APPENDIX - General Instructions and Helpful Hints

Reprinted from: R. Rollefson and H. Richards, "Laboratory Experiments in Elementary Physics," Waveland Press, Inc. (Copied by permission.)

In developing a grasp of physics, the beginning student can draw on a variety of resources: textbooks, lectures and demonstrations, problem solving, discussion sessions, and the laboratory.

The laboratory furnishes a unique opportunity for testing physical principles in a quantitative fashion against the actual behavior of nature. Laboratory experience provides the student a more realistic feeling for the origin and limitations of physical concepts; it teaches him to be aware of experimental errors, of ways to minimize them and how to estimate the reliability of the result of an experiment; it introduces him to the need for keeping clear and accurate records of experimental investigations.

The first requirement of good laboratory work is a good notebook. We recommend a large bound or spiral notebook with paper of good enough quality to stand occasional erasures (needed most commonly in improving pencil sketches or graphs . . . correction of numbers should always be done by crossing them out, thus $\stackrel{ 3.1461}{3.1416}$ , because occasionally one finds later that the original number was right; the correction a mistake). Coarse (1/4") cross-ruled pages are more versatile than blank or lined pages, being useful for tables, crude graphs, and sketches, while still providing the horizontal lines needed for plain writing. Put everything that you commit to paper right into your notebook. Avoid any scribbling of notes on loose pages; such scraps are always getting lost. A good plan is to write initially only on the right-hand pages, leaving the left page for afterthoughts you may decide to put in later, and also for the kind of exploratory calculations that you would be tempted to do on scratch paper.

Heading of Experiment. Copy from manual number and name of experiment. Write down date of experiment and name of partner.

Original data in notebook. Original data must always go directly into your notebook, as they are gathered. By "original data" is meant the actual readings you have taken. For example, suppose you know your micrometer caliper is in error by .006 mm. You should record the actual readings taken on the caliper, and then correct the average. In this way, it will be clear that you have made the corrections. Also, when you take 5 or 6 successive readings of a measurement, record each reading, not just the average. From the scatter of the readings, the reader can get an estimate of the precision of the measurement. When both partners record data, errors of recording can be discovered. Record serial numbers of equipment, if given, so that you can find the same equipment if you have to check results later. The data should be arranged in tabular form when appropriate and each item or table should be properly labeled.

Housekeeping deletions. You may think that a notebook combining all work would soon become quite a mess, with a proliferation of erroneous and superseded material. Indeed it might, but you can improve matters greatly with a little housekeeping work every hour or so. Jut draw a box around any erroneous or unnecessary material and hatch three or four parallel diagonal lines across this box. (This way you can come back and rescue the deleted calculations later if you should discover that the first idea was right after all. It occasionally happens.)

You are expected to keep your notes up, from minute to minute, as you go along. You are not encouraged to take a lot of time at home to "write it up"-you probably have more important things to do than merely making a beautiful notebook.

Remarks and sketches. "Pictorial" sketches of apparatus are normally not recommended. On the other had, a simple diagrammatic sketch is often the simplest and clearest way to define the various quantities indicated in a table of data: a phrase or sentence introducing each table or calculation is essential for making sense out of the notebook record. When a useful result occurs at any stage, describe it with at least a word or phrase. At the end of an experiment, you need some sort of written comment usually including a brief summary table of results.

Graphs. Create graphs directly upon the cross- hatched paper of your notebook. Show points as circles or crosses, i.e., $\bigcirc, \odot$ , or X. Instead of connecting points by straight lines, draw a smooth curve which may actually miss most of the points but which shows the functional relationship between the plotted quantities. Original data in graphic form (such as the equipotential contours of the potentials and fields lab) you should also fasten directly into your book.

Units; coordinate labels. Physical quantities always require a number and a unit to have meaning. Likewise, graphs have abscissas and ordinates which always need labeling.

Data and results. A neat summary of data and results will make your notebook more meaningful to both you and your instructor.

Limitations of space and equipment usually require that one works with a partner. In addition, it is often stimulating to discuss your work with someone as you go along.

Independent calculations; checks. If possible both partners should perform completely independent calculations. Mistakes in calculation are inevitable, and the more complete the independence of the two calculations, the more complete is the check against these mistakes. Poor results on experiments are often the result of computational errors.

Your instructor will check your work before you leave the lab, and may write down some comments, suggestions, or questions at the end of your work.

Your instructor can help deepen your understanding and "feel" for the subject. Take advantage of opportunities to talk over your work with him.

The value of measurements is greatly enhanced by accompanying them with an estimate of their reliability. Thus, saying that the average diameter of a cylinder is 10.00 + 0.02 mm tells much more than the statement that the cylinder is a centimeter in diameter.

The term "error" is used in physics interchangeable with uncertainty. "Error" in physics does not have the same meaning as mistake. Mistakes, such as errors in calculations, should be corrected before estimating the experimental error. In estimating the reliability of a single quantity such as the diameter of a cylinder, sources of error of a number of different kinds must be taken into account. Firstly there may be an actual variation of the quantity being measured, i.e., the diameter of the cylinder may actually be different in different places. One must then specify where the measurement was made, or, if he wants the diameter in order to calculate the volume, he must find the average diameter by means of a number of measurements at carefully selected places, and the scatter of the measurements will give a first estimate of the reliability of the average diameter.

Secondly, the micrometer caliper with which the measurement is made may be in error. The errors introduced in this way will of course not be distributed equally on both sides of the true value so that averaging a large number does no good. They may be eliminated (or at least reduced) by calibrating the measuring instrument: in the case of the micrometer caliper by taking the zero error (the reading when the jaws are closed) and the readings on selected precision gauges, of dimensions approximately equal to those of the cylinder to be measured. Errors of this type are called systematic.

There is another type of systematic error which occurs in the measurement of a cylinder. The micrometer will always measure the largest diameter between its jaws, so that if there are small bumps or depressions on the cylinder, the average of a large number of measurements will not give the true average diameter but a quantity somewhat larger. (This error can of course be reduced by making the jaws of the caliper smaller in cross section.)

Finally, if one is measuring something of definite size with a calibrated instrument, there are still errors of measurement which, one hopes, are as often positive as negative and will therefore average out, if a large number of trials is taken. For example, the reading of the micrometer caliper may vary because it is not closed with the same force every time, and because the observer's estimate of the fraction of the smallest division varies from trial to trial. It is clear that the average of a number of these measurements should be closer to the true value than any one measurement, and the deviations of the individual measurements from the average give an indication of the reliability of that average value.

If one finds the average of the absolute values of the deviations, this "average deviation from the mean" is often taken as a measure of reliability. For example, let column 1 represent 10 readings of the diameter of a cylinder taken at one place so that variations in the cylinder do not come into consideration.

Measurements	Deviation from Ave.
9.943 mm	.000
9.942	.001
9.944	.001
9.941	.002
9.943	.000
9.943	.000
9.945	.002
9.943	.000
9.941	.002
9.942	.001
Ave = 9.943	Ave = .0009
Diameter = 9.943 + .001mm

Expressed algebraically, the average deviation from the mean is _m = |x_i - x|n where $x_{i}$ is the ith measurement of n taken and $\ave{x}$ is the mean or arithmetic average of the readings.

A more useful measure of the spread in a set of measurements is the standard deviation s (or root mean square deviation). One defines s as s = [ _i=1^n (x_i - x)^2n ]^12 , where $\ave{x}$ is the mean or arithmetical average of the set of n measurements and $x_{i}$ is the ith measurement.

The standard deviation s clearly weighs large deviations more heavily than the average deviation thus giving a less optimistic estimate of the reliability. Careful analysis shows that even a better estimator (and appreciably less optimistic for small sets of measurements) is n/(n-1)s = [ _i=1^n (x_i - x)^2n - 1 ]^12 ,

This estimator $\sigma$ is the ``universe" or ``parent" standard deviation and its square, $\sigma^{2}$ , the variance. (For an explanation of the $\sqrt{ n/(n-1)}$ factor see e.g. Parratt, "Probability and Experimental Errors in Science," Wiley and Sons, 1961, p. 89f.)

If the errors are distributed about the true value in what is called a "normal" manner (i.e. the errors have a gaussian distribution, $e^{-x^{2}}$ , about zero), then on average 68% of a large number of measurements will lie closer than $\sigma$ to the true value.

Let $\epsilon$ be the error in a measurement whose value is a. Then $\epsilon/a$ is called the relative error of the measurement, and 100 $\epsilon/a$ is the percentage error. These terms are useful in laboratory work.

If the desired result is the sum or difference of two measurements, the absolute uncertainties can be added. Let x and y be the errors in $\Delta x$ and $\Delta y$ respectively. For the sum we have $z = x + \Delta x + y + \Delta y = x + y + \Delta x + \Delta y$ and the relative error is $\Delta x + \Delta y / (x + y)$ . Since the signs of $\Delta x$ and $\Delta y$ can be opposite, adding the absolute values gives a pessimistic estimate of the uncertainty. It can be shown that if errors have a normal or Gaussian distribution and are independent, they should be added by taking the square root of the sum of the squares, i.e., z = x^2 + y^2.

For the difference of two measurements we obtain a relative error of $\Delta x + \Delta y / (x - y)$ which can of course become very large if x is nearly equal to y. This result shows that it is wise to avoid, if possible, designing an experiment where one measures two large quantities and takes their difference to obtain the desired quantity.

If the result wanted is a power of the quantity measured, one computes the uncertainty of the result as follows. Let a be the measurement, $\epsilon$ the error and $z = an^{2}$ the result desired. Then z = (a ±)^n = a^2 ±na^n-1 + ... where the remaining terms of higher power in $\epsilon$ and therefore negligible. The relative error in z is zz = na^n-1 a^n = n a, while the relative error in a was $\epsilon/a$ . For example, let $d \pm \epsilon$ be the measured diameter of a disk, for which we want the area A. The relationship between A and d is $A = \pi d^{2}/4$ . For an uncertainty $\epsilon$ in d, A = (d ±)^2/4 = (d^2 ±2d+ ^2)/4. The true value is $\pi d^{2}/4$ giving an error of $2d\epsilon/4$ if one neglects the term quadratic in $\epsilon$ (and one typically does). This makes the relative error of $\Delta A/A = 2\epsilon /d$ , twice the relative error in d.

If the desired result is the product of two variables, we follow a similar procedure. Let z (actual) = xy be the desired result, with $x + \Delta x$ and $y + \Delta y$ the measured quantities. Then introducing the uncertainties z (measured) = (x + x) (y + y) = xy + x y + y x + ... and the relative error is, neglecting the terms quadratic in the uncertainties, zz = xx + yy,or the sum of the relative errors of the individual measurements.

These results follow in more general form from calculus. Let R = f(x, y, z) be the functional relationship between three measurements and the desired result. If one differentiates R, then dR = fxdx + fydy + fzdz, gives the uncertainty in R, when the uncertainties dx, dy, and dz are known.

For example, consider the density of a solid. The functional relationship is = mr^2 l, where m = mass, r = radius, l = length, are the three measured quantities and $\rho$ = density. Hence, m = 1r^2 l; r = -2mr^3 l; l = -mpi r^2 l^2, and so d = 1r^2 l dm + -2mr^3 l dr + -2mr^3 l dl.

To get the relative error divide by $\rho = m/(\pi r^{2} l)$ obtaining d = dmm + 2drr + dll, where the negative signs have been dropped to estimate the worst possible combination of errors. We can then express the relative error in the density so: = mm + 2rr + ll, and if one can assume that the errors are distributed normally, then we can say that = ( mm )^2 + ( 2rr )^2 + (ll)^2 , or = ( mm )^2 + ( 2rr )^2 + (ll)^2 .

Suppose you have measured the diameter of a circular disc and wish to compute its area. A = d^24 = r^2.

Let the average value of the diameter by $24.326 \pm 0.003$ mm; dividing d by 2 to get r we obtain $12.163 \pm .0015$ mm with a relative error $(\Delta r/r)$ of 0.00012. Squaring r we find, $12.163 \times 12.163 = 148.938569$ .

Are we going to use all these digits? NO! Beyond the uncertainty in r and $r^{2}$ , one has no experimental reason for including them, they are insignificant. The relative error in $r^{2}$ is twice that in r so that the value of $r^{2}$ is uncertain to about 4 in the 5th digit since $(2 \Delta r/r) \times r$ is $0.00024 \times 147.94 = 0.0355 \approx .04$ , i.e. $r^{2} = 147.94 \pm .04$ . There is clearly no point in carrying figures beyond the 5th in further calculation. Here, the first five figures are called significant figures. Now in computing the area from $A = \pi r^{2}$ , how many digits of $\pi$ must be used? r^2 = 147.94 ×3.1415926535897....? which gives $A = 464.7668246 mm^{2}$ . But the relative error in A is just $(2 \Delta r/r) = 0.00024$ which implies that $\Delta A = 0.1 mm^{2}$ . A is uncertain in the first decimal place. It makes no sense to include digits for $\pi$ beyond the second decimal place since those numbers will not affect the final result.

A rule of rather general applicability is to use one more digit in constants than is available in your measurements, and to save not more than one more digit in computations than the number of significant figures in the data.

For the experiments in this manual these systematic errors are usually negligible compared to other uncertainties. An exception sometimes occurs for the larger masses especially the 100 gram, the 500 gram and 1 kg masses. Some contain drilled holes into which lead shot and a plug have been added to adjust the mass to within tolerance (typically $1.000 \pm 0.003 kg)$ . Occasionally a plug works loose and the calibration lead shot is lost. You can check the assigned mass values by weighing them on the triple beam balances. Report any deviations greater than 0.4% to the instructor.

The line drawings were done by Professor Severn, and were, essentially, mere reproductions of sketches done by Professors Estberg, White, and Warren. The first figure in the DC Circuits experiment comes from Guide to Electronic Measurements and Laboratory Practice, by Stanley Wolf. The figures for the Oscilloscope experiment are reprinted from The Art of Electronics, Cambridge University Press, by Horowitz and Hill (the generic oscilloscope), and Laboratory Experiments in Elementary Physics, Waveland Press, by Richards and Rollefson (the Lissajous figures). This document was prepared using L^ATEX.