Physics 150. Brigham Young University. Spring Ross L. Spencer

Transcription

1 Brigham Young University Spring 006 Ross L. Spencer

2 Objectives of the course Physics To become acquainted with observational techniques and equipment used to make physical measurements and to develop skills for assembling and using this equipment in a practical way. This objective involves learning how to make equipment operate in the most favorable manner for a particular use.. To gain skills for experimental inquiry. This involves conceptualizing and planning in detail the steps to be taken in a new experiment. This will be done for short single measurements in the lab experiments then on a larger scale when you plan for and perform an experiment of your own design later in the course. 3. To develop the practice of careful observation. You will document these observations along with procedures and data in a laboratory notebook while you carry out the experiment. 4. To gain experience in transferring information or data reliably from apparatus to numerical or graphical form with as much precision as possible and to learn to mathematically, numerically, and graphically organize and manipulate data to clarify and give insight into a phenomenon under study. A detailed list of specific experimental concepts and skills objectives follows. Many of these will be unfamiliar to you at this point but should become familiar as the semester progresses. The list should be periodically reviewed throughout the course as an aid in remembering experimental issues that are crucial to success in many situations. They are grouped into five areas though there is overlap in several cases. The areas are: Measurement Issues, Transducers, Electrical, Data Acquisition, and Analysis

3 Measurements Determination of experimental limitations and improvement of measurement Sensitivity/ Precision Accuracy/Standards Repeatability Errors: systematic vs. statistical Isolation of single variable Comparison vs. absolute measurement Data Acquisition Computer interfacing/ LabVIEW Digitization Noise/ Signal level/ Amplification Electrical / Optical Circuit basics: Ohms law, voltage divider VOM meters Power Supplies Amplification-- preamplifiers, power amplifiers Timer/Counters LED s, photodiodes, lasers Optical alignment Transducers Sensitivity, instrumental resolution Calibration: linear, nonlinear Time response Saturation Hysteresis Stability (drift) Other dependencies (eg. temperature, pressure) Many different physical quantities can be measured Analysis Data plotting Curve fitting: linear, nonlinear, goodness of fit, number of free variables Statistical analysis Propagation of error Numerical differentiation, Numerical Integration Averaging; data smoothing a single curve, averaging repeated curves, time response Experimental Physics Mindset You will need to develop a different mindset when approaching experimental physics than you have become accustomed to using in your other courses. One of the AIMS of a BYU education is to educate students within a learning environment that is intellectually enlarging. I strongly feel that development of this experimental mindset can truly be intellectually enlarging. The following comments are to start you working on the mindset. It will take real work but can be the most valuable thing you take away. la. It is the nature of equipment used in experimental laboratory work that it not be automatic. Automation generally implies specialization and total lack of versatility. If a piece of equipment is to be adaptable to a variety of uses, there must be available a variety of adjustments. The experimenter (you, the student) must make the adjustments and optimize the apparatus for use in the particular situation. This optimization often requires detailed knowledge of the apparatus and demands constant mental attention. THINK. Your success in this type of activity will depend greatly on how well you learn to manipulate the equipment. The equipment used in this laboratory is not designed to avoid adjustment pitfalls. Overcoming the pitfalls is where the learning comes; the natural pitfalls are purposely allowed to occur. Students often say that the equipment doesn't work." It is not supposed to work by itself; you must "make" it work.

4 lb. Most of the apparatus is relatively simple and straightforward, but occasionally we will use more complex equipment which the student is not prepared to understand in detail. Research scientists commonly labor in this same situation. One must take the "black-box" approach in which the function of the black box is known and the function of each adjustment is known, but the details of operation are not known. This approach is particularly valid when using electronic equipment, but it is also true when using something like a stopwatch. a. Science in general, and specifically physics, is based upon the assumptions that nature is predictable, that interrelationships of precise mathematical form exist between various measurable quantities and that detailed cause and effect can be understood by studying these relationships. Confusion often arises due to the simultaneous operation of several causes. To clarify individual effects, controlled experimentation is necessary, and idealizations are often made in order to isolate clearly a single cause and study the mathematical relationships involved. Complex effects often imply multiple causes but may also imply complex causes. We would like to develop some ability in data analysis to provide insight into this type of situation. 3a. The uninitiated often think that the "scientific method" (postulate, test, evaluate) is used only in making momentous discoveries; whereas, in reality, it is used on a continuous basis, hour by hour or minute by minute when investigating the unknown in many fields. The development of this attitude in your approach to knowledge is one of the major objectives of the laboratory, but we realize that this development cannot be instantaneous but takes time. 3b. The ultimate use of this laboratory experience would be that the student would be able to use the techniques learned here to study physical parameters of personal interest either now or at a later time. To encourage this, some of our laboratories will be "open-ended." In these labs introductory material will be provided, and the student will be asked to extend the technique to measurements of personal choosing. Note: The verification or illustration of physical laws discussed in the lectures of Physics 11, 13, and 0 is not one of the objectives of this laboratory. This purpose is fulfilled by the short walk-in lab experiments associated with those classes. The emphasis in this laboratory is on learning to handle equipment and to manipulate experimental data. Laboratory Procedures The procedures for this laboratory are outlined below and represent our attempt to meet the objectives outlined for the laboratory without being a burden on the student. 1. Students will work in groups of two or three, but each will document the experimental process in a personal notebook.. Brief outlines for each laboratory period are provided. These outlines will not provide a step-by-step procedure for making measurements but will outline desired data. We will rely on the textbook of 11 and 13 for the theory associated with the laboratory studies and will supply apparatus instruction details and some background on experimental technique.

5 3. Each student will keep a laboratory notebook in which notes relating to each experiment will be recorded during the laboratory period. Grading of the individual lab notebooks will be preformed on the fly by the TA during the class period. See section on notebooks and grading for details. Lab scores will be available online on Blackboard. Please check your scores frequently and strive to improve them. The letter grades for the class will be made out by the faculty member in charge of the class from these scores and the student designed experiment write-up. Letter grades will be based on your relative standing in the class, not on an absolute scale. 4. A culminating emphasis on experimentation is found in a few laboratory periods designated as "Projects or Student-Designed Experiments." These are scheduled at the middle and at the end of the semester. For these self-designed experiments, the two or three students working as a group will decide on something they would like to measure using the equipment and techniques developed up to that point in the laboratory (or any equipment available in our equipment stockroom which would enhance the measurement). The students will plan and conduct the experiment taking notes in their lab notebooks but will then write a formal report describing the experiment and presenting their results. Each student will write his or her individual report. These projects will require a significant amount of time out of class in preparation and report writing as outlined later. No final exam will be given, which compensates for the extra student time and effort required in this project. For details see the section on student designed experiments.

6 LABORATORY NOTEBOOK -- WEEKLY LAB GRADING Purpose The purpose of the notebook is to create a complete, accurate record of each experiment as you are performing it. Engineers or scientists working in an experimental situation must learn to write down in a permanent way the things they do in sufficient detail that after several months they (or someone else) could reconstruct the significant details of the work done. The information recorded in the lab notebook is used as the basis for reports or publications of the work. Often personal thoughts and insights at the time of the experiment are as significant as the collected data. In the spirit of training for this "real-life" situation we ask you to keep a record of what you do in the laboratory each week. Additionally it is a valuable aid in clarifying your thinking before, during, and after the experiment. Audience The primary audience for this assignment is a peer student in Physics 150 that has not done this experiment. The secondary audience is the course instructor who will grade this assignment during the lab. Content The notebook should contain experimental details, procedures, data, observations, and reflections on what you are doing. Organization The entry for each lab should be organized into three sections with the following section headings: Introduction and Experimental Plan, Procedures, Data and Observations, Discussion of Results and Conclusions. The Introduction and Experimental Plan will be written before the lab period. For all but the last experiment in this course it will be brief, just a few sentences, but it is essential. This section will include any background comments and a plan of attack for the experiment. This plan may change during the experiment-- that is fine. It is nevertheless valuable to think through the experiment beforehand and to plan in writing your approach. The Procedures, Data and Observations section will be completed as the experiment is carried out. This is the heart of the lab notebook. This section will be organized in the following way. Divide your pages into two columns by drawing a vertical line down the page one third of the way from the right edge of the paper. Label the larger (left) column Procedures and Data and the smaller (right) column Observations. In the left column you will write a step-by-step description of the experimental procedures you are using and the data you collect. Use sketches, tables and graphs where needed to make this as clear as possible. Include any calculations you

7 make in order to perform the experiment. The right column will contain observations on the procedures and data at the time of the experiment. The section is to be written entirely while you are performing the experiment. Write down what happens when it happens. Keep your observations reasonably short; follow up with more in-depth analysis in the discussion section. The Discussion of Results and Conclusions is where you think in the notebook about the experiment you have just completed and the comment on the results obtained. Discuss things like what worked, what didn t work and why, and what surprises you encountered. The conclusion is a sentence or two summarizing what you have learned. Medium The assignment will be written in black ball-point pen in your laboratory notebook. The choice of pen is for permanence, black ball-point pen is consistently the most permanent. The lab notebook should be bound (not spiral) so that sheets stay put. Never tear a sheet out of a lab notebook. If a section is found to be in error, you can carefully place a big X over the section, still leaving it readable. Weekly Lab Grading Your work and your notebook will be graded by the TA during the lab period with a strong emphasis on notebook keeping. Grading will be as follows. Participation: ( points) Active in discussion, hands on the equipment, engaged with experiment, clean up. Notebook keeping ( points) Well thought out introduction and plan of experiment (3 points) Clear, complete procedures and careful data logging (3 points) Insightful observations ( points) Discussion and Conclusions, graded the following week

8 AN INTRODUCTION TO EXPERIMENTATION: AN OVERVIEW Before describing any detailed experiments, we give here an overview of the activities involved in any experimental laboratory work in the physical sciences. These ideas will be referred to in most of the weekly laboratory outlines and should be understood well by the student. Four essential elements or steps of any experimental study are: 1. Design or formulate procedures and apparatus to make measurements on or about a phenomenon or system one desires to understand more than it is presently understood.. Make measurements and record the data, generally in numerical form. 3. Analyze, decipher, organize, summarize, dissect, correlate, and in any other way study the data to find all the direct information as well as any hidden insights that the data may contain. 5. Ascertain the accuracy or reliability with which the measurements have been made, identify sources of error, and make corrections for known extraneous effects. Design Before initiating an experimental study it is of prime importance to clarify and solidify in your mind (and usually on paper) precisely what you want to measure and the data you expect to get. Only then can you give meaningful direction to the process. Experiment Experimental measurements fall into two broad categories: 1. Measurement of a specific physical quantity which can be specified by one number (or three numbers in the case of vectors).. Measurements of functional relationships in which a physical quantity varies with and depends upon one or more other physical quantities. The second type measurement, in an experimental sense, is simply a set of measurements of the first kind. The fact that there is implied a functional relationship, known or unknown, makes the analysis more difficult, more intriguing, more challenging, and more significant. Most of science and engineering is associated with such functional relationships. For example, the fundamental laws of physics are simply statements of functional relationships, which have been measured. The understanding of any phenomenon of science is a direct consequence of understanding such functional relationships. (Note: A functional relationship does not imply mathematical equations. A function can be represented by a graph taken directly from experimental measurements.)

9 Analysis Physics 150 For measurements of a single value, the only analysis of the data possible is the determination of anticipated accuracy of the measurements. For measurements involving functional relationships, the analysis is more rich and interesting. Even when studying physical quantities that are functions of only one variable, the analysis can be divided into several steps as follows: 1. Displaying the data in either graphical or tabular form.. Searching for mathematical equations that "fit" or "describe" the experimentally observed graphical function. Such equations are known as empirical equations in contrast to equations derived from fundamental laws. Note: Many of the fundamental laws themselves are simply empirical equations that have been "canonized" (accepted by the scientific community). 3. Comparison with theoretical predictions that are based on fundamental laws or postulates. This aspect of analysis may involve the creation of a theory to explain your results. To formulate a theory, one must propose a simplified "model" and then apply known principles in an attempt to predict the data measured. An analysis at the level of step (1) has value only as a predictive tool to predict similar measurements at points of the independent variable between or in regions near the positions of the measured points (interpolations or extrapolation). The analysis at the level of step () is the heart of data reduction and analysis, for it is a search for mathematical relationships that will give insight into the fundamental physical laws that are operating. Details on how to curve fit are given in a separate section later. When the student has completed steps (1) and (), he or she will find the real understanding and meaning behind the experiment in step (3). Only when you have made some comparison between what you actually measured and what you anticipated or postulated based on some model (imperfect as it may be) do you gain insight and understanding of the phenomena involved. Measurement of "numbers" with no thoughts or ideas behind them is not the work of a scientist but rather the work of a technician (a technical recorder). You may note that step (1) above is easy; step () is more demanding, and step (3) requires prior background, knowledge, experience, and understanding. A fundamental postulate of modern physical science is that nature can be described in mathematical terms and that the simplest mathematical form that adequately describes the phenomenon is the preferred analysis. The theoretical derivation of such empirically discovered mathematical relationships provides challenges and goals for theoretical analysis or, in rare cases, for new laws of science. Note: We have outlined the hard road to success as an experimental scientist without showing you the details. Error Analysis The analysis of errors is required to validate the arguments in the analysis of steps () and (3) above and often precedes and /or is mingled with the analysis. All empirical equations will have

10 undetermined parameters. The accuracy to which these parameters can be determined will always be in question and constitutes a major effort in data reduction and error analysis. In many experiments there will be known extraneous effects for which corrections can be made, and the corrected data can be analyzed as if the extraneous effect were not present. When differences are found between theory and experiment, one must ask whether the errors are in the theory, in the apparatus, or in the data-taking process. When considering the accuracy of any measurement, one may think of the errors associated with the measurement in one of two ways, statistical or systematic. These types of errors will be discussed in the following pages. 1. Statistical errors--in this case one assumes that repeated measurements of the same physical parameter would differ from each other in a random manner such that the range of several similar measurements would scatter around the average value, and the average would be more correct than any single measurement. Such random errors can be analyzed using statistical techniques discussed later.. Systematic errors--these errors are, in one sense, not errors in the experiment but errors in our thinking about the experiment. We must be careful in all experimental work to distinguish carefully between the actual apparatus and readings taken from the apparatus and the idealized thinking which fills our minds when we design and perform experiments. For example, in physics we talk about massless ropes, point masses, point charges, perfect electrical and thermal insulators, frictionless surfaces, etc. Such idealizations serve a very important role in developing the theoretical structure of science, but they are only important in the development, not in the final product. After the idealized situations are studied in the most elementary treatment, the more meaningful science is developed (in each branch of physics) in which extended mass, extended charge, heat loss through conduction, surfaces with friction, and other more realistic conditions are analyzed. A systematic error is simply an effect which you have not included in your analysis. Your thinking was too idealized, or you had no indication that an extraneous effect was operating. If you consider the non-idealized effects and carry out the necessary analysis or if you can recognize extraneous effects in your apparatus, corrections to the data suggested by this understanding can be made and their influence on the data removed; but if you do not recognize them, they constitute a serious error that cannot be eliminated by repeated measurement. Removal of systematic errors by correction or by redesign of techniques and apparatus is one of the greatest challenges of the experimentalist and demands constant attention. It is often possible to make internal consistency checks to see if some systematic error exists. Such efforts, if possible, constitute an essential part of good data analysis and reduction. Before experimentation begins, it is highly beneficial for students to realize the interwoven nature of experimental design, data collection, data reduction and analysis, and error analysis. Success in an experimental laboratory will depend upon this understanding.

11 FITTING EMPIRICAL EQUATIONS TO EXPERIMENTAL DATA If a mathematical equation can be found to describe (very accurately or only approximately) an experimental set of data, not only greater insight into the data, but also greater utility of the data is obtained. For example, one can algebraically manipulate the data and can do computer calculations with much greater ease. The process of finding an empirical equation is simply to make an intelligent "guess" of the "form" of the equation. This form will have two or more undetermined parameters that can be adjusted to obtain a desired or bet "fit" to the data. The two most commonly encountered general "forms" of equations are given below with some special cases as illustrations: 3 4 A. The polynomial expansion, y a + bx + cx + dx + ex This form includes the linear equation ( y a + bx ), the quadratic equation q ( y a + bx + cx ), and the simple power equation ( y px, when q is an integer) as special cases. The simple power equation with q not an integer is also common. (b) The exponential function y h exp(kx), Where h and k are constants and exp is the natural log exponential. Other more elaborate equations are used in special situations. Computer programs have been developed to fit very complex (as well as simple) equations to prescribed data. Here the concept is only introduced. This procedure generally consists of two steps: (1) Plot the raw data to get a "feel" for what type curve might fit and then make an intelligent guess of the form of the equation. Often some theoretical arguments will suggest the form of the equation. () Determine the undetermined parameters by using one of the following two techniques: A. Manipulate each data point algebraically to produce a linear curve from which parameters can be extracted. Here is an example of this process: If one anticipated the data would fit an equation of the form y a + bx m, where m is any real number (not necessarily an integer), one should recognize that a is the y-intercept of the y(x) data which have been previously plotted and that a can usually be obtained with reasonable accuracy from a smooth hand-drawn curve through the data points extended to the y-axis. One would then note that

12 log (y-a) log b + mlog x. Thus, if for each data point the values log (y-a) and log x are determined, a plot of this modified data with variables u log (y-a) and v log x should be a straight line, u log b + mv, with a y-intercept of log b and a slope of m. The example given is rather complex, but it illustrates that some creativity mixed with logic is required to know what type of manipulations need to be made to the data before plotting a modified curve. Scientists often manipulate the data to obtain a straight line. If one desires to demonstrate that a particular form is appropriate, the approach illustrated here has the advantage that the final curve gives one a conceptual "feel" for validity of the fit. B. A second technique known as a "least-squares fit" involves calculations of the deviation of each data point from the proposed curve and minimization of the sum of the squares of these deviations by changing the undetermined parameters. This procedure involves computer programming or the use of a computer program such as Microsoft Excel but gives the best fit and is the procedure generally used in any serious research work. Technique A provides a good conceptual verification of the goodness of fit of a proposed form of the equation and is, therefore, very valuable in the research. Technique B is the standard research level approach to curve fitting and is used following technique A to refine the fit to obtain the "best" parameters associated with the chosen form of an equation. Very elaborate programs are available for curve fitting, some of which will be introduced as needed in the course. INTRODUCTION TO STATISTICAL ANALYSIS Multiple Measurements and Mean Values Consider the following example of using statistical methods to evaluate the uncertainties associated with random errors. Think of making several measurements of the distance between two points along a straight line. These measurements, which could be made by the same individual or by several different people with different measuring instruments or various procedures, may yield a variety of values of that single distance. If we assume the errors in the individual measurements could increase or decrease the determined length with equal probability (that is, if the errors are random), the average of all N measured values would be the best estimate of the true value. If systematic errors exist, the errors are really not random, and you must separately consider such errors. Proceeding with assumed random errors, designate the i th measured distance as x i and the mean or average of these measured values by the symbol x. In mathematical form: ( x1+ x + x3 + K+ xn ) xi x N N ( 1 )

13 where the symbol Σ implies a summation that runs from 1 to N. Now consider a second example of the use of statistical analysis in experimental studies. Think of measuring the diameters of N individual hairs in your head. If you make precise measurements on several individual hairs, these measurements will not yield the same value but rather a distribution of values because each hair is actually different in size. If you designate each individual measurement as x i, you can calculate a mean x as above. The value x, however, now represents your best estimate of the size of any specific hair of your head selected at random; however, there is no such thing as a true value. In both examples given, and in fact in all measurements in which you use statistical methods, the amount of variance or scatter in the measured values as well as the determined mean value, has great significance. The variance is an indication of the expected deviation of a single measurement from the mean value x. Histograms and Probability Distributions When you make a reasonably large number N of measurements, you obtain a conceptual understanding of the size of the scatter and then check for the existence of any abnormalities in the overall distribution by making a so-called histogram of the measurements. The histogram is a bar graph obtained by grouping together all measurements that fall within predetermined small intervals of the variable, x. You must judiciously select the size of the interval, x, such that a suitably large number of measurements fall in each interval. You can understand the histogram by studying the example shown in Fig. 1. There must be several intervals in the histogram, but also there must be several measurements in each interval. As you make more and more measurements, the size of the intervals can be reduced to retain approximately the same number of counts in each interval. As this process continues, the histogram becomes more meaningful. Now consider making an enormously large number of Number of Events in interval x x x Figure 1 x Figure

14 measurements of a value x under similar conditions. You could construct a histogram with very small intervals, x, and a large number of measurements will still fall in each interval. Now the tops of the bars of the histogram will appear almost as a smooth curve. If you were to increase N indefinitely, a smooth curve called the parent probability distribution would result. Many physical processes that have natural random variations exhibit a probability distribution that is a very symmetrical, bell-shaped distribution known as the Gaussian or normal distribution given by the formula P( x) 1 e σ π x / σ Fig. shows a Gaussian distribution and also an abnormal probability distribution for comparison. The parent distribution provides the most detailed information available about a statistical process. In practice, however, the distribution is generally not known unless a very large number of measurements have been made. An experimental histogram represents the best approximation to the distribution that can be obtained from the data. Faced with the lack of knowledge of the parent distribution, it is common to assume in experimental work that random errors follow a Gaussian distribution. Standard Deviation and Probable Error The standard deviation is a single numerical value that expresses the quantitative spread or scatter of the measurements around the average value x. You can calculate the standard deviation for a given set of N measurements and relate this value to the width of the distribution peak as discussed below. To develop the idea of the standard deviation, return to the example of N individual measurements. You can easily calculate x and then the individual deviation d i x - xi of each measurement from the average. Some numerical measure of the average deviation is desirable, but the average of the deviations d i is precisely zero To obtain a numerical value that describes quantitatively the scatter, the standard practice is to square the deviations, average the squares, and then take the square root of the average. From this process we obtain the so called root-mean-square (rms) or standard deviation, σ N, which we write mathematically as ( x - x1 ) +( x - x ) +( x - x3 ) + K d i σ N. ( ) N N For a set of measurements taken on items that have an actual variation of the variable within the group, as in the example of hairs given above, the standard deviation σ N is appropriate. However for multiple measurements of a single item in which the variations are associated with statistical errors, a more thorough analysis than given here would tell us to use the following standard deviation.

15 The use of the (N-1) value rather than N in the rms calculation is associated with the fact that we have no indication of the uncertainty in the measurement if we make only one measurement. d i σ N -1. ( 3 ) N - 1 We can determine the standard deviation for any set of measurements by calculating the average values and then calculating each deviation. Knowing the deviations can then calculate the standard deviation. In order to proceed in this discussion, the assumption is made that the deviations d i are Gaussian in nature. In other words, it is assumed that a very, very large number of measurements would yield deviations that follow a Gaussian distribution and that the average x of all these measurements is the desired true value. In most measurements there is no way to justify such an assumption without making a very large number of independent measurements and determining the actual probability distribution. Nevertheless, it is common to make this assumption if there is no reason to believe that the deviations are not Gaussian. Listed below are some results that follow when you assume the distribution to be Gaussian. These results, which are proved in formal statistics courses, are very easy to use and have great utility in evaluating experimental data. You must realize, however, that the associated predictions are only valid statistically if the parent distribution is truly a Gaussian distribution. 1. If a large set of measurements are taken, 68.3 percent of the x i values lie within the range x - σ to x + σ, and 31.7 percent lie outside this range. This result implies that a single measurement is within σ of the true value 68.3 percent of the time.. If we define the probable error P.E. to be the value such that 50% of the x values are within the range x - P.E. to x + P.E then it can be shown that the probable error P.E σ. This relationship implies that if you calculate σ, you immediately know the probable error. Thus, if you were to make a single measurement, you could have a 50 percent confidence that this measured value was only one P.E. away from the true value. 3. The Gaussian distribution also has the property that 95.5 percent of the x i values are within the range x - σ to x + σ, and 99.7 percent of the x i values are within the range x - 3σ to x + 3σ). In simpler terms, it is commonly stated that you can have a 95.5 percent confidence limit that a single measurement will be within σ of the true value or a 99.7 percent confidence that it will be within 3σ of the true value. Quoting Experimental Uncertainties In reporting experimental results, data are usually listed with a numerical value for the uncertainty. For example, the best value of the mass of an electron currently is ( ± ) x 10-4 atomic mass units. Since uncertainty can be defined in many ways, how do we know what number to assign?

16 For the purposes of most analyses, and in particular for the purposes of this class, the rules for assigning uncertainty follow: 1. For statistical uncertainty, use one standard deviation (unless otherwise specified).. For instrumental uncertainty, estimate the uncertainty as well as you can. Although this uncertainty isn't strictly Gaussian, it is usually treated as if it were Gaussian in subsequent error analysis. 3. For known systematic errors, the data should be appropriately corrected and no systematic uncertainty quoted.

17 Propagation of Errors due to Mathematical Manipulations Physics 150 You may often desire to calculate a quantity mathematically from other values that have been obtained experimentally. Each of these experimental values will have associated uncertainties, and you must consider the effect of these uncertainties on the calculated quantity. For example, the calculated density ρ of a circular cylindrical object is given by But M, R, and L might be measured values with uncertainties the uncertainty ρ in ρ? mass M ρ. ( 4 ) volume π R L M, R, and L. What then is For statistical or random errors, a detailed analysis allows you to determine the uncertainty in a calculated quantity in terms of the uncertainties in the measured quantities. If we take a general function f of the variables a, b, c,, it can be shown that the uncertainty in f can be calculated by the relationship: f f f ( f ) ( a ) + ( b ) + ( c ) + K. ( 5 ) a b c Again, this relationship assumes all errors are random and are Gaussian in nature. We see that there are two essential contributions from each variable to the uncertainty in f: the partial derivative of f with respect to the variable, and the uncertainty in the variable. Clearly, the greater the uncertainty in a given variable, the greater will be the uncertainty in the derived quantity, f. Also if f depends strongly on a given variable, the uncertainty in that variable will be more important to the uncertainty in f; hence, the partial derivative appears as a multiplicative factor in each term. Furthermore, note that the uncertainties add in quadrature like the components of a vector or the sides of a right triangle when we use the Pythagorean theorem. In case you have never before seen a partial derivative, the concept is really quite simple. The function f depends on several variables; however, in each term of equation (6) we are only interested in one variable at a time. Hence, when we evaluate f/ a, we treat b, c, as constants while taking the derivative of f with respect to a. This general formula can be used for any relationship, such as the one for density above. There ρ M 1 π R L ρ M ρ L - M π R L ρ - L ρ R - M 3 π R L ρ -. R

18 Hence Physics 150 ρ ρ M M L + L R + 4 R Now let s take a concrete example. You are asked to measure the density of a metal cylinder. You obtain the following data using a vernier caliper and an electronic scale:. ( 6 ) M 185. g L 6.3 cm R 1.05 cm Based on the precision of the measuring instruments, you deduce that: M 0.1 g L 0.01 cm R 0.01 cm Solving for ρ, we have ρ 8.58 g/cm 3. Then we may use equation (7) to solve for ρ : ρ ρ M M + L L + R 4 R 8.58 g / cm g / cm 3 Thus we would say that the density of the cylinder is ρ (8.58 ± 0.16) g/cm 3. (By looking at the above expressions, which quantity would you like to measure with greater accuracy? Why?)

19 Although equation (6) is a general formula that can be applied to any equation, the following expressions are often useful: 1. If f ± a ± b ± c. If f abc or f a / bc, etc. ( f ) ( a ) +( b ) +( c ) ( 7 ) f f a a b + b c + c ( 8 ) The quantity f / f is termed the fractional or relative error in f. As we proceed with the laboratories this semester look for opportunities to use these ideas to better understand the meaning and accuracy of the experimental results you obtain.