Name:      Date:

College of St. Catherine

Dept. of Physics

Dr. Terrence F. Flower

Revised 1995

Converted to HTML 1997

ACKNOWLEDGEMENTS

Special thanks is in order to the many who have made this information useable and up to date. Many students provided important suggestions to improve and alter the laboratory experiences. Ms. Janice Mosley did much of the original word processing and provided many helpful hints on organization and layout. Additionally, the College of St. Catherine Alumnae Association supported work on this manual through the Sr. Marie Ursula Award, once in 1987 for the original compilation and again in 1994 to allow computer upgrades to be included.

Several others provided direct information to make this more complete. Estes Industries of Penrose, Colorado granted us permission to reprint rocketry information. Mr. Grant Hermanson contributed many helpful ideas and did much of the actual hypertext conversion. Many of the errors which might be found herein are attributable to the main author himself.

Thanks in advance to the students who will use it this year and help to make more helpful changes for future classes. Have a good academic year!

TABLE of CONTENTS

Acknowledgements

The Laboratory

Laboratory Procedures

Unity Constants

Computer Techniques

Appendix A - Laboratory Safety

Appendix B - Handling of Radioactive Materials

Appendix C - Calculator Use

Appendix D - Rocketry Data

Bibliography

THE LABORATORY - Activity Oriented

There are two major reasons that the laboratory section is part of a traditional introductory physics course. One is to reenforce concepts and ideas studied in the lecture and secondly to put into practice the processes of science itself. It is not a place where Nobel Prize experiments are originally performed (although we do have the opportunity to replicate, in essence, some such activities as learning exercises, primarily for the sake of the first reason above.) Rather, we can concentrate on the processes of science that are globally applicable to all of the sciences.

In 1958 Thomas Kuhn published his seminal work on The Structure of Scientific Revolutions. The basic premise of his work suggested that practicing scientists use existing paradigms or models that best explain the world. When anomalies in these paradigms result in crisis a new paradigm is ultimately developed. This suggests that the body of knowledge we call science is not absolute, but rather depends upon the consensus of the practitioners of science. The progress we have seen most lately in science and technology would suggest otherwise - that we are indeed moving closer and closer to a comprehensive view of the world that is correct. Scientists today are trained by immersion. Graduate students at the universities are placed into doing science. That is the key word - doing! And that is what we practice in the laboratory, doing science.

It matters not so much what we do in the laboratory but more so how we do it, i.e. the process by which we do it. We will, for the sake of reenforcing the concepts in the lecture, correlate activities as closely as is possible, with the content material. This is accomplished readily with the blended, integrated laboratory and lecture. In fact, in our course we call the traditional lecture the Content while the traditional laboratory learning is called Activity. Rather than meeting a specific separate time for the activity or lab, (which sometimes could be a week later) we do what is appropriate when it comes up. In other words, we are engaged in a continuous interactive learning experience. The content is accomplished interactively in small collaborative groups. Lectures are not the normal mode of learning. The activity, or lab part will still concentrate on PROCESS. The processes of observing, collecting data, analyzing data and testing our models is applicable to all of the disparate branches of science. The computer will be the interactive tool/device that will make this possible.

The processes of science are viewed by many as the most important aspect of the discipline. These are best developed in the laboratory, but your success will depend in large part on your attitude in the lab. The following describes the objectives, the lab procedures, data collection and analysis, and the lab report:

Bloom's taxonomy of educational objectives includes three domains: the cognitive, the affective, and the psychomotor.

OBJECTIVES:

In the realm of Cognitive Knowledge the goals are:

1. To supplement factual knowledge from the lecture by hands-on manipulation of apparatus and application of ideas discussed in class.
2. To develop the processes of science, including observation, measurement, classification, communication, hypothesizing, predicting, and data analysis.
3. To develop data analysis skills, including methods of recording data, manipulating data, interpreting and reporting results.

In the Affective Domain the goals are:

1. To understand the attitude and method of science.
2. To develop an appreciation for and a values system concerning the physical sciences.
3. To develop character and a sense of social responsibility.

In the Psychomotor Domain the objective is:

1. To develop the physical skill to operate instruments and manipulate apparatus.

LABORATORY PROCEDURES

Normally, the lab handout or summary describes the procedure, appropriate theory, and necessary apparatus for each investigation. The student is expected to come prepared for the lab session, having read the experiment guide and necessary sections of the textbook. By doing this, maximum benefit can be gained from the laboratory experience.

APPARATUS:

Most of the equipment found in the physics laboratory is delicate and expensive. Please use care in handling the apparatus and report malfunctions immediately. At times equipment will have to be shared. It is expected that students will cooperate in the use of equipment. When the experiment is complete, please leave the apparatus as it was or place it in storage as directed by your instructor.

THE LABORATORY REPORT:

Good reports are easy to construct. It is suggested that each student collaborative group purchase a standard size three-ring loose leaf binder to be used for the activities only. This will allow for easy insertion of the activity prinouts and additions for each activity session, replacement of pages, and the addition of the appropriate kind of graph paper where needed. It is necessary to use care in the handling of the notebook since pages can be lost readily. A notebook with a pocket in the cover will permit the placement of a computer diskette which can be used in the course. You may wish to purchase a diskette for use in the class. Diskettes can also be lost so a zippered pocket is adviseable.

The Experiment Guide will normally include a Title, Objectives, Theory, Description of Apparatus, and Procedures. The student is expected to insert this into the notebook as part of the lab, and then record data analysis. Be neat in recording data and in the write-up. All write-ups will be completed on the Word Processor available. No hand written reports will be accepted. Do not plan to record data and copy it later more neatly. Many mistakes occur this way. If mistakes are made in recording data, do not be alarmed; simply draw a line through the error and write the correct figure on the next line. Record your data in tabular form rather than in rows. Above all, be clear and neat!

Limitations to the experiment should be included at this point. Discuss problems with the equipment, yourself, etc. This should be the first part of the formal report, assuming the above directions, objectives, etc. are given.

Data Analysis is next. This should include tabulation and recording of data and appropriate calculations, graphs, computer printouts, and a summary of results with uncertainties and % error and/or % uncertainty as described later in this handout. This is the place to discuss the experiment, make comments, etc. Carry on a dialogue with the reader.

After a complete analysis of this experiment, Summary and Conclusions are necessary. This section should include a summary of the problems and successes encountered in performing the experiment as well as an explanation of what the results mean. Discuss difficulties and recommendations to the instructor concerning improvement of the experiment for future use. Be frank. State your individual ideas, not someone else's. Report your results honestly. If they disagree with another's results or with those expected, don't worry. You are graded on what you did and how you did it. It could be that your result is valid.

Please conclude with a final paragraph or Recommendations to help us make this manual better next year. Thanks.

The report is due at the lecture session on Monday following the experiment. This is necessary so that the reports may be graded and returned at the earliest time.

MEASUREMENT TECHNIQUES AND DATA ANALYSIS:

In a structured laboratory experiment frequently the theory has been studied earlier and the expected results are known. Many times, however, the outcome of an experiment is only suspected. It is necessary in either situation to make careful measurements and record the data properly so that deviations from expected results are small and predictions can be made from the conclusions.

The physical measurements made in an experiment are never exact. They are limited by two factors called precision and accuracy. Precision refers to the number of significant figures in a measurement. Accuracy is concerned with closeness of the measurement to the true value. Consider the example of a marksman firing a rifle at a target. Precision would determine the consistency of the rifle, whether it would shoot in the same spot repeatedly. Accuracy refers rather to the closeness of the shot to the center of the bullseye. A value may be very precise (many decimal places) but be highly inaccurate.

Every measurement involves reading some kind of scale, whether it be a meter on an electronic instrument or a mark on some kind of physical apparatus. The fineness of the graduations on the scale determines the precision or how many significant figures are reported. In every case, however, the final digit must be estimated. This estimation is significant in that it gives some meaningful information about the quantity being measured. Only one estimated digit should be read. It should be a reasonable value.

There is a degree of uncertainty in estimating the last digit. Suppose the sides of a block of wood are to be measured in order to compute its volume. Consider the following five observations. The more times measurements are taken, the closer the average is to the correct value. The results are shown in Table 1.

Observation Number	Length Cm	Width Cm	Depth Cm
1	10.05	6.23	4.01
2	9.96	6.29	4.05
3	10.10	6.19	3.99
4	10.03	6.25	4.03
5	10.12	6.27	4.04
Average Cm	10.05	6.25	4.02

Table 1: Individual Measurements of a Block

None of the measurements was exactly the same as another. Which value is best to use in computing the volume? Statistically, the value that has the highest probability of being correct is the Arithmetic Mean, or the Average.

(1-1)

where i is the observation number, N the total number of observations and X the value recorded on the ith observation.

How accurate are these? Or, what is the degree of uncertainty in these measurements? A theorem in mathematics suggests a method to determine the degree of uncertainty. It is called the average deviation, (a.d.). It gives a range of accuracy for any individual measurement and is found by adding up the absolute value of the difference between each measurement and the Mean, and dividing by the total number of observations.

di =| xi - x |

(1-2)

and

a.d. = di / N

(1-3)

But it is known in statistics that the arithmetic mean computed from N equally reliable measurements is more accurate than any individual measurement by a factor of N. Thus the Average Deviation of the Mean, A.D. is:

A.D. = a.d./N

(1-4)

For the length of the block of wood:

Observation Number	Length CM	DeviationCM
1	10.05	0.00
2	9.96	0.09
3	10.10	0.05
4	10.03	0.02
5	10.12	0.07
Average	10.05 Cm	di=0.23

a.d. = .23/5 = .05

(1-5)

A.D. = a.d./N = .05/5 = .02

(1-6)

Thus the best value to report for the data is:

10.05 + 0.02 cm

All scientific values are reported this way, even the most fundamental constants such as the speed of light.

Let us return now to our original problem of determining the volume of the block of wood. We have three values (length, width, and depth) each with some uncertainty:

Length = 10.05 ± 0.02 cm

Width = 6.25 ± 0.02 cm

Depth = 4.02 ± 0.01 cm

Consider the block: We know its dimensions within some limits.

Figure 1.1: Uncertainty in Volume of a Box

The volume could be as little as 10.03 x 6.26 x 4.01 cm or as large as 10.07 x 6.26 x 4.03 cm (the mean values plus or minus their uncertainties).

Thus we need some rules for mathematical manipulation of values with uncertainties. Consider the simple case where some quantity F is determined by a single variable x (which is the quantity measured) then:

dF/dx = F/x

(1-7)

where F is the uncertainty in F as X varies from X to X+ X. If the X is small enough (which it can be made so by many measurements), then

dF/dx f/x

(1-8)

or

F df/dx x

(1-9)

It is necessary to know the functional relationship of F; i.e., F as a function of X, so that its deviation may be calculated.

For example, suppose we had a cube for which we measured the side as

x = 5.00 ± 0.01 cm

The volume is then merely

The volume is then merely

F = X3

where	dF/dx = 3X2
Then	F dF/dx x
and	F 3x2x
or	F 3 (5.00)2 (.01)
so that	F 0.75 cm3
while	F = x3 = 125.00 cm3
Then you would report	F = 125.00 ± 0.75 cm

If F depends on several variables x,y,z,... we need multivariable calculus to find F. By the chain rule:

F = F/x x + F/y y + F/z z + . . .

(1-10)

where the partial derivatives F/x , F/y ... are merely simple derivatives of the function F with respect to one variable at a time. To eliminate problems with signs, it can be shown that:

(F)2 = (F/x x)2 + (F/y y)2 + (F/z z)2 + . . .

(1-11)

F = (F/x x)2 + (F/y y)2 + (F/z z)2 + . . .

(1-12)

Then, for our block of wood, the volume can be found:

or

F = .64

The result is reported as :

Volume = 251.25 ± .64 cm

This may seem quite involved, but it is usually only done once at the end when final results are calculated.

Percent Uncertainties can also be calculated:

% Uncertainty = F / F x 100%

(1-13)

If a result is known for your experiment and you wish to compare your answer with the known or accepted result:

% Error = ( | True Value - Experimental Value | / True Value ) x 100%

(1-14)

GRAPHING

Graphs are clear ways to represent data collected in an experiment, or merely in the calibration of equipment to perform experiments. A graph has two variables, one dependent on the other. The independent variable is usually plotted on the x-axis (abscissa) and the dependent variable on the y-axis (ordinate).

Graphs can be used to make predictions for missing values or verify expected results. They clearly show relationships between variables.

It is expected that experiments use graphs whenever possible following tables of data collected. The axes should be clearly labeled indicating appropriate units. This is a necessary part of data analysis.

In this course you will use several different types of graph paper:

LINEAR

SEMI-LOG

LOG-LOG

Use of these particular types of paper will be explained by your instructor during the course of labs.

Choose a scale that is appropriate to the data. For example, if you are measuring distances in the range of 200 to 400 metres, do not start at zero and extend the axis to 1000 metres. You want to include as much detail as possible in your graph.

Label the axes properly. This should include the quantity, such as time, and the appropriate units, such as microseconds.

It is proper to encircle your data points with a small circle. Draw your graph line up to, but not through the circle's edge. This way the data point is unobstructed.

Circled points are experimental or measured points. Uncircled points are points used to calculate the slope. They are not experimental points. Note: The line does not necessarily go through all points.

The slope m = y / x =(y₂ - y₁) / (x₂ - x₁)

Figure 1.2: Calculating the Slope of a Line

Many times the graph gives an adequate picture of the relationship between two variables measured in the lab. Frequently, you will be able to detect an empirical relationship just by looking at the graph. An equation is the most powerful and succinct form of expressing an observed phenomena.

If the relationship is linear (straight-line) finding the empirical equation is a relatively easy task. Much of the time, even is the line wanders (so to speak), at least for some segments it is straight. So being able to describe a linear relationship is important. An algebraic technique called linear regression can be used to give the slope and y-intercept of the line. This gives the straight line that best fits your data. The researcher then can determine if there is indeed a cause and effect relationship between two variables and just how they are connected. To avoid time-consuming calculations herein, your instructor will show you perform this on the computer using EXCEL.

The results of a set of paired data points (independent variable and dependent variable) yields a linear equation of the form:

y = mx + yo

(1-15)

The computer printout will also give a correlation coefficient, indicating how well the x value (independent variable) predicts the y-value (dependent variable) with a straight line. Values range from -1 to +1. A value close to +1 or -1 indicates strong correlation and a value near zero indicates poor correlation or a very wide scattering of values.

LOGARITHMS

Recall that a logarithm to the base a of a number x is the exponent of y to which the base a must be raised.

In other words:

x = ay

(1-16)

Then we write

y = loga x

(1-17)

The algebraic properties of logarithms should be reviewed by the student. However, some of the necessary results are:

loga AB = loga A + loga B

(1-18)

loga A/B = loga A - loga B

(1-19)

loga An = n loga A

(1-20)

SEMI-LOG PLOTS

In this course we will encounter several situations where the dependent variable follows an exponential decay. For example, the charge or voltage on a capacitor during discharge and the decay of radioactive materials do this. The voltage across the capacitor in an R-C circuit during decay can be written as:

V(t) = Vo e-t/

(1-21)

where Vo is the initial voltage just prior to discharge and is the time constant, RC. At t = 0, V(t) = Vo. By the time t = 7 (seven time constants) the voltage has decayed to:

V(t = 7 ) = Voe-7 = .000912 Vo 10-3 Vo

(1-22)

When is of the order of milliseconds, this is a short time to decay to 1/1000 of the starting value; i.e., it has decayed three orders of magnitude! While complete analysis of this problem will be performed in the laboratory, we can get a plot of V(t) on linear graph paper or with EXCEL:

Figure 1.3: Voltage as a function of time during discharge of an R-C circuit.

Obviously this is not helpful, so consider that according to equation (1-19), if we take the logarithm of both sides:

ln V(t) = ln Vo - 1/ t

(1-23)

where ln V(t) is the dependent variable and t is the independent variable. Then, ln V_o is the y-intercept and - 1/ is the slope of the curve. Note that the negative sign can be interpreted as consistent with decay, decreasing voltage with time, or negative slope. We could plot this on linear graph paper or use EXCEL. See Figure 1.4.

Figure 1.4: Voltage as a function of time on linear graph paper.

However, this result is cumbersome since we really don't care about the logarithm of the voltage. Semi-log graph paper overcomes this problem by "automatically" scaling the dependent variable by its logarithm. We can then directly plot (and read) the voltage from the chart. Still, the slope and y-intercepts are the same. Since, in this case, we are traversing three orders of magnitude, we use graph paper with three repeating scales. This paper is normally available in two or three scales. Choose the one appropriate to your data.

Figure 1.5: Voltage as a function of time

LOG-LOG PLOTS

Sometimes we have a situation where both the dependent and independent variables vary by several orders of magnitude. Trajectories in a gravitational field are written in simple terms as:

d = 1/2 at2

(1-24)

From our backgrounds in algebra we recognize this to be a parabolic curve on linear graph paper. Suppose we were to take the logarithm of both sides again:

log d = log (1/2 at2 )

(1-25)

log d = log 1/2 + log a + 2 log t

(1-26)

The log 1/2 and log a result in a constant value. Again this constant can represent the y-intercept in a linear equation. Log d is the dependent variable and log t is the independent variable. The coefficient of log t is +2. This is the slope of the line.

See Figure 1.6 for a plot of this on a log-log graph. This should also be plotted on linear paper, since logarithms are not helpful in this case. The real values can be more meaningful. Again, EXCEL makes it easier.

SUMMARY and CONCLUSIONS

The lab report summarizes your experiences in the laboratory. It is critical that the analysis be completed properly. This includes both calculations and graphing.

Other techniques of graphing and analysis will be presented by your instructor in the laboratory. Good luck and have fun!!!

Figure 1.6:Gravitational Force as a Function of Distance between masses.

UNITS

In these discussions and in the following laboratory activities, the International System of units (SI) is used throughout, both for theoretical discussions and for actual measurements. In this system there are four basic units: the metre for length, the kilogram for mass, the second for time, and the ampere for electric current. All other mechanical and electrical quantities are derived and expressed in terms of these four units and are given in the table of Units.

Often it is convenient to use units related to these basic units by some power of 10. For example, we may measure length in metres, kilometres (10³ metres), centimetres (10^-2 metres), millimetres (10^-3 metres), microns (10^-6 metres), or angstroms (10^-10 metres), depending on the scale of the corresponding physical situation. Ordinarily, with a few exceptions, related units are indicated by attaching a prefix to the basic unit. For example, kilo always means 10³ , and 1 kilometre = 10³ metres. The prefixes in common use, with some examples of each are given in the table of Unit Prefixes.

UNIT PREFIXES

Power of ten	Prefix	Abbreviation	Examples
1012	tera-	T
109	giga-	G	gigahertz (GHz)
106	mega-	M	megahertz (MHz)
			megohm (M )
			megawatt (MW)
103	kilo-	k	kilovolt (kV)
			kilowatt (kW)
10-2	centi-	c	centimetre (cm)
10-3	milli-	m	milliampere (mA)
10-6	micro-	u	microvolt (uV)
10-9	nano-	n	nanosecond (nsec)
10-12	pico-	p	picofarad (pF)

UNITS

Physical Quantity	SI
length	metre (m)
mass	kilogram (kg)
time	second (sec)
force	newton (N) = kg-m/sec
energy	joule (J) = N-m
power	watt (W) = J/sec
electric charge	coulomb (C)
electric current	ampere (A) = C/sec
electric potential	volt (V) = J/C
electric field	volt/metre or newton/coulomb
magnetic field (B)	Tesla or Webers/meter
resistance	ohm ( ) = volt/ampere
capacitance	farad (F) = coulomb/volt
inductance	henry (H) = volt-sec/ampere

CONSTANTS

A list of physical constants which may be needed for your laboratory work is given in the table of Fundamental Physical Constants. The fundamental constants are given in SI units. In practical calculations, other units such as electron volts or atomic mass units are sometimes more convenient to use than the basic SI units. A few commonly used conversion factors are also given.

Also check out Yahoo's page of scientific constants, especially the NIST Fundamental Physical Constants page

FUNDAMENTAL PHYSICAL CONSTANTS

Name	Symbol	Value
Speed of light	c	2.998 x 108 m/sec
Charge of electron	e	1.602 x 10-19 coul
Mass of electron	m	9.109 x 10-31 kg
Mass of neutron	m	1.675 x 10-27 kg
Mass of proton	m	1.672 x 10-27 kg
Planck's constant	h	6.626 x 1023 J-sec
	h = h/2	1.054 x 1023 j-sec
Permittivity of free space		8.854 x 10-12 F/m
	1/4	8.988 x 109 m/F
Permeability of free space	u	4 x 10-7 Wb/amp-m
Boltzmann's constant	k	1.380 x 10-23 J/K
Gas constant	R	8.314 J/mole-K
Avogadro's number	N	6.023 x 1023 molecules/mole
Mechanical equivalent of heat	J	4.186 J/cal
Gravitational constant	G	6.67 x 10-11 N-m/Kg2

OTHER USEFUL CONSTANTS

Name	Symbol	Value
Planck's constant	h	4.136 x 10-15 eV sec
Boltzmann's constant	k	8.617 x 10-5 eV/K
Coulomb constant	e /4	14.42 ev A
Electron rest energy	mec2	0.5110 MeV
Proton rest energy	Mpc2	938.3 MeV
Energy equivalent of 1 amu	Mc2	931.5 MeV
Electron magnetic moment	u = eh/2me	0.9273 x 10-24 J-m/Wb
Bohr radius	a = 4 h/me	0.5292 x 10-11 m
Electron Compton wavelength	= h/mc	2.426 x 10- m
Fine-structure constant	= e/4 hc	1/137.0
Classical electron radius	r = e/4 mc	2.818 x 10-10 m
Rydberg constant	R	1.097 x 107 m

CONVERSION FACTORS

1 eV = 1.602 x 10^-19 joule

1 A = 10^-10 m

1 amu = 1.661 x 10^-27 kg = 931.5 MeV

COMPUTER TECHNIQUES

The Department of Physics currently has installed in the laboratory a terminal connected to the campus VAX 11-780 computer system located at the Computing Center in the Library. This computer system is accessible at any time during night and day from any terminal on campus or even from off campus via modem. Additionally, microcomputer systems are currently in place or being installed in Mendel Hall for your use in the various science and math departments. The following information applies to the VAX computer available to all.

You will have to secure a computer account. All students registered for this course are eligible for an account even if you are a visiting student from another campus. You will need to contact the secretary in the computing center. An account and password will be assigned as soon as possible.

You may wish to take one of the many short courses on the use of the VAX. These courses are offered at frequent times at no charge through the Computing Center. The day and time of the next course is posted in the Computing Center. Several programs have been written exclusively for this course. They include:

STATS

This is a statistical program designed to calculate Mean, Average Deviation, and Average Deviation of the Mean. These three quantities have been explained earlier for Data Analysis. The user merely inputs the number of data points and the computer calculates the appropriate quantities directly. Additionally, it prints out the data values so the user can check her/his work.

LINREG

This statistical program calculates a linear regression best fit to pairs of data values. In this course we will concentrate mostly on experiments that fit straight lines. Higher order data sets follow naturally and are available in many standard statistical packages. In addition to giving an Equation of Best Fit, it also gives the Means, Standard Deviations, for each set of data and the Correlation Coefficient between the pairs. This quantity tells us how "straight" the straight line is.

RSROCK

This program allows the user to calculate and simulate a rocket trajectory based on initial values. The user must input thrust as a function of time (in Newtons, every .02 seconds,) the engine mass which includes fuel, fuel mass, parachute mass, rocket mass, wadding mass, aerodynamic drag coefficient, etc. It then plots the entire trajectory.

These programs are in the DEPT:PHYSICS account and are accessible to all users. To run them, merely type in, at the prompt character $:

$ RUN CSC$DEPT:[PHYSICS.PHYSIC]PROGRAM NAME

These programs were written so that input must be in capital letters, so it is wise to depress the CAP LOCK when beginning. The output to the printer is performed in serial with the terminal. The terminal in the Physics Lab is connected this way, but not all terminals and printers on campus are connected this way. If you have difficulty, check with your instructor or computer consultant.

St. Kate's Home Page	Physics Home Page