Electric Potential. otherwise to move the object from initial point i to final point f


 Brian Wade
 4 years ago
 Views:
Transcription
1 PHY2061 Enched Physcs 2 Lectue Notes Electc Potental Electc Potental Dsclame: These lectue notes ae not meant to eplace the couse textbook. The content may be ncomplete. Some topcs may be unclea. These notes ae only meant to be a study ad and a supplement to you own notes. Please epot any naccuaces to the poesso. Wok and Potental Enegy Applyng a oce ove a dstance eues wok: W = Fd Fand dae constant W 12 = F ds othewse to move the object om ntal pont to nal pont The wok done by a oce on an object to move t om pont to pont s opposte to the change n the potental enegy: ( ) W = Δ U = U U In othe wods, the wok expended by the oce s postve, the potental enegy o the object s loweed. Fo example, an apple s dopped om the banch o a tee, the oce o gavty does wok to move (acceleate actually) the apple om the banch to the gound. The apple now has less gavtatonal potental enegy. These concepts ae ndependent o the type o oce. So the same pncpal also apples to the electc eld actng on an electc chage. We dene the electc potental as the potental enegy o a postve test chage dvded by the chage 0 o the test chage. U V = 0 It s by denton a scala uantty, not a vecto lke the electc eld. The SI unt o electc potental s the Volt (V) whch s 1 Joule/Coulomb. The unts o the electc eld, whch ae N/C, can also be wtten as V/m (dscussed late). Changes n the electc potental smlaly elate to changes n the potental enegy: ΔU Δ V = 0 D. Acosta Page 1 9/12/2006
2 PHY2061 Enched Physcs 2 Lectue Notes Electc Potental So we can compute the change n potental enegy o an object wth chage cossng an electc potental deence: Δ U = Δ V Ths motvates anothe unt o potental enegy, snce oten we ae nteested n the potental enegy o a patcle lke the electon cossng an electc potental deence. Consde an electon cossng a potental deence o 1 volt: ( )( ) C 1 V J = 1 ev Δ U = Δ V = eδ V = = Ths s a tny numbe, whch we can dene as one electonvolt (abbevated ev ). It s a basc unt used to measue the tny eneges o subatomc patcles lke the electon. You can easly convet back to the SI unt Joules by just multplyng by the chage o the electon, e. A common conventon s to set the electc potental at nnty (.e. nntely a away om any electc chages) to be zeo. Then the electc potental at some pont just ees to the change n electc potental n movng the chage om nnty to pont. Δ V = V V V The wok done by the electc eld n movng an electc chage om nnty to pont s gven by: ( ) W = Δ U = Δ V = V V = V whee the last step s done by ou conventon. But keep n mnd that t s only the deences n electc potental that have any meanng. A constant oset n electc potental o potental enegy does not aect anythng. Electc Potental om Electc Feld Consde the wok done by the electc eld n movng a chage 0 a dstance ds: dw = F ds= E ds 0 The total wok done by the eld n movng the chage a macoscopc dstance om ntal pont to nal pont s gven by a lne ntegal along the path: W = 0 E d s Ths wok s elated to the negatve change n potental enegy o electc potental: D. Acosta Page 2 9/12/2006
3 PHY2061 Enched Physcs 2 Lectue Notes Electc Potental W 0 ( ) = Δ V = V V Δ V = V V = E d s= E d s The last step changes the decton o the ntegaton and eveses the sgn o the ntegal. Eupotental Suaces Eupotental suaces ae suaces (not necessaly physcal suaces) whch ae at eual electc potental. Thus, between any 2 ponts on the suace ΔV=0. Ths mples that no wok can be done by the electc eld to move an object along the suace, and thus we must have E ds = 0 Theeoe, eupotental suaces ae always pependcula to the decton o the electc eld (the eld lnes). Feld lnes Eupotental lnes The potental lnes ndcate suaces at the same electc potental, and the spacng s a measue o the ate o chage o the potental. The lnes themselves have no physcal meanng. Potental o a Pont Chage Let s calculate the electc potental at a pont a dstance away om a postve chage. That s, let us calculate the electc potental deence when movng a test chage om nnty to a pont a dstance away om the pmay chage. Δ V = V V = E d s D. Acosta Page 3 9/12/2006
4 PHY2061 Enched Physcs 2 Lectue Notes Electc Potental Let us choose a adal path. Then E ds= E ds snce the eld ponts n the opposte decton o the path. Howeve, we choose ntegatng vaable d, then ds = d snce ponts adally outwad lke the eld. We thus have: Δ V = E ds= Ed d d = K = K = K = K Snce the electc potental s chosen (and shown hee) to be zeo at nnty, we can just wte o the electc potental a dstance away om a pont chage : ( ) V = K It looks smla to the expesson o the magntude o the electc eld, except that t alls o as 1/ athe than 1/ 2. We also could ntegated n the opposte sense: Δ V = V V = E d s Then E ds= E d Δ V = V = E ds= Ed d d 1 = K = K = K = K 2 2 V = K Potental o Many Pont Chages By the supeposton pncpal, the electc potental asng om many pont chages s just: V = K whee s the chage o the th chage, and s the dstance om the chage to some pont P whee we wsh to know the total electc potental. The advantage o ths calculaton s that you only have to lnealy add the electc potental asng om each pont chage, athe than addng each vecto component sepaately as n the case o the electc eld. D. Acosta Page 4 9/12/2006
5 PHY2061 Enched Physcs 2 Lectue Notes Electc Potental Electc Dpole y + + P Let s see how to calculate the electc potental at pont P due to an electc dpole. By the supeposton pncple, the total potental s: V = V+ V = K + + V = K + d θ x whee + s the dstance om the postve chage to pont P, and  the dstance om the negatve chage. Now o lage dstances, electc dpole. d dcosθ, whee d s the sepaaton o the + dcosθ pcosθ V = K = K 2 2 whee = + and p d s the electc dpole moment. Potental o Contnuous Chage Dstbutons Potental between 2 Paallel Plates Let s calculate the electc potental deence between 2 lage paallel conductng plates sepaated by a dstance d, wth the uppe plate (denoted + ) at hghe electc potental than the lowe. + Fom what we leaned by Gauss s Law and conductos, we know that the electc eld σ asng om a conducto wth a chage densty σ s E =. It s thus a constant between ε 0 the two plates n ths example. The electc potental deence s gven by a lne ntegal: E ds D. Acosta Page 5 9/12/2006
6 PHY2061 Enched Physcs 2 Lectue Notes Electc Potental + Δ V = V+ V = E d s E d s = E ds Δ V = E d opposte dectons Anothe way to vew ths esult s that we apply an electc potental deence between two conductng plates (lage compaed to the sepaaton d), the magntude o the electc eld between them s: E ΔV = d Ths motvates the altenate unts o electc eld o V/m. See moe examples n the textbook on contnuous chage dstbutons! Electc Feld om Electc Potental We have seen n the pevous example o the electc potental between two paallel plates, that E ΔV = Δ s whee Δs s the spacng between the plates, whee the path s paallel to the eld decton (and pependcula to eupotental suaces). In act, the eld ponts n decton opposte to nceasng electc potental deence along path s: ΔV E= s ˆ Δ s Now n the nntesmal lmt, dv E= s ˆ ds whch apples to the eld calculated n any egon, unom o not. Wtng ths nto the usual Catesan coodnates: E = V whee s the gadent opeato. It s a shothand o: D. Acosta Page 6 9/12/2006
7 PHY2061 Enched Physcs 2 Lectue Notes Electc Potental V Ex = x V Ey = y V Ez = z So the electc eld s elated to the negatve ate o change o the electc potental. Ths s a specc manestaton o a moe geneal elaton that a oce s elated to the ate o change o the coespondng potental enegy: F = U ( n one dmenson: du F = ) dx Fo the case o the electc eld, F= E and U = V, so E= V E = V D. Acosta Page 7 9/12/2006
8 PHY2061 Enched Physcs 2 Lectue Notes Electc Potental Conductos and Electc Potental Recall that the valence electons n a conducto ae ee to move, but that n electostatc eulbum they have no net velocty. Anothe conseuence o ths s that: ΔV = 0 acoss a conducto I not, electons would move om hghe to lowe potental, and thus not be n statc eulbum. Ths mples that the suace o the conducto, no matte what shape, s also an eupotental suace. We leaned aleady that the electc eld s pependcula to the suace o a conducto (othewse chages would acceleate along the suace), and eupotental lnes ae always pependcula to the electc eld lnes. Eupotental lne Suace o conducto Example: Let s consde as an example 2 conductng sphees connected by a thn conductng we. One sphee has a smalle adus ( 1 ) than the othe ( 2 ); and the chages on the two sphees ae 1 and 2 espectvely. By the above agument, all suaces ae at the same electc potental. Let s ase the ente system to potental V wth espect to a pont nntely a away. The two sphees must have the same potental, so by euatng the potental enegy o each chaged sphee (whch s the same as that o a pont chage at the cente o the sphee) we get: 1 2 K = K = 2 2 D. Acosta Page 8 9/12/2006
9 PHY2061 Enched Physcs 2 Lectue Notes Electc Potental Now let s detemne the suace chage denstes. Snce σ =, the last euaton can 2 4π be wtten: σ 4π σ = 2 24π2 2 σ1 2 = σ 2 1.e. the suace chage densty s nvesely popotonal to the adus o the sphee. Now the magntude electc eld at the suace o the sphee s: E 2 1 σ 4π σ = K = = 4πε ε Thus, the eld stength s popotonal to the suace chage densty, whch s nvesely popotonal to the adus o the sphee. Fo a lage enough chage 1 and small enough adus 1, the beakdown electc eld 6 stength n a could be exceeded ( 3 10 V/m) and a dschage (lghtenng bolt) occu. Ths s the bass o a lghtenng od. Let the lage sphee epesent a lage suace, such as the Eath, and the small sphee a small naow pont such as a od. I the two suaces accue a lage chage, such as dung a lghtenng stom, the electc eld s stongest at the naow od and a beakdown s most lkely to occu thee. Do not stand nea tall naow objects (lke tees!) n an electcal stom! D. Acosta Page 9 9/12/2006
Gravitation. Definition of Weight Revisited. Newton s Law of Universal Gravitation. Newton s Law of Universal Gravitation. Gravitational Field
Defnton of Weght evsted Gavtaton The weght of an object on o above the eath s the gavtatonal foce that the eath exets on the object. The weght always ponts towad the cente of mass of the eath. On o above
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 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 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 informationPerturbation Theory and Celestial Mechanics
Copyght 004 9 Petubaton Theoy and Celestal Mechancs In ths last chapte we shall sketch some aspects of petubaton theoy and descbe a few of ts applcatons to celestal mechancs. Petubaton theoy s a vey boad
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 informationDrag force acting on a bubble in a cloud of compressible spherical bubbles at large Reynolds numbers
Euopean Jounal of Mechancs B/Fluds 24 2005 468 477 Dag foce actng on a bubble n a cloud of compessble sphecal bubbles at lage Reynolds numbes S.L. Gavlyuk a,b,,v.m.teshukov c a Laboatoe de Modélsaton en
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 informationContinuous Compounding and Annualization
Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem
More 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 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 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 informationAdditional File 1  A modelbased circular binary segmentation algorithm for the analysis of array CGH data
1 Addtonal Fle 1  A modelbased ccula bnay segmentaton algothm fo the analyss of aay CGH data FangHan Hsu 1, HungI H Chen, MongHsun Tsa, LangChuan La 5, ChCheng Huang 1,6, ShhHsn Tu 6, Ec Y Chuang*
More informationOrbit dynamics and kinematics with full quaternions
bt dynamcs and knematcs wth full quatenons Davde Andes and Enco S. Canuto, Membe, IEEE Abstact Full quatenons consttute a compact notaton fo descbng the genec moton of a body n the space. ne of the most
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 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 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 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 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 information(Semi)Parametric Models vs Nonparametric Models
buay, 2003 Pobablty Models (Sem)Paametc Models vs Nonpaametc Models I defne paametc, sempaametc, and nonpaametc models n the two sample settng My defnton of sempaametc models s a lttle stonge than some
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 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 information12.1. FÖRSTER RESONANCE ENERGY TRANSFER
ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 11 1.1. FÖRSTER RESONNCE ENERGY TRNSFER Föste esonance enegy tansfe (FR) efes to the nonadiative tansfe of an electonic excitation fom a dono molecule to
More 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 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 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 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 Coverage Gap Filling Algorithm in Hybrid Sensor Network
A Coveage Ga Fllng Algothm n Hybd Senso Netwok Tan L, Yang Mnghua, Yu Chongchong, L Xuanya, Cheng Bn A Coveage Ga Fllng Algothm n Hybd Senso Netwok 1 Tan L, 2 Yang Mnghua, 3 Yu Chongchong, 4 L Xuanya,
More informationAN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM
AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,
More 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 informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More 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 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 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 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 informationGreen's function integral equation methods for plasmonic nanostructures
Geens functon ntegal equaton methods fo plasmonc nanostuctues (Ph Couse: Optcal at the Nanoscale) Thomas Søndegaad epatment of Phscs and Nanotechnolog, Aalbog Unvest, Senve 4A, K9 Aalbog Øst, enma. Intoducton
More informationAREA COVERAGE SIMULATIONS FOR MILLIMETER POINTTOMULTIPOINT SYSTEMS USING STATISTICAL MODEL OF BUILDING BLOCKAGE
Radoengneeng Aea Coveage Smulatons fo Mllmete PonttoMultpont Systems Usng Buldng Blockage 43 Vol. 11, No. 4, Decembe AREA COVERAGE SIMULATIONS FOR MILLIMETER POINTTOMULTIPOINT SYSTEMS USING STATISTICAL
More informationReview C: Work and Kinetic Energy
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department o Physcs 8.2 Revew C: Work and Knetc Energy C. Energy... 2 C.. The Concept o Energy... 2 C..2 Knetc Energy... 3 C.2 Work and Power... 4 C.2. Work Done by
More informationAn Algorithm For Factoring Integers
An Algothm Fo Factong Integes Yngpu Deng and Yanbn Pan Key Laboatoy of Mathematcs Mechanzaton, Academy of Mathematcs and Systems Scence, Chnese Academy of Scences, Bejng 100190, People s Republc of Chna
More informationA New replenishment Policy in a Twoechelon Inventory System with Stochastic Demand
A ew eplenshment Polcy n a woechelon Inventoy System wth Stochastc Demand Rasoul Haj, Mohammadal Payesh eghab 2, Amand Babol 3,2 Industal Engneeng Dept, Shaf Unvesty of echnology, ehan, Ian (haj@shaf.edu,
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 informationJoint Virtual Machine and Bandwidth Allocation in Software Defined Network (SDN) and Cloud Computing Environments
IEEE ICC 2014  NextGeneaton Netwokng Symposum 1 Jont Vtual Machne and Bandwdth Allocaton n Softwae Defned Netwok (SDN) and Cloud Computng Envonments Jonathan Chase, Rakpong Kaewpuang, Wen Yonggang, and
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 informationSELFINDUCTANCE AND INDUCTORS
MISN0144 SELFINDUCTANCE AND INDUCTORS SELFINDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction.............................................. 1 A 2. SelfInductance L.........................................
More informationUniform Rectilinear Motion
Engineeing Mechanics : Dynamics Unifom Rectilinea Motion Fo paticle in unifom ectilinea motion, the acceleation is zeo and the elocity is constant. d d t constant t t 111 Engineeing Mechanics : Dynamics
More informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2  Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of noncoplanar vectors Scalar product
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 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 informationREAL INTERPOLATION OF SOBOLEV SPACES
REAL INTERPOLATION OF SOBOLEV SPACES NADINE BADR Abstact We pove that W p s a eal ntepolaton space between W p and W p 2 fo p > and p < p < p 2 on some classes of manfolds and geneal metc spaces, whee
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 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 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 informationMoment and couple. In 3D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r
Moment and couple In 3D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M + + + + M M + O A Moment about an abita
More informationPCA vs. Varimax rotation
PCA vs. Vamax otaton The goal of the otaton/tansfomaton n PCA s to maxmze the vaance of the new SNP (egensnp), whle mnmzng the vaance aound the egensnp. Theefoe the dffeence between the vaances captued
More informationPREVENTIVE AND CORRECTIVE SECURITY MARKET MODEL
REVENTIVE AND CORRECTIVE SECURITY MARKET MODEL Al Ahmadhat Rachd Cheaou and Omd Alzadeh Mousav Ecole olytechnque Fédéale de Lausanne Lausanne Swzeland al.hat@epfl.ch achd.cheaou@epfl.ch omd.alzadeh@epfl.ch
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 informationValuation of Floating Rate Bonds 1
Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned
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 informationAMB111F Financial Maths Notes
AMB111F Financial Maths Notes Compound Inteest and Depeciation Compound Inteest: Inteest computed on the cuent amount that inceases at egula intevals. Simple inteest: Inteest computed on the oiginal fixed
More informationREAL TIME MONITORING OF DISTRIBUTION NETWORKS USING INTERNET BASED PMU. Akanksha Eknath Pachpinde
REAL TME MONTORNG OF DSTRBUTON NETWORKS USNG NTERNET BASED PMU by Akanksha Eknath Pachpnde A Thess submtted to the Faculty of the Gaduate School of the Unvesty at Buffalo, State Unvesty of New Yok n patal
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 informationI = Prt. = P(1+i) n. A = Pe rt
11 Chapte 6 Matheatcs of Fnance We wll look at the atheatcs of fnance. 6.1 Sple and Copound Inteest We wll look at two ways nteest calculated on oney. If pncpal pesent value) aount P nvested at nteest
More informationest using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.
9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,
More 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 informationKeywords: Transportation network, Hazardous materials, Risk index, Routing, Network optimization.
IUST Intenatonal Jounal of Engneeng Scence, Vol. 19, No.3, 2008, Page 5765 Chemcal & Cvl Engneeng, Specal Issue A ROUTING METHODOLOGY FOR HAARDOUS MATIALS TRANSPORTATION TO REDUCE THE RISK OF ROAD NETWORK
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 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 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 informationPrejudice and the Economics of Discrimination
Pelmnay Pejudce and the Economcs of Dscmnaton Kewn Kof Chales Unvesty of Chcago and NB Jonathan Guyan Unvesty of Chcago GSB and NB Novembe 17, 2006 Abstact Ths pape eexamnes the ole of employe pejudce
More informationIlona V. Tregub, ScD., Professor
Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation
More informationRotation Kinematics, Moment of Inertia, and Torque
Rotaton Knematcs, Moment of Inerta, and Torque Mathematcally, rotaton of a rgd body about a fxed axs s analogous to a lnear moton n one dmenson. Although the physcal quanttes nvolved n rotaton are qute
More informationRisk Sensitive Portfolio Management With CoxIngersollRoss Interest Rates: the HJB Equation
Risk Sensitive Potfolio Management With CoxIngesollRoss Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,
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 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 informationDeterminants of Borrowing Limits on Credit Cards Shubhasis Dey and Gene Mumy
Bank of Canada Banque du Canada Wokng Pape 20057 / Document de taval 20057 Detemnants of Boowng mts on Cedt Cads by Shubhass Dey and Gene Mumy ISSN 11925434 Pnted n Canada on ecycled pape Bank of Canada
More informationAN EQUILIBRIUM ANALYSIS OF THE INSURANCE MARKET WITH VERTICAL DIFFERENTIATION
QUIIRIUM YI OF T IUR MRKT WIT VRTI IFFRTITIO Mahto Okua Faculty of conomcs, agasak Unvesty, 4 Katafuch, agasak, 8508506, Japan okua@net.nagasaku.ac.p TRT ach nsuance poduct pe se s dentcal but the nsuance
More informationOn the Efficiency of Equilibria in Generalized Second Price Auctions
On the Effcency of Equlba n Genealzed Second Pce Auctons Ioanns Caaganns Panagots Kanellopoulos Chstos Kaklamans Maa Kyopoulou Depatment of Compute Engneeng and Infomatcs Unvesty of Patas and RACTI, Geece
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 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 informationNUCLEAR MAGNETIC RESONANCE
19 Jul 04 NMR.1 NUCLEAR MAGNETIC RESONANCE In this expeiment the phenomenon of nuclea magnetic esonance will be used as the basis fo a method to accuately measue magnetic field stength, and to study magnetic
More informationMechanics 1: 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 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 informationSimultaneous Detection and Estimation, False Alarm Prediction for a Continuous Family of Signals in Gaussian Noise
Sultaneous Detecton and Estaton, False Ala Pedcton fo a Contnuous Faly of Sgnals n Gaussan Nose D Mchael Mlde, Robet G Lndgen, and Mos M Bean Abstact New pobles ase when the standad theoy of jont detecton
More informationModeling and computing constrained
F EAURE A RICLE HE COMPUAION OF CONSRAINED DYNAMICAL SYSEMS: MACHING PHYSICAL MODELING WIH NUMERICAL MEHODS Reseaches have nvestgated modelng and computaton of constaned dynamcal systems, but scentsts
More informationNontrivial lower bounds for the least common multiple of some finite sequences of integers
J. Numbe Theoy, 15 (007), p. 393411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations White Pape 48 Revision by Victo Avela > Executive summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance
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 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 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 informationSymmetric polynomials and partitions Eugene Mukhin
Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation
More informationAP Physics C: Mechanics 2011 FreeResponse Questions
AP Phyc C: Mechanc FeeRepone Queton About the College Boa The College Boa a monven notfopoft oganzaton that connect tuent to college ucce an oppotunty. Foune n 9, the College Boa wa ceate to epan acce
More informationNURBS Drawing Week 5, Lecture 10
CS 43/585 Compute Gaphics I NURBS Dawing Week 5, Lectue 1 David Been, William Regli and Maim Pesakhov Geometic and Intelligent Computing Laboato Depatment of Compute Science Deel Univesit http://gicl.cs.deel.edu
More informationCHAPTER 8 Potential Energy and Conservation of Energy
CHAPTER 8 Potental Energy and Conservaton o Energy One orm o energy can be converted nto another orm o energy. Conservatve and nonconservatve orces Physcs 1 Knetc energy: Potental energy: Energy assocated
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 informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations By Victo Avela White Pape #48 Executive Summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance the availability
More informationLesson 8 Ampère s Law and Differential Operators
Lesson 8 Ampèe s Law and Diffeential Opeatos Lawence Rees 7 You ma make a single cop of this document fo pesonal use without witten pemission 8 Intoduction Thee ae significant diffeences between the electic
More informationDegrees of freedom in HLM models
Degees o eedom n HLM models The vaous degees o eedom n a HLM2/HLM3 model can be calculated accodng to Table 1 and Table 2. Table 1: Degees o Feedom o HLM2 Models Paamete/Test Statstc Degees o Feedom Gammas
More informationQuantity Formula Meaning of variables. 5 C 1 32 F 5 degrees Fahrenheit, 1 bh A 5 area, b 5 base, h 5 height. P 5 2l 1 2w
1.4 Rewite Fomulas and Equations Befoe You solved equations. Now You will ewite and evaluate fomulas and equations. Why? So you can apply geometic fomulas, as in Ex. 36. Key Vocabulay fomula solve fo a
More informationHow To Find The Optimal Stategy For Buying Life Insuance
Life Insuance Puchasing to Reach a Bequest Ehan Bayakta Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, 48109 S. David Pomislow Depatment of Mathematics, Yok Univesity Toonto, Ontaio,
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 information