INTRODUCTION TO LABORATORY
PURPOSE AND OBJECTIVES
The purpose of the laboratory exercises is to provide you with some direct experience with the concepts you will study in the lecture portion of the course. In addition you will be exposed to the techniques that are used to obtain and analyze the experimental data which are used to construct or test theories. In this view, these exercises should be regarded more as self-performed demonstrations rather than out-and-out experiments. That is, the results are well-established experimental facts. You are asked to demonstrate for yourself that the results are consistent with the general theoretical view presented in the course. In accomplishing this goal you should also come to appreciate the approximate effect that uncertainties have on the result. Finally, you should gain experience in drawing logical conclusions from your data, and what relation these conclusions bear with the nature of our modern world.
Laboratory assignments appear on the course syllabus. Prior to coming to the laboratory it is expected that the student will have become acquainted with the purpose of the experiment and will have planned his method of doing the experiment by reading the lab manual.
Generally, students will work in pairs in doing experiments. Each laboratory exercise consists of three main parts: Prelab, or Introduction, Procedure and Analysis. Each of these parts is viewable using the browser on your lab computer. The Prelab has a set of questions for you to answer before coming to the lab session. The completed Question sheets should be torn out of the manual and handed in to the Lab Instructor. They will comprise 5 points out of a total of 25 points for each lab report. During the Procedure segment, the data will be taken and analyzed on a computer Spreadsheet. In many cases, the data analysis is also completed on the Spreadsheet. The steps in the Analysis section can be completed in the lab, or later at home when the final report is prepared. In the latter case, the required numerical values may be written in the appropriate spaces on the Spreadsheet.
During the lab period, both partners will collaborate in filling out the Spreadsheet, and then obtain two print-outs to use for completing the lab report at home. All other parts of the experiment will be written individually by each partner in his or her own words. Discussion of the parts of the experiment between partners is encouraged prior to writing the report. The actual writing of the laboratory report is to be done outside of the scheduled laboratory, however it is strongly advised that the student do as many of the analysis calculations as is possible before leaving the laboratory. This procedure often identifies data that the student forgot to take and helps him or her identify parts of the experiment that he or she does not fully understand. At the end of the laboratory period the laboratory table should be cleaned and the apparatus arranged approximately as you found it at the beginning of the period. Ask the laboratory instructor to check the equipment and the table. If everything is satisfactory, the instructor will initial your Spreadsheet print-out.
Treat your apparatus with respect. Many students must use this laboratory after you. Please make sure that they are not condemned to work with inferior equipment on your account. Care in the handling of any apparatus is a valuable skill. Your instructor will try to see that conspicuous success, or lack of it, in doing this is reflected in your laboratory grade.
THE LABORATORY REPORT
The report is to be written following your regular laboratory period and submitted for grading at the beginning of the next laboratory meeting. Students who make efficient use of their two-hour laboratory period should require only one or two additional hours outside of the laboratory to complete the report. Each report is graded on a 25 point basis, 5 points of which is allocated to the Prelab Questions. Points will be deducted for late laboratory reports at the rate of:
up to 1 week late -4 points
up to 2 weeks late -10 points
up to 3 weeks late -20 points
later than that forget it-----0 points credit
The first page(s) of the report [usually the Spreadsheet print-out(s)] will always have a heading . Make sure that the heading information on your data sheet is completed accurately.
A complete or full-length report will have the following subheadings:
|A. Data||the spreadsheet print-out|
|B. Purpose||usually provided in experiment write-ups|
|C. Diagram of Apparatus||usually provided in experiment write-ups|
|D. Procedure||usually provided in experiment write-ups|
|E. Sample Computations||a simple run through of each calculation|
|F. Graphs, if applicable|
|G. Results||discussion in your own words|
Parts B, C, and D are usually provided as part of the materials that describe the experiment to you in the lab manual. Therefore you will be required to do only sections A, E, F, G, and H, as applicable.
A brief description of the subheadings is given below:
The data necessary to complete the report are all recorded on the Spreadsheet. Make sure that all the necessary data has been taken by referring to the Procedure section of the laboratory exercise.
The purpose tells why the data is to be taken; it states briefly the objectives of the experiment.
C. Diagram of Apparatus
This step should include clear labeling.
This step includes the procedures you followed to fulfill the purpose of the experiment. Because of time limitations, often rather specific suggestions will be made in the laboratory manual about the procedure.
E. Sample Calculations
There should be one sample computation for each major type of calculation. For example, suppose the cross sectional area of a cylindrical wire is required.
The sample computation should have the form:
A = π r2 = π (2.186 mm)2 = 15.01 mm2
Note that the symbolic relationship shows the grader what quantity is being calculated, and whether you have the correct relationship. Proper units and the correct number of significant figures are necessary. Actual calculations should be made on your calculator or by the Spreadsheet itself; however, use only the correct number of significant figures. The results of other similar calculations should be displayed on the Spreadsheet.
The construction of a graph is often the easiest way to display data in a compact form for interpretation and analysis. A graph also will point out inconsistent readings, which may need rechecking or further study in the experiment. A graph should have enough information on it so that even if it were separated from the rest of the report, it would still be intelligible to a student of physics. Most of the graphs required will be constructed using the Spreadsheet Graph Wizard. However, it may be necessary for you to construct a graph by hand. Each hand-drawn graph should contain the following information:
A title, telling exactly what is being represented
A choice of scales such that the graph will fill most of the graph paper and so that the points may be located with reasonable exactness.
Each axis marked with the quantity, its symbol and unit. It is customary to plot the independent variable along the x-axis (abscissa), and the dependent variable along the y-axis (ordinate).
Each plotted point on the graph is to be indicated by a sharp dot and surrounded by a small circle centered on the dot. If more than one curve is plotted on the same page, open triangles, squares, etc., may be used to distinguish the various curves.
To facilitate analysis and interpretation, the plot should, whenever practical, be a straight line. If the form of the function is y = mx + b, the quantity x should be plotted as the independent variable and the quantity y as the dependent variable. The slope is m, and the intercept is b. The best straight line should be drawn, visually averaging the circles about the line. A transparent ruler is helpful in doing this. Sometimes the line will meet only a few of the circles. Where the curve (line) does meet a circle, the circle should be left open. That is, the line should be broken at the circle and continued on the other side of the circle. The error bars show the uncertainties in the quantities plotted. Uncertainties are discussed later in this introduction. In any case, the curve should be a smooth curve, averaging the points. In general, as many points should lie on one side of the curve as on the other.
Your results will generally appear on the Spreadsheet, either in the form of a single value, together with the appropriate estimate of uncertainty, or in the form of a graph or table. A graph is to be preferred whenever a choice can be made. Sometimes tables or graphs are used to present intermediate results and the final result is a single value with an estimate of certainty. In this section, you should discuss the reasonableness of your results, given the uncertainties expected in the measurement.
The experiment was planned and performed with a purpose in mind. The conclusion should indicate how well the purpose was fulfilled by the experiment. To do this you may need to analyze the various sources of uncertainty and how much each source contributes to the overall uncertainty of your final result. Also indicate any further experimentation that might have come to mind. That is, if you had more time, how would you go further with this apparatus?
The representation of a physical quantity should have a unit to tell what was counted, an order of magnitude and a statement about its reliability. This fact brings us to a consideration of significant figures. A significant figure is any digit in the numerical part of a measurement, which does not overstate the reliability of the measurement.
For example, suppose we are measuring the width of a door by using a meter stick having only centimeter marks on it. We would be reasonably certain of the door width to the nearest centimeter. Let us assume the width to be between 71 and 72 cm. We could estimate to the tenth of a centimeter between the two readings. We might write this reading as 71.3 cm, with the 3 underlined to show that not all people would agree on the exact tenth. However, if we have any ability to estimate, the 3 has some significance since the correct value is more apt to be 3 than, say, 9. To write a fourth digit would require ten times more accuracy, and would be very misleading for this case. On your data Spreadsheet, always use the correct number of significant figures. If the Spreadsheet will not let you write the correct number of decimal places in a particular cell, the Number Format needs to be changed. Select the cell where the result is entered, and go to the Format...Cells menu. On the Number tab, choose Number and enter the requisite number of decimal places in the indicator box.
It is customary to write large and small numbers as powers of 10 with the first part of the number indicating the number of significant digits. For example, 3 x 105 would indicate one significant figure, while 2.65 x 105 and 2.00 x 105 would both indicate three significant figures.
It should be clear that the number of significant figures in the result of any calculation depends directly on the number of significant figures in each factor entering into the calculation, being limited in general by the factor with the least number of significant figures. In calculations, discard superfluous digits as you go along, increasing by one the last significant digit if the succeeding digit is 5 or more (there is another convention relating to the rounded off digit being odd or even, which will not be discussed here).
UNCERTAINTY IN MEASUREMENTS
No measurement is perfectly precise. A possible exception to this rule is the case where the result of the measurement is an integer, such as the atomic number of a given atom. The precision of a simple measurement of length, for example, is limited by the construction of the meter stick used to make the measurement--one can only say that the true length of the measured object is somewhere between the values that correspond to the two marks on the meter stick between which the end of the object lies.
Even if it looks at first as if the end of the object coincides with one of the marks, closer examination always reveals that the coincidence is not exact. It may be possible to refine the measurement by estimating the relative placement of the end of the object between the two marks, but there will always remain a non-vanishing uncertainty in the measured length.
It is customary to include the uncertainty in a given measurement in the written results of the measurement as:
(best value of the quantity) ± (uncertainty).
The length of a laboratory table, for example, might be written:
2.354 ± 0.002 meters
Which means that the person measuring the table could read the meter stick to the nearest two millimeters. The lab table might be anywhere between exactly 2.352 meters and 2.356 meters long, with 2.354 meters being the best estimate of the true length that the measurer could come up with using that particular meter stick.
The uncertainty may be regarded as an estimate of the maximum amount of unavoidable error in the measurement. One knows, for example, that the exact length of the above table is not 2.354 meters; in writing 2.354 ± .002, the experimenter in effect states that the greatest error (that is, the greatest difference between the true value of the length and 2.354 meters) that he thinks could exist in his measurement is 0.002 meters. He cannot avoid making some sort of error of about this amount without using a different measuring instrument or a different technique. Thus the problem of finding the uncertainty in a measurement is the same as the problem of estimating the error in that measurement.
PROPAGATION OF ERROR
Measurements in physics are only occasionally as direct as the measurement of the lab table where the length of the table was measured by direct comparison with the length of the meter stick. A measurement of velocity, for example, usually involves measurements of length and time, so both measurements will contribute to the uncertainty in a measurement of the velocity.
A propagation of error calculation is the calculation of the uncertainty in an indirect measurement from the known uncertainties in the direct measurements on which it is based. If a car goes 1.50 ± .05 miles in 0.0402 ± .0006 hours, for example, the actual speed of the car might be anywhere between:
1.55 mi/0.0396 h = 39.1 mi/h and 1.45 mi/0.0408 h = 35.5 mi/h
So we write the measured speed as 37.3 ± 1.8 mi/hr.
The calculation of propagation of errors can be simplified by remembering three rules for how errors propagate in arithmetic operations. These rules are given below:
1. If two (or more) quantities are added or subtracted, then the uncertainty in the result is the sum of the uncertainties in the original quantities. For example, if a boy is 2.005 ± .003 meters tall and a girl is 1.673 ± .005 meters tall, the difference in their heights is 0.332 ± .008 meters.
2. If two quantities are multiplied together, or if one quantity is divided by another, then the relative uncertainty (or the percentage uncertainty) in the result is the sum of the relative uncertainties (or the percentage uncertainties) in the original quantities. If a large number of quantities are multiplied and/or divided the relative uncertainty in the result is the sum of all the relative uncertainties in the original quantities.
3. If a number is squared (cubed, taken to the fourth power, etc.), then the relative error in the result is twice (three times, four times, etc.) the relative error in the number.
Relative and percentage uncertainty of a quantity are defined below:
relative uncertainty = (uncertainty) / (best value)
percentage uncertainty = (relative uncertainty) x 100%
As an exercise, you should apply this rule to the propagation of error in the speed of the car calculated above.
SUGGESTIONS FOR ESTIMATING UNCERTAINTIES
Uncertainties in measurements will arise from different sources in different experiments, depending on the type of quantity being measured, the measuring instrument used, the technique used in the measurement, the skill of the experimenter, etc. It is usually possible, however, to classify uncertainties as being of one of the two following types:
1. Uncertainties due to limitations of measuring instruments or technique: One can usually reduce the uncertainty in a measurement of length by using micrometer calipers instead of a meter stick; the uncertainty in a measurement of length using a meter stick, therefore, is due to a limitation in the measuring instrument.
2. Uncertainties due to random errors or to the statistical nature of the measured quantity: The thickness of a sheet of metal may not be uniform, so that measurements of the thickness made at random over the sheet will give a number of different values. A large number of people asked to measure the same quantity will usually come up with a set of different answers due to small random individual variations in technique. The number of counts coming from a given radioactive sample measured on a Geiger counter in equal periods of time will vary in a random fashion from the average value. All of these are examples of measurements where this second type of uncertainty is important.
Uncertainties of the first type can usually be estimated by the experimenter from the least count of the measuring instrument, the "least count" being the value represented by the distance between two adjacent scale readings. Depending on a number of factors (the presence or absence of parallax, the widths of meter pointers, the skill of the experimenter in estimating fractions of the distance between scale readings, etc.), the uncertainty might be anywhere between one-tenth to one times the least count.
To estimate the uncertainty in a particular reading, you should ask yourself, "between what limits can it reasonably be said that the true value lies?" The uncertainty will be one half of the difference between these limits.
If you have understood everything in the lab notes up to now, you know how to make a propagation of error calculation and how to make estimates of the uncertainties of directly measured quantities, so that you should be able to determine the uncertainty in any measurement you make in the laboratory. You may wish to determine how successful you have been in making a measurement; you should do this by comparing the actual error in your measurement with the uncertainty in it. Of course, you cannot find the error in your measurement unless you know the true value of the quantity you have measured. In most cases (but by no means all) encountered in the laboratory, very accurate values of the quantities measured are known because they have been measured before by skilled experimenters using the best available equipment. You can usually accept these values, which may be found in your textbook, in handbooks, or supplied to you in the laboratory, as probably better than your own and determine the error in your measurement by:
Amount of error = | (measured value) - (accepted value) |
The percentage error in a measurement is:
Percentage error = 100% x (amount of error) / (accepted value) .
You should be concerned if the amount of error is greater than the uncertainty in the measurement. If, for example, you measure the boiling point of water to be 105. ± 1 °C, you should know that something is wrong. If something like this happens, you should look for two kinds of errors:
Personal errors: In this category is collected a wide variety of mistakes on the part of the experimenter. Some of these are: misreading the instruments, arithmetic errors, calculator errors, etc. All of these errors can be avoided and/or corrected, so they are not acceptable excuses for error in the results of a measurement. Your lab instructor may be able to help if you cannot find some source of personal error yourself.
Systematic errors: In this category are placed all errors, due to measuring instruments or technique that can be corrected if they can be discovered. Examples are errors due to improper calibration of instruments, zero readings not being taken into account, viewing a scale at an angle when parallax is present, etc. A clue that systematic error is present occurs when a number of measurements are all in error by about the same amount and in the same direction. In the above example of the boiling point of water the thermometer was probably improperly calibrated, so that all measurements of temperatures near 100 °C would be about four degrees too high. Systematic errors can be very difficult to find; a true test of a good experimental scientist is his ability to find and eliminate all the systematic errors in a complicated experiment.
If, at the end of an experiment in the laboratory, your measurement is not in agreement with the accepted value and you cannot find any personal errors, you might include a list of possible sources of systematic error with your results, along with some quantitative idea of how much each source might contribute to the error in your results.
The difference between unavoidable errors and avoidable errors is reflected
in the differences in meaning of the words "precise", "accurate", and "exact":
A precise measurement has little or no unavoidable error (small uncertainty). An accurate measurement has little or no avoidable error (systematic or personal). An exactmeasurement has neither avoidable nor unavoidable error; that is, it is infinitely precise and infinitely accurate.