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### HW-10s

Course: PHYSICS 112, Summer 2009
School: Iowa State
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Word Count: 685

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1. C t F ext B The magnitude F of the magnetic field is increasing. Flux: increasing due to increasing F . t F loop must be down ( OE ) to counteract upward increasing flux; current is then clockwise. C 2. t OE F ext B The magnitude F of the magnetic field is increasing. Flux: increasing due to increasing F . t F loop must be up ( ) to counteract downward increasing flux; current is then counterclockwise. C...

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1. C t F ext B The magnitude F of the magnetic field is increasing. Flux: increasing due to increasing F . t F loop must be down ( OE ) to counteract upward increasing flux; current is then clockwise. C 2. t OE F ext B The magnitude F of the magnetic field is increasing. Flux: increasing due to increasing F . t F loop must be up ( ) to counteract downward increasing flux; current is then counterclockwise. C 3. t F ext B The magnitude F of the magnetic field is decreasing. Flux: decreasing due to decreasing F . t F loop must be up ( ) to counteract upward decreasing flux; current is then counterclockwise. C 4. t OE F ext B The magnitude F of the magnetic field is decreasing. Flux: decreasing due to decreasing F . t F loop must be down ( OE ) to counteract downward decreasing flux; current is then clockwise. C 5. t oq F ext B The magnitude F of the magnetic field is increasing. Flux: zero at all times, since ) oe 90; no current flows C 6. t qpF ext B The magnitude F of the magnetic field is decreasing. Flux: zero at all times, since ) oe 90; no current flows C 7. t F ext (initial) B t F is rotating through 90 to line up with the B axis. Flux: decreasing since ) increases from 0 to 90 so cos ) is always decreasing. t F loop must be up ( ) to counteract upward decreasing flux; current is then counterclockwise. C 8. t F ext (initial) B t F is rotating through 90 to line up with the C axis. Flux: decreasing since ) increases from 0 to 90 so cos ) is always decreasing. t F loop must be up ( ) to counteract upward decreasing flux; current is then counterclockwise. C 9. t qpF ext (initial) B t F is rotating through 90 to line up with the C axis. Flux: zero at all times, since ) oe 90; no current flows C 10. t oq F ext (initial) B t F is rotating through 90 to line up with the D direction. Flux: increasing in magnitude from zero (because ) oe 90 so cos ) oe 0) to some finite value. t F loop must be up ) ( to counteract downward increasing flux; current is then counterclockwise. C 11. t F ext (initial) B t F rotates through 90 to line up with the B direction and then continues through another 90 until it is lined up with the D direction. Flux: decreasing during first half of the rotation since ) increases from 0 to 90, and reversing but increasing in magnitude during second half of the rotation as ) decreases from 90 to 0. t F loop must be up ( ) during both halves; current is then counterclockwise. C 12. t OE F ext (initial) B t F is rotating through 90 to line up with the B direction and also decreasing in magnitude. t Flux: decreasing since the magnitude of F is decreasing and since ) increases from 0 to 90 so cos ) is also decreasing. t F loop must be down ( OE ) to counteract downward decreasing flux; current is then clockwise. C 13. t OE F ext (initial) B t F is rotating through 90 to line up with the C direction and also increasing in magnitude. t Flux: the increase in the magnitude of F causes the flux to increase, while the increase in ) from 0 to 90 causes cos ) to decrease. More information would be necessary to know, at least at the beginning, which of these effects is greater, and which direction the current flows. However, overall, we can see that the flux decreases from some initial value which is not zero to a final value which is zero. Therefore, overall, the loop has gone from a situation in which there was downward flux to a situation in which there is zero flux. Therefore the average t value of F loop must be down (in the original direction of the external magnetic field), so the average current is clockwise. Depending on how fast the magnitude of the external field is increasing compared to the increase in angle, the current may have been clockwise the whole time or it may have been counterclockwise for a while and then clockwise for most of the change in the external magnetic field.
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Iowa State - PHYSICS - 112
HOMEWORK FOR TUESDAY, JULY 19, 2005 In this problem we will consider the path of light rays from air into glass and vice versa. The glass has a flat, horizontal surface, and the air is above it. We want to determine the angles for the reflected and refrac
Iowa State - PHYSICS - 112
HOMEWORK SOLUTIONS FOR JULY 21 1. An object is placed 10 cm from the vertex of a concave mirror of radius of curvature 12 cm. Find the image distance and the magnification, and characterize the image in the usual way. Check your results by ray tracing usi
Iowa State - PHYSICS - 112
HOMEWORK ASSIGNMENT FOR JULY 26, 2005Consider the interference between two slits separated by a distance . on which light of wavelength - shines, producing a pattern of dark and bright fringes on a screen a distance H away from the slits. First draw a sk
Iowa State - PHYSICS - 112
Consider the interference between two slits separated by a distance . on which light of wavelength - shines, producing a pattern of dark and bright fringes on a screen a distance H away from the slits. First draw a sketch of this physical situation, showi
Iowa State - PHYSICS - 112
HOMEWORK FOR THURSDAY, JULY 28, 20051. A diffraction grating has lines spaced 2.00 .m apart. Let's consider white visible light (wavelengths ranging from 0.400 .m for violet light to 0.700 .m for red light) shining on it. (a) Determine all the angles for
Iowa State - PHYSICS - 112
SOLUTIONS 1. A diffraction grating has lines spaced 2.00 .m apart. Let's consider white visible light (wavelengths ranging from 0.400 .m for violet light to 0.700 .m for red light) shining on it. (a) Determine all the angles for which interference maxima
Iowa State - PHYSICS - 112
HOMEWORK FOR TUESDAY, AUGUST 2, 20051. Consider the atom denoted by 183 Au 79 The common name of this element is _. It has _ electrons, _ protons, _ neutrons, and _ quarks. This atom is radioactive and decays by alpha decay with a half-life of 49 seconds
Iowa State - PHYSICS - 112
1. Consider the atom denoted by 183 Au 79 The common name of this element is: gold It has 79 electrons, 79 protons, and 183 79 oe 104 neutrons. This atom is radioactive and decays by alpha decay with a half-life of 49 seconds. Determine the fraction of it
Iowa State - PHYSICS - 112
REFRACTION WORKSHEET &quot;Triangle&quot; Diagram: On other pages in this set you will find a large triangle that represents a triangular prism. We want to follow the path of a light ray striking one of the surfaces as it passes through the prism and exits one of t
Iowa State - PHYSICS - 112
PHYSICS 112 - REVIEW FOR EXAM 1ELECTRIC CHARGES Electric charges are positive, negative, or zero; they are measured in coulombs (symbol: C). Electric charges are quantized: every object in the universe has a electric charge ; oe , 8/, where 8 oe 0, 1, 2,
Iowa State - PHYSICS - 112
PHYSICS 112 - REVIEW FOR EXAM 2CURRENT M rate at which electic charge is flowing in a wire M oe ?UX RESISTANCE V and POTENTIAL DIFFERENCE ?Z related by Ohm's law: ?Z oe MV . The power dissipated in a resistor has magnitude T oe M ?Z oe M 2 V oe ?Z 2 V .
Iowa State - PHYSICS - 112
PHYSICS 112 - REVIEW FOR EXAM 3t MAGNETIC FLUX: The magnetic flux through a plane circuit is F oe FE cos ), where ) is the angle between F 2 and the normal ( perpendicular) to the plane of the circuit. Units: T-m . MAGNETIC INDUCTION The current induced
Iowa State - PHYSICS - 112
PHYSICS 112 - SPRING 2003 - EXAM 1 - February 12, 2003 Name: _ Recitation Section Number: _ SHOW YOUR WORK! Although some of these problems are multiple-choice, full credit will be given only if you explain how you arrived at your answer. Either show your
Iowa State - PHYSICS - 112
PHYSICS 112 - SPRING 2003 - EXAM 3 SHOW YOUR WORK! Full credit will be given only if you explain how you arrived at your answer. Either show your work (especially in a calculation) or give a short explanation. Nothing elaborate is required, but the grader
Iowa State - PHYSICS - 112
PHYSICS 112 - EXAM 4 (first part of final exam) - SPRING 2003Use only a right-handed coordinate system ( B C D along thumb, forefinger, and middle finger of your right hand when these three fingers are perpendicular to one another). Show your calculation
Iowa State - PHYSICS - 112
PHYSICS 112 - SPRING 2004 - EXAM 2 NO CALCULATORS. SHOW YOUR WORK! Full credit will be given only if you explain how you arrived at your answer. Either show your work (especially in a calculation) or give a short explanation. Nothing elaborate is required
Iowa State - PHYSICS - 112
PHYSICS 112 - FINAL EXAM (second, comprehensive part) - SPRING 2005cos2 0 oe 1 cos2 30 oe 0.75 cos2 45 oe 0.50 cos2 60 oe 0.25 cos2 90 oe 0.1. [9 points total] An electric charge of 2.0 .C is placed at a point 3.0 m from a fixed electric charge of 3.0 .
Iowa State - PHYSICS - 112
PHYSICS 112 - SUMMER 2005 - EXAM 3 Name: _ Recitation Section: _NO CALCULATORS ALLOWED - BUT, LUCKILY, NO CALCULATORS NEEDED. SHOW YOUR WORK! Full credit will be given only if you explain how you arrived at your answer. Either show your work (especially
Iowa State - PHYSICS - 112
USEFUL INFORMATIONIMPORTANT PHYSICAL CONSTANTS - 299,792,458 m/s 300 , 10) m/s K oe 6.673 , 10&quot; N m# /kg# 5/ oe1 41%0(speed of light) (gravitational constant) (electrical constant) (Planck's constant) (permittivity of free space) (fundamental electric
Iowa State - PHYSICS - 112
TOPIC 1. REVIEWINTRODUCTIONIn Physics 112 we will be making heavy use of vectors, so we will start with a review of vectors. The important vector quantities we will be discussing are electric forces, electric fields, electric currents, magnetic forces,
Iowa State - PHYSICS - 112
TOPIC 2. ELECTRICITY Topic 2A. Electric Charges and ForcesELECTRIC CHARGEObjects, including elementary particles like the electron and proton, have a number of properties. The one we have studied the most so far is mass. The mass of an object is importa
Iowa State - PHYSICS - 112
TOPIC 3. Electric Currents ELECTRIC CURRENTSWhat causes charges to flow, and what hinders the free flow of charge? The most important practical applications of electrical phenomena are in the innumerable forms of electronic devices. In these, it is the m
Iowa State - PHYSICS - 112
Topic 4. Magnetic Forces and FieldsDo currents exert forces on each other in the same way that charges do? Our study of electric forces and electric fields began with a simple experimental observation: charged particles exert forces on each other. From t
Iowa State - STAT - 511
IntroductionOne possible model: yij = i +ijIntroduction (continued)The first part of Stat 511 re-examines methods from Stat 500 from a linear models perspective.A simple study: does a proprietary food additive increase milk production in dairy cows?
Iowa State - STAT - 511
IntroductionThe first part of Stat 511 re-examines methods from Stat 500 from a linear models perspective. A simple study: does a proprietary food additive increase milk production in dairy cows? 6 cows, housed one per stall. Randomly choose 3 to get foo
Iowa State - STAT - 511
Geometry of the Gauss-Markov Linear ModelX is a linear combination of the columns of X: 1 . X = [x1 , . . . , xp ] . = 1 x1 + + p xp . . p The set of all possible linear combinations of the columns of X is called the column space of X and is denoted by C
Iowa State - STAT - 511
Geometry of the Gauss-Markov Linear ModelReminder from the last section of the notes: y = X + We saw two possible X matrices for the t-test. This section focuses on the Q: does it matter which X we use? Important pieces of information for what follows: X
Iowa State - STAT - 511
Estimating Estimable Functions of In the Gauss-Markov or Normal Theory Gauss-Markov Linear Model, the distribution of y depends on only through X, i.e., y (X, 2 I) or y N(X, 2 I)The Response Depends on Only through XWe now shift attention from E(y) to
Iowa State - STAT - 511
Estimating Estimable Functions of We now shift attention from E(y) to the parameter vector . Remember our t-test questions about 1 - 2 and 1 - 2 ? y y Those are questions about or linear combinations of . We've seen some models where there is a unique so
Iowa State - STAT - 511
Proof of the Gauss-Markov TheoremSuppose dy is any linear unbiased estimator other than the OLS ^ estimator C. ^ Need to show Var(dy) &gt; Var(C). ^ ^ Can relate the two Var by writing Var(dy) = Var(dy - C + C) ^ ^ Var(dy) = Var(dy - C + C) ^ ^ ^ ^ = Var(dy
Iowa State - STAT - 511
Proof of the Gauss-Markov TheoremGauss-Markov Th'm: ^ The OLS estimator, C, is the unique BLUE of C in GM model: y = X + , N(0, 2 I) ^ Need to show Var(C) is strictly less than the variance of any other linear unbiased estimator of C for all IRp and 2 IR
Iowa State - STAT - 511
Estimating Estimable Functions of : 11 12 22 23 33 41 42An Examplecustomer1 2 3 4 Which movie is best? y= X +movie 1 2 3 4 1 ? ? 3 5 ? ? 3 3 1 ? Can we guess ratings for customer/movie combinations not in the dataset? = + 4 1 3 5 3 3 11 1 1 1 1 1
Iowa State - STAT - 511
Estimating Estimable Functions of :An Example movie 1 2 3 4 1 ? ? 3 5 ? ? 3 3 1 ? customer i's rating of movie j + Ci + mj +ijcustomer1 2 3 4 =Can we guess ratings for customer/movie combinations not in the dataset? Which movie is best?YijYij=Cop
Iowa State - STAT - 511
Alternative ParameterizationsFor example yij i = 1, 2, 3 j = 1, 2Recall that the Gauss-Markov Linear Model simply says that E(y) C(X) and Var(y) = 2 I for some 2 &gt; 0.Treatment Effects E(yij ) = + i 1 2 3 Cell Means E(yij ) = iThus, as long as C(X) = C
Iowa State - STAT - 511
Inference Under the Normal Theory Gauss-Markov Linear Model Inference (cont.)Remember our dairy cow study (2 treatments, 3 reps per trt) and our questions (Introduction, slide 3) We've answered all questions except the last one: y 3) When does t = [(1 -
Iowa State - STAT - 511
Inference Under the Normal Theory Gauss-Markov Linear ModelRemember our dairy cow study (2 treatments, 3 reps per trt) and our questions (Introduction, slide 3) We've answered all questions except the last one: 3) When does t = [(1 - 2 ) - (1 - 2 )] / s2
Iowa State - STAT - 511
Practical Data AnalysisA not uncommon situation: A client brings you data from a food development study: 3 treatments:Old &quot;on market&quot; formulation of soup new formulation, Same salt content as old new formulation, Reduced salt contentThey recruited 25 p
Iowa State - STAT - 511
Practical Data AnalysisPractical Data AnalysisHow would you analyze the data? (N.B. all analyses account for blocking / pairing of obs. within subject) 1. ANOVA F test of O = S = R , report p-value and trt. means 2. 3 paired t-tests: O = S , O = R , and
Iowa State - STAT - 511
POWER OF THE F-TESTA very common consulting question: &quot;I'm planning a study to do .&quot;. How many replicates (per treatment) should I use? I know 5 ways that can be used to determine an appropriate sample sizeAs many as you can afford (time, money) n = 3 p
Iowa State - STAT - 511
POWER OF THE F-TESTSuppose C is a q p matrix such that C is testable. Earlier, we established that the quadratic form incorporating ^ C - d has a non-central F distribution ^ F = (C - d) Fq,n-k where 2 = (C - d) [C(X X)- C ] 2-1 ( 2 )A very common cons
Iowa State - STAT - 511
REDUCED vs. FULL MODEL F-TESTTests of C - d = 0 lead to F tests, but that's not the only way to an F test 500/402: big emphasis on model comparison SS in ANOVA tables usually explained in terms of model comparison e.g. SS for AB interaction is the differ
Iowa State - STAT - 511
REDUCED vs. FULL MODEL F-TESTWhy does model comparison lead to F tests? If you test Ho using a C test or using a model comparison test, do you get the same answer? Again, will answer using a general setup (Normal GM model) y = X + , N(0, 2 I)Questions a
Iowa State - STAT - 511
Equivalence of model comparison and Cb F testsSummary of results from Cb estimates and testsContinuation of the Storage time example from Part 9. Data: Storage Temperature 20 C 30 C 2 5 Time Ho: 6 6 7 7 Temp Ho: 16 9 12 15 These correspond to tests of:
Iowa State - STAT - 511
Equivalence of model comparison and Cb F testsContinuation of the Storage time example from Part 9. Data: Storage Time 3 months 6 months Storage Temperature 20 C 30 C 2 5 6 6 7 7 9 12 15 16Copyright c 2011 Dept. of Statistics (Iowa State University)Sta
Iowa State - STAT - 511
ANalysis Of VAriance (ANOVA) for a sequence of models Some examplesMultiple RegressionModel comparison can be generalized to a sequence of models (not just one full and one reduced model) N(0, 2 I)X1 = 1, X2 = [1, x1 ], X3 = [1, x1 , x2 ], . . . Xm = [
Iowa State - STAT - 511
ANalysis Of VAriance (ANOVA) for a sequence of modelsModel comparison can be generalized to a sequence of models (not just one full and one reduced model) Context: usual nGM model: y = X + , Let X1 = 1 and Xm = X. But now, we have a sequence of models &quot;i
Iowa State - STAT - 511
THE AITKEN MODELAnalysis of averagesExamples - 1y = X + , (0, 2 V)Identical to the Gauss-Markov Linear Model except that Var ( ) = 2 V instead of 2 I.V is assumed to be a known nonsingular Variance matrix.The Normal Theory Aitken Model adds an assu
Iowa State - STAT - 511
THE AITKEN MODELy = X + , (0, 2 V)Identical to the Gauss-Markov Linear Model except that Var ( ) = 2 V instead of 2 I. V is assumed to be a known nonsingular Variance matrix. The Normal Theory Aitken Model adds an assumption of normality: N(0, 2 V) Obs
Iowa State - STAT - 511
the bootstrapProcess optimization Many physical processes can be described (at least approximately) as quadratic functions of input variable(s)ExamplesWe've seen a lot about inference on C in a nGM (or nAitken) modelA huge number of questions can be a
Iowa State - STAT - 511
the bootstrapWe've seen a lot about inference on C in a nGM (or nAitken) model A huge number of questions can be answered by appropriate choice of C key point is that C is a linear function of y What if quantity of interest is not a linear function of y?
Iowa State - STAT - 511
Randomization/permutation testsBootstrapping preserves the fixed effectsRandomization / Permutation testsResampling from a single pool of observations tests HoR : F1 (x) = F2 (x)Notice a subtle point: HoR is slightly more general than Ho:1 = 2H0R is
Iowa State - STAT - 511
Randomization / Permutation testsBootstrapping preserves the fixed effectsDifference of two means: resample Y1i and resample Y2i bootstrap estimates, 1B - 2B , are centered on/near Y1 - Y2 , ^ ^ which estimates 1 - 2 Regression bootstrap: ^ resample ^i
Iowa State - STAT - 511
LINEAR MIXED-EFFECT MODELSSeedling weight in 2 genotype study from Aitken model section. Seedling weight measured on each seedling. Two (potential) sources of variation: among flats and among seedlings within a flat. Yijk = + i + Tij + Tij ijk ijkExamp
Iowa State - STAT - 511
LINEAR MIXED-EFFECT MODELSStudies / data / models seen previously in 511 assumed a single source of &quot;error&quot; variation y = X + . are fixed constants (in the frequentist approach to inference) is the only random effect What if there are multiple sources of
Iowa State - STAT - 511
Experimental Designs and LME'sOne example:LME models provide one way to model correlations among observationsVery useful for experimental designs where there is more than one size of experimental unitOr designs where the observation unit is not the sa
Iowa State - STAT - 511
Experimental Designs and LME'sLME models provide one way to model correlations among observations Very useful for experimental designs where there is more than one size of experimental unit Or designs where the observation unit is not the same as the exp
Iowa State - STAT - 511
THE ANOVA APPROACH TO THE ANALYSIS OF LINEAR MIXED EFFECTS MODELSThis is the commonly-used model for a CRD with t treatments, n experimental units per treatment, and m observations per experimental unit. We can write the model as y = X + Zu + , where X=[
Iowa State - STAT - 511
THE ANOVA APPROACH TO THE ANALYSIS OF LINEAR MIXED EFFECTS MODELSA model for expt. data with subsampling yijk = + i + uij + eijk , (i = 1, ., t; j = 1, ., n; k = 1, ., m) = (, i , ., t ) , u = (u11 , u12 , ., utn ) , = (e111 , e112 , ., etnm ) , IRt+1 ,
Iowa State - STAT - 511
Two approaches for E MSRCBD with random blocks and multiple obs. per blockijkYijk = + i + j + ij +where i cfw_1, . . . , B, j cfw_1, . . . , T, k cfw_1, . . . , N.with ANOVA table:Expected Mean Squares from two different sources Source 1: Searle (19
Iowa State - STAT - 511
Two approaches for E MSRCBD with random blocks and multiple obs. per block Yijk = + i + j + ij +ijkwhere i cfw_1, . . . , B, j cfw_1, . . . , T, k cfw_1, . . . , N. with ANOVA table: Source Blocks Treatments BlockTrt Error C. total df B-1 T-1 (B-1)(T-1
Iowa State - STAT - 511
ANOVA ANALYSIS OF A BALANCED SPLIT-PLOT EXPERIMENTFieldBlock 1 0 100 150 50 150 100 50 100 150 0 50Plot Genotype B0Genotype CGenotype AExample: the corn genotype and fertilization response studyBlock 2 150 100 Block 3 100 50 0 0 150 50 0 0Main pl