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8 Pages

### Lab3

Course: MATH 216, Fall 2011
School: Michigan
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Word Count: 1751

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3: Lab Higher-Order Numerical Methods Goals In this lab you will compare the order of accuracy of various numerical methods; that is, you will test-drive some new solvers. You will implement an improved Eulers method and a fourth-order Runge-Kutta method, in addition to your Eulers method program from Lab 2. The example from Lab 2 of an RC circuit with AC voltage provides a case-study for comparing these three...

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Michigan - MATH - 216
Name:Section/Time of lecture:Professor/GSI:MATH 216 WINTER 2009MIDTERM ITo get full score you need to carefully explain what you did. No calculators allowed.1Problem Points Score1102283224262TOTAL882Problem 1.a-4pt Verify that the fun
Michigan - MATH - 216
MATH 216 FIRST MIDTERM EXAMOctober 12, 2009Please write your name:Section:The test contains 7 problems worth 100 points total. To get the full credit youhave to show your work.Problems1234567Points15121515151612ScoreTypeset by AMS-T
Michigan - MATH - 216
MATH 216 FIRST MIDTERM EXAMFebruary 15, 2010Please write your name:Section:The test contains 8 problems worth 100 points total. To get the full credit youhave to show your work.ProblemPoints1122123144125126147128Score12Total100T
Michigan - PHYS - 140
Michigan - PHYS - 140
Physics 140, Fall 2010Second Midterm ExamPhysics Department, University of MichiganNovember 4, 2010FORM 1Please print your name: SAUL LOU SHENYour INSTRUCTOR:_INSTRUCTIONS AND INFORMATION1. Fill in YOUR NAME, INSTRUCTOR and the EXAM FORM NUMBER on
Michigan - PHYS - 140
Physics 140, Winter 2010Second Midterm Exam: March 11, 2010Physics Department, University of MichiganFORM 1Please print your name: _Your DISCUSSION SECTION or INSTRUCTOR:_INSTRUCTIONS AND INFORMATION1.Fill in YOUR NAME, SECTION NUMBER and EXAM FOR
Michigan - PHYS - 140
Physics 140, Fall 2010Second Midterm ExamPhysics Department, University of MichiganNovember 4, 2010FORM 1Please print your name: _Your INSTRUCTOR:_INSTRUCTIONS AND INFORMATION1. Fill in YOUR NAME, INSTRUCTOR and the EXAM FORM NUMBER on the scantro
Michigan - PHYS - 140
Physics 140 Winter 2011lecture #13 21 Feb#13 21Midterm evaluations on ctoolsTodays topic: torqueReading Quiz: Which of the following does not directlyrelated to the torque acting on an object?A. The magnitude of the forceB. The lever armC. The an
Michigan - ENGIN - 101
Mississippi State - PH - 2233
PHYSICS - III10:00 10:50 MWF, 350 Hilbun (Sec -1)Dipangkar DuttaOffice: 010 C Hilbun Hall,email: d.dutta@msstate.edu,Ph: 325-3105Office hours: 2:30 3:30 MWF, or by appointmentCourse website: on WebCT/Blackboard Learning Systems(http:/oncampus.msst
Mississippi State - PH - 2233
Recap of last lectureChapter 13,Periodic motionA restoring force which is directly proportional to the displacementfrom equilibrium will cause a periodic motion called SHM.Example a spring driven glider:Restoring force = Fx = -kx or ax = d2x/dt2 = -
Mississippi State - PH - 2233
Recap of last lectureChapter 13,Periodic motionSHM can be described as uniform circular motion projected on a planeThis gives x = A cos andax = -2xHence for the spring driven glider : = (k/m)and1kmf=T = 2frequency and period are independent2
Mississippi State - PH - 2233
Recap of last lectureChapter 13,Periodic motionA simple pendulums period is given byThe period of a physical pendulum isT = 2LgDamped oscillationsThe decrease in the amplitude due to dissipative forcesis called damping and these oscillations are
Mississippi State - PH - 2233
Recap of last lectureChapter 13,Periodic motionFor damped spring with a damping force of Fx = -bvxd2 xkb dx= xdt 2mm dtWith:solutionx = AeForced (driven)oscillator (b / 2 m )tA=cos( t + )we have&amp;=kb2m 4m 2Fmax22(k md ) 2 + b2 d
Mississippi State - PH - 2233
Recap of last lectureWaves are transverse or longitudinal or a combinationIf the oscillations producing the wave is a SHMvelocity v = f where is the wavelengthWave function or amplitude of wave atAny arbitrary position x and time t is:y(x,t) = Acos(
Mississippi State - PH - 2233
Recap of last lectureThe wave equation for a periodic wave is: 2y 1 2y=222xv tThis relates the curvature of the wave to the transverse acceleration.The velocity of a wave in string is given byWhere F is the tension and is its linear mass density
Mississippi State - PH - 2233
Recap of last lectureInstantaneous power =Max Power = F 2A2F 2A2 sin 2 (kx t)Average power = Max powerIntensity = Power per unit areaW/m2I1/I2 = r22/r12Waves reflect at boundaries(changes in medium)If Particles at the boundary are:Fixed (feel a
Mississippi State - PH - 2233
Recap of last lectureSound waves:y(x,t) = Acos(kx-t) for wave traveling in the +x direction(same as any other wave).But can also be described in terms of pressure fluctuation.P = BkAsin(kx-t)Pmax = BkAPressure varies sinusoidally but has a /2 (90)
Mississippi State - PH - 2233
Recap of last lecturev=v fluid =restoring forceinertia resisting changeBvsolid =Yv gas = RTMIntensity = &lt;Power&gt;av/area = BkA2= B 2A2= P2max/B= (10dB)log(I/I0)where I0 = 1x10-12W/m2(I0 threshold for human hearing at 1000Hz)Applications suc
Mississippi State - PH - 2233
Recap of last lectureStanding waves in longitudinal waves (sound waves)open columnsf1 = v/2L, f2 = v/L, f3 = 3v/2Lhttp:/The_Gas_Tubeclosed columnsf1 = v/4L, f3 = 3v/4L, f5 = 5v/4LWave interferenceDestructive interference when the distance between
Mississippi State - PH - 2233
Recap of last lectureJust like other waves sound waves interfereWhen the frequency of two sound waves is the same:Destructive interference when the distance between the speaker is /2, 3/2, 5/2Constructive interference when the distance between the spe
Mississippi State - PH - 2233
Recap of last lectureDoppler Effectfront = (v vS)/fSbehind = (v + vS)/fSfL = (v +/- vL)fS/(v +/- vS)Electromagnetic waves occur overa wide range Where wavelength is large, frequency is small. The range extends from low energy and frequency(radio
Mississippi State - PH - 2233
Recap of last lectureElectromagnetic waves do not need a medium to propagateTime varying E and B fields produce EM waves such as E &amp; B fieldsassociated with accelerating charged particles.They satisfy the wave equation and v = f, in vacuum v = c = 3x1
Mississippi State - PH - 2233
Recap of last lectureEM waves also satisfy the wave eqn. and the E &amp; B fields vary sinusoidallyWhen propagation through a medium the velocity changes from c tov = c/n where n is called the refractive index = KKm where =K0and = Km0Recap of last lectur
Mississippi State - PH - 2233
Recap of Mondays lectureEM radiation have both particle like and wave like properties. TheQuantum description of EM radiation takes into account both typeof behavior.Laws of reflection i= rReflection can be specular anddiffuse. When studying reflect
Mississippi State - PH - 2233
Recap of last lectureFor light traveling from a to b where na &gt; nb, as the angle ofincidence becomes more and more acute, the light ceases to betransmitted, only reflected. crit = sin-1(nb/na)Frustrated TIRTIRIn which of the followingsituations is
Mississippi State - PH - 2233
Recap of last lectureThe wavelength dependence of refraction is called dispersionna = c/va = 0/aThere are two states of linear polarization of lightCertain materials block EM waves polarized light in a particulardirectionIf the angle between the pol
Mississippi State - PH - 2233
Recap of last lectureLight can also be polarized by reflection. It is 100% polarized inthe direction parallel to the reflecting surface when the angle ofincidence = brewster angle = tan-1 (nb/na)Scattered light perpendicular to the direction of propag
Mississippi State - PH - 2233
Recap of last lectureparticlesdsin = m (maximas)dsin = (m+ ) (minimas)For small sin ~ tanor d.ymax /R = mor ymax = mR/dI = 0cE2p= I0 cos2/2wavesIn Youngs experiment, coherent light passing throughtwo slits (S1 and S2) produces a pattern of dark
Mississippi State - PH - 2233
Recap of last lectureFor small the maximas are at 2t = mn wheren = 0/nMaxima for 2t=(m+ ) m=0,1,2Minima for 2t = mPhase shift of /2An air wedge between two glass plates Just like the thin film, two waves reflect back from the air wedgein close pro
Mississippi State - PH - 2233
Recap of last lectureBright rings when net phase shift = 2t+ /2 =m or 2t = (m+ )dark rings when 2t = m m=0,1,2,Thin film coatings can make perfectreflectors or perfect absorbersDiffractionandFraunhofer DiffractionP will be a dark band ifasin = O
Mississippi State - PH - 2233
Recap of last lecturew1/a85% ofthe powerP will be a dark band ifasin = Orsin = m/a m =1, 2, For small sin ~ ~ tany = xm/a with m =1, 2, Intensity in a single-slit patternEp = E0sin(/2)/(/2) sin/2 I = I0 /2 2 = (2/)*[path difference] = (2/
Mississippi State - PH - 2233
Recap of last lecture sin/2 I =I0/2 2 = (2 / )asinThe first minima is at 1 = / 2 /( Two slit interference sin /2 I =I0 /2 d = 4adsin = m (maximas)dsin = (m+ ) (minimas)I = I0cos2/22 sin /2 I = I 0 cos 2 /2 = (2 /)2 = ( 2 /)2Diffra
Mississippi State - PH - 2233
Recap of last lectureRevisit 2 slit interference: actually diffraction pattern from eachslit interfere to produce the pattern seen which is a little differentfrom the idealized 2 slit pattern we saw in the last chapter.I I0 cos2 sin /2 22 /2 f = (
Mississippi State - PH - 2233
Recap of last lecture.When light has a wavelength l much smaller than objects that itinteracts with, we can treat light as composed of straight-line rays.This regime is called geometric optics.Reflections from a spherical mirrorsa + b = 2ftan a = h
Mississippi State - PH - 2233
Recap of last lectureSpherical reflecting surfaces can be concave orconvex. (concave is when the mirror is silveredon the outer surface while convex is the otherkind.)Under the paraxial approximation (smallangles or light almost parallel to the opti
Mississippi State - PH - 2233
Recap of last lecturena/s + nb/s = (nb-na)/Rm = y/y = -nas/nbsA pair of spherical surfaces can form concave orconvex lenses. Light refracting through thesespherical surfaces tend to converge in convex lensesand diverge in concave lenses.For thin le
Mississippi State - PH - 2233
Recap of last lectureHuman eye: most of the bending (refraction) due to the cornea which has asmall focal length.The lens is used to get additional variable focusing so that near and far objectscan be on focus at the retinaNormal eye near point = 25
Mississippi State - PH - 2233
Recap of last lectureMagnificationby a simplemagnifier'Mh/ fh/ NNfIf eye focuses at near p oint,achieve slightly greater power :MN1fN = near point , 25 cmMicroscope magnification = M1*M2 = s25/f1f2Telescope magnification = -f1/f2Microsc
Mississippi State - PH - 2233
Recap of last lecturePostulate of Classical relativity:Laws of mechanics (Newtons laws ) are invariant in all inertial reference framesCrisis in classical relative: In classical relativity the velocity of light would bedifferent in a frame moving with
Mississippi State - PH - 2233
Recap of last lectureConsequences of the postulates of special relativity:1. Speed of light is a constant2. Simultaneity of events is frame dependent3. Time intervals are frame dependent4. spatial intervals (length) is frame dependent5. Space-time i
Mississippi State - PH - 2233
Recap of last lectureConsequences of the postulates of special relativity:1. Time in moving frames is dilated t = t2. Length in moving frames is contracted l = l/3. = 1/(1-u2/c2)1/24. The complete Lorentz transforms are given by:t = (t - ux/c2)x= (
Mississippi State - PH - 2233
Recap of last lectureLooked at 4 phenomena which are problematic for classical physicsLine spectra seen for different elements when they are heated/energizedPhoto-electric effect (electrons emitted from metal when light withFrequency above a threshold
Mississippi State - PH - 2233
Recap of last lectureLooked at 4 phenomena which are problematic for classical physicsLine spectra and black body radiation spectrum can be explained interms of discrete energy levels in atoms where E1 E2 = hf;Ephoton = hfThe same particle like prope
Mississippi State - PH - 2233
Recap of last lectureEvidence for wave like behavior of particles:Diffraction and two slit interference patternsThese are a consequence of the uncertainty principleDpDx hParticles can be described by wave-functions y (x,y,z,t)Particle wave-functions
Mississippi State - PH - 2233
Recap of last lectureSolving the wave equation (Schrdinger Eq.) for the most simpleproblem a particle in a box (infinite potential well) shows thatthe particle can only occupy discrete energy levels. The wave fn.for each of these energy levels resembl
Mississippi State - CH - 1213
John W. MooreConrad L. StanitskiPeter C. Jurswww.cengage.com/chemistry/mooreChapter 12Fuels, Organic Chemicals andPolymersPetroleumPetroleum is a complex mixture of: alkanes cycloalkanes alkenes aromatic hydrocarbonsIts composition and color
Mississippi State - CH - 1213
Chapter 13Chemical Kinetics: Rates ofReactionsChemical KineticsThe study of speeds of reactions and the nanoscalepathways or rearrangements by which atoms andmolecules are transformed from reactants toproducts.Chemical kinetics is also called reac
Mississippi State - CH - 1213
Chapter 14Chemical EquilibriumCharacteristics of Chemical EquilibriumMany reactions fail to go to completion.[Reactants] stop decreasing, and[Products] stop increasing.The reaction reaches equilibrium.If there are more:products than reactants = pr
Mississippi State - CH - 1213
Chapter 15The Chemistry of Solutes andSolutionsSolubility &amp; Intermolecular ForcesSolution = homogeneous mixture of substances.It consists of: solvent - component in the greatest amount. solute - all other components (may be &gt;1).Solvent-solute inte
Mississippi State - CH - 1213
Chapter 16Acids and BasesArrhenius DefinitionArrhenius: any substance which ionizes in water toproduce: Protons is an Acid. Hydroxide ions is a Base.Better version of the Arrhenius definition:Acid: Hydronium ions (H3O+) in water are acidic.Base:
Mississippi State - CH - 1213
Chapter 17Additional Aqueous EquilibriaBuffer SolutionsBuffer = chemical system that resists changes in pH.ExampleAdd 0.010 mol of HCl or NaOH to:1 L SolutionInitialpHafter HClpure H2O7.002.0012.00[CH3COOH] = 0.5 M+ [CH3COONa] = 0.5 M4.74
Mississippi State - CH - 1213
Chapter 18Thermodynamics:Directionality of ChemicalReactionsReactant- &amp; Product-Favored Processes Why are equilibria product- or reactant- favored? Why do some reactions occur spontaneously? Why do others require help (heat, spark)? Exothermic rea
Mississippi State - CH - 1213
Chapter 19Electrochemistry &amp; itsApplicationsElectrochemistryElectrochemistry is the study and use of e- flow inchemical reactions.Redox reactions generate (and use) e Those e- can be harnessed (batteries). Corrosion is an electrochemical reaction.
Mississippi State - CH - 1213
Chapter 20Nuclear ChemistryThe Nature of RadioactivityHenri Becquerel (1896): U salts emitted rays that fog a photographic plate. U metal was a stronger emitter.Marie and Pierre Curie: Isolated Po and Ra that did the same. Marie Curie called the p
UCLA - MATH - 3C
UCSB - ECON - 134a
25.00%20.00%17.50%15.00%10.00%5.50%5.00%0.00%0.00%5.00%10.00%15.00%20.00%25.00%30.00%35.00%
UCSB - ECON - 134a
25.00%20.00%15.00%Correlation -1Correlation -0.5Correlation 010.00%Correlation 0.5Correlation 15.00%0.00%0.00%5.00%10.00%15.00%20.00%25.00%30.00%35.00%40.00%
UCSB - ECON - 134a
Useful FormulasQuadratic Formula 2 + + = 0, = 2 42Time Value of Money1) Present Value of a Single Cash Flow occurring T periods from now =1 + 2) Present Value of an Annuity, paying C, with T payments and discounted at r%1=11 + 3) Present Val
UCSB - ECON - 134a
Econ134a-Financial Management Finance is the Interaction of TIME, MONEYand UNCERTAINTYBusiness Organizations1. Sole Proprietorship2. Partnerships and Limited Partnerships3. CorporationsSole Proprietorship No Corporate Income Tax Unlimited Liabili
UCSB - ECON - 134a
Time Value of MoneyTime Value of Money Is Money received Today worth more or lessthan the same received in the Future? Usually MORE Why?Time Value of Money We could invest the money received today in ariskless asset and have more in the future. E