2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,


 Magnus Newman
 2 years ago
 Views:
Transcription
1 3.4. KEPLER S LAWS 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 move aound in the wold is a diect consequence of these consevation laws athe than being the esult of some detailed mechanism. It is nice to give an example of how consevation of angula momentum has similaly poweful (and pehaps moe famous) consquences. We ecall that oughly 500 yeas ago, Keple made one of the geat beakthoughs (not just in physics, but in human thought), poviding evidence that planet motions as descibe by Tycho Bahe ae much moe simply descibed in a wold model with the sun (athe than the eath) at the cente. I don t think we can ovestate the impotance of ealizing that we ae not at the cente of the univese. Quantitatively, Keple noticed seveal things (Keple s laws): The obits of planets aound the sun ae elliptical, the peiods of the obits ae elated to thei adii, and as the obit poceeds it sweeps out equal aea in equal times. Of these, the equal aea law is the one which is elated to consevation of angula momentum. If the obits wee cicula it would be tivial that they sweep out aea at a constant ate. The equal aea law is, in a sense, all that is left of the pefection that people had sought with cicula obits. If we ae at a distance fom the cente of ou coodinate system (the sun), and we move by an angle θ, then fo small angles the aea that is swept out is A = 1 ( ) 1 dθ 2 2 θ = 2 t. (3.59) 2 The equal aea law is the statement that the tem in paentheses, da = 1 dθ 2 2, (3.60) is a constant, independent of time. We know that angula momentum is conseved, so let s see if this has something to do with the equal aea law. The vecto position of the planet can always be witten as = ˆ, whee ˆ is a unit vecto pointing outwad towad the cuent location. The velocity consists of components in the ˆ diection and in the ˆθ diection, aound the cuve, d = d dθ + ˆ ˆθ. (3.61)
2 146 CHAPTER 3. WE ARE NOT THE CENTER OF THE UNIVERSE Hence p m d = m d dθ + m ˆ ˆθ (3.62) ( L p = (ˆ ) m d ) ( ˆ +(ˆ ) m dθ ) ˆθ. (3.63) To finish the calculation we pull all the scalas out of the coss poducts, ( L = m d ) ( (ˆ ˆ )+ m 2 dθ ) (ˆ ˆθ), (3.64) and then we note that ˆ ˆ = 0, (3.65) ˆ ˆθ = ẑ. (3.66) Thus we find that the angula momentum is given by ( L = m 2 dθ ) ẑ. (3.67) Compaing Eq. (3.60) with Eq. (3.67), we see that da = 1 2m ( L ẑ ), (3.68) so that consevation of angula momentum ( L = constant) implies that da/ is a constant the equal aea law. To go futhe in deiving Keple s laws we need to know about the actual foces between the sun and the planets. You pobably know that one of Newton s geat tiumphs was to ealize that if gavity obeys the invese squae law, then the ate at which the moon is falling towad the eath as it obits is consistent with the ate at which objects we can hold in ou hands fall towad the gound. In moden language we say that the potential enegy fo two masses M and m sepaated by a distance is given by V () = GMm, (3.69) whee G is (appopiately enough) known as Newton s constant. We ae inteested in the case whee m is the mass of a planet and M is the mass of the sun. We choose a coodinate system in which the sun is fixed at the oigin. To undestand what happens it is useful to wite down the total enegy of the system. We have the potential enegy explicitly, so we need the kinetic
3 3.4. KEPLER S LAWS 147 enegy. We know that the velocity has two components, one in the adial diection and one in the angula diection, d = d dθ + ˆ ˆθ, (3.70) so that 1 2 mv2 1 2 m d 2 = 1 2 m [ (d ) 2 ( + dθ ) ] 2. (3.71) So the total enegy of the system, kinetic plus potential, is given by [ (d E = 1 ) 2 ( 2 m + dθ ) ] 2 GMm. (3.72) But we know that angula momentum is conseved, so we can say something about the tem that has dθ/ in it: L z = m 2 dθ (3.73) dθ = L z m 2. (3.74) Substituting into ou expession fo the total enegy this becomes [ (d E = 1 ) 2 ( 2 m + L ) ] 2 z m 2 GMm (3.75) = 1 ( ) d 2 2 m + L2 z 2m 2 GMm (3.76) = 1 ( ) d 2 2 m + V eff(), (3.77) whee in the last step we have intoduced an effective potential V eff () = L2 z 2m 2 GMm. (3.78) Notice that by doing this ou expession fo the total enegy comes to look like the enegy fo motion in one dimension (), with a potential enegy that has one pat fom gavity and one pat fom the indiect effect of the angula momentum. Notice that the contibution fom angula momentum is positive, and vaies as 1/ 2. This means that the coesponding foce F = V/
4 148 CHAPTER 3. WE ARE NOT THE CENTER OF THE UNIVERSE gavitational potential!!1/ centifugal potential! +1/ 2 total effective potential hamonic appoximation 40 potential V() 20 0!20!40!60!80! adius (abitay units) Figue 3.1: Effective potential enegy fo planetay motion, fom Eq (3.78).
5 3.4. KEPLER S LAWS 149 1/ is positive it pushes outwad along the adius. This foce is what we expeience when we sit in a ca going aound a cuve: the centifugal foce. Imagine that we tie a weight on the end of a sting and swing it in a cicle ove ou heads. The sting will stay taught, and this must be because thee is a foce pulling outwad; again this is the centifugal foce, and is geneated by this special tem in the effective potential. Notice that we have eliminated any mention of the angle θ, and in the pocess have changed the potential enegy fo motion along the adial diection. This is a much moe geneal idea. We often eliminate coodinates in the hope of simplifying things, and ty to take account of thei effects though an effective potential fo the coodinates that we do keep in ou desciption. This is vey impotant in big molecules, fo example, whee we don t want to keep tack of evey atom but hope that we can just think about a few things such as the distance between key esidues o the angle between two big ams of the molecule. It s not at all obvious that this should wok, even as an appoximation, although in the pesent case it s actually exact. Recall that the total enegy is the sum of kinetic and potential, and this total is conseved o constant ove time. Thee is a minimum effective potential enegy fo adial motion, as can be seen in Fig 3.1, If the total enegy is equal to this minimum, then thee can be no kinetic enegy associated with the coodinate, hence d/ = 0. Thus fo minimum enegy obits, the adius is constant the planet moves in a cicula obit. If we look at obits that have enegies just a bit lage than the minimum, we can appoximate V eff () as being like a hamonic oscillato. Then the adius should oscillate in time, but time is being maked by going aound the obit, so eally the adius will be a sine o cosine function of the angle, and this is the desciption of an ellipse if it is not too eccentic. In fact if you wok hade you can show that the obits ae exactly ellipses fo any value of the enegy up to some maximum. This is anothe of Keple s laws. Once you have the ellipse you can elate its size (the analog of adius fo a cicle) to the peiod of the obit, and this is the last of Keple s laws. Notice that if the enegy is positive then it is possible fo the planet to escape towad at finite velocity, and then the obit is not bound. But if the total enegy is negative, thee is no escape, and the adius moves between two limiting values, namely the points whee the total enegy intesects the effective potential. We eally should say moe about all this, but it is teated in many standad texts.
6 150 CHAPTER 3. WE ARE NOT THE CENTER OF THE UNIVERSE
Chapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43
Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.
More informationCh. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth
Ch. 8 Univesal Gavitation Pat 1: Keple s Laws Objectives: Section 8.1 Motion in the Heavens and on Eath Objectives Relate Keple s laws of planetay motion to Newton s law of univesal gavitation. Calculate
More informationRevision Guide for Chapter 11
Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams
More informationFXA 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 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 informationmv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2 " 2 = GM . Combining the results we get !
Chapte. he net foce on the satellite is F = G Mm and this plays the ole of the centipetal foce on the satellite i.e. mv mv. Equating the two gives = G Mm i.e. v = G M. Fo cicula motion we have that v =!
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 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 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 informationA) 2 B) 2 C) 2 2 D) 4 E) 8
Page 1 of 8 CTGavity1. m M Two spheical masses m and M ae a distance apat. The distance between thei centes is halved (deceased by a facto of 2). What happens to the magnitude of the foce of gavity between
More informationF G r. Don't confuse G with g: "Big G" and "little g" are totally different things.
G1 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 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 informationSo we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1)
Lectue 17 Cicula Motion (Chapte 7) Angula Measue Angula Speed and Velocity Angula Acceleation We ve aleady dealt with cicula motion somewhat. Recall we leaned about centipetal acceleation: when you swing
More informationChapter 13. VectorValued Functions and Motion in Space 13.6. Velocity and Acceleration in Polar Coordinates
13.6 Velocity and Acceleation in Pola Coodinates 1 Chapte 13. VectoValued Functions and Motion in Space 13.6. Velocity and Acceleation in Pola Coodinates Definition. When a paticle P(, θ) moves along
More informationResources. Circular Motion: From Motor Racing to Satellites. Uniform Circular Motion. Sir Isaac Newton 3/24/10. Dr Jeff McCallum School of Physics
3/4/0 Resouces Cicula Motion: Fom Moto Racing to Satellites D Jeff McCallum School of Physics http://www.gapsystem.og/~histoy/mathematicians/ Newton.html http://www.fga.com http://www.clke.com/clipat
More information81 Newton s Law of Universal Gravitation
81 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event,
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 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 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 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 information2. Orbital dynamics and tides
2. Obital dynamics and tides 2.1 The twobody 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 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 informationMechanics 1: Work, Power and Kinetic Energy
Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).
More informationGeneral Physics (PHY 2130)
Geneal Physics (PHY 130) Lectue 11 Rotational kinematics and unifom cicula motion Angula displacement Angula speed and acceleation http://www.physics.wayne.edu/~apetov/phy130/ Lightning Review Last lectue:
More informationProblem Set 6: Solutions
UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 164 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente
More information12. Rolling, Torque, and Angular Momentum
12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.
More informationChapter 26  Electric Field. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University
Chapte 6 lectic Field A PowePoint Pesentation by Paul. Tippens, Pofesso of Physics Southen Polytechnic State Univesity 7 Objectives: Afte finishing this unit you should be able to: Define the electic field
More informationL19 Geomagnetic Field Part I
Intoduction to Geophysics L191 L19 Geomagnetic Field Pat I 1. Intoduction We now stat the last majo topic o this class which is magnetic ields and measuing the magnetic popeties o mateials. As a way o
More informationOrbital Motion & Gravity
Astonomy: Planetay Motion 1 Obital Motion D. Bill Pezzaglia A. Galileo & Fee Fall Obital Motion & Gavity B. Obits C. Newton s Laws Updated: 013Ma05 D. Einstein A. Galileo & Fee Fall 3 1. Pojectile Motion
More informationSolutions to Homework Set #5 Phys2414 Fall 2005
Solution Set #5 1 Solutions to Homewok Set #5 Phys414 Fall 005 Note: The numbes in the boxes coespond to those that ae geneated by WebAssign. The numbes on you individual assignment will vay. Any calculated
More informationSamples of conceptual and analytical/numerical questions from chap 21, C&J, 7E
CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known
More informationVoltage ( = Electric Potential )
V1 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 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 informationPhysics 202, Lecture 4. Gauss s Law: Review
Physics 202, Lectue 4 Today s Topics Review: Gauss s Law Electic Potential (Ch. 25Pat I) Electic Potential Enegy and Electic Potential Electic Potential and Electic Field Next Tuesday: Electic Potential
More informationUNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Approximate time two 100minute sessions
Name St.No.  Date(YY/MM/DD) / / Section Goup# UNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Appoximate time two 100minute sessions OBJECTIVES I began to think of gavity extending to the ob of the moon,
More information2008 QuarterFinal Exam Solutions
2008 Quatefinal Exam  Solutions 1 2008 QuateFinal Exam Solutions 1 A chaged paticle with chage q and mass m stats with an initial kinetic enegy K at the middle of a unifomly chaged spheical egion of
More informationAlgebra and Trig. I. A point is a location or position that has no size or dimension.
Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite
More informationCoordinate Systems L. M. Kalnins, March 2009
Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean
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 informationExam I. Spring 2004 Serway & Jewett, Chapters 15. Fill in the bubble for the correct answer on the answer sheet. next to the number.
Agin/Meye PART I: QUALITATIVE Exam I Sping 2004 Seway & Jewett, Chaptes 15 Assigned Seat Numbe Fill in the bubble fo the coect answe on the answe sheet. next to the numbe. NO PARTIAL CREDIT: SUBMIT ONE
More informationESCAPE VELOCITY EXAMPLES
ESCAPE VELOCITY EXAMPLES 1. Escape velocity is the speed that an object needs to be taveling to beak fee of planet o moon's gavity and ente obit. Fo example, a spacecaft leaving the suface of Eath needs
More informationCHAPTER 9 THE TWO BODY PROBLEM IN TWO DIMENSIONS
9. Intoduction CHAPTER 9 THE TWO BODY PROBLEM IN TWO DIMENSIONS In this chapte we show how Keple s laws can be deived fom Newton s laws of motion and gavitation, and consevation of angula momentum, and
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 informationClassical Lifetime of a Bohr Atom
1 Poblem Classical Lifetime of a Boh Atom James D. Olsen and Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ 85 (Mach 7, 5) In the Boh model of the hydogen atom s gound state,
More informationChapter F. Magnetism. Blinn College  Physics Terry Honan
Chapte F Magnetism Blinn College  Physics 46  Tey Honan F.  Magnetic Dipoles and Magnetic Fields Electomagnetic Duality Thee ae two types of "magnetic chage" o poles, Noth poles N and South poles S.
More informationSection 53 Angles and Their Measure
5 5 TRIGONOMETRIC FUNCTIONS Section 5 Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles,
More informationTORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION
MISN034 TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION shaft TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION by Kiby Mogan, Chalotte, Michigan 1. Intoduction..............................................
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 informationPY1052 Problem Set 3 Autumn 2004 Solutions
PY1052 Poblem Set 3 Autumn 2004 Solutions C F = 8 N F = 25 N 1 2 A A (1) A foce F 1 = 8 N is exeted hoizontally on block A, which has a mass of 4.5 kg. The coefficient of static fiction between A and the
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 plutonium39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming
More information14. Gravitation Universal Law of Gravitation (Newton):
14. Gavitation 1 Univesal Law of Gavitation (ewton): The attactive foce between two paticles: F = G m 1m 2 2 whee G = 6.67 10 11 m 2 / kg 2 is the univesal gavitational constant. F m 2 m 1 F Paticle #1
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 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 ighthand end. If H 6.0 m and h 2.0 m, what
More informationHour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and
Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon
More informationSimple Harmonic Motion
Simple Hamonic Motion Intoduction Simple hamonic motion occus when the net foce acting on an object is popotional to the object s displacement fom an equilibium position. When the object is at an equilibium
More informationVoltage ( = Electric Potential )
V1 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 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 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 informationPower and Sample Size Calculations for the 2Sample ZStatistic
Powe and Sample Size Calculations fo the Sample ZStatistic James H. Steige ovembe 4, 004 Topics fo this Module. Reviewing Results fo the Sample Z (a) Powe and Sample Size in Tems of a oncentality Paamete.
More information6.2 Orbits and Kepler s Laws
Eath satellite in unstable obit 6. Obits and Keple s Laws satellite in stable obit Figue 1 Compaing stable and unstable obits of an atificial satellite. If a satellite is fa enough fom Eath s suface that
More informationCarterPenrose diagrams and black holes
CatePenose 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 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 informationMultiple choice questions [60 points]
1 Multiple choice questions [60 points] Answe all o the ollowing questions. Read each question caeully. Fill the coect bubble on you scanton sheet. Each question has exactly one coect answe. All questions
More information2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90
. Tigonometic Ratios of An Angle Focus on... detemining the distance fom the oigin to a point (, ) on the teminal am of an angle detemining the value of sin, cos, o tan given an point (, ) on the teminal
More informationReview Module: Dot Product
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics 801 Fall 2009 Review Module: Dot Poduct We shall intoduce a vecto opeation, called the dot poduct o scala poduct that takes any two vectos and
More informationTheory and measurement
Gavity: Theoy and measuement Reading: Today: p11  Theoy of gavity Use two of Newton s laws: 1) Univesal law of gavitation: ) Second law of motion: Gm1m F = F = mg We can combine them to obtain the gavitational
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 informationClassical Mechanics (CM):
Classical Mechanics (CM): We ought to have some backgound to aeciate that QM eally does just use CM and makes one slight modification that then changes the natue of the oblem we need to solve but much
More information7 Circular Motion. 71 Centripetal Acceleration and Force. Period, Frequency, and Speed. Vocabulary
7 Cicula Motion 71 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 information2  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1
 ELECTROSTATIC POTENTIAL AND CAPACITANCE Page. Line Integal of Electic Field If a unit positive chage is displaced by `given by dw E. dl dl in an electic field of intensity E, wok done is Line integation
More informationCh. 14: Gravitation (Beta Version 7/01) 14 Gravitation
Ch. 14: Gavitation (Beta Vesion 7/01) 14 Gavitation The Milky Way galaxy is a diskshaped collection of dust, planets, and billions of stas, including ou Sun and sola system. The foce that binds it o any
More informationIntroduction to Electric Potential
Univesiti Teknologi MARA Fakulti Sains Gunaan Intoduction to Electic Potential : A Physical Science Activity Name: HP: Lab # 3: The goal of today s activity is fo you to exploe and descibe the electic
More informationSpirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project
Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.
More informationGauss Law. Physics 231 Lecture 21
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 informationSemipartial (Part) and Partial Correlation
Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated
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 informationrotation  Conservation of mechanical energy for rotation  Angular momentum  Conservation of angular momentum
Final Exam Duing class (13:55 pm) on 6/7, Mon Room: 41 FMH (classoom) Bing scientific calculatos No smat phone calculatos l ae allowed. Exam coves eveything leaned in this couse. Review session: Thusday
More informationSAMPLE CHAPTERS UNESCO EOLSS THE MOTION OF CELESTIAL BODIES. Kaare Aksnes Institute of Theoretical Astrophysics University of Oslo
THE MOTION OF CELESTIAL BODIES Kaae Aksnes Institute of Theoetical Astophysics Univesity of Oslo Keywods: celestial mechanics, twobody obits, theebody obits, petubations, tides, nongavitational foces,
More informationChapter 22 The Electric Field II: Continuous Charge Distributions
Chapte The lectic Field II: Continuous Chage Distibutions 1 [M] A unifom line chage that has a linea chage density l equal to.5 nc/m is on the x axis between x and x 5. m. (a) What is its total chage?
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 informationPhysics 111 Fall 2007 Electrostatic Forces and the Electric Field  Solutions
Physics 111 Fall 007 Electostatic Foces an the Electic Fiel  Solutions 1. Two point chages, 5 µc an 8 µc ae 1. m apat. Whee shoul a thi chage, equal to 5 µc, be place to make the electic fiel at the
More informationFluids Lecture 15 Notes
Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2D, this velocit
More informationPHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013
PHYSICS 111 HOMEWORK SOLUTION #5 Mach 3, 2013 0.1 You 3.80kg physics book is placed next to you on the hoizontal seat of you ca. The coefficient of static fiction between the book and the seat is 0.650,
More informationMagnetic Field in a TimeDependent Capacitor
Magnetic Field in a TimeDependent Capacito 1 Poblem Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ 8544 (Octobe 3, 23) Reconside the classic example of the use of Maxwell s displacement
More informationIn the lecture on double integrals over nonrectangular domains we used to demonstrate the basic idea
Double Integals in Pola Coodinates In the lectue on double integals ove nonectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example
More informationChapter 8, Rotational Kinematics. Angular Displacement
Chapte 8, Rotational Kinematics Sections 1 3 only Rotational motion and angula displacement Angula velocity and angula acceleation Equations of otational kinematics 1 Angula Displacement! B l A The length
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 informationA couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance, d. F A F B (= F A
5 Moment of a Couple Ref: Hibbele 4.6, edfod & Fowle: Statics 4.4 couple is a pai of foces, equal in magnitude, oppositely diected, and displaced by pependicula distance, d. d (=  ) Since the foces ae
More informationGravity and the figure of the Earth
Gavity and the figue of the Eath Eic Calais Pudue Univesity Depatment of Eath and Atmospheic Sciences West Lafayette, IN 479071397 ecalais@pudue.edu http://www.eas.pudue.edu/~calais/ Objectives What is
More informationLINES AND TANGENTS IN POLAR COORDINATES
LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Polacoodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and
More informationForces & Magnetic Dipoles. r r τ = μ B r
Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent
More informationReview of Vectors. Appendix A A.1 DESCRIBING THE 3D WORLD: VECTORS. 3D Coordinates. Basic Properties of Vectors: Magnitude and Direction.
Appendi A Review of Vectos This appendi is a summa of the mathematical aspects of vectos used in electicit and magnetism. Fo a moe detailed intoduction to vectos, see Chapte 1. A.1 DESCRIBING THE 3D WORLD:
More informationChapter 30: Magnetic Fields Due to Currents
d Chapte 3: Magnetic Field Due to Cuent A moving electic chage ceate a magnetic field. One of the moe pactical way of geneating a lage magnetic field (.11 T) i to ue a lage cuent flowing though a wie.
More informationGravitation and Kepler s Laws
3 Gavitation and Keple s Laws In this chapte we will ecall the law of univesal gavitation and will then deive the esult that a spheically symmetic object acts gavitationally like a point mass at its cente
More informationTrigonometric Functions of Any Angle
Tigonomet Module T2 Tigonometic Functions of An Angle Copight This publication The Nothen Albeta Institute of Technolog 2002. All Rights Reseved. LAST REVISED Decembe, 2008 Tigonometic Functions of An
More informationGauss Law in dielectrics
Gauss Law in dielectics We fist deive the diffeential fom of Gauss s law in the pesence of a dielectic. Recall, the diffeential fom of Gauss Law is This law is always tue. E In the pesence of dielectics,
More informationChapter 10 Angular Momentum
Chapte 0 Angula Momentum Conceptual Poblems 5 A paticle tavels in a cicula path and point P is at the cente o the cicle. (a) the paticle s linea momentum p is doubled without changing the adius o the cicle,
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 informationExperiment MF Magnetic Force
Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuentcaying conducto is basic to evey electic moto  tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating
More information