Physics 2010/2110 Laboratory
Purpose and Objectives
The laboratory is an essential part of this course. If
it is to describe nature, physics has to be tied to experiments.
Naturally, you will not be making any
earth-shattering new discoveries in the lab.
The main purpose of the lab is to give you experience with
how physical measurements are made and the care that needs to be taken in
interpreting data. You will be exposed to the techniques that are used to obtain
and analyze the experimental data that are used to construct or test theories.
In addition, the laboratory exercises will provide you with some direct
experience with the concepts you will study in the lecture part of the course.
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 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.
Each laboratory experiment is designed to enable you to attain good
understanding of the material through
three distinct activities: Prelab/Introduction,
Procedure and Analysis. These
portions of each experiment are described below in detail:
Prelab/Introduction is the material that you should carefully
review before coming to lab. It consists of an
introduction to the material that will be covered along with prelab questions
that must be turned in when you get to the laboratory.
Procedure is the description of the experiment that must be
performed when in the laboratory. It will involve the
steps necessary to actually perform the experiment.
For each experiment that you do, you will
use the computer to bring up a Worksheet,
which is a Microsoft Excel document.
At the end of the experiment, you
should print out this Worksheet. It
will constitute yours and your partner's data sheet, which you should take with
you to perform the analysis and to write up the lab report on the experiment.
Analysis involves the development and manipulation of the measurement
results in such a way as to test the assumptions of the experiment.
This process can involve calculations, graphing, and drawing. At the
end of the Analysis segment, there
will usually be several questions concerning your experimental results,
the measurement error and the comparison with what is expected from the
experiment. Unless your instructor
says otherwise the product of these steps should be submitted at the beginning
of the next weekÕs lab period along with the data sheet and a conclusion about
your results as your lab report.
Laboratory assignments appear on the course syllabus.
Prior to coming to the laboratory you are expected to become
acquainted with the purpose of the experiment by reading and
understanding the Prelab for the
experiment. When you come to lab, treat both the computer and 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.
We will try to see that conspicuous success, or lack of it,
in doing this is reflected in your laboratory grade.
The following topics refer to general issues that are
encountered in almost every experiment that you will perform in the
laboratory. Make yourself familiar
with these ideas, and complete the Prelab
for the Introduction below.
They are to be handed in at the beginning of the very
first lab meeting.
Significant Figures
The representation of a physical quantity
should have units 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 that 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 Worksheet,
always use the correct number of significant figures.
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 would indicate three significant figures. The
expression 2.00 x 105 would also 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, rounding the last
significant digit up if the succeeding digit is 5 or more.
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 lies 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 some uncertainty in the measured length.
It is customary to include the uncertainty in
a given measurement in the written results of the measurement as:
Taking and Handling Data
(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 or she thinks could exist in his measurement is 0.002 meters. One 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 a lab table where the length of the table is 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
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, 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 uncertainty in the result is twice (three times, four times, etc.) the relative uncertainty in the number.
Relative and percentage uncertainty of a quantity are defined as:
Percentage uncertainty = (relative uncertainty) x 100%
As an exercise, you should apply these rules 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.
For instance, one can usually reduce the uncertainty in a length measurement 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 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 could it reasonably be said that the true value lies. The uncertainty will be one half of the difference between these limits.
Definite procedures exist for estimating uncertainties of the second type. Their application is based on the assumptions that the errors in the measurement are truly random and that enough measurements have been made that an analysis by use of statistical methods is valid. These procedures are outlined below.
Suppose a set of measurements has been made in which the errors are random. The best value of the physical quantity is the average, the mean of the set of measured values. The deviation of a given reading is simply the difference between the reading and the mean value. A quantity called the Standard Deviation (abbreviated s) is a measure of the average error of an individual measurement. The Standard Deviation is defined to be the square root of the quotient of the sum of all the deviations squared and the number of readings less one (see the formula below).
For example, this table shows the deviations calculated for a set of measurements:
|
Measurement number |
Measured length (cm) |
Deviation (cm) |
| 1 | 6.54 | .01 |
| 2 | 6.49 | .04 |
| 3 | 6.53 | .00 |
| 4 | 6.50 | .03 |
| 5 | 6.55 | .02 |
| 6 | 6.52 | .01 |
| 7 | 6.51 | .02 |
| 8 | 6.54 | .01 |
| 9 | 6.52 | .01 |
Deviation in measurement No. 1: (x1 -
)
= |6.53 - 6.54| cm = .01 cm
At this point you may think it would be reasonable to write the result of a measurement as (best value) ± (s) so that the length measured above would be written 6.53 ± .02 cm. However, the larger the number of individual measurements there are in a set of measurements of the same quantity, the more confidence one should place in the mean of the measured values as being close to the true value of the quantity. The quantity s is likely to be about the same for a large number of measurements as for a small number. If the errors are truly random, a better measurement of the uncertainty in the best value of the measurement is the Standard Deviation of the Mean (abbreviated sm), which is defined as
where N is the number of measurements in the set. It can be shown that one can be about 68% confident that the true value is within one sm of the mean value, and about 95% confident that the true value is within two sm's of the mean.
In this laboratory we shall choose the convention that the uncertainty due to random errors is two times the standard deviation of the mean (95% confidence):
Uncertainty due to random errors = 2 x sm.
In the above example, then
so that the result of the measurement is 6.53 ± 0.01 cm.
Note that in the above calculation uncertainties are usually written with only one significant figure. This is a good "rule of thumb" to follow, reflecting the fact that a great amount of uncertainty is involved in the estimate of the uncertainty itself. Note also that the uncertainty in any measurement determines the number of significant figures one can give to the best value.
It should be emphasized that both types of uncertainty will almost always be present in any measurement, but one type will usually be larger, and the larger type gives the true uncertainty in the measurement. A good way of seeing whether random errors might be important enough to provide appreciable uncertainty without going through a long calculation of the sm is to quickly test whether the measurement is reproducible. If a few independent measurements give the same answer, then the sm will most likely be very small compared with the least count of the measuring instrument.
Avoidable Errors
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 this 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 |  |
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 since the error (5°C) is greater than the uncertainty (1°C). 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 five degrees too high. Systematic errors can be very difficult to find; a true test of a good experimental scientist is his or her 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), i.e. a small error.
An exact measurement has neither avoidable nor unavoidable
error; that is, it is infinitely precise and infinitely accurate.
Prelab:
Print out and complete the
Prelab questions
and turn them in at the beginning of your first lab.