**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**

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.

**LABORATORY PROCEDURE**

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

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.

**THE APPARATUS**

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 pointsup 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*

**H. **Conclusions, including
a Statement of Reliability of the Results

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:

**A. Data**

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.

**B. Purpose**

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.

**D. Procedure**

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 r^{2}
= π
(2.186 mm)^{2}
= 15.01 mm^{2}

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**.

**F. Graphs**

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:

Your __name__

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.

**G. Results**

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.

**H. Conclusion**

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?

**SIGNIFICANT FIGURES**

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 **FormatÖCells
**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 10^{5}would
indicate one significant figure, while 2.65 x 10^{5}
would indicate three significant figures. 2.00 x 10^{5}
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, 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:

**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 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:

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.

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.

**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 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 *exact*measurement
has neither avoidable nor unavoidable error; that is, it is infinitely
precise and infinitely accurate.