Universal Gravitation


 Jemima Ward
 10 days ago
 Views:
Transcription
1 Univesal Gavitation Keple s Thee Laws of Planetay Motion IB 1 Law 1: All planets obit the Sun in elliptical paths with the Sun at one focus. Law : An imaginay line joining any planet to the Sun sweeps out equal aeas in equal time intevals. Law 3: The squae of the obital peiod of any planet is popotional to the cube of its aveage obital adius. T T Fomula 3 k 3 T 3 Newton s Law of Univesal Gavitation Evey paticle in the Univese attacts evey othe paticle with a foce that is diectly popotional to the poduct of the masses and that is invesely popotional to the squae of the distance between them. Two appoximations used in deiving the law: 1. Masses ae consideed to be point masses. Point mass: infinitely small object (adius = 0) whose mass is m. The foce between two spheical masses whose sepaation is lage compaed to thei adii is the same as if the two sphees wee point masses with thei masses concentated at the centes of the sphees. Sun Mean EathSun distance = 1.50 x m Eath Mean adius = 6.96 x 10 8 m Mean adius = 6.37 x 10 6 m 1
2 Fomula m1 m Fg Gm m Fg = 1 Gavitational Constant : G = 6.77 x N m /kg Point mass Extended spheical body IB 1 Re Re 3Re 4Re Newton s Deivation of Keple s Thid Law What povides the centipetal foce fo obital motion? gavitation Deivation F c ma c GMm mv GM v but v T GM T GM 4 T T 4 3 GM Application weighing the Sun T yea days hous s 3 4 GM x10 m 6.67x10 30 MS.0x10 kg accepted value=1.99x10 30 kg M s What is the esultant gavitational foce on the Eath fom the Sun and Moon, as shown below? Aveage EathSun distance = 1.50 x m Aveage EathMoon distance = 3.84 x 10 8 m Sun Mass = 1.99 x kg Eath Mass = 5.98 x 10 4 kg Moon Mass = 7.36 x 10 kg
3 Gavitational Field Stength IB 1 Gavitational field stength at a point in a gavitational field: the gavitational foce exeted pe unit mass on a small/point mass Symbol: g Fomula: g = F g / m Units: N/kg (m/s ) Deiving fomula fo gavitational field stength at any point above the suface of a planet g = F g /m g = (GMm/ )/m g = GM/ Point mass Type: vecto Deiving fomula fo gavitational field stength at the suface of a planet 1. What is the gavitational field stength of the Eath at its suface? g = GM/ g o = G M p / R p g at the suface of the Eath g o = G M E /R e. What is the gavitational field stength at an altitude equal to the adius of Eath? Extended spheical body g o g atio g/g o = R E / Re Re 3Re 4Re 3
4 Aveage EathMoon distance = 3.84 x 10 8 m IB 1 Eath Mass = 5.98 x 10 4 kg Moon Mass = 7.36 x 10 kg 3. a) What is the esultant gavitational field stength at a point midway between the Eath and Moon? b) What is the esultant gavitational foce acting on a kg space pobe at this location? Aveage EathMoon distance = 3.84 x 10 8 m Eath Mass = 5.98 x 10 4 kg Moon Mass = 7.36 x 10 kg 4. a) Is thee a point whee the esultant gavitational field stength of the Eath and Moon is zeo? If so, whee? b) What is the esultant gavitational foce acting on a kg space pobe at this location? 4
5 Gavitational Potential Enegy Diffeence in gavitational potential enegy between any two points in a gavitational field: IB 1 ΔE P = mgδh This diffeence is path independent. 1. Same ΔE P between any two points no matte what path is taken between them.. Wok done in moving a mass between two points in a gavitational field is independent of the path taken. 3. ΔE P is zeo between any two points at the same level no matte what path is taken. 4. ΔE P is zeo fo any closed path (a path that begins and ends at same point). Old fomula fo gavitational potential enegy: E p = mgh Base level: infinity E P = 400 J discuss poblems with definition 1) g vaies above suface ) abitay base level E P = 100 J Gavitational PE at infinity: zeo E P = 0 Gavitational Potential Enegy of a mass at a point in a gavitational field: the wok done in binging a small point mass in fom infinity to that point in the gavitational field Deivation of gavitational potential enegy fomula 5
6 E p =  GMm/ E P = Gm 1 m / Fomula not valid inside planet Symbol: V Units: J Fomula: Type: scala What is the gavitational potential enegy of a 5000 kg satellite: Potential enegy vs. distance: IB 1 E p at suface E p = GM p m/r p a) on the suface of the Eath? b) obiting the Eath at an altitude of 00 km? c) How much does the potential enegy of the satellite incease when it is put into this obit? Gavitational Potential Gavitational potential at a point in a gavitational field: wok done pe unit mass to bing a small point mass in fom infinity to that point in the gavitational field Fomulas: Diffeence in gavitational potential: Gavitational Potential vs. distance ΔV = W/m ΔV = ΔE p /m Gavitational potential at a point: V = E p /m V = GM/ Symbol: V Units: J/kg Type: scala V at suface V o = GM p /R p 6
7 1. What is the gavitational potential due to the Eath s gavitational field: IB 1 a) at the suface of the Eath? b) At a location thee Eath adii fom the cente of the Eath? c) What is the change in potential in moving fom the suface to this new location? d) What is the minimum amount of enegy needed to lift a 5000 kg satellite to this location?. What is the net gavitational potential at a spot midway between the Eath and the Sun? Sun Eath 3. Deive an expession fo the gavitational potential at the suface of a planet in tems of the gavitational field stength. 7
8 Escape Speed IB 1 Escape Speed: minimum initial speed an object must have at the suface of a planet in ode to escape the gavitational attaction of the planet Tavel to infinity Just make it means velocity is zeo at infinity means E K is zeo at infinity as well as E P E o = E p + E k E f = E p Planet Assumptions: planet is isolated ignoe ai esistance Deivation: Note: 1. Diection of tavel is ielevant ΔE p is path independent. independent of mass of ocket 3. Moe speed (E K ) is needed in eal life since ai fiction is not negligible at lowe altitudes 1. What is the escape speed fo Eath?. If the Eath became a black hole, how lage would it be? 8
9 Satellite Motion IB 1 Peiod of a satellite: Natual Satellites Acceleation of a satellite: Atificial Satellites Obital speed of a satellite: T T 3 k 3 T 3 a c = v / v = π /T a c = 4π /T a c = g = GM/ v = π /T F c ma c GMm mv GM v GM v Weightlessness Fee fall Obital motion Deep space 1. Compae the motion of satellites A and B. A faste, less time (smalle peiod) B slowe, moe time (longe peiod) T /R 3 = constant tue fo all satellites T = kr 3/. What happens to the equied obital speed if: a) the mass of the satellite inceases? Nothing speed is independent of mass b) the satellite is boosted into a highe obit? Satellite would obit at a slowe (tangential) speed 3. What would happen to a satellite if it encounteed appeciable ai esistance? Slow down, dop to lowe obit, and speed up, encounte even moe ai molecules (dense), cycle continues spial to Eath 9
10 Enegy of Obiting Satellites Compae the enegies of the two obiting satellites. IB 1 Gavitational Potential Enegy Kinetic Enegy Total Enegy Enegy Deivations Gaphs of the enegies of an obiting satellite Gavitational Potential Enegy Kinetic Enegy Total Enegy R E Compaisons: A 1500 kg satellite is to be put into obit aound the Eath at an altitude of 00 km. a) How much potential enegy will the satellite have at this altitude? d) What is the obital speed of the satellite? b) How much kinetic enegy will the satellite need to obit at this altitude? e) What is the minimum amount of enegy needed to lift the satellite fom the suface of the Eath to this altitude? c) What is the total amount of enegy the satellite has at this altitude? 10
11 Compaisons IB 1 Equipotential suface: a suface on which the potential is the same eveywhee 1. The gavitational foce does no wok as a mass moves on along equipotential suface.. The wok done in moving a mass between equipotential sufaces is path independent. 3. The wok done in moving a mass along a closed path is zeo. one point mass two point masses 11
12 On the diagam at ight: IB 1 a) Sketch the gavitational field aound the point mass. b) Sketch equipotential sufaces aound the point mass. What is the elationship between the gavitational field and the equipotential sufaces? Pependicula Field lines point in diection of deceasing potential Gavitational Potential Gadient gadient: ate of change with espect to something slope o deivative gavitational potential gadient: the gavitational field is the negative gadient of the gavitational potential with espect to distance Fomula deive g = ΔV/Δ A B 80 J/kg 70 J/kg What is the aveage gavitational field stength between equipotential sufaces A and B if they ae 5.0 m apat? 1
13 Pactice Questions IB 1 1. a) Calculate the gavitational foce the Sun exets on the Eath b) Compae this to the gavitational foce that the Eath exets on the Sun.. a) Calculate the stength of the gavitational field of the Sun at a location one million kilometes fom the Sun. b) What is the Sun s gavitational foce at this point? 3. a) Calculate the stength of the Sun s gavitational field at the suface of the Eath. b) Explain why the net gavitational field stength at the suface of the Eath can be appoximated as due solely to the Eath s gavitational field. 13
14 IB 1 4. a) Calculate the esultant gavitational field at a spot midway between the Eath and Sun. b) Compae the contibutions fom the Sun and the Eath to this esultant field. c) What is the gavitational foce acting on a 5000 kg space pobe at this location? 5. A 5000kg satellite obits Mas at a distance of 1000 km. a) What is the gavitational potential at the suface of Mas? Mass of Mas: 6.4 x 10 3 kg Mean planetay adius: 3.37 x 10 6 m Mas b) How much gavitational potential enegy does the satellite have on the suface of Mas? c) What is the gavitational potential at obiting altitude? d) How much gavitational potential enegy does the satellite have at this altitude? e) What is the minimum enegy needed to lift the satellite to this altitude? 14
15 IB 1 6. A kg satellite is placed in a low altitude obit. a) If the altitude is sufficiently low, what is the appoximate adius of the satellite s obit? b) Calculate the satellite s obital speed. c) Calculate the obital peiod of the satellite. d) Calculate the gavitational potential enegy of the satellite. e) Calculate the kinetic enegy of the satellite. f) Calculate the total enegy of the satellite. g) What is the minimum amount of enegy needed to lift the satellite into this obit? 15
FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.
Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing
More 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 informationChapter 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 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 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 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 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 information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More 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 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 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 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 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 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 informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More informationPhysics 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 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 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 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 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 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 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 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 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 informationThe Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = W/q 0 1V [Volt] =1 Nm/C
Geneal Physics  PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit
More 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 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 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 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 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 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 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 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 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 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 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 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 information4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first nonzero digit to
. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate
More informationDisplacement, Velocity And Acceleration
Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,
More informationMultiple choice questions [70 points]
Multiple choice questions [70 points] Answe all of the following questions. Read each question caefull. Fill the coect bubble on ou scanton sheet. Each question has exactl one coect answe. All questions
More informationChapter 4. Electric Potential
Chapte 4 Electic Potential 4.1 Potential and Potential Enegy... 43 4.2 Electic Potential in a Unifom Field... 47 4.3 Electic Potential due to Point Chages... 48 4.3.1 Potential Enegy in a System of
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 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 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 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 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 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 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 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 informationSolution Derivations for Capa #8
Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass
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 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 informationSolutions for Physics 1301 Course Review (Problems 10 through 18)
Solutions fo Physics 1301 Couse Review (Poblems 10 though 18) 10) a) When the bicycle wheel comes into contact with the step, thee ae fou foces acting on it at that moment: its own weight, Mg ; the nomal
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 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 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 informationAP Physics Electromagnetic Wrap Up
AP Physics Electomagnetic Wap Up Hee ae the gloious equations fo this wondeful section. F qsin This is the equation fo the magnetic foce acting on a moing chaged paticle in a magnetic field. The angle
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 informationProblems on Force Exerted by a Magnetic Fields from Ch 26 T&M
Poblems on oce Exeted by a Magnetic ields fom Ch 6 TM Poblem 6.7 A cuentcaying wie is bent into a semicicula loop of adius that lies in the xy plane. Thee is a unifom magnetic field B Bk pependicula to
More informationOn the Relativistic Forms of Newton's Second Law and Gravitation
On the Relativistic Foms of Newton's Second Law and avitation Mohammad Bahami,*, Mehdi Zaeie 3 and Davood Hashemian Depatment of physics, College of Science, Univesity of Tehan,Tehan, Islamic Republic
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 informationThe Gravity Field of the Earth  Part 1 (Copyright 2002, David T. Sandwell)
1 The Gavity Field of the Eath  Pat 1 (Copyight 00, David T. Sandwell) This chapte coves physical geodesy  the shape of the Eath and its gavity field. This is just electostatic theoy applied to the Eath.
More informationChapter 2. Electrostatics
Chapte. Electostatics.. The Electostatic Field To calculate the foce exeted by some electic chages,,, 3,... (the souce chages) on anothe chage Q (the test chage) we can use the pinciple of supeposition.
More informationLab #7: Energy Conservation
Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 14 Intoduction: Pehaps one of the most unusual
More informationIntroduction to Fluid Mechanics
Chapte 1 1 1.6. Solved Examples Example 1.1 Dimensions and Units A body weighs 1 Ibf when exposed to a standad eath gavity g = 3.174 ft/s. (a) What is its mass in kg? (b) What will the weight of this body
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 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 informationAn Introduction to Omega
An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei iskewad chaacteistics? The Finance Development Cente 2002 1 Fom
More informationNewton s Shell Theorem
Newton Shell Theoem Abtact One of the pincipal eaon Iaac Newton wa motivated to invent the Calculu wa to how that in applying hi Law of Univeal Gavitation to pheicallyymmetic maive bodie (like planet,
More informationVISCOSITY OF BIODIESEL FUELS
VISCOSITY OF BIODIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use
More 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 informationChapter 9 Rotation. Conceptual Problems
Chapte 9 otation Conceptual Poblems 7 Duing a baseball game, the pitche has a blazing astball. You have not been able to swing the bat in time to hit the ball. You ae now just tying to make the bat contact
More informationLearning Objectives. Decreasing size. ~10 3 m. ~10 6 m. ~10 10 m 1/22/2013. Describe ionic, covalent, and metallic, hydrogen, and van der Waals bonds.
Lectue #0 Chapte Atomic Bonding Leaning Objectives Descibe ionic, covalent, and metallic, hydogen, and van de Waals bonds. Which mateials exhibit each of these bonding types? What is coulombic foce of
More informationChapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6
Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe
More 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 informationChapter 4: Fluid Kinematics
Oveview Fluid kinematics deals with the motion of fluids without consideing the foces and moments which ceate the motion. Items discussed in this Chapte. Mateial deivative and its elationship to Lagangian
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 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 informationGravitational Mechanics of the MarsPhobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning
Gavitational Mechanics of the MasPhobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This
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 informationLab M4: The Torsional Pendulum and Moment of Inertia
M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disklike mass suspended fom a thin od o wie. When the mass is twisted about the
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 information10. Collisions. Before During After
10. Collisions Use conseation of momentum and enegy and the cente of mass to undestand collisions between two objects. Duing a collision, two o moe objects exet a foce on one anothe fo a shot time: F(t)
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 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 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 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 informationGauss s law relates to total electric flux through a closed surface to the total enclosed charge.
Chapte : Gauss s Law Gauss s Law is an altenative fomulation of the elation between an electic field and the souces of that field in tems of electic flu. lectic Flu Φ though an aea ~ Numbe of Field Lines
More informationStructure and evolution of circumstellar disks during the early phase of accretion from a parent cloud
Cente fo Tubulence Reseach Annual Reseach Biefs 2001 209 Stuctue and evolution of cicumstella disks duing the ealy phase of accetion fom a paent cloud By Olusola C. Idowu 1. Motivation and Backgound The
More informationA r. (Can you see that this just gives the formula we had above?)
241 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down  you can pedict (o contol) motion
More informationLesson 7 Gauss s Law and Electric Fields
Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual
More informationPhys 2101 Gabriela González. cos. sin. sin
1 Phys 101 Gabiela González a m t t ma ma m m T α φ ω φ sin cos α τ α φ τ sin m m α τ I We know all of that aleady!! 3 The figue shows the massive shield doo at a neuton test facility at Lawence Livemoe
More informationJapan s trading losses reach JPY20 trillion
IEEJ: Mach 2014. All Rights Reseved. Japan s tading losses each JPY20 tillion Enegy accounts fo moe than half of the tading losses YANAGISAWA Akia Senio Economist Enegy Demand, Supply and Foecast Goup
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 informationMagnetic Bearing with Radial Magnetized Permanent Magnets
Wold Applied Sciences Jounal 23 (4): 495499, 2013 ISSN 18184952 IDOSI Publications, 2013 DOI: 10.5829/idosi.wasj.2013.23.04.23080 Magnetic eaing with Radial Magnetized Pemanent Magnets Vyacheslav Evgenevich
More informationNuclear models: FermiGas Model Shell Model
Lectue Nuclea mode: FemiGas Model Shell Model WS/: Intoduction to Nuclea and Paticle Physics The basic concept of the Femigas model The theoetical concept of a Femigas may be applied fo systems of weakly
More informationEconomics 326: Input Demands. Ethan Kaplan
Economics 326: Input Demands Ethan Kaplan Octobe 24, 202 Outline. Tems 2. Input Demands Tems Labo Poductivity: Output pe unit of labo. Y (K; L) L What is the labo poductivity of the US? Output is ouhgly
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 information3.02 Potential Theory and Static Gravity Field of the Earth
3.02 Potential Theoy and Static Gavity Field of the Eath C. Jekeli, The Ohio State Univesity, Columbus, OH, USA ª 2007 Elsevie B.V. All ights eseved. 3.02. Intoduction 2 3.02.. Histoical Notes 2 3.02..2
More information