GEN PHYSICS A STUDIO
GEN PHYSICS A STUDIO PHY 2048C
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Appendices Appendix I Estimates for the Reliability of Measurements In any measurement there is always some error or uncertainty in the result This uncertainty may be large or small sometimes small enough to completely neglect but it is always there Uncerminties and errors can come from any number of sources such as human errors in reading and recording data uncertainties of instrument resolution changes in the environment of the experiment and many others In any experiment one of the main jobs of the experimenter is to determine the size of the uncertainty in the measurement and when possible to identify the causes of any possible errors In general when one makes a number of measurements of the same quantity in an experiment one usually obtains different results What we need then is a method of determining from the different results the best value of the measured quantity and with what certainty we are able to call this value the best These various measurements may be the results of completely different experiments or they may be the results of the same experiment repeated several times Differences between the measurements can be due systematic such as those errors resulting from the method of the measurement or random such as those errors resulting from the limited accuracy of the equipment As an experimenter you try to eliminate the former and minimize the latter The discussion that follows presents two methods for estimating errors The first is used when it is not possible to make repeated measurements In such cases one must resort to making a reasonable estimate of the uncertainties When multiple measurements are possible a well defined mathematical formalism can be used to estimate the errors in the measured quantities This appendix also discusses the presentation of results which should be guided by the size of the certainty in the final result Finally this appendix presents the mathematical formalism needed to estimate how uncertainties in measured quantities effect uncertainties in quantities deduced from measurements This propagation of error is necessary whenever measured results must be combined to determine the quantity of interest Estimated Uncertainties or Guesstimated Uncertainties It is not always possible to calculate the uncertainty in the result of an experiment using the results of a number of separate measurements of the same quantity One39s ability to make multiple measurements is often limited by time and money In such cases an uncertainty can be estimated or guemtz39mated by observation of how close a measurement can be made or by a crude determination from variations in similar measurements Guesstimated uncertainties are usually a good estimate of the random uncertainties eg the standard deviation Whenever possible during this course uncertainties calculated from multiple measurements should be used However you will find in most cases this will not be possible due to time limitations In these cases an estimated uncertainty should always be made and the basis for this estimate should be smted Calculated Uncertainties Assume that you have made n different measurements of a quantity X Usually the results of these measurements will vary call them X1 X2 X We define the mean or average of these measurements as For each measurement you can now calculate the deviation from the average namely X 7 9 From the de nition of the mean it follows that the sum of these deviations must be zero There are several ways of estimating uncertainty First we will discuss a very simple method then we will present the most commonly used method This latter method provides the basis for determining uncertainties in most of our experiments More details concerning this method can be found in the section on Elements antatz39xtz39eal Inferenee Another common method is to simply guess the error this also has it place in our laboratory and it is important that you learn to estimate the error in a simply by I 39 how the t was made 1 A simple estimate of the uncertainty of your measurements can be obtained by adding the sum of the absolute values of the differences between the individual measurements and the average value of the measurements This sum divided by the number of measurements is called the average absolute deviation 0 blip pa 7 11 The paraIneter 06 is an approximate measure of the typical deviation of any one of the measurements from the average The uncertainty in the average result of a set of n measurements can be estimated by computing the value of 0c and dividing it by W where again n is the number of measurements Thus one can write the result ofn experimental determinations ofXas x i us 2 The most commonly used method of estimating uncerminty and the method you should use in this course is based on statistical considerations In this system uncertainty is defined in terms of the root Ineanixaaare or standard deviation 7 which is related to the square of the difference of each measured value from the mean The division by n is discussed in the Elernentx afftatz39xtz39eal Iiyerenee section As stated in that section the chances are 49 roughly two out of three more exactly 6826 that the mean of a sample of 71 measurements of a quantity differs by less than from the true mean value for that quantity The quantity is the 71 71 standard deviation of the mean Once you have computed the mean and standard deviation ofa set of measurements the results are quoted as follows 7 remit 2 i 7 J Example The following example demonstrates how this is done in practice Assume you have five measurements of distance X The table below shows how to compute 9 06 and 7 and how to present the final result Length Measurements n X 9795 lEle y Xf x10 1 4512 40002 0002 004 2 4509 40032 0032 1023 3 4514 0018 0018 324 4 4516 0038 0038 1442 5 4510 40022 0022 448 mm 22561 000 0112 3277 First calculate the average of the measurements 9 7 1 X 7 22561 5 9 45122 Next calculate the average deviation 0 1 06 g0112 00224 1 00101 J5 Then calculate the standard deviation 7 3277104 2 81910 5 1 a 28010 2 0 2 712810 J5 From these values we can state the result of the measurement in one of two ways Using the average absolute deviation X 45122 i 0010 wt and using the standard deviation X 45122 i 0013 wt Methods of Stating Error In giving the result of an experiment it is clearly meaningless to smte the result to much greater precision than is indicated by your estimate of its error Thus it is nonsensical to give a result like the following A 1325432 i 0104372 mt The meaningful result would be smted A 133 i 01 wt or perhaps A 1325 r 010m The digits 132 are said to be significant because they lie within the range of reliability as measured by the stated error A rule of thumb regarding the carrying of signi cant figures though a series of arithmetic operations is that you should carry one more than the minimum number of signi cant digits in any of the contributing factors It should be noted that zeroes give rise to some ambiguity here since they are used to indicate the position of the decimal point and may not be significant digits at all Therefore you should learn to state results in terms of number between one and ten times an appropriate power often Thus 125 i004105 125000 r 4000 One final word of caution ln taking and recording individual measurements do not round off the numbers according to your estimate or guess as to the reliability of each measurement If you do you will force 0 to be larger than you estimate ie carry at least one more figure than you think significant Only round off the final results in your reports Propagation of Uncertainty We will rst describe a simple method of calculating the uncertainty in the nal result of an experiment which involves several measurements from the uncertainties in the measurements of the contributions To take a very simple example suppose we have measured two lengths and that the nal result of our experiment is to be the sum or difference of these two lengths Let the measured lengths and their respective actual uncertainties be A i AA and B 1 AB Then the nal result will have an uncertainty which may lie anywhere between AA AB Thus the sum is written as X A B 1 AA AB while the difference is written as D AiBi AA AB Notice the fractional uncertainty in the difference D is much greater than that in either A or B alone if A and B are nearly equal lfwe want to nd the uncerminty in the product ofA and B we proceed as follows P A i AB that is P AB 1 AAB BAA i AAAB Since AA and AB are usually small compared with A and B we can neglect the product AAAB By neglecting this term we nd P zAB i AAB BAA Instead of giving the absolute uncertainty as written above one usually gives the fractional uncertainty by dividing the uncertainty terms by AB We thus de ne the fractional uncertainty AABBAAamp P AB A B and write the product as P AB i AP It can be shown that the same relationship will hold for the quotienth AB so that one can write 3 These formulas overestimate the uncerminty somewhat since the probability that in a given experiment the uncertainty in A and B would each have the same sign is only 50 A better estimate 52 and the one which you should use in this course can be obtained by using the differentiating procedure outlined below A detailed derivation of these formulae is presented at the end of the section for those who are familiar with multivariable calculus In any event you should use the formulas given at the end of this writeiup whenever possible Mathematical Derivation of Error Propagation Formulae Small uncertainties in a function of one Variable Suppose we have a function P which depends on only one variable X ie F Now when the uncertainty in X which we call AX is small we can use differentials to estimate the relationship between the uncertainty in X and the uncertainty in F designated as AB From the definition of a differential dF dF idX 1 dX where g is the first derivative owaith respect to X evaluated at X By definition dX is infinitely small as is 01F However ifAX is small compared to X then to a good approximation we can replace dX and 01F in the expression above by AX and AF which are finite in size In addition we can assume that W E does not vary over the interval AX even though X and Fmay vary For example suppose F 1X2 where a is a consmnt Then 01F 21X 01X 2 Converting this to finite changes gives AP 251 XAX In terms ofuncertainty this says If the quantity X has an uncertainty AX then the uncertainty in P where F IX will be 21X AX Quite often it is desirable to talk about fractional uncertainty AF For the example above 2 F aX g 2m 2a X This equations states the fractional uncertainty in F is twice the fractional uncertainty in X Small uncertainties in functions of several Variables Suppose F is a function of several variables ie F Fag2 and each of these variables has its own uncertainty AX A and A2 The problem can be analyzed using the same procedure as described above except that we must take the differential owaith respect to several variables In analogy to the procedure for one variable one gets ajmajAyajAz 4 6X 6 6 AF 6F where T and 2 are pariz39al derivatives A partial derivative means that the derivative is taken with x 6F 3 E 3 respect to one variable while all the other variables are considered constant For eXaInple take F axgjjz where a is a consmnt 6F 251993 6F 3ax2y2 d 6F 1ch3 i7772n 77 5X Z a Z 5 zz Using these partial derivatives in Equation 4 one obtains 21993 3ax2y2 519623 AF7Ax7Ay 72AZ 5 Z Z Z or expressed in terms of fractional uncertainties AF2Mampamp F X j Z Propagation of the Mean Square Uncertainty The method described in Equations 1 through 6 gives us a way of estimating the uncertainty in a function of several variables each with its own uncertainty However as we argued when discussing the statistical treatment of data it is the meanisquare uncertainty which in general gives the best estimate In order to calculate this using the present method we rst square Equation 4 AF2 63 my 11 ijz 63 AZJZ 6X 6 6g 7 26iaiMAy26iajMAz26jajAyAz 6X 6 6X 6 6 6 Now let AX A and AZ take on all possible values within their allowed ranges of variation The 6F 6F derivatives 7 ex a7 and 1 are essentially constants ifAX A and AZ are small If we further assume that no change in one variable will affect the change in any other variable then the crossterms will be zero This can be seen by noting that the uncertainty when not squared has a sign associated with it For each AX there is a 7 Axvalue and for each Ay a 7A Averaging over these four possibilities to nd the average AXAJ crossterm gives AXA AXQAJ 7AX7AJ iApdAy AXA171171 0 A similar argument can be made for the other crossterms Then the general form for the mean square uncertainty and the one used in propagating uncertainties is 54 AF2 gimj J63 my J63 My 8 EXaInples Suppose F X 7 Then differentiating we get AP AX 7 A From above we see that the uncertainty in F is given by A132 M2 M or AP M2 M As a second eXaInple suppose F 99 then FAX PA 7 AFJlAXXAj7 X J 2 2 AFZ FAX FA X J The uncertainty in F becomes 2 2 AP F Q X J Another way of solving this same problem is to first take the natural log of each side ln F ln x In then differentiate amp F X y neXt square each side dropping the crossterms 2 A132 FfEJZ F2 E X y and nally obtaining 2 2 AP F El X y As a final eXaInple suppose F 96 then taking natural log ofboth sides lnFlnxilny Differentiating E E F X y Squaring and summing 2 AF2F2 2F2 Q X y w emf This result is the same as the result for F 99 then Formulae for Standard Cases Following is a list of some formulae for propagating uncertainties 11le w emf 2 F Cx jmze in which C n m and e are constants 3 F CXV in which Cis a consmnt AFF y2 Aylnx2 4 F CXlogmy in which C is a consmnt 2 2 AFzF Aylog10 e V X ylogmy 5 F CX sin in which Cis a constant Akamzmj 6 F CX cosy sin in which Cis a consmnt A AZ 2 2 Am a i X cosy tang I