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# Engr Physics I PHYS 1210

UW

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This 10 page Class Notes was uploaded by Eloy Orn on Wednesday October 28, 2015. The Class Notes belongs to PHYS 1210 at University of Wyoming taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/230355/phys-1210-university-of-wyoming in Physics 2 at University of Wyoming.

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Date Created: 10/28/15

Curve Fitting Linear and Nonlinear Least Squares Physics 1210 Notes Appendix D 1 PREFACE Appendix C detailed the major components that comprise an effective graph and also discussed the functional relationships which produce straight lines on linear semilog or loglog graphs This Appendix demonstrates the use of regression analysis to obtain a best fit functional expression for set of experimental data 2 LINEAR REGRESSION Engineers utilize two types of formulas to mathematically describe the relationship between a dependent variable and its independent variables 1 Theoretical relationships 2 Empirical relationships In order to show the difference between these two relationships let39s consider the following example You are given the job to determine an accurate force versus velocity functional relationship for a certain 05 kg object as it undergoes a vertical free fall through a given fluid Table D1 presents a typical acceleration versus velocity experimental data set and the corresponding force values which was computed from Newton s second law Table D1 Dynamics of Free dV Fall Experiment F mam 031 Velocity Accel Force ms In542 N where F net force on the object N 0 98 49 mass the object s mass 05 kg 138 96 48 V velocity ms 199 92 46 dVdt acceleration msz 22 91 46 251 90 45 5 306 88 44 F04612V 56518 377 83 42 5 gt 0913913 409 81 41 465 75 38 4 551 71 36 g 621 60 30 a 3 7 722 52 26 g 788 45 23 U 2 r 853 39 20 979 21 11 1031 15 08 1 1093 09 05 1121 03 02 0 1137 00 00 0 Velocity This tabular data between the net force and its corresponding velocity is also graphed in Figure D1 If there is an available theoretical or an Figure D1 Linear Curve Fit of Force vs Velocitv Data accepted empirical relationship that supposedly described this phenomena you should use it to see how well it matches the experimental results One must however always remember that valid experimental results re ect reality whereas theoretical or empirical correlations are usually very restricted in their range of applicability In cases where the physics of the phenomena or the system s properties are not well understood the form of satisfactory empirical relationship must be deduced from the experimental data alone In this case a simple linear relationship between the force and velocity is the simplest produces a somewhat reasonable fit That is FmVb 032 where m slope of the resulting straight line N sm b y intercept of the straight line N or Yo The best least square linear fit to the above data set can be easily obtained by superimposing a trendline as shown in Figure D1 The Excel procedure to affix a trendline and its corresponding equation that has any arbitrary Y0 intercept and the coefficient of determination r2 to a graph is described in the ES 1060 text I A mathematical relationship for r2 is given by Equation D6 and its physical significance is described in the ES 1060 text A value of 1 indicates a perfect match between the data and the corresponding predicted values while a value of 0 implies that the trendline is no better than just using the mean value as the predicted value for all points The ESlO60 text states that values of 1 above 08 or so are considered good By this criterion the linear trendline depicted in Figure Dl should be an excellent fit to the data since it has a very high r2 of 09698 Despite this it is fairly obvious from Figure Dl that the force versus velocity relationship is not linear 3 Estimated Errors Knowledge of the estimated error in m and Y0 is vital to permit propagation of error calculations whenever m and Y0 are used in subsequent calculations The above trendline method does not produce this information but for linear relationship only Excel provide a convenient way to calculate the least squares estimators error in the least squares estimators and the coefficient of determination I Click on Tools in the main menu bar 2 Click on Data Analysis in the pull down menu If Data Analysis is not an option then a Click on AddIns in the pull down menu b Click on Analysis ToolPak in the AddIns dialog box the check box must be checked c Click OK d Click on Data Analysis 3 Click on Regression in the Data Analysis dialog box 4 Click on OK The above procedure loads the data analysis tools addin and launches the regression analysis tool The regression analysis dialog box should now be visible on the screen This dialog box contain four regions Input Output Residuals and Normal Probability The Residuals and Normal Probability regions should not be changed unless you have an understanding of advanced statistics In the Input region you must provide the appropriate cell references for the dependent variable Y in this case the force and the independent variable X velocity In addition you need to specify where to put the results of the regression analysis and this information is conveyed in the Output region 2D The statistical relationship should be forced to comply with any known boundary conditions of the functional relationship The most common boundary condition is that the intercept value Y0 must be zero This boundary condition is easily handled by checking the box labeled Constant is Zero in the Input region of the regression analysis dialog box The output results of a linear regression analysis of the data in Table Dl are presented in Table D2 Table D2 Regression statistics summary table from Excel on the data in Table D l SUMNIARY OUTPUT Regression Statistics Multiple R 09848057 R Square 09698422 Adjusted R 09680682 Square Standard 03080752 Error Observations 19 AN OVA df SS MS F Signi cance F Regression 1 51887577 51887577 54670112 22971E14 Residual 17 16134754 00949103 Total 18 53501053 Coef cients Standard t Stat Pvalue Lower 95 Upper 95 Lower Upper Error 95 0 95 0 Intercept 56517517 01366055 41372802 1661E18 536353892 59399645 53635389 59399645 XVariable 1 704611782 00197239 23381641 2297E14 05027922 04195643 05027922 04195643 There are five items of interest in the regression statistics summary table given in Table DZ 1 R square r2 096 98 2 coefficient of the intercept b N 5652N 3 standard error of the intercept Ab N 01366 N 4 coefficient ole m Nm 04612 Nm 5 standard error ole Am Nm 001972 Nm Note that the above b and m coefficients agree with the trendline value given in Figure D1 and that their relative standard errors are both fairly small The small relative standard coefficient errors and the high r2 value would normally imply an excellent fit It should be noted that Excel used the default number of significant digits and that it is the students39 responsibility to determine the appropriate number of significant digits 3D 4 Nonlinear Trendline Besides linear trendline Excel has the capability of fitting logarithmic polynomial of arbitrary order power or exponential functions to data In the case presented in Figure 1D it appears that a quadratic relationship should produce an excellent fit Figure D2 substantiates this in that 5 this quadratic trendline has a r2 of 09986 as compared to a value of 09698 for the linear fit Higher order polynomials 4 7 may be used but any increase in r2 that is obtained by this increased complexity is rather superficial 3 7 5 Nonlinear Optimizing Solver If we start over on this problem and apply some basic dynamics to this free fall problem the summation of forces in this case must be equal to the gravitational body force massg in the downward direction plus a drag force in the upward direction that is some unknown function of velocity Therefore theory implies that the force versus velocity relationship must have the following general form Force N 7 F 00271VZ 01278V 4977 4 6 8 Velocity mls F mass 0 g DragV 033 but it does not supply any information about how the drag varies with velocity Our own personal experience indicates that the drag force increases with velocity and extensive experimental testing over the years has shown that power laws can be used frequently to correlate velocitydrag data over limited velocity ranges If this is assumed to be the case here then F mass 0 g aVb 034 Theory and some empirical insight has therefore been combined to obtain a possible function form between velocity and force in terms of two arbitrary constants a b that is based upon the physics of phenomena and not just blind curve fitting as was done in the linear and quadratic curve fit examples The values of a and b that give the best fit with the experimental data can be determine through the use of the Excel nonlinear optimizing solver which was also covered in ES 1060 l The first requirement of using the nonlinear optimizing solver is the development of a regression function that you what to optimize in terms of minimizing or maximizing its value or obtaining a specified value The trendlines that are presented in the previous two curve fits are based upon least square regression in which the following regression function is minimize n 2 2E E 095 11 where E is the measured force and F is the corresponding predicted value in the data set that contains n values In this case Equation D4 would be subsituted for F Instead of doing this lets maximize r2 That is n 2 2 E E 2 11 r 1 n 036 2 F F 2 11 where F is the mean force of the experimental data set Excel provides a nonlinear optimizing solver for minimizing functions such as Equation D6 However the problem must be prepared properly to obtain an appropriate solution Table D3 presents a copy of the spreadsheet see file Appendix Dxls for the actual spreadsheet that was used to 4D determine a amp b This table contains five columns column 1 is the independent variable velocity column 3 is the dependent variable force column 4 is the predicted dependent variable the force calculated from Equation D4 column 5 is the square of the difference between columns 3 and 4 and column 6 is the square of the difference between column 3 and the average force which is calculated at the end of column 3 The columns 5 amp 6 are then summed and these values are used to calculate the I value for a guess set of coefficients a b For instance the guess of 11 produces a very poor rz value of 588 Appendix Dxls a 00852316 Nms0b g 98 ms02 b 16632532 m 05 kg Force N Velocity Accel Measured Predictedquot Ei Fiquot2 Fav Fiquot2 ms msAZ Fi Fi N A2 N A2 0 98 49 49 000E00 393 138 96 48 48 208E03 354 199 92 46 46 104E03 283 22 91 46 46 113E03 266 251 90 45 45 375E05 250 306 88 44 44 227E03 220 377 83 42 41 616E04 152 409 81 41 40 139E03 128 465 75 38 38 267E03 069 551 71 36 34 114E02 040 621 60 30 31 151E02 001 722 52 26 26 278E04 010 788 45 23 23 830E05 045 853 39 20 19 397E03 094 979 21 11 11 372E03 349 1031 15 08 08 417E04 470 1093 09 05 03 102E02 609 1121 03 02 02 133E05 766 1137 00 00 00 163E03 852 Fav 29 Sum 580E02 5350 R92 0998916 1 SUMQi FiA2SUMFav F042 Ei see Module ForcemgVab ForcemgVab Table D3 Excel table used to perform nonlinear regression Excel uses an iterative approach to solve the nonlinear regression problem once it has an initial guess set to start this iterative process In this case the program will systematically vary a and b to determine the local gradient of r2 and thereby determine how the a b set should be varied to maximize r2 In order to use the solver tool the tool must be loaded into Excel The solver can be loaded by 1 Click on Tools in the main menu bar 2 Click on Solver in the pull down menu If Solver is not an option then a Click on AddIns in the pull down menu b click on Solver AddIn in the AddIns dialog box the check box must be checked 5D c Click OK d Click Solver The Solver dialog box is now visible The first menu item is the target cell which is I in this case The second item delineates what action is to be perform on the target cell In this example we wish to maximize the target cell The third item specifies which cells may have their values varied to accomplish the objective which in this case are cells containing the guess values of the regression parameters a and b Note that named cells can be utilized in specifying the cell locations of the target cell and the adjustable cells As an option you can set numerical constraints on the adjustable cells A little thought about the physics of this problem indicates that a and b are both positive and these constraints may be added In some problems you may wish to change the default Precision and Tolerance values by first clicking the Options button Now click OK and Excel will attempt to find the optimum solution and replace the guess values of the regression parameters with the optimum values Table D3 indicates that combined theoreticalempirical correlation F 49 00852V1663 CD7 produces a r2 of 09989 which is slightly better than the quadratic but has more physically significant Instead of basing the curve fit on r2 try using the least squares regression method to compute the coefficients and compare your results One word of caution nonlinear functions often contain more than one solution and that a given guess set may produce a local solution in this case a local maximum instead of a global solution Highly nonlinear problems may also require a fairly accurate initial guess to obtain a global solution or any solution and you may have to resort to plots to produce an accurate initial guess See Nonlinear Regressionxls for another example References 1 Introduction to Engineering Computing B R Dewy McGrawHill Primus 1994 pg XLlXLZO Your ES 1060 text 2 Physics 12101310 Laboratory Manual University of Wyoming Department of Physics and Astronomy KendallHunt Publishing 1992 pg 106114 6D Classwork and review Turn in one per person at the end of class Name 7 i 777 and at least one other person you completed this with i i if I A few practice bits from early in Chapter 17 on Temperature and Heat 1 When temperature is measured in Kelvin the pressure in a constant volume gas thermometer is directly proportional to the temperature 2 amp Tr F1 The Kelvin scale is related to the Celsius scale as TKTC27315 If the pressure in a room is zero C at 1 atmosphere of pressure at what temperature C would the pressure be doubled Answer 273 C or 273x2 Kelvin 2 The zeroth law of thermodynamics says that if some object C is in thermal equilibrium with objects A and B then A and B are also in thermal equilibrium with each other Give an example of two objects that are out of thermal equilibrium with each other Offer a way to bring them into thermal equilibrium You take a pan out of the oven set it on the counter Heat ows from the pan onto the counter and into the air until the counter and the air and all the things in your kitchen are at the same temperature ie in thermal equilibrium 11 Physics review For each of the following problems discuss and identify the primary physical principle or approach needed to solve the problem A Linear kinematics B Angular kinematics C Fma or 39cIx D Conservation of linear momentum E Conservation of angular momentum F Conservation of Energy G Archimedes principle H WorkEnergy theorem 1 Bernoulli39s equation A Cd player spinning at 500 rpm is subject to an angular acceleration of 2 rads2 How long does it take to come to a stop Xena the warrior ess pushes with a force of 100 N perpendicular to a dungeon door of mass 200 kg and width 1 In How long does it take to close the door C also B perhaps A simple pendulum of mass 1 kg and length 1 m hanging at rest is struck by a bullet of mass 10 g traveling at 300 ms What is the speed of the bulletpendulum combined mass immediately after impact of the bullet embeds itself in the pendulum A simple pendulum of mass 1 kg and length 1 m hanging at rest is struck by a bullet of mass 10 g traveling at 300 ms If the bullet embeds itself in the pendulum how high does the pendulum rise from its equilibrium position A space ship orbits the earth in an elliptical orbit If it is traveling 100 ms when it is 108 m from earth how fast is it traveling when it is 107 from earth F Xena the warrior ess fires a 200 g arrow at 80 ms into a dungeon door of mass 200 kg and width 1 m The arrow embeds itself in the door at 05 m from the hinges How long does it take the door to close through a distance of 90 degrees E and BC are needed Xena the warrior ess fires a 200 g arrow at 80 ms into a rodent of unusual size which falls off a 10 m high ledge How fast is the rat falling when it hits the floor F A 10 kg box slides initially at 5 ms across a wooden floor with coefficient of kinetic friction 03 How far does it slide before coming to rest H A 10 kg wooden box is falling at 20 ms when it lands on one side of a teeter totter with radius 3 m A rodent of unusual mass 20 sits on the other side of the teeter totter With what speed is the rat initially launched into the air as it becomes airborne F and E A motionless 10000 kg space station shaped like a square slab is struck at its center by a 5 kg meteor travelling at 1000 ms perpendicular to the slab The meteor is vaporized by the impact What is the change in velocity of the space station A motionless 10000 kg space station shaped like a square slab is struck at one of its comers by a 5 kg meteor travelling at 1000 ms perpendicular to the slab The meteor is vaporized by the impact What is the angular velocity of the space station E What is the linear velocity A distant 10000000 kg asteroid initially at rest 1010 km away with respect to the moon is attracted toward the moon by their mutual gravitational force With what speed does the astroid hit the moon F A 10000000 kg asteroid moving at 11 kms hits the moon at its equator tangentially How much does the rotation period of the moon change How much does the velocity of the moon change Xena the warrior ess fires a 200 kg rodent at a velocity of 100 kms up at 30 degree incline over a massless string and massless pulley toward a distant 1000000 kg asteroid with a coefficient of friction 02 If the air density is 1 kg cubic meter what color is the rat ZZ III In science and engineering it is very important to be able to verbalize the meaning of mathematical expressions Write a couple of grammatically correct sentences that tell what these expressions mean 1 1 X1x v tEat2 2 FfuFNumgc050 4 K1U1W K2U2 other 6 xAe 2blquot tcoswtj 7 P1pgy1ll2pviP2pgy2ll2pvi IV Describe from your perspective the most challenging aspect of physics class this semester Again from your perspective and keeping in mind the effort you have put into this class offer one idea for making learning physics more effective

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