Planck Quantization of Newton and Einstein Gravitation
|
|
- Marian Wright
- 6 years ago
- Views:
Transcription
1 Plank Quantization of Newton and Einstein Gavitation Espen Gaade Haug Nowegian Univesity of Life Sienes Mah 0, 06 Abstat In this pape we ewite the gavitational onstant based on its elationship with the Plank length and, based on this, we ewite the Plank mass in a slightly di eent fom (that gives exatly the same value). In this way we ae able to uantize a seies of end esults in Newton and Einstein s gavitation theoies. The fomulas will still give exatly the same values as befoe, but eveything elated to gavity will then ome in uanta. Numeially this only has impliations at the uantum sale; fo mao objets the disete steps ae so tiny that they ae lose to impossible to notie. Hopefully this an give additional insight into how well o not so well (ad ho) uantized Newton and Einstein s gavitation ae potentially linked with the uantum wold. Key wods: Quantized gavitation, gavitational onstant, esape veloity, gavitational time dilation, Shwazshild adius, Plank length, bending of light, Plank mass, Plank length. Foundation We suggest that the gavitational onstant should be witten as a funtion of Plank s edued onstant G p () whee is the edued Plank s onstant and is the well tested ound-tip speed of light. We ould all this Plank s fom of the gavitational onstant. The is an unknown onstant that is alibated so that G p mathes ou best estimate (measuement) fo the gavitational onstant. As shown by Haug (06), the Plank fom of the gavitational onstant enables us to ewite the Plank length as and the Plank mass as l p = m p = Gp = s s = p () = Using the gavitational onstant in the Plank fom, as well as the ewitten Plank units, we ae easily able to modify a seies of end esults fom Newton and Einstein s gavitational theoies to ontain uantization as well. Newton Univesal Gavitational Foe The Newton gavitational foe is given by F G = G p m m (4) espenhaug@ma.om. Thanks to Vitoia Tees fo helping me edit this manusipt. In vesion 5 a mathematial typo in the gavitational aeleation and Newtons vesion of Kelles thid law was fixed. If you find this pape of inteest you will possibly also find my eent pape in the Relativity and Cosmology setion The Collapse of the Shwazshild Radius: The End of Blak Holes of inteest. ()
2 Using the gavitational onstant of the fom G p ewite the Newton gavitational foe fo two Plank masses as and the Plank mass of m p = we an In the speial ase whee we get F GP = G p m pm p F GP = (5) be F Gp It seems fom this that gavity ould be intepeted as hits pe seond. Fo lage masses the fom will F Gp = G p N m pn m p F Gp = G p N N m p F Gp = N N (7) whee N is the numbe of Plank masses in objet one and N is the numbe of Plank masses in objet two. In the ase when the two masses ae of eual size we have F GP = N (8) Esape Veloity at the Quantum Sale The taditional esape veloity is given by GM v e = (9) whee G is the taditional gavitational onstant and M is the mass of the objet we ae tying to esape fom, and is the adius of that objet. In othe wods, we stand at the sufae of the objet, fo example a hydogen atom o a planet. Based on the gavitational onstant witten in the Plank fom we an find the esape veloity at Plank sale; see also the Appendix fo a deivation fom sath. It must be GpNm p v e,p = s v e,p = v e,p = v e,p = whee N is the numbe of Plank masses in the planet o mass in uestion. A patiulaly inteesting ase is when we only have one Plank mass N =and =@ (this is atually the Shwazshild adius of a Plank mass objet). This gives us (0) v v e,p = () as the esape veloity fo a patile with Plank mass with is. Next we will see if the fomula above an also be used to alulate the esape veloity of Eath. The Eath s mass is kg. We must onvet this to the numbe of Plank masses. The Plank mass is
3 m p = The Eath s mass in tems of the numbes of Plank masses must be Futhe the adius of the Eath is metes. We an now just plug this into the Plank sale esape veloity: v e,p = v e,p = metes/seond whih is eual to 40,69 km/h, the well-known esape veloity fom the Eaths gavitational field. We think ou new way of looking at gavity ould have onseuenes fo the undestanding of gavity. Gavitation must ome in disete steps and the esape veloity must also ome in disete steps fo a given adius; this is beause the amount of matte likely omes in disete steps. 4 Obital Speed The obital speed is given by GM v o We an ewite this in the fom of the Plank gavitational onstant and the Plank mass as () This an also be witten as v o,p GpNm p N v o,p N@ v o,p. () v o,p p ve N@ = (4) 5 Gavitational Aeleation The gavitational aeleation field in moden physis is given by This an be ewitten in uantized fom as g = GM (5) g = GpM g N g = N@ (6)
4 4 6 Gavitational Paamete The standad gavitational paamete is given by This an be ewitten in uantized fom as µ = GM (7) µ p = G pm µ p = G pnm p 7 Keple s Thid Law of Motion µ p N µ p = N@ (8) The Newton mehanis vesion of Keple s thid law of motion fo a iula obit is given by a = 4 G(M s + m) Whee M s is the mass of the Sun, m the mass of the planet, P is the peiod, and a is the semi-majo axis. This an be e-witten as (9) a = a = a = 4 G p(n m p + N m 4 N + N (N + N ) (0) whee N is the numbe of Plank masses in the mass of the Sun M s and N is the numbe of Plank mass of the planet m. IntheasetheplanetsmassismuhsmallethantheSunsmass,weanusethe following appoximation a N whee N is now the numbe of Plank masses in the Sun. () 8 Gavitational Time Dilation at Plank Sale Einstein s gavitational time dilation is given by t 0 = t f GM ve = t f () whee v e is the taditional esape veloity. We an ewite this in the fom of uantized esape veloity (deived above). v u t Let s see if we an alulate the time dilation at, fo example, the sufae of the Eath fom Plank sale gavitational time dilation. The Eath s mass is kg. And again, the Eath s mass in ve,p ()
5 tems of the Plank mass must be Futhe, the adius of the Eath is metes. We an now just plug this into the uantized gavitational time dilation t f That is fo evey seond that goes by in oute spae (a lok fa away fom the massive objet), seonds goes by on the sufae of the Eath. That is fo evey yea in in oute spae (vey fa fom the Eath), thee ae about milliseonds left to eah an Eath yea. This is natually the same as we would get with Einstein s fomula. Still, the new way of witing the fomula gives additional insight. Ciula obits gavitational time dilation The time dilation fo a lok at iula obit is given by GM ve t 0 = t f = (4) whee v e is the taditional esape veloity. We an ewite this in the fom of uantized esape veloity (deived above). v u t ve,p N@ (5) 9 The Shwazshild Radius The Shwazshild adius of a mass M is given by s = GM (6) Rewitten into the uantum ealm as desibed in this atile it must be s = GpM s = GpNmp Fo a lok at the Shwazshild adius we get a time dilation of s N s = (7) =0. (8) At the Shwazshild adius, time stands still. Fo a adius shote than that the gavitational time dilation euation above beaks down. At obital adius lage then s Exept if we assume epesents the adius of an indivisible patile. Thus if we move away fom the point patile onept, this would simply mean that we ould not go below the Plank sale Shwazshild Radius.
6 6 Mass in Shwazshild mete The Shwazshild mass in tems of metes is given by This an be e-witten as mete = GM (9) mete = GpNmp mete mete = N@ (0) 0 Quantized Gavitational Bending of Light The angle of defletion in Einstein s Geneal elativity theoy is given by This an be ewitten as GR = 4GM GRH = 4GpM GRH = 4GpNmp GRH = GRH = N () whee N is the numbe of Plank masses making up the mass we ae inteested in. Fom the fomula above, this means that the defletion of angles omes in uanta. Lets also ontol that ou Plank sale defletion ooted in Plank and GR is onsistent fo lage bodies like the Sun, fo example. The sola mass is M s kg. The Sun s mass in tems of the numbe of Plank masses must be Futhe, the adius of the Sun is s metes. We an just plug this into the Plank sale defletion: GRH = 4N@ = () If we multiply this by we get a bending of light of about.75 aseonds o.75 of a degee. 600 This is the same as has been onfimed by expeiments and helped make Einstein famous, as Newton gavitation supposedly only pedited half of the bending of light. Newton bending of light is given by Newton = GM () See fo example Soaes (009) and Momeni (0) fo deivations of bending of light unde Newton gavitation. Gavitational Redshift The Einstein gavitational edshift is given by lim z() =!+ (4) GM R whee R e is the distane between the ente of the mass of the gavitating body and the point at whih the photon is emitted. This we an ewite as
7 7 lim z() =!+ lim z() = s!+ lim z() =!+ GM R R e (5) Futhe in the Newtonian limit when R e is su an appoximate the above expession with iently lage ompaed to the Shwazshild adius we GM lim z()!+ R e lim z() N!+ R e Einstein s Field Euation And finally we get to Einstein s field euation. It is given by N@ lim z() (6)!+ R e 8 G R µv gµvr = Tµv (7) 4 I am fa fom an expet on Einstein s field euation, but based on the Plank gavitational onstant given in this pape we an ewite it as 8 Gp R µv gµvr = T 4 8 R µv gµvr = T 4 µv R µv gµvr = Tµv (8) Bea in mind = h and based on this we an altenatively wite Einstein s field euation as R µv gµvr = T µv (9) h The potential intepetation and usefulness of this ewitten vesion of Einstein s field euation we leave up to othe expets to onside. An inteesting uestion is natually whethe o not it is onsistent with some of the deivations given above in this fom. Table Summay The table below summaizes ou ewiting of some gavitational fomulas. The output is still the same, but based on this view of gavity, masses, gavitational time dilation, and even esape veloity all ome in disete steps. 4 Conlusion By making the gavitational onstant a funtion fom of the edued Plank onstant one an easily ewite many of the end esults fom Newton and Einstein s gavitation in uantized fom. Even if this is seen as an ad ho method, it ould still give new insight into what degee uantized Newton s gavitation and Geneal elativity ae onsistent with the uantum ealm.
8 8 Table : The table shows some of the standad gavitational elationships given by Newton and Einstein and thei expession in uantized fom. Units: Newton and Einstein fom: Quantized-fom: Gavitational onstant G G p Newton s gavitational foe F G = G MM Newton s gavitational foe F G = G Keple s thid law Newton s Esape veloity fom any mass v e = Obital veloity fo any mass 4 a = G(M s+m) GM v o GM F G = G p m pm p F G = N @ 4 a (N +N ) v e,p = v o,p N Gavitational aeleation field g = GM g p = N@ Gavitational paamete µ = GM µ p = N@. GM v Gavitational time dilation t 0 = t f = e t o = t f Obital time dilation t 0 = t f GM = v e N@ t o = t f Shwazshild adius s = GM s = Bending of light GR = 4GM GRH = 4N@ Blak holes Possible Depends on uantum intepetation Appendix: Esape veloity Deivation of the esape veloity fom Plank sale E mv GmM E GN m pn m p Nmpv E N N v E N N N N N (40) whee N is the numbe of Plank masses in the smalle mass m (fo example a oket) and N is the numbe of Plank masses in the othe mass. This we have to set to 0 and solve with espet to v to find the esape veloity: N v N N = 0 v = NN N v = v = This is a uantized esape veloity. Sine N anels out we an simply all N fo N and wite the esape veloity as v = whee N is the numbe of Plank masses in the mass we ae tying to esape fom. Refeenes Einstein, A. (96): Näheungsweise Integation de Feldgleihungen de Gavitation, Sitzungsbeihte de Königlih Peussishen Akademie de Wissenshaften Belin. (4) (4)
9 9 Haug, E. G. (06): The Gavitational Constant and the Plank Units. A Deepe Undestanding of the Quantum Realm, Mah 06. Momeni, D. (0): Bending of Light a Classial Analysis, Woking pape, pp. 4. Newton, I. (686): Philosophiae Natualis Pinipia Mathematis. London. Plank, M. (90): Uebe das Gesetz de Enegieveteilung im Nomalspetum, Annalen de Physik, 4. Shwazshild, K. (96a): Übe das Gavitationsfeld eine Kugel aus Inkompessible Flussigkeit nah de Einsteinshen Theoie, Sitzungsbeihte de Deutshen Akademie de Wissenshaften zu Belin, Klasse fu Mathematik, Physik, und Tehnik, p.44. (96b): Übe das Gavitationsfeld eines Massenpunktes nah de Einsteinshen Theoie, Sitzungsbeihte de Deutshen Akademie de Wissenshaften zu Belin, Klasse fu Mathematik, Physik, und Tehnik, p.89. Soaes, D. S. L. (009): vesion 4. Newtonian Gavitational Defletion of Light Revisited, Woking Pape
From E.G. Haug Escape Velocity To the Golden Ratio at the Black Hole. Branko Zivlak, Novi Sad, May 2018
Fom E.G. Haug Esape eloity To the Golden Ratio at the Blak Hole Banko Zivlak, bzivlak@gmail.om Novi Sad, May 018 Abstat Esape veloity fom the E.G. Haug has been heked. It is ompaed with obital veloity
More informationFinding the Planck Length Independent of Newton s Gravitational Constant and the Planck Constant The Compton Clock Model of Matter
Finding the Plank Length Independent of Newton s Gravitational Constant and the Plank Constant The Compton Clok Model of Matter Espen Gaarder Haug Norwegian University of Life Sienes September 9, 08 In
More informationAnswers to Coursebook questions Chapter 2.11
Answes to Couseook questions Chapte 11 1 he net foe on the satellite is F = G Mm and this plays the ole of the entipetal foe on the satellite, ie mv mv Equating the two gives π Fo iula motion we have that
More informationTime Dilation in Gravity Wells
Time Dilation in Gavity Wells By Rihad R. Shiffman Digital Gaphis Asso. 038 Dunkik Ave. L.A., Ca. 9005 s@isi.edu This doument disusses the geneal elativisti effet of time dilation aused by a spheially
More informationThe Planck Mass Particle Finally Discovered! The True God Particle! Good bye to the Point Particle Hypothesis!
The Plank Mass Patile Finally Disoveed! The Tue God Patile! Good bye to the Point Patile Hypothesis! Espen Gaade Haug Nowegian Univesity of Life Sienes Septembe, 06 Abstat In this pape we suggest that
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 informationThe Mystery of Mass as Understood from Atomism
The Mystey of Mass as Undestood fom Atomism Esen Gaade Haug June, 08 Abstat Ove the ast few yeas I have esented a theoy of moden atomism suoted by mathematis [, ]. In eah aea of analysis undetaken in this
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 informationHistory of Astronomy - Part II. Tycho Brahe - An Observer. Johannes Kepler - A Theorist
Histoy of Astonomy - Pat II Afte the Copenican Revolution, astonomes stived fo moe obsevations to help bette explain the univese aound them Duing this time (600-750) many majo advances in science and astonomy
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 informationASuggestedBoundaryforHeisenberg suncertaintyprinciple
ASuggestedBoundayfoHeisenbeg sunetaintypinile Esen aade Haug Nowegian Univesity of Life Sienes Januay 9, 08 Abstat In this ae we ae ombining Heisenbeg s unetainty inile with Haug s suggested maimum veloity
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 informationMass- and light-horizons, black holes' radii, the Schwartzschild metric and the Kerr metric
006-010 Thiey De Mees Mass- and light-hoizons, blak holes' adii, the Shwatzshild meti and the Ke meti mpoved alulus. (using gavitomagnetism) T. De Mees - thieydm@pandoa.be Abstat Blak holes geneally ae
More informationEscape Velocity. GMm ] B
1 PHY2048 Mach 31, 2006 Escape Velocity Newton s law of gavity: F G = Gm 1m 2 2, whee G = 667 10 11 N m 2 /kg 2 2 3 10 10 N m 2 /kg 2 is Newton s Gavitational Constant Useful facts: R E = 6 10 6 m M E
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 information10. Universal Gravitation
10. Univesal Gavitation Hee it is folks, the end of the echanics section of the couse! This is an appopiate place to complete the study of mechanics, because with his Law of Univesal Gavitation, Newton
More informationThe Mystery Behind the Fine Structure Constant Contracted Radius Ratio Divided by the Mass Ratio? APossibleAtomistInterpretation
The Mystey Behind the Fine Stuctue Constant Contacted Radius Ratio Divided by the Mass Ratio? APossibleAtomistIntepetation Espen Gaade Haug Nowegian Univesity of Life Sciences June 10, 017 Abstact This
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 investigation
More informationCentral Force Motion
Cental Foce Motion Cental Foce Poblem Find the motion of two bodies inteacting via a cental foce. Examples: Gavitational foce (Keple poblem): m1m F 1, ( ) =! G ˆ Linea estoing foce: F 1, ( ) =! k ˆ Two
More informationAppendix B The Relativistic Transformation of Forces
Appendix B The Relativistic Tansfomation of oces B. The ou-foce We intoduced the idea of foces in Chapte 3 whee we saw that the change in the fou-momentum pe unit time is given by the expession d d w x
More informationBetween any two masses, there exists a mutual attractive force.
YEAR 12 PHYSICS: GRAVITATION PAST EXAM QUESTIONS Name: QUESTION 1 (1995 EXAM) (a) State Newton s Univesal Law of Gavitation in wods Between any two masses, thee exists a mutual attactive foce. This foce
More informationThe Kerr-metric, mass- and light-horizons, and black holes' radii.
006 Thiey De Mees The Ke-meti, mass- and light-hoizons, and blak holes' adii. (using the Analogue Maxwell theoy) T. De Mees - thieydm @ pandoa.be Abstat Blak holes an geneally be defined as stella objets
More informationGravitoelectromagnetism. II. Speed of Light in Gravitational Field
Zbigniew Osiak aitoeletomagnetism. II. May 9, 8 aitoeletomagnetism. II. peed of Light in aitational Field Zbigniew Osiak E-mail: zbigniew.osiak@gmail.om http://oid.og/--57-36x http://ixa.og/autho/zbigniew_osiak
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 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 information(conservation of momentum)
Dynamis of Binay Collisions Assumptions fo elasti ollisions: a) Eletially neutal moleules fo whih the foe between moleules depends only on the distane between thei entes. b) No intehange between tanslational
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 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 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 informationKEPLER S LAWS AND PLANETARY ORBITS
KEPE S AWS AND PANETAY OBITS 1. Selected popeties of pola coodinates and ellipses Pola coodinates: I take a some what extended view of pola coodinates in that I allow fo a z diection (cylindical coodinates
More informationRevised Newtonian Formula of Gravity and Equation of Cosmology in Flat Space-Time Transformed from Schwarzschild Solution
Intenational Jounal of Astonomy and Astophysis,,, 6-8 http://dx.doi.og/.46/ijaa.. Published Online Mah (http://www.sip.og/jounal/ijaa) evised Newtonian Fomula of Gavity and Equation of Cosmology in Flat
More informationExtra Examples for Chapter 1
Exta Examples fo Chapte 1 Example 1: Conenti ylinde visomete is a devie used to measue the visosity of liquids. A liquid of unknown visosity is filling the small gap between two onenti ylindes, one is
More informationPhotographing a time interval
Potogaping a time inteval Benad Rotenstein and Ioan Damian Politennia Univesity of imisoaa Depatment of Pysis imisoaa Romania benad_otenstein@yaoo.om ijdamian@yaoo.om Abstat A metod of measuing time intevals
More informationPhysics 218, Spring March 2004
Today in Physis 8: eleti dipole adiation II The fa field Veto potential fo an osillating eleti dipole Radiated fields and intensity fo an osillating eleti dipole Total satteing oss setion of a dieleti
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 informationSuppose you have a bank account that earns interest at rate r, and you have made an initial deposit of X 0
IOECONOMIC MODEL OF A FISHERY (ontinued) Dynami Maximum Eonomi Yield In ou deivation of maximum eonomi yield (MEY) we examined a system at equilibium and ou analysis made no distintion between pofits in
More informationSolution to HW 3, Ma 1a Fall 2016
Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.
More informationThe Schwartzchild Geometry
UNIVERSITY OF ROCHESTER The Schwatzchild Geomety Byon Osteweil Decembe 21, 2018 1 INTRODUCTION In ou study of geneal elativity, we ae inteested in the geomety of cuved spacetime in cetain special cases
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 10 Solutions. r s
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.033 Decembe 5, 003 Poblem Set 10 Solutions Poblem 1 M s y x test paticle The figue above depicts the geomety of the poblem. The position
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 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 informationJournal of Theoretics
Jounal of Theoetis Volume 6-1, Feb-Mah 4 An Altenative Exlanation of the Cosmologial Redshift by the Tahyon Plasma Field in Integalati Sae Takaaki Musha musha@jda-tdi.go.j, musha@jg.ejnet.ne.j MRI, -11-7-61,
More informationSolving Problems of Advance of Mercury s Perihelion and Deflection of. Photon Around the Sun with New Newton s Formula of Gravity
Solving Poblems of Advance of Mecuy s Peihelion and Deflection of Photon Aound the Sun with New Newton s Fomula of Gavity Fu Yuhua (CNOOC Reseach Institute, E-mail:fuyh945@sina.com) Abstact: Accoding to
More informationGENERAL RELATIVITY: THE GEODESICS OF THE SCHWARZSCHILD METRIC
GENERAL RELATIVITY: THE GEODESICS OF THE SCHWARZSCHILD METRIC GILBERT WEINSTEIN 1. Intoduction Recall that the exteio Schwazschild metic g defined on the 4-manifold M = R R 3 \B 2m ) = {t,, θ, φ): > 2m}
More informationUniform Circular Motion
Unifom Cicula Motion constant speed Pick a point in the objects motion... What diection is the velocity? HINT Think about what diection the object would tavel if the sting wee cut Unifom Cicula Motion
More informationRed Shift and Blue Shift: A realistic approach
Red Shift and Blue Shift: A ealisti appoah Benhad Rothenstein Politehnia Uniesity of Timisoaa, Physis Dept., Timisoaa, Romania E-mail: benhad_othenstein@yahoo.om Coina Nafonita Politehnia Uniesity of Timisoaa,
More informationω = θ θ o = θ θ = s r v = rω
Unifom Cicula Motion Unifom cicula motion is the motion of an object taveling at a constant(unifom) speed in a cicula path. Fist we must define the angula displacement and angula velocity The angula displacement
More informationF g. = G mm. m 1. = 7.0 kg m 2. = 5.5 kg r = 0.60 m G = N m 2 kg 2 = = N
Chapte answes Heinemann Physics 4e Section. Woked example: Ty youself.. GRAVITATIONAL ATTRACTION BETWEEN SMALL OBJECTS Two bowling balls ae sitting next to each othe on a shelf so that the centes of the
More informationAY 7A - Fall 2010 Section Worksheet 2 - Solutions Energy and Kepler s Law
AY 7A - Fall 00 Section Woksheet - Solutions Enegy and Keple s Law. Escape Velocity (a) A planet is obiting aound a sta. What is the total obital enegy of the planet? (i.e. Total Enegy = Potential Enegy
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 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 informationCorrespondence Analysis & Related Methods
Coespondene Analysis & Related Methods Oveview of CA and basi geometi onepts espondents, all eades of a etain newspape, osstabulated aoding to thei eduation goup and level of eading of the newspape Mihael
More informationPaths of planet Mars in sky
Section 4 Gavity and the Sola System The oldest common-sense view is that the eath is stationay (and flat?) and the stas, sun and planets evolve aound it. This GEOCENTRIC MODEL was poposed explicitly by
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 informationKEPLER S LAWS OF PLANETARY MOTION
EPER S AWS OF PANETARY MOTION 1. Intoduction We ae now in a position to apply what we have leaned about the coss poduct and vecto valued functions to deive eple s aws of planetay motion. These laws wee
More informationTutorial Exercises: Central Forces
Tutoial Execises: Cental Foces. Tuning Points fo the Keple potential (a) Wite down the two fist integals fo cental motion in the Keple potential V () = µm/ using J fo the angula momentum and E fo the total
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 informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More informationProblem 1. Part b. Part a. Wayne Witzke ProblemSet #1 PHY 361. Calculate x, the expected value of x, defined by
Poblem Pat a The nomal distibution Gaussian distibution o bell cuve has the fom f Ce µ Calculate the nomalization facto C by equiing the distibution to be nomalized f Substituting in f, defined above,
More informationCentral Force Problem. Central Force Motion. Two Body Problem: Center of Mass Coordinates. Reduction of Two Body Problem 8.01 W14D1. + m 2. m 2.
Cental oce Poblem ind the motion of two bodies inteacting via a cental foce. Cental oce Motion 8.01 W14D1 Examples: Gavitational foce (Keple poblem): 1 1, ( ) G mm Linea estoing foce: ( ) k 1, Two Body
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 informationAP Physics - Coulomb's Law
AP Physics - oulomb's Law We ve leaned that electons have a minus one chage and potons have a positive one chage. This plus and minus one business doesn t wok vey well when we go in and ty to do the old
More informationPhysics 161 Fall 2011 Extra Credit 2 Investigating Black Holes - Solutions The Following is Worth 50 Points!!!
Physics 161 Fall 011 Exta Cedit Investigating Black Holes - olutions The Following is Woth 50 Points!!! This exta cedit assignment will investigate vaious popeties of black holes that we didn t have time
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 informationRepresentation of gravity field equation and solutions, Hawking Radiation in Data General Relativity theory
epesentation of gavity field equation and solutions Hawking adiation in Data Geneal elativity theoy Sangwha-Yi Depatment of Math aejon Univesity 300-76 ABSAC In the geneal elativity theoy we find the epesentation
More informationHomework 7 Solutions
Homewok 7 olutions Phys 4 Octobe 3, 208. Let s talk about a space monkey. As the space monkey is oiginally obiting in a cicula obit and is massive, its tajectoy satisfies m mon 2 G m mon + L 2 2m mon 2
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 informationWhat Form of Gravitation Ensures Weakened Kepler s Third Law?
Bulletin of Aichi Univ. of Education, 6(Natual Sciences, pp. - 6, Mach, 03 What Fom of Gavitation Ensues Weakened Keple s Thid Law? Kenzi ODANI Depatment of Mathematics Education, Aichi Univesity of Education,
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 informationEspen Gaarder Haug Norwegian University of Life Sciences April 4, 2017
The Mass Gap, Kg, the Plank Constant and the Gravity Gap The Plank Constant Is a Composite Constant One kg Is 85465435748 0 36 Collisions per Seond The Mass Gap Is.734 0 5 kg and also m p The Possibility
More information1 Fundamental Solutions to the Wave Equation
1 Fundamental Solutions to the Wave Equation Physial insight in the sound geneation mehanism an be gained by onsideing simple analytial solutions to the wave equation One example is to onside aousti adiation
More informationPhysics 161: Black Holes: Lecture 5: 22 Jan 2013
Physics 161: Black Holes: Lectue 5: 22 Jan 2013 Pofesso: Kim Giest 5 Equivalence Pinciple, Gavitational Redshift and Geodesics of the Schwazschild Metic 5.1 Gavitational Redshift fom the Schwazschild metic
More informationCOMPARING MORE THAN TWO POPULATION MEANS: AN ANALYSIS OF VARIANCE
COMPARING MORE THAN TWO POPULATION MEANS: AN ANALYSIS OF VARIANCE To see how the piniple behind the analysis of vaiane method woks, let us onside the following simple expeiment. The means ( 1 and ) of
More informationA Modified Newtonian Quantum Gravity Theory Derived from Heisenberg s Uncertainty Principle that Predicts the Same Bending of Light as GR
A Modified Newtonian Quantum Gravity Theory Derived from Heisenberg s Unertainty Priniple that Predits the Same Bending of Light as GR Espen Gaarder Haug Norwegian University of Life Sienes Marh 6, 208
More informationA Motion Paradox from Einstein s Relativity of Simultaneity
Motion Paradox from Einstein s Relativity of Simultaneity Espen Gaarder Haug Norwegian University of Life Sienes November 5, 7 bstrat We are desribing a new and potentially important paradox related to
More informationSimple Considerations on the Cosmological Redshift
Apeiron, Vol. 5, No. 3, July 8 35 Simple Considerations on the Cosmologial Redshift José Franiso Garía Juliá C/ Dr. Maro Mereniano, 65, 5. 465 Valenia (Spain) E-mail: jose.garia@dival.es Generally, the
More informationOn a quantity that is analogous to potential and a theorem that relates to it
Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) 34-39 On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich
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 informationExperiment 1 Electric field and electric potential
Expeiment 1 Eleti field and eleti potential Pupose Map eleti equipotential lines and eleti field lines fo two-dimensional hage onfiguations. Equipment Thee sheets of ondutive papes with ondutive-ink eletodes,
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 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 informationRecitation PHYS 131. must be one-half of T 2
Reitation PHYS 131 Ch. 5: FOC 1, 3, 7, 10, 15. Pobles 4, 17, 3, 5, 36, 47 & 59. Ch 5: FOC Questions 1, 3, 7, 10 & 15. 1. () The eloity of a has a onstant agnitude (speed) and dietion. Sine its eloity is
More informationTidal forces. m r. m 1 m 2. x r 2. r 1
Tidal foces Befoe we look at fee waves on the eath, let s fist exaine one class of otion that is diectly foced: astonoic tides. Hee we will biefly conside soe of the tidal geneating foces fo -body systes.
More informationLecture 1a: Satellite Orbits
Lectue 1a: Satellite Obits Outline 1. Newton s Laws of Motion 2. Newton s Law of Univesal Gavitation 3. Calculating satellite obital paametes (assuming cicula motion) Scala & Vectos Scala: a physical quantity
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 informationASuggestedBoundaryforHeisenberg suncertaintyprinciple
ASuggestedBoundayfoHeisenbeg suncetaintypincile Esen aade Haug Nowegian Univesity of Life Sciences Januay, 07 Abstact In this ae we ae combining Heisenbeg s uncetainty incile with Haug s suggested maimum
More informationSingle Particle State AB AB
LECTURE 3 Maxwell Boltzmann, Femi, and Bose Statistics Suppose we have a gas of N identical point paticles in a box of volume V. When we say gas, we mean that the paticles ae not inteacting with one anothe.
More informationCHAPTER 5: Circular Motion; Gravitation
CHAPER 5: Cicula Motion; Gavitation Solution Guide to WebAssign Pobles 5.1 [1] (a) Find the centipetal acceleation fo Eq. 5-1.. a R v ( 1.5 s) 1.10 1.4 s (b) he net hoizontal foce is causing the centipetal
More informationObjective Notes Summary
Objective Notes Summay An object moving in unifom cicula motion has constant speed but not constant velocity because the diection is changing. The velocity vecto in tangent to the cicle, the acceleation
More informationAuchmuty High School Mathematics Department Advanced Higher Notes Teacher Version
The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!
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 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 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 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 informationarxiv: v4 [physics.class-ph] 14 Jul 2018
Noname manusipt No. will be inseted by the edito Long-Range Longitudinal Eleti Wave in Vauum Radiated by Eleti Dipole: Pat I Altay Zhakatayev, Leila Tlebaldiyeva axiv:7.v4 [physis.lass-ph] 4 Jul 8 Reeived:
More informationThe Radii of Baryons
Jounal Heading Yea; Vol. (No.): page ange DOI: 0.592/j.xxx.xxxxxxxx.xx The Radii of Bayons Maio Evealdo de Souza Depatmento de Físia, Univesidade Fedeal de Segipe, São Cistovão, 4900-000, Bazil Astat Consideing
More informationCircular Orbits. and g =
using analyse planetay and satellite motion modelled as unifom cicula motion in a univesal gavitation field, a = v = 4π and g = T GM1 GM and F = 1M SATELLITES IN OBIT A satellite is any object that is
More informationPhysics 201 Homework 4
Physics 201 Homewok 4 Jan 30, 2013 1. Thee is a cleve kitchen gadget fo dying lettuce leaves afte you wash them. 19 m/s 2 It consists of a cylindical containe mounted so that it can be otated about its
More informationGeneralized Vapor Pressure Prediction Consistent with Cubic Equations of State
Genealized Vapo Pessue Pedition Consistent with Cubi Equations of State Laua L. Petasky and Mihael J. Misovih, Hope College, Holland, MI Intodution Equations of state may be used to alulate pue omponent
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 information