOS LESSON 10-1 DESCRIBING FIELDS Essential Idea: Electic chages and masses each influence the space aound them and that influence can be epesented
More information7.2. Coulomb s Law. The Electric Force
Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat
More informationGravitation. Chapter 12. PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman. Lectures by James Pazun
Chapte 12 Gavitation PowePoint Lectues fo Univesity Physics, Twelfth Edition Hugh D. Young and Roge A. Feedman Lectues by James Pazun Modified by P. Lam 5_31_2012 Goals fo Chapte 12 To study Newton s Law
More informationChapter 5 Force and Motion
Chapte 5 Foce and Motion In chaptes 2 and 4 we have studied kinematics i.e. descibed the motion of objects using paametes such as the position vecto, velocity and acceleation without any insights as to
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 informationUniversal Gravitation
Chapte 1 Univesal Gavitation Pactice Poblem Solutions Student Textbook page 580 1. Conceptualize the Poblem - The law of univesal gavitation applies to this poblem. The gavitational foce, F g, between
More informationChapter 4. Newton s Laws of Motion
Chapte 4 Newton s Laws of Motion 4.1 Foces and Inteactions A foce is a push o a pull. It is that which causes an object to acceleate. The unit of foce in the metic system is the Newton. Foce is a vecto
More informationHW Solutions # MIT - Prof. Please study example 12.5 "from the earth to the moon". 2GmA v esc
HW Solutions # 11-8.01 MIT - Pof. Kowalski Univesal Gavity. 1) 12.23 Escaping Fom Asteoid Please study example 12.5 "fom the eath to the moon". a) The escape velocity deived in the example (fom enegy consevation)
More informationVainshtein mechanism in second-order scalar-tensor theories
Vainshtein mechanism in second-ode scala-tenso theoies A. De Felice, R. Kase, S. Tsujikawa Phys. Rev. D 85 044059 (202)! Tokyo Univesity of Science Antonio De Felice, Ryotao Kase and Shinji Tsujikawa Motivation
More information1 Spherical multipole moments
Jackson notes 9 Spheical multipole moments Suppose we have a chage distibution ρ (x) wheeallofthechageiscontained within a spheical egion of adius R, as shown in the diagam. Then thee is no chage in the
More informationA thermodynamic degree of freedom solution to the galaxy cluster problem of MOND. Abstract
A themodynamic degee of feedom solution to the galaxy cluste poblem of MOND E.P.J. de Haas (Paul) Nijmegen, The Nethelands (Dated: Octobe 23, 2015) Abstact In this pape I discus the degee of feedom paamete
More informationPendulum in Orbit. Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ (December 1, 2017)
1 Poblem Pendulum in Obit Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ 08544 (Decembe 1, 2017) Discuss the fequency of small oscillations of a simple pendulum in obit, say,
More informationOur Universe: GRAVITATION
Ou Univese: GRAVITATION Fom Ancient times many scientists had shown geat inteest towads the sky. Most of the scientist studied the motion of celestial bodies. One of the most influential geek astonomes
More informationProblems with Mannheim s conformal gravity program
Poblems with Mannheim s confomal gavity pogam June 4, 18 Youngsub Yoon axiv:135.163v6 [g-qc] 7 Jul 13 Depatment of Physics and Astonomy Seoul National Univesity, Seoul 151-747, Koea Abstact We show that
More informationGalilean Transformation vs E&M y. Historical Perspective. Chapter 2 Lecture 2 PHYS Special Relativity. Sep. 1, y K K O.
PHYS-2402 Chapte 2 Lectue 2 Special Relativity 1. Basic Ideas Sep. 1, 2016 Galilean Tansfomation vs E&M y K O z z y K In 1873, Maxwell fomulated Equations of Electomagnetism. v Maxwell s equations descibe
More informationII. Electric Field. II. Electric Field. A. Faraday Lines of Force. B. Electric Field. C. Gauss Law. 1. Sir Isaac Newton ( ) A.
II. Electic Field D. Bill Pezzaglia II. Electic Field. Faaday Lines of Foce B. Electic Field C. Gauss Law Updated 08Feb010. Lines of Foce 1) ction at a Distance ) Faaday s Lines of Foce ) Pinciple of Locality
More information= 4 3 π( m) 3 (5480 kg m 3 ) = kg.
CHAPTER 11 THE GRAVITATIONAL FIELD Newton s Law of Gavitation m 1 m A foce of attaction occus between two masses given by Newton s Law of Gavitation Inetial mass and gavitational mass Gavitational potential
More informationSolution of a Spherically Symmetric Static Problem of General Relativity for an Elastic Solid Sphere
Applied Physics eseach; Vol. 9, No. 6; 7 ISSN 96-969 E-ISSN 96-9647 Published by Canadian Cente of Science and Education Solution of a Spheically Symmetic Static Poblem of Geneal elativity fo an Elastic
More informationGeneral momentum equation
PY4A4 Senio Sophiste Physics of the Intestella and Integalactic Medium Lectue 11: Collapsing Clouds D Gaham M. Hape School of Physics, TCD Geneal momentum equation Du u P Dt uu t 1 B 4 B 1 B 8 Lagangian
More informationNewton s Laws, Kepler s Laws, and Planetary Orbits
Newton s Laws, Keple s Laws, and Planetay Obits PROBLEM SET 4 DUE TUESDAY AT START OF LECTURE 28 Septembe 2017 ASTRONOMY 111 FALL 2017 1 Newton s & Keple s laws and planetay obits Unifom cicula motion
More informationPhysics 312 Introduction to Astrophysics Lecture 7
Physics 312 Intoduction to Astophysics Lectue 7 James Buckley buckley@wuphys.wustl.edu Lectue 7 Eath/Moon System Tidal Foces Tides M= mass of moon o sun F 1 = GMm 2 F 2 = GMm ( + ) 2 Diffeence in gavitational
More informationPressure Calculation of a Constant Density Star in the Dynamic Theory of Gravity
Pessue Calculation of a Constant Density Sta in the Dynamic Theoy of Gavity Ioannis Iaklis Haanas Depatment of Physics and Astonomy Yok Univesity A Petie Science Building Yok Univesity Toonto Ontaio CANADA
More information17.1 Electric Potential Energy. Equipotential Lines. PE = energy associated with an arrangement of objects that exert forces on each other
Electic Potential Enegy, PE Units: Joules Electic Potential, Units: olts 17.1 Electic Potential Enegy Electic foce is a consevative foce and so we can assign an electic potential enegy (PE) to the system
More informationProblems with Mannheim s conformal gravity program
Poblems with Mannheim s confomal gavity pogam Abstact We show that Mannheim s confomal gavity pogam, whose potential has a tem popotional to 1/ and anothe tem popotional to, does not educe to Newtonian
More informationTo Feel a Force Chapter 7 Static equilibrium - torque and friction
To eel a oce Chapte 7 Chapte 7: Static fiction, toque and static equilibium A. Review of foce vectos Between the eath and a small mass, gavitational foces of equal magnitude and opposite diection act on
More informationCh 13 Universal Gravitation
Ch 13 Univesal Gavitation Ch 13 Univesal Gavitation Why do celestial objects move the way they do? Keple (1561-1630) Tycho Bahe s assistant, analyzed celestial motion mathematically Galileo (1564-1642)
More informationBut for simplicity, we ll define significant as the time it takes a star to lose all memory of its original trajectory, i.e.,
Stella elaxation Time [Chandasekha 1960, Pinciples of Stella Dynamics, Chap II] [Ostike & Davidson 1968, Ap.J., 151, 679] Do stas eve collide? Ae inteactions between stas (as opposed to the geneal system
More informationDEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS TSOKOS LESSON 6- THE LAW OF GRAVITATION Essential Idea: The Newtonian idea of gavitational foce acting between two spheical bodies and the laws of mechanics
More informationAaa Hal ARC 103 Haq Mou Hill 114 Mug Seh PHY LH Sen Zzz SEC 111
The Fist Midtem!!! The fist common hou midtem exam will be held on Thusday Octobe 5, 9:50 to 11:10 PM (at night) on the Busch campus. You should go to the oom coesponding to the fist 3 lettes of you last
More informationChapter 4. Newton s Laws of Motion. Newton s Law of Motion. Sir Isaac Newton ( ) published in 1687
Chapte 4 Newton s Laws of Motion 1 Newton s Law of Motion Si Isaac Newton (1642 1727) published in 1687 2 1 Kinematics vs. Dynamics So fa, we discussed kinematics (chaptes 2 and 3) The discussion, was
More informationA New Approach to General Relativity
Apeion, Vol. 14, No. 3, July 7 7 A New Appoach to Geneal Relativity Ali Rıza Şahin Gaziosmanpaşa, Istanbul Tukey E-mail: aizasahin@gmail.com Hee we pesent a new point of view fo geneal elativity and/o
More informationUnit 6 Test Review Gravitation & Oscillation Chapters 13 & 15
A.P. Physics C Unit 6 Test Review Gavitation & Oscillation Chaptes 13 & 15 * In studying fo you test, make sue to study this eview sheet along with you quizzes and homewok assignments. Multiple Choice
More information3. Electromagnetic Waves II
Lectue 3 - Electomagnetic Waves II 9 3. Electomagnetic Waves II Last time, we discussed the following. 1. The popagation of an EM wave though a macoscopic media: We discussed how the wave inteacts with
More informationis the instantaneous position vector of any grid point or fluid
Absolute inetial, elative inetial and non-inetial coodinates fo a moving but non-defoming contol volume Tao Xing, Pablo Caica, and Fed Sten bjective Deive and coelate the govening equations of motion in
More informationRigid Body Dynamics 2. CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2018
Rigid Body Dynamics 2 CSE169: Compute Animation nstucto: Steve Rotenbeg UCSD, Winte 2018 Coss Poduct & Hat Opeato Deivative of a Rotating Vecto Let s say that vecto is otating aound the oigin, maintaining
More informationF 12. = G m m 1 2 F 21 = F 12. = G m 1m 2. Review. Physics 201, Lecture 22. Newton s Law Of Universal Gravitation
Physics 201, Lectue 22 Review Today s Topics n Univesal Gavitation (Chapte 13.1-13.3) n Newton s Law of Univesal Gavitation n Popeties of Gavitational Foce n Planet Obits; Keple s Laws by Newton s Law
More informationkg 2 ) 1.9!10 27 kg = Gm 1
Section 6.1: Newtonian Gavitation Tutoial 1 Pactice, page 93 1. Given: 1.0 10 0 kg; m 3.0 10 0 kg;. 10 9 N; G 6.67 10 11 N m /kg Requied: Analysis: G m ; G m G m Solution: G m N m 6.67!10 11 kg ) 1.0!100
More informationF(r) = r f (r) 4.8. Central forces The most interesting problems in classical mechanics are about central forces.
4.8. Cental foces The most inteesting poblems in classical mechanics ae about cental foces. Definition of a cental foce: (i) the diection of the foce F() is paallel o antipaallel to ; in othe wods, fo
More informationGravity: Newtonian, post-newtonian Relativistic
Gravity: Newtonian, post-newtonian Relativistic X Mexican School on Gravitation & Mathematical Physics Playa del Carmen, 1 5 December, 2014 Clifford Will Distinguished Professor of Physics University of
More informationAdvanced Newtonian gravity
Foundations of Newtonian gravity Solutions Motion of extended bodies, University of Guelph h treatment of Newtonian gravity, the book develops approximation methods to obtain weak-field solutions es the
More informationAH Mechanics Checklist (Unit 2) AH Mechanics Checklist (Unit 2) Circular Motion
AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) Cicula Motion No. kill Done 1 Know that cicula motion efes to motion in a cicle of constant adius Know that cicula motion is conveniently descibed
More informationPhysics 181. Assignment 4
Physics 181 Assignment 4 Solutions 1. A sphee has within it a gavitational field given by g = g, whee g is constant and is the position vecto of the field point elative to the cente of the sphee. This
More informationLecture 22. PE = GMm r TE = GMm 2a. T 2 = 4π 2 GM. Main points of today s lecture: Gravitational potential energy: Total energy of orbit:
Lectue Main points of today s lectue: Gavitational potential enegy: Total enegy of obit: PE = GMm TE = GMm a Keple s laws and the elation between the obital peiod and obital adius. T = 4π GM a3 Midtem
More informationAST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1
Please ead this fist... AST S: The oigin and evolution of the Univese Intoduction to Mathematical Handout This is an unusually long hand-out and one which uses in places mathematics that you may not be
More informationTheWaveandHelmholtzEquations
TheWaveandHelmholtzEquations Ramani Duaiswami The Univesity of Mayland, College Pak Febuay 3, 2006 Abstact CMSC828D notes (adapted fom mateial witten with Nail Gumeov). Wok in pogess 1 Acoustic Waves 1.1
More informationPHY2061 Enriched Physics 2 Lecture Notes. Gauss Law
PHY61 Eniched Physics Lectue Notes Law Disclaime: These lectue notes ae not meant to eplace the couse textbook. The content may be incomplete. ome topics may be unclea. These notes ae only meant to be
More informationIs there a magnification paradox in gravitational lensing?
Is thee a magnification paadox in gavitational ing? Olaf Wucknitz wucknitz@asto.uni-bonn.de Astophysics semina/colloquium, Potsdam, 6 Novembe 7 Is thee a magnification paadox in gavitational ing? gavitational
More informationChapter 13: Gravitation
v m m F G Chapte 13: Gavitation The foce that makes an apple fall is the same foce that holds moon in obit. Newton s law of gavitation: Evey paticle attacts any othe paticle with a gavitation foce given
More informationProjection Gravitation, a Projection Force from 5-dimensional Space-time into 4-dimensional Space-time
Intenational Jounal of Physics, 17, Vol. 5, No. 5, 181-196 Available online at http://pubs.sciepub.com/ijp/5/5/6 Science and ducation Publishing DOI:1.1691/ijp-5-5-6 Pojection Gavitation, a Pojection Foce
More informationDerivation of the Gravitational Red Shift from the Theorem of Orbits
1 Deivation of the Gavitational Red Shift fom the Theoem of Obits by Myon W. Evans, Alpha Institute fo Advanced Study, Civil List Scientist. emyone@aol.com and www.aias.us Abstact The expeimentally obsevable
More informationJ. N. R E DDY ENERGY PRINCIPLES AND VARIATIONAL METHODS APPLIED MECHANICS
J. N. E DDY ENEGY PINCIPLES AND VAIATIONAL METHODS IN APPLIED MECHANICS T H I D E DI T IO N JN eddy - 1 MEEN 618: ENEGY AND VAIATIONAL METHODS A EVIEW OF VECTOS AND TENSOS ead: Chapte 2 CONTENTS Physical
More informationKinetic energy, work, and potential energy. Work, the transfer of energy: force acting through distance: or or
ENERGETICS So fa we have been studying electic foces and fields acting on chages. This is the dynamics of electicity. But now we will tun to the enegetics of electicity, gaining new insights and new methods
More informationPrinciples of Physics I
Pinciples of Physics I J. M. Veal, Ph. D. vesion 8.05.24 Contents Linea Motion 3. Two scala equations........................ 3.2 Anothe scala equation...................... 3.3 Constant acceleation.......................
More informationChapter 4: The laws of motion. Newton s first law
Chapte 4: The laws of motion gavitational Electic magnetic Newton s fist law If the net foce exeted on an object is zeo, the object continues in its oiginal state of motion: - an object at est, emains
More informationPHYS 705: Classical Mechanics. Small Oscillations
PHYS 705: Classical Mechanics Small Oscillations Fomulation of the Poblem Assumptions: V q - A consevative system with depending on position only - The tansfomation equation defining does not dep on time
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 informationQuestion 1: The dipole
Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite
More informationGeometry of the homogeneous and isotropic spaces
Geomety of the homogeneous and isotopic spaces H. Sonoda Septembe 2000; last evised Octobe 2009 Abstact We summaize the aspects of the geomety of the homogeneous and isotopic spaces which ae most elevant
More informationTAMPINES JUNIOR COLLEGE 2009 JC1 H2 PHYSICS GRAVITATIONAL FIELD
TAMPINES JUNIOR COLLEGE 009 JC1 H PHYSICS GRAVITATIONAL FIELD OBJECTIVES Candidates should be able to: (a) show an undestanding of the concept of a gavitational field as an example of field of foce and
More information1 Fundamental Solutions to the Wave Equation
1 Fundamental Solutions to the Wave Equation Physical insight in the sound geneation mechanism can be gained by consideing simple analytical solutions to the wave equation. One example is to conside acoustic
More informationEELE 3331 Electromagnetic I Chapter 4. Electrostatic fields. Islamic University of Gaza Electrical Engineering Department Dr.
EELE 3331 Electomagnetic I Chapte 4 Electostatic fields Islamic Univesity of Gaza Electical Engineeing Depatment D. Talal Skaik 212 1 Electic Potential The Gavitational Analogy Moving an object upwad against
More informationLecture 3. Basic Physics of Astrophysics - Force and Energy. Forces
Foces Lectue 3 Basic Physics of Astophysics - Foce and Enegy http://apod.nasa.gov/apod/ Momentum is the poduct of mass and velocity - a vecto p = mv (geneally m is taken to be constant) An unbalanced foce
More informationPhysics: Work & Energy Beyond Earth Guided Inquiry
Physics: Wok & Enegy Beyond Eath Guided Inquiy Elliptical Obits Keple s Fist Law states that all planets move in an elliptical path aound the Sun. This concept can be extended to celestial bodies beyond
More informationCircular Motion & Torque Test Review. The period is the amount of time it takes for an object to travel around a circular path once.
Honos Physics Fall, 2016 Cicula Motion & Toque Test Review Name: M. Leonad Instuctions: Complete the following woksheet. SHOW ALL OF YOUR WORK ON A SEPARATE SHEET OF PAPER. 1. Detemine whethe each statement
More informationPACS: c ; qd
1 FEEDBACK IN GRAVITATIONAL PROBLEM OF OLAR CYCLE AND PERIHELION PRECEION OF MERCURY by Jovan Djuic, etied UNM pofesso Balkanska 8, 11000 Belgade, ebia E-mail: olivedj@eunet.s PAC: 96.90.+c ; 96.60.qd
More information7.2.1 Basic relations for Torsion of Circular Members
Section 7. 7. osion In this section, the geomety to be consideed is that of a long slende cicula ba and the load is one which twists the ba. Such poblems ae impotant in the analysis of twisting components,
More information1) Consider an object of a parabolic shape with rotational symmetry z
Umeå Univesitet, Fysik 1 Vitaly Bychkov Pov i teknisk fysik, Fluid Mechanics (Stömningsläa), 01-06-01, kl 9.00-15.00 jälpmedel: Students may use any book including the tetbook Lectues on Fluid Dynamics.
More informationPHYSICS NOTES GRAVITATION
GRAVITATION Newton s law of gavitation The law states that evey paticle of matte in the univese attacts evey othe paticle with a foce which is diectly popotional to the poduct of thei masses and invesely
More informationSIO 229 Gravity and Geomagnetism. Lecture 6. J 2 for Earth. J 2 in the solar system. A first look at the geoid.
SIO 229 Gavity and Geomagnetism Lectue 6. J 2 fo Eath. J 2 in the sola system. A fist look at the geoid. The Thee Big Themes of the Gavity Lectues 1.) An ellipsoidal otating Eath Refeence body (mass +
More information[Griffiths Ch.1-3] 2008/11/18, 10:10am 12:00am, 1. (6%, 7%, 7%) Suppose the potential at the surface of a hollow hemisphere is specified, as shown
[Giffiths Ch.-] 8//8, :am :am, Useful fomulas V ˆ ˆ V V V = + θ+ φ ˆ and v = ( v ) + (sin θvθ ) + v θ sinθ φ sinθ θ sinθ φ φ. (6%, 7%, 7%) Suppose the potential at the suface of a hollow hemisphee is specified,
More informationarxiv: v2 [gr-qc] 18 Aug 2014
Self-Consistent, Self-Coupled Scala Gavity J. Fanklin Depatment of Physics, Reed College, Potland, Oegon 970, USA Abstact A scala theoy of gavity extending Newtonian gavity to include field enegy as its
More informationm1 m2 M 2 = M -1 L 3 T -2
GAVITATION Newton s Univesal law of gavitation. Evey paticle of matte in this univese attacts evey othe paticle with a foce which vaies diectly as the poduct of thei masses and invesely as the squae of
More informationPotential Energy and Conservation of Energy
Potential Enegy and Consevation of Enegy Consevative Foces Definition: Consevative Foce If the wok done by a foce in moving an object fom an initial point to a final point is independent of the path (A
More informationEarth and Moon orbital anomalies
Eath and Moon obital anomalies Si non è veo, è ben tovato Ll. Bel axiv:1402.0788v2 [g-qc] 18 Feb 2014 Febuay 19, 2014 Abstact A time-dependent gavitational constant o mass would coectly descibe the suspected
More informationAP-C WEP. h. Students should be able to recognize and solve problems that call for application both of conservation of energy and Newton s Laws.
AP-C WEP 1. Wok a. Calculate the wok done by a specified constant foce on an object that undegoes a specified displacement. b. Relate the wok done by a foce to the aea unde a gaph of foce as a function
More informationClassical Mechanics Homework set 7, due Nov 8th: Solutions
Classical Mechanics Homewok set 7, due Nov 8th: Solutions 1. Do deivation 8.. It has been asked what effect does a total deivative as a function of q i, t have on the Hamiltonian. Thus, lets us begin with
More informationChapter 2: Basic Physics and Math Supplements
Chapte 2: Basic Physics and Math Supplements Decembe 1, 215 1 Supplement 2.1: Centipetal Acceleation This supplement expands on a topic addessed on page 19 of the textbook. Ou task hee is to calculate
More informationElectrostatics (Electric Charges and Field) #2 2010
Electic Field: The concept of electic field explains the action at a distance foce between two chaged paticles. Evey chage poduces a field aound it so that any othe chaged paticle expeiences a foce when
More informationPHYS Dynamics of Space Vehicles
PHYS 4110 - Dynamics of Space Vehicles Chapte 3: Two Body Poblem Eath, Moon, Mas, and Beyond D. Jinjun Shan, Pofesso of Space Engineeing Depatment of Eath and Space Science and Engineeing Room 55, Petie
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 9
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.033 Novembe 17, 2006 Poblem Set 9 Due: Decembe 8, at 4:00PM. Please deposit the poblem set in the appopiate 8.033 bin, labeled with name
More informationPhysics 2B Chapter 22 Notes - Magnetic Field Spring 2018
Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field
More informationVoltage ( = Electric Potential )
V-1 of 10 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 informationLecture 19 Angular momentum. Chapter
PHYS 172H: Moden Mechanics Fall 2010 Lectue 19 ngula momentum Chapte 11.4 11.7 The angula momentum pinciple dp = F dl =? net d ( p ) d dp = p+ = v γ mv = = 0 The angula momentum pinciple fo a point paticle
More informationExam 3: Equation Summary
MAACHUETT INTITUTE OF TECHNOLOGY Depatment of Physics Physics 8. TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t = Exam 3: Equation ummay = Impulse: I F( t ) = p Toque: τ =,P dp F P τ =,P
More informationStress, Cauchy s equation and the Navier-Stokes equations
Chapte 3 Stess, Cauchy s equation and the Navie-Stokes equations 3. The concept of taction/stess Conside the volume of fluid shown in the left half of Fig. 3.. The volume of fluid is subjected to distibuted
More informationForce between two parallel current wires and Newton s. third law
Foce between two paallel cuent wies and Newton s thid law Yannan Yang (Shanghai Jinjuan Infomation Science and Technology Co., Ltd.) Abstact: In this pape, the essence of the inteaction between two paallel
More information15 Solving the Laplace equation by Fourier method
5 Solving the Laplace equation by Fouie method I aleady intoduced two o thee dimensional heat equation, when I deived it, ecall that it taes the fom u t = α 2 u + F, (5.) whee u: [0, ) D R, D R is the
More information