Diffusion and Transport. 10. Friction and the Langevin Equation. Langevin Equation. f d. f ext. f () t f () t. Then Newton s second law is ma f f f t.
|
|
- Doreen Garrison
- 6 years ago
- Views:
Transcription
1 Diffusion and Tanspot 10. Fiction and the Langevin Equation Now let s elate the phenomena of ownian motion and diffusion to the concept of fiction, i.e., the esistance to movement that the paticle in the fluid expeiences. These concepts wee developed by Einstein in the case of micoscopic motion unde themal excitation, and macoscopically by Geoge Stokes who was the fathe of hydodynamic theoy. Langevin Equation Conside the foces acting on a paticle as we pull it though a fluid. We pull the paticle with an extenal foce fext, which is opposed by a dag foce fom the fluid, fd. The dag o damping acts as esistance to motion of the paticle, which esults fom tying to move the fluid out of the way. f d v A dag foce equies movement, so it is popotional to the velocity of the paticle vdx/ dt x and the fiction coefficient is the popotionality constant that descibes the magnitude of the damping. Newton s second law would say the acceleation of this paticle is popotional to the sum of these foces: ma f d f ext. Now micoscopically, we also ecognize that thee ae timedependent andom foces that the molecules of the fluid exet on a molecule (f). So that the specific molecula details of solute solvent collisions can be aveaged ove, it is useful to think about a nanoscale solute in wate (e.g., biological macomolecules) with dimensions lage enough that its position is simultaneously influenced by many solvent molecules, but is also light enough that the constant inteactions with the solvent leave an unbalanced foce acting on the solute at any moment in time: f () t f () t. Then Newton s second law is ma f f f t. i i d ext The dag foce is pesent egadless of whethe an extenal foce is pesent, so in the absence of extenal foces (fext=0) the equation of motion govening the spontaneous fluctuations of this solute is detemined fom the foces due to dag and the andom fluctuations: d ma f f t mx x f ( t) 0 Andei Tokmakoff 5/3/2017
2 This equation of motion is the Langevin equation. An equation of motion such as this that includes a time-dependent andom foce is known as stochastic. Inseting a andom pocess into a deteministic equation means that we need to use a statistical appoach to solve this equation. We will be looking to descibe the aveage and oot-meansquaed position of the paticle. Fist, what can we say about the andom foce? Although thee may be momentay imbalances, on aveage the petubations fom the solvent on a lage paticle will aveage to zeo at equilibium: f ( t) 0 Equation seems to imply that the dag foce and the andom foce ae independent, but in fact they oiginate in the same molecula foces. If the molecule of inteest is a potein that expeiences the fluctuations of many apidly moving solvent molecules, then the aveaged foces due to andom fluctuations and the dag foces ae elated. The fluctuation dissipation theoem is the geneal elationship that elates the fiction to the coelation function fo the andom foce. In the Makovian limit this is f ( t) f ( t) 2 k T ( t t) Makovian indicates that no coelation exists between the andom foce fo t t > 0. Moe geneally, we can ecove the fiction coefficient fom the integal ove the coelation function fo the andom foce 1 dt fr fr( t) 2kT To descibe the time evolution of the position of ou potein molecule, we would like to obtain an expession fo mean-squae displacement x 2 (t). The position of the molecule can be descibed by integating ove its time-dependent velocity: mean-squae displacement in tems of the velocity autocoelation function t t 2 x t dt dt x t x t 0 0 ( ) ( ) ( ) t x() t dtx ( t), so we can expess the 0 Ou appoach to obtaining x 2 (t) stats by multiplying eq. by x and then ensemble aveaging. d m x x xx x f () t 0 dt Fom eq., the last tem is zeo, and fom the chain ule we know d ( xx) x d x dx x dt dt dt Theefoe we can wite eq. as 2
3 d m xxxx xx0 dt Futhe, the equipatition theoem states that fo each tanslational degee of feedom the kinetic enegy is patitioned as 1 2 kt mx 2 2 d So, m xx xx kt dt Hee we ae descibing motion in 1D, but when fluctuations and displacement ae included fo 3D motion, then we switch x and kt 3 kt. Integating eq. twice with espect to time, and using the initial condition x = 0, we obtain 2kT 2 m x t exp t 1 m To investigate this equation, let s conside two limiting cases. Fom eq. we see that m/ζ has units of time, and so we define the damping time and investigate time scale shot and long compaed to τc: C m / 1) Fo t C, we can expand the exponential in eq. and etain the fist thee tems, which leads to kt m 2) Fo t C, eq. is dominated by the leading tem: 2kT 2 x t (long time: diffusive) x t v t (shot time: inetial) In the diffusive limit the behavio of the molecule is govened entiely by the fluid, and its mass does not matte. The diffusive limit in a stochastic equation of motion is equivalent to setting m 0. We see that τc is a time-scale sepaating motion in the inetial and diffusive limits. It is a coelation time fo the andomization of the velocity of the ballistic diffusive 3
4 paticle due to the andom fluctuations of the envionment. Fo vey little fiction o shot time, the paticle moves with taditional deteministic motion xms = vms t, whee oot-mean-squae displacement xms = x 2 1/2 and vms comes fom the aveage tanslational kinetic enegy of the paticle. Fo high-fiction o long times, we see diffusive behavio with xms~t 1/2. Futhemoe, by compaing eq. to ou ealie continuum esult, x 2 = 2Dt, we see that the diffusion constant can be elated to the fiction coefficient by D kt (in 1D) This is the Einstein fomula. Fo 3D poblems, we eplace kt with 3kT in the expessions above and find D3D = 3kT/ How long does it take to appoach the diffusive egime? Vey fast. Conside a 100 kda potein with R = 3 nm in wate at T = 300 K, we find a chaacteistic coelation time fo andomizing 12 velocities of τc 3 10 s, which coesponds to a distance of about 10 2 nm befoe the onset of diffusive behavio. We can find othe elationships. Noting the elationship of x 2 to the velocity autocoelation function in eq., we find that the paticle velocity is descibed by 2 tm / 2 t/ C x x( ) v e x v e x x v v t v x which can be integated ove time to obtain the diffusion constant. kt vx vx( t) dt D 0 This expession is the Geen Kubo elationship. This is a pactical way of analyzing molecula tajectoies in simulations o using paticle-tacking expeiments to quantify diffusion constants o fiction coefficients. 4
5 Fiction and Viscosity How is the micoscopic fiction oiginating in andom foces elated to macoscopic expeimental obsevables that measue a fluid s esistance to moving an object? ζ is elated to the dynamic viscosity of the fluid and factos descibing the size and shape of the object (but not its mass). Viscosity measues the esistance to shea. A fluid is placed between two plates of aea a sepaated along z, and one plate is moved elative to anothe by applying a foce along x. Since the velocity of the fluid at the inteface with a plate is taken to be the velocity of the plate (noslip bounday conditions: v ( z 0) 0 ), this sets up a velocity gadient along z. The elationship between the shea velocity gadient and the foce is dv x x fx a dz whee η, the dynamic viscosity (kg m 1 s 1 ), is the popotionality facto. shea stess f x a Shape Mattes A sphee, od, o cube with the same mass and suface aea will espond diffeently to flow. Stokes detemined the elationships between dag coefficient and fluid viscosity. Specifically, consideing the case whee a sphee of adius R is esisted by lamina flow of the fluid, one finds that the dag foce on the sphee is fd 6Rv and the viscous foce pe unit aea is entiely unifom acoss the suface of the sphee. This gives us Stokes Law 6R 5
6 Hee R is efeed to as the hydodynamic adius of the sphee, which efes to the adius at which one can apply the no-slip bounday condition, but which on a molecula scale may include wate that is stongly bound to the molecule. Combining eq. with eq. gives the Stokes Einstein elationship fo the tanslation diffusion constant of the sphee 1 D tans k T 6R One can obtain a simila a Stokes Einstein elationship fo oientational diffusion of a sphee in a viscous fluid. Relating the oientational diffusion constant and the dag foce that aises fom esistance to shea, one obtains V=4πR 3 /3 is the volume of the sphee. Reynolds Numbe D ot k T 6V The Reynolds numbe is a dimensionless numbe that indicates whethe the motion of a paticle in a fluid is dominated by inetial o viscous foces. 2 inetial foces R viscous foces When R 1, the paticle moves feely, expeiencing only weak esistance to its motion by the fluid. If R 1, it is dominated by the esistance and intenal foces of the fluid. Fo the latte case, we can conside the limit m 0 in eq., and find that the velocity of the paticle is popotional to the andom fluctuations: vt ( ) f( t) /. Hydodynamically, fo a sphee of adius moving though a fluid with dynamic viscosity η and density ρ at velocity v, v v( dv / dz) R Using pictue above: R 2 2 ( dv/ dz) Conside fo an object with adius 1 cm moving at 10 cm/s though wate: R Now compae to a potein with adius 1 nm moving at 10 m/s: R = J. ene and R. Pecoa, Dynamic Light Scatteing: With Applications to Chemisty, iology, and Physics. (Wiley, New Yok, 1976), pp. 78, E. M. Pucell, Life at low Reynolds numbe, Am. J. Phys. 45, 3-11 (1977). 6
7 Dag Foce in Hydodynamics The dag foce on an object is detemined by the foce equied to displace the fluid against the diection of flow, fd Cv d a This expession takes the fom of a pessue (tem in backets) exeted on the coss-sectional aea of the object along the diection of flow, a. Cd is the dag coefficient, a dimensionless popotionality constant that depends on the shape of the object. In the case of a sphee of adius : a=π 2 in the tubulent flow egime ( R 1000 ) Cd = Detemination of Cd is somewhat empiical since it depends on R and the type of flow aound the sphee. The dag coefficient fo a sphee in the viscous/lamina/stokes flow egimes (R <1) is Cd 24 / R. This comes fom using the Stokes Law fo the dag foce on a sphee f 6v and the Reynolds numbe R vd. d 7
8 Readings 1. R. Zwanzig, Nonequilibium Statistical Mechanics. (Oxfod Univesity Pess, New Yok, 2001). 2.. J. ene and R. Pecoa, Dynamic Light Scatteing: With Applications to Chemisty, iology, and Physics. (Wiley, New Yok, 1976). 8
V7: Diffusional association of proteins and Brownian dynamics simulations
V7: Diffusional association of poteins and Bownian dynamics simulations Bownian motion The paticle movement was discoveed by Robet Bown in 1827 and was intepeted coectly fist by W. Ramsay in 1876. Exact
More informationOSCILLATIONS AND GRAVITATION
1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,
More informationMobility of atoms and diffusion. Einstein relation.
Mobility of atoms and diffusion. Einstein elation. In M simulation we can descibe the mobility of atoms though the mean squae displacement that can be calculated as N 1 MS ( t ( i ( t i ( 0 N The MS contains
More information, and the curve BC is symmetrical. Find also the horizontal force in x-direction on one side of the body. h C
Umeå Univesitet, Fysik 1 Vitaly Bychkov Pov i teknisk fysik, Fluid Dynamics (Stömningsläa), 2013-05-31, kl 9.00-15.00 jälpmedel: Students may use any book including the textbook Lectues on Fluid Dynamics.
More information2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum
2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known
More informationChapter 5 Force and Motion
Chapte 5 Foce and Motion In Chaptes 2 and 4 we have studied kinematics, i.e., we descibed the motion of objects using paametes such as the position vecto, velocity, and acceleation without any insights
More informationChapter 5 Force and Motion
Chapte 5 Foce and Motion In chaptes 2 and 4 we have studied kinematics i.e. descibed the motion of objects using paametes such as the position vecto, velocity and acceleation without any insights as to
More informationFalls in the realm of a body force. Newton s law of gravitation is:
GRAVITATION Falls in the ealm of a body foce. Newton s law of avitation is: F GMm = Applies to '' masses M, (between thei centes) and m. is =. diectional distance between the two masses Let ˆ, thus F =
More information1 Similarity Analysis
ME43A/538A/538B Axisymmetic Tubulent Jet 9 Novembe 28 Similaity Analysis. Intoduction Conside the sketch of an axisymmetic, tubulent jet in Figue. Assume that measuements of the downsteam aveage axial
More informationSection 11. Timescales Radiation transport in stars
Section 11 Timescales 11.1 Radiation tanspot in stas Deep inside stas the adiation eld is vey close to black body. Fo a black-body distibution the photon numbe density at tempeatue T is given by n = 2
More informationSubstances that are liquids or solids under ordinary conditions may also exist as gases. These are often referred to as vapors.
Chapte 0. Gases Chaacteistics of Gases All substances have thee phases: solid, liquid, and gas. Substances that ae liquids o solids unde odinay conditions may also exist as gases. These ae often efeed
More informationChapter 7-8 Rotational Motion
Chapte 7-8 Rotational Motion What is a Rigid Body? Rotational Kinematics Angula Velocity ω and Acceleation α Unifom Rotational Motion: Kinematics Unifom Cicula Motion: Kinematics and Dynamics The Toque,
More informationAP-C WEP. h. Students should be able to recognize and solve problems that call for application both of conservation of energy and Newton s Laws.
AP-C WEP 1. Wok a. Calculate the wok done by a specified constant foce on an object that undegoes a specified displacement. b. Relate the wok done by a foce to the aea unde a gaph of foce as a function
More informationChapter 13 Gravitation
Chapte 13 Gavitation In this chapte we will exploe the following topics: -Newton s law of gavitation, which descibes the attactive foce between two point masses and its application to extended objects
More informationNewton s Laws, Kepler s Laws, and Planetary Orbits
Newton s Laws, Keple s Laws, and Planetay Obits PROBLEM SET 4 DUE TUESDAY AT START OF LECTURE 28 Septembe 2017 ASTRONOMY 111 FALL 2017 1 Newton s & Keple s laws and planetay obits Unifom cicula motion
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More informationOn a quantity that is analogous to potential and a theorem that relates to it
Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) 34-39 On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich
More informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationCBE Transport Phenomena I Final Exam. December 19, 2013
CBE 30355 Tanspot Phenomena I Final Exam Decembe 9, 203 Closed Books and Notes Poblem. (20 points) Scaling analysis of bounday laye flows. A popula method fo measuing instantaneous wall shea stesses in
More informationEM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1)
EM- Coulomb s law, electic field, potential field, supeposition q ' Electic field of a point chage ( ') E( ) kq, whee k / 4 () ' Foce of q on a test chage e at position is ee( ) Electic potential O kq
More informationis the instantaneous position vector of any grid point or fluid
Absolute inetial, elative inetial and non-inetial coodinates fo a moving but non-defoming contol volume Tao Xing, Pablo Caica, and Fed Sten bjective Deive and coelate the govening equations of motion in
More informationReview: Electrostatics and Magnetostatics
Review: Electostatics and Magnetostatics In the static egime, electomagnetic quantities do not vay as a function of time. We have two main cases: ELECTROSTATICS The electic chages do not change postion
More informationPES 3950/PHYS 6950: Homework Assignment 6
PES 3950/PHYS 6950: Homewok Assignment 6 Handed out: Monday Apil 7 Due in: Wednesday May 6, at the stat of class at 3:05 pm shap Show all woking and easoning to eceive full points. Question 1 [5 points]
More informationKEPLER S LAWS OF PLANETARY MOTION
EPER S AWS OF PANETARY MOTION 1. Intoduction We ae now in a position to apply what we have leaned about the coss poduct and vecto valued functions to deive eple s aws of planetay motion. These laws wee
More informationLiquid gas interface under hydrostatic pressure
Advances in Fluid Mechanics IX 5 Liquid gas inteface unde hydostatic pessue A. Gajewski Bialystok Univesity of Technology, Faculty of Civil Engineeing and Envionmental Engineeing, Depatment of Heat Engineeing,
More informationBetween any two masses, there exists a mutual attractive force.
YEAR 12 PHYSICS: GRAVITATION PAST EXAM QUESTIONS Name: QUESTION 1 (1995 EXAM) (a) State Newton s Univesal Law of Gavitation in wods Between any two masses, thee exists a mutual attactive foce. This foce
More informationChapter 4. Newton s Laws of Motion
Chapte 4 Newton s Laws of Motion 4.1 Foces and Inteactions A foce is a push o a pull. It is that which causes an object to acceleate. The unit of foce in the metic system is the Newton. Foce is a vecto
More informationIn the previous section we considered problems where the
5.4 Hydodynamically Fully Developed and Themally Developing Lamina Flow In the pevious section we consideed poblems whee the velocity and tempeatue pofile wee fully developed, so that the heat tansfe coefficient
More informationAn Exact Solution of Navier Stokes Equation
An Exact Solution of Navie Stokes Equation A. Salih Depatment of Aeospace Engineeing Indian Institute of Space Science and Technology, Thiuvananthapuam, Keala, India. July 20 The pincipal difficulty in
More informationWhat molecular weight polymer is necessary to provide steric stabilization? = [1]
1/7 What molecula weight polyme is necessay to povide steic stabilization? The fist step is to estimate the thickness of adsobed polyme laye necessay fo steic stabilization. An appoximation is: 1 t A d
More informationDEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS LSN 10-: MOTION IN A GRAVITATIONAL FIELD Questions Fom Reading Activity? Gavity Waves? Essential Idea: Simila appoaches can be taken in analyzing electical
More informationOn the integration of the equations of hydrodynamics
Uebe die Integation de hydodynamischen Gleichungen J f eine u angew Math 56 (859) -0 On the integation of the equations of hydodynamics (By A Clebsch at Calsuhe) Tanslated by D H Delphenich In a pevious
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 informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G-type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this investigation
More informationA 1. EN2210: Continuum Mechanics. Homework 7: Fluid Mechanics Solutions
EN10: Continuum Mechanics Homewok 7: Fluid Mechanics Solutions School of Engineeing Bown Univesity 1. An ideal fluid with mass density ρ flows with velocity v 0 though a cylindical tube with cosssectional
More informationChem 453/544 Fall /08/03. Exam #1 Solutions
Chem 453/544 Fall 3 /8/3 Exam # Solutions. ( points) Use the genealized compessibility diagam povided on the last page to estimate ove what ange of pessues A at oom tempeatue confoms to the ideal gas law
More informationChapter 5. Applying Newton s Laws. Newton s Laws. r r. 1 st Law: An object at rest or traveling in uniform. 2 nd Law:
Chapte 5 Applying Newton s Laws Newton s Laws st Law: An object at est o taveling in unifom motion will emain at est o taveling in unifom motion unless and until an extenal foce is applied net ma nd Law:
More informationNumerical solution of diffusion mass transfer model in adsorption systems. Prof. Nina Paula Gonçalves Salau, D.Sc.
Numeical solution of diffusion mass tansfe model in adsoption systems Pof., D.Sc. Agenda Mass Tansfe Mechanisms Diffusion Mass Tansfe Models Solving Diffusion Mass Tansfe Models Paamete Estimation 2 Mass
More informationEXAM NMR (8N090) November , am
EXA NR (8N9) Novembe 5 9, 9. 1. am Remaks: 1. The exam consists of 8 questions, each with 3 pats.. Each question yields the same amount of points. 3. You ae allowed to use the fomula sheet which has been
More informationPhysics 2212 GH Quiz #2 Solutions Spring 2016
Physics 2212 GH Quiz #2 Solutions Sping 216 I. 17 points) Thee point chages, each caying a chage Q = +6. nc, ae placed on an equilateal tiangle of side length = 3. mm. An additional point chage, caying
More information= 4 3 π( m) 3 (5480 kg m 3 ) = kg.
CHAPTER 11 THE GRAVITATIONAL FIELD Newton s Law of Gavitation m 1 m A foce of attaction occus between two masses given by Newton s Law of Gavitation Inetial mass and gavitational mass Gavitational potential
More informationB. Spherical Wave Propagation
11/8/007 Spheical Wave Popagation notes 1/1 B. Spheical Wave Popagation Evey antenna launches a spheical wave, thus its powe density educes as a function of 1, whee is the distance fom the antenna. We
More informationPhysics C Rotational Motion Name: ANSWER KEY_ AP Review Packet
Linea and angula analogs Linea Rotation x position x displacement v velocity a T tangential acceleation Vectos in otational motion Use the ight hand ule to detemine diection of the vecto! Don t foget centipetal
More informationPhysics 111. Ch 12: Gravity. Newton s Universal Gravity. R - hat. the equation. = Gm 1 m 2. F g 2 1. ˆr 2 1. Gravity G =
ics Announcements day, embe 9, 004 Ch 1: Gavity Univesal Law Potential Enegy Keple s Laws Ch 15: Fluids density hydostatic equilibium Pascal s Pinciple This week s lab will be anothe physics wokshop -
More informationPhysics 2B Chapter 22 Notes - Magnetic Field Spring 2018
Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field
More informationThe geometric construction of Ewald sphere and Bragg condition:
The geometic constuction of Ewald sphee and Bagg condition: The constuction of Ewald sphee must be done such that the Bagg condition is satisfied. This can be done as follows: i) Daw a wave vecto k in
More informationDEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS TSOKOS LESSON 10-1 DESCRIBING FIELDS Essential Idea: Electic chages and masses each influence the space aound them and that influence can be epesented
More informationNuclear and Particle Physics - Lecture 20 The shell model
1 Intoduction Nuclea and Paticle Physics - Lectue 0 The shell model It is appaent that the semi-empiical mass fomula does a good job of descibing tends but not the non-smooth behaviou of the binding enegy.
More information7.2.1 Basic relations for Torsion of Circular Members
Section 7. 7. osion In this section, the geomety to be consideed is that of a long slende cicula ba and the load is one which twists the ba. Such poblems ae impotant in the analysis of twisting components,
More informationRelated Rates - the Basics
Related Rates - the Basics In this section we exploe the way we can use deivatives to find the velocity at which things ae changing ove time. Up to now we have been finding the deivative to compae the
More informationVoltage ( = Electric Potential )
V-1 of 10 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More informationPhysics 107 TUTORIAL ASSIGNMENT #8
Physics 07 TUTORIAL ASSIGNMENT #8 Cutnell & Johnson, 7 th edition Chapte 8: Poblems 5,, 3, 39, 76 Chapte 9: Poblems 9, 0, 4, 5, 6 Chapte 8 5 Inteactive Solution 8.5 povides a model fo solving this type
More informationRigid Body Dynamics 2. CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2018
Rigid Body Dynamics 2 CSE169: Compute Animation nstucto: Steve Rotenbeg UCSD, Winte 2018 Coss Poduct & Hat Opeato Deivative of a Rotating Vecto Let s say that vecto is otating aound the oigin, maintaining
More informationPHYS 1444 Section 501 Lecture #7
PHYS 1444 Section 51 Lectue #7 Wednesday, Feb. 8, 26 Equi-potential Lines and Sufaces Electic Potential Due to Electic Dipole E detemined fom V Electostatic Potential Enegy of a System of Chages Capacitos
More informationSection 26 The Laws of Rotational Motion
Physics 24A Class Notes Section 26 The Laws of otational Motion What do objects do and why do they do it? They otate and we have established the quantities needed to descibe this motion. We now need to
More informationm1 m2 M 2 = M -1 L 3 T -2
GAVITATION Newton s Univesal law of gavitation. Evey paticle of matte in this univese attacts evey othe paticle with a foce which vaies diectly as the poduct of thei masses and invesely as the squae of
More informationPHYS 1444 Lecture #5
Shot eview Chapte 24 PHYS 1444 Lectue #5 Tuesday June 19, 212 D. Andew Bandt Capacitos and Capacitance 1 Coulom s Law The Fomula QQ Q Q F 1 2 1 2 Fomula 2 2 F k A vecto quantity. Newtons Diection of electic
More informationPhysics Fall Mechanics, Thermodynamics, Waves, Fluids. Lecture 18: System of Particles II. Slide 18-1
Physics 1501 Fall 2008 Mechanics, Themodynamics, Waves, Fluids Lectue 18: System of Paticles II Slide 18-1 Recap: cente of mass The cente of mass of a composite object o system of paticles is the point
More informationPhysics 221 Lecture 41 Nonlinear Absorption and Refraction
Physics 221 Lectue 41 Nonlinea Absoption and Refaction Refeences Meye-Aendt, pp. 97-98. Boyd, Nonlinea Optics, 1.4 Yaiv, Optical Waves in Cystals, p. 22 (Table of cystal symmeties) 1. Intoductoy Remaks.
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 10 Solutions. r s
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.033 Decembe 5, 003 Poblem Set 10 Solutions Poblem 1 M s y x test paticle The figue above depicts the geomety of the poblem. The position
More informationLecture 24 Stability of Molecular Clouds
Lectue 4 Stability of Molecula Clouds 1. Stability of Cloud Coes. Collapse and Fagmentation of Clouds 3. Applying the iial Theoem Refeences Oigins of Stas & Planetay Systems eds. Lada & Kylafis http://cfa-www.havad.edu/cete
More informationTidal forces. m r. m 1 m 2. x r 2. r 1
Tidal foces Befoe we look at fee waves on the eath, let s fist exaine one class of otion that is diectly foced: astonoic tides. Hee we will biefly conside soe of the tidal geneating foces fo -body systes.
More informationForce can be exerted by direct contact between bodies: Contact Force.
Chapte 4, Newton s Laws of Motion Chapte IV NEWTON S LAWS OF MOTION Study of Dynamics: cause of motion (foces) and the esistance of objects to motion (mass), also called inetia. The fundamental Pinciples
More informationThree dimensional flow analysis in Axial Flow Compressors
1 Thee dimensional flow analysis in Axial Flow Compessos 2 The ealie assumption on blade flow theoies that the flow inside the axial flow compesso annulus is two dimensional means that adial movement of
More informationQualifying Examination Electricity and Magnetism Solutions January 12, 2006
1 Qualifying Examination Electicity and Magnetism Solutions Januay 12, 2006 PROBLEM EA. a. Fist, we conside a unit length of cylinde to find the elationship between the total chage pe unit length λ and
More information20-9 ELECTRIC FIELD LINES 20-9 ELECTRIC POTENTIAL. Answers to the Conceptual Questions. Chapter 20 Electricity 241
Chapte 0 Electicity 41 0-9 ELECTRIC IELD LINES Goals Illustate the concept of electic field lines. Content The electic field can be symbolized by lines of foce thoughout space. The electic field is stonge
More informationON THE TWO-BODY PROBLEM IN QUANTUM MECHANICS
ON THE TWO-BODY PROBLEM IN QUANTUM MECHANICS L. MICU Hoia Hulubei National Institute fo Physics and Nuclea Engineeing, P.O. Box MG-6, RO-0775 Buchaest-Maguele, Romania, E-mail: lmicu@theoy.nipne.o (Received
More informationMONTE CARLO SIMULATION OF FLUID FLOW
MONTE CARLO SIMULATION OF FLUID FLOW M. Ragheb 3/7/3 INTRODUCTION We conside the situation of Fee Molecula Collisionless and Reflective Flow. Collisionless flows occu in the field of aefied gas dynamics.
More informationChapter 2: Basic Physics and Math Supplements
Chapte 2: Basic Physics and Math Supplements Decembe 1, 215 1 Supplement 2.1: Centipetal Acceleation This supplement expands on a topic addessed on page 19 of the textbook. Ou task hee is to calculate
More informationElectrostatics. 3) positive object: lack of electrons negative object: excess of electrons
Electostatics IB 12 1) electic chage: 2 types of electic chage: positive and negative 2) chaging by fiction: tansfe of electons fom one object to anothe 3) positive object: lack of electons negative object:
More informationThe second law of thermodynamics - II.
Januay 21, 2013 The second law of themodynamics - II. Asaf Pe e 1 1. The Schottky defect At absolute zeo tempeatue, the atoms of a solid ae odeed completely egulaly on a cystal lattice. As the tempeatue
More informationSAMPLE QUIZ 3 - PHYSICS For a right triangle: sin θ = a c, cos θ = b c, tan θ = a b,
SAMPLE QUIZ 3 - PHYSICS 1301.1 his is a closed book, closed notes quiz. Calculatos ae pemitted. he ONLY fomulas that may be used ae those given below. Define all symbols and justify all mathematical expessions
More informationDEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS TSOKOS LESSON 6- THE LAW OF GRAVITATION Essential Idea: The Newtonian idea of gavitational foce acting between two spheical bodies and the laws of mechanics
More informationLecture 5 Solving Problems using Green s Theorem. 1. Show how Green s theorem can be used to solve general electrostatic problems 2.
Lectue 5 Solving Poblems using Geen s Theoem Today s topics. Show how Geen s theoem can be used to solve geneal electostatic poblems. Dielectics A well known application of Geen s theoem. Last time we
More informationSupplementary Figure 1. Circular parallel lamellae grain size as a function of annealing time at 250 C. Error bars represent the 2σ uncertainty in
Supplementay Figue 1. Cicula paallel lamellae gain size as a function of annealing time at 50 C. Eo bas epesent the σ uncetainty in the measued adii based on image pixilation and analysis uncetainty contibutions
More informationDo not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Seies UG Examination 2015 16 FLUID DYNAMICS WITH ADVANCED TOPICS MTH-MD59 Time allowed: 3 Hous Attempt QUESTIONS 1 and 2, and THREE othe questions.
More informationPROBLEM SET #3A. A = Ω 2r 2 2 Ω 1r 2 1 r2 2 r2 1
PROBLEM SET #3A AST242 Figue 1. Two concentic co-axial cylindes each otating at a diffeent angula otation ate. A viscous fluid lies between the two cylindes. 1. Couette Flow A viscous fluid lies in the
More information( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx.
9. LAGRANGIAN OF THE ELECTROMAGNETIC FIELD In the pevious section the Lagangian and Hamiltonian of an ensemble of point paticles was developed. This appoach is based on a qt. This discete fomulation can
More informationModule 9: Electromagnetic Waves-I Lecture 9: Electromagnetic Waves-I
Module 9: Electomagnetic Waves-I Lectue 9: Electomagnetic Waves-I What is light, paticle o wave? Much of ou daily expeience with light, paticulaly the fact that light ays move in staight lines tells us
More informationLecture 8 - Gauss s Law
Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.
More information17.1 Electric Potential Energy. Equipotential Lines. PE = energy associated with an arrangement of objects that exert forces on each other
Electic Potential Enegy, PE Units: Joules Electic Potential, Units: olts 17.1 Electic Potential Enegy Electic foce is a consevative foce and so we can assign an electic potential enegy (PE) to the system
More informationPHYSICS NOTES GRAVITATION
GRAVITATION Newton s law of gavitation The law states that evey paticle of matte in the univese attacts evey othe paticle with a foce which is diectly popotional to the poduct of thei masses and invesely
More informationGalilean Transformation vs E&M y. Historical Perspective. Chapter 2 Lecture 2 PHYS Special Relativity. Sep. 1, y K K O.
PHYS-2402 Chapte 2 Lectue 2 Special Relativity 1. Basic Ideas Sep. 1, 2016 Galilean Tansfomation vs E&M y K O z z y K In 1873, Maxwell fomulated Equations of Electomagnetism. v Maxwell s equations descibe
More information1) Consider an object of a parabolic shape with rotational symmetry z
Umeå Univesitet, Fysik 1 Vitaly Bychkov Pov i teknisk fysik, Fluid Mechanics (Stömningsläa), 01-06-01, kl 9.00-15.00 jälpmedel: Students may use any book including the tetbook Lectues on Fluid Dynamics.
More information10. Force is inversely proportional to distance between the centers squared. R 4 = F 16 E 11.
NSWRS - P Physics Multiple hoice Pactice Gavitation Solution nswe 1. m mv Obital speed is found fom setting which gives v whee M is the object being obited. Notice that satellite mass does not affect obital
More informationRoger Pynn. Basic Introduction to Small Angle Scattering
by Roge Pynn Basic Intoduction to Small Angle Scatteing We Measue Neutons Scatteed fom a Sample Φ = numbe of incident neutons pe cm pe second σ = total numbe of neutons scatteed pe second / Φ dσ numbe
More informationHydroelastic Analysis of a 1900 TEU Container Ship Using Finite Element and Boundary Element Methods
TEAM 2007, Sept. 10-13, 2007,Yokohama, Japan Hydoelastic Analysis of a 1900 TEU Containe Ship Using Finite Element and Bounday Element Methods Ahmet Egin 1)*, Levent Kaydıhan 2) and Bahadı Uğulu 3) 1)
More informationω = θ θ o = θ θ = s r v = rω
Unifom Cicula Motion Unifom cicula motion is the motion of an object taveling at a constant(unifom) speed in a cicula path. Fist we must define the angula displacement and angula velocity The angula displacement
More informationElectricity Revision ELECTRICITY REVISION KEY CONCEPTS TERMINOLOGY & DEFINITION. Physical Sciences X-Sheets
Electicity Revision KEY CONCEPTS In this session we will focus on the following: Stating and apply Coulomb s Law. Defining electical field stength and applying the deived equations. Dawing electical field
More informationr cos, and y r sin with the origin of coordinate system located at
Lectue 3-3 Kinematics of Rotation Duing ou peious lectues we hae consideed diffeent examples of motion in one and seeal dimensions. But in each case the moing object was consideed as a paticle-like object,
More informationI( x) t e. is the total mean free path in the medium, [cm] tis the total cross section in the medium, [cm ] A M
t I ( x) I e x x t Ie (1) whee: 1 t is the total mean fee path in the medium, [cm] N t t -1 tis the total coss section in the medium, [cm ] A M 3 is the density of the medium [gm/cm ] v 3 N= is the nuclea
More information7.2. Coulomb s Law. The Electric Force
Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat
More informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More informationATMO 551a Fall 08. Diffusion
Diffusion Diffusion is a net tanspot of olecules o enegy o oentu o fo a egion of highe concentation to one of lowe concentation by ando olecula) otion. We will look at diffusion in gases. Mean fee path
More informationChapter 12. Kinetics of Particles: Newton s Second Law
Chapte 1. Kinetics of Paticles: Newton s Second Law Intoduction Newton s Second Law of Motion Linea Momentum of a Paticle Systems of Units Equations of Motion Dynamic Equilibium Angula Momentum of a Paticle
More informationUniversal Gravitation
Chapte 1 Univesal Gavitation Pactice Poblem Solutions Student Textbook page 580 1. Conceptualize the Poblem - The law of univesal gavitation applies to this poblem. The gavitational foce, F g, between
More information16.1 Permanent magnets
Unit 16 Magnetism 161 Pemanent magnets 16 The magnetic foce on moving chage 163 The motion of chaged paticles in a magnetic field 164 The magnetic foce exeted on a cuent-caying wie 165 Cuent loops and
More informationCh 13 Universal Gravitation
Ch 13 Univesal Gavitation Ch 13 Univesal Gavitation Why do celestial objects move the way they do? Keple (1561-1630) Tycho Bahe s assistant, analyzed celestial motion mathematically Galileo (1564-1642)
More informationAST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1
Please ead this fist... AST S: The oigin and evolution of the Univese Intoduction to Mathematical Handout This is an unusually long hand-out and one which uses in places mathematics that you may not be
More informationPROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.
POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and
More information