York University
Department of Physics and Astronomy
PHYS 1011/1411/1421 3.0
Laboratory Manual
Fall 2025
Contents
0 Introductory Information
0.1 Why do Laboratory Work? . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.2 General Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.2.1 Lab Schedules and Attendance . . . . . . . . . . . . . . . . . . . . .
0.2.2 Prelab Preparation and Reports . . . . . . . . . . . . . . . . . . . .
0.2.3 Lab Marks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.2.4 Lab Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.2.5 Academic Honesty . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.2.6 Cleanliness and Care of Equipment . . . . . . . . . . . . . . . . . . .
0.2.7 Lab Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.3 Measurement and Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . .
0.3.1 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.3.2 Uncertainties (Errors) . . . . . . . . . . . . . . . . . . . . . . . . . .
0.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.4.1 Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.4.2 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.5 Fitting a Straight Line to the Data . . . . . . . . . . . . . . . . . . . . . . .
0.5.1 Estimate “by eye” . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.5.2 The Method of Least Squares . . . . . . . . . . . . . . . . . . . . . .
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1 Data Analysis using Logger Pro
1.1 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Part I: Installing Logger Pro on your computer . . . . . . . . . . . . . . . .
1.2.1 Logger Pro 3.16.2 Downloads . . . . . . . . . . . . . . . . . . . . . .
1.3 Part II: Entering Data and Creating a Graph with Error Bars . . . . . . . .
1.3.1 Adding a Calculated Column . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Adding and Graphically Displaying Uncertainties . . . . . . . . . . .
1.4 Part III: Using the fitting tool in Logger Pro . . . . . . . . . . . . . . . . .
1.4.1 Questions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Explore Logger Pro Capabilities . . . . . . . . . . . . . . . . . . . . . . . .
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2 Uniformly Accelerated Linear Motion
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Experimental Setup and Measurements . . . . . . . . . . . . . . . . . . . .
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I
CONTENTS
2.2.1 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calculating the Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Questions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measurement of g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Questions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Thrust and Friction
3.1 Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Setup and Explore . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Quantify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Questions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Measuring Static and Kinetic Coefficients of Friction . . . . . . . . .
3.2.3 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Data Collection Procedure . . . . . . . . . . . . . . . . . . . . . . . .
3.2.6 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.7 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Oscillatory Motion and Conservation of Energy
4.1 Spring-Mass System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Measurements and Calculations . . . . . . . . . . . . . . . . . . . . .
4.1.3 Questions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Setup and Data Collection . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Questions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Angular Motion
5.1 Centripetal Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 About the Experimental Setup . . . . . . . . . . . . . . . . . . . . .
5.1.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4 Questions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Rotational Inertia (Moment of Inertia) . . . . . . . . . . . . . . . . . . . . .
5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Measurement and Calculations . . . . . . . . . . . . . . . . . . . . .
5.2.4 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.5 Questions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.4
II
CONTENTS
6 Standing Waves
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Standing Waves in a Rope and a Spring . . . . . . . . . . . . . . . . . . . .
6.2.1 Transverse Standing waves in a rope . . . . . . . . . . . . . . . . . .
6.2.2 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.3 Longitudinal Standing Waves in a Spring . . . . . . . . . . . . . . .
6.2.4 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Sound standing waves in the air tube . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Measurements and Calculations . . . . . . . . . . . . . . . . . . . . .
6.3.3 Questions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A Sample Lab Report
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B Statistical Treatment of Random Error
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C Method of Least Squares
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D Uncertainty Calculations using Differentials
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III
Chapter 0
Introductory Information
0.1
Why do Laboratory Work?
“It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you
are. If it doesn’t agree with experiment, it’s wrong.” – Richard Feynman
Physics is driven by experiment. The Physical Laws of Nature are mathematical constructs
derived to describe the results of observations. This mathematical description of physical
phenomena (usually referred to as “theory”) needs to not only fit the data but also to make
predictions that can be tested with new experiments and new observations. And the cycle
continues between new observations and new theoretical models. So it is crucial for any
physicist to understand the methodology and limitations of experimentation as it is the
bedrock of physics.
This first laboratory course is an introduction to experimental research. Sometimes a lab
will probe the accuracy of a particular equation presented in the lectures while other times
the laboratory will be a medium for teaching some new material which will not be covered
in the lecture course. The key element is to train your brain, eyes, and hands in good
experimental techniques and data analysis (particularly error analysis), while familiarizing
yourself with some of the instruments used in experimental science.
The key element of experimental research is precision, both in the sense of minimizing the
uncertainty in a measured quantity as well as designing experiments to observe an effect for
the first time (which is, in effect, improving the precision to the point where the signal can
be separated from the inherent experimental backgrounds, i.e., the noise). For example, say
you had two experimental groups, A and B, which were measuring the acceleration due to
gravity, g, for the first time. The results of their experiments were gA = 9.82 ± 0.08 m/s2
and gB = 9.79 ± 0.04 m/s2 . Even though subsequent, more refined measurements showed
that g = 9.81 m/s2 and so experiment A found a value closer to the “correct” value, it is
still true that experiment B was better since it was more precise. It was a bit of a statistical
fluke that group A found a result closer to the actual value. Physics researchers spend the
bulk of their time examining and minimizing the uncertainties in their measurements since,
by definition, when doing research you don’t know a priori what is the “correct” answer –
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Introductory Information
that is what you are trying to find! This is all to say that in doing these labs, you must
always keep in mind the uncertainties (generally called “errors” although this is a bit of a
misnomer) in the measurements, and improvements to a lab are almost always referring to
changes that would reduce these uncertainties.
0.2
General Information
You will need:
• the lab manual (to be posted on eClass)
• the usual writing materials for personal notes (writing utensils, paper, ruler)
• a scientific calculator (non-programmable)
0.2.1
Lab Schedules and Attendance
The laboratories are located in 102C and 102D Bethune College.
During registration you selected a particular 3-hour laboratory time. If you would like to
change the section you are registered in, this can ONLY be done through the York online
registration system and must be done before September 16, when the first in-person labs
start. Students are required to attend all laboratory sessions to which they are assigned.
Attendance will be monitored by the demonstrator. A schedule for your laboratories
will be posted on the Labs eClass site. Lab section 99 is reserved for those who have already
successfully completed the labs within the past two years during a previous attempt at
completing a first-year physics course at York University. You must complete the online
form in eClass to be allowed to enroll in lab section 99.
0.2.2
Prelab Preparation and Reports
From the posted schedule you will know which experiment you will be doing on which date.
Before coming to do the experiment, you are expected to read the appropriate
section of the manual. Be sure you understand the theory involved, consult your textbook, and plan your practical work. A set of prelab exercises for each lab will be posted
on eClass and must be completed and submitted via eClass before you come to the lab.
This preparation is most important as it is unlikely that you will be able to finish the
experiments satisfactorily or learn from them if you do not prepare beforehand. There may
be short, unannounced quizzes on the experiment during some labs.
We do not require you to write an elaborate report for each experiment1 . A laptop is provided as part of the experimental setup to complete your lab report. You will be using
Word to complete your lab report and Logger Pro to collect and analyze data. The report should include your name, title and date. The experimental data, whenever possible,
should be summarized in the form of a table, with title, column headings, units and experimental uncertainties. Graphs should have titles, axes labeled and units included. Error
1
A sample lab report is included in the manual (Appendix A).
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Introductory Information
bars on graphs should be used to indicate measured quantities with known uncertainties.
Calculations should be shown and organized in a logical way, with short comments and explanations. Just formulas with substituted data are not acceptable, ensure you include all
calculation steps. Calculations of uncertainties is an important part of the lab report. The
next section in the manual provides more information regarding uncertainty calculations
and rounding of final result and its uncertainty.
You are encouraged to record in your report for future reference any comments regarding
the theory or method or apparatus which enhance your understanding. Your report should
resemble a research scientist’s day-to-day experimental log rather than a polished scientific
paper.
If you come to the lab properly prepared as outlined above, the three-hour session should be
sufficient for the taking of measurements, conducting calculations and completion of your
laboratory report. Be punctual - latecomers will find it difficult to complete the assignment.
All lab reports, finished or unfinished, must be submitted to eClass by the end
of the three-hour lab session.
Your report will be marked by the assigned demonstrator for your lab section.
0.2.3
Lab Marks
The final lab mark will contribute 10-20% (the exact percentage depends on the course) to
your total grade. It will take into consideration prelab questions and quizzes, and the lab
reports. All your lab marks will be posted on the lab eClass site for you to check. You
must pass the lab portion to pass the course.
0.2.4
Lab Layout
The laboratory rooms (BC 102C and BC 102D) are setup up with six group work stations.
At each work station the experimental equipment will be setup and four students will be
randomly assigned per station. Each student will be provided with a laptop to complete their
lab report. During most experiments you will work in pairs and will share the apparatus
and data collection. However, each student will be conducting their own data analysis and
writing their lab report individually and submitting via eClass.
0.2.5
Academic Honesty
Students will certainly discuss and talk about their studies with their friends and this can
be very useful; but any work that you hand in must have been done by yourself. This is
the only way to test your own competence and to prepare yourself for positions of responsibility after graduation. If scientists are dishonest, they are useless. THE UNIVERSITY
CONSIDERS ALL FORMS OF COPYING AND CHEATING TO BE SERIOUS OFFENCES. For information on York University’s academic honesty policy please
read the Academic Honesty Policy.
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Introductory Information
0.2.6
Cleanliness and Care of Equipment
We do not charge you for accidental breakages, but please report them to the demonstrator
or lab technologist immediately, so that equipment can be replaced or repaired.
Students must leave their place of work in the lab neat with all the apparatus complete.
Each experimental set-up will be used by approximately forty students before it is retired
for the year, so leave it for the next student in the state in which you would like to find
it.
When a student submits their report and is ready to leave, the demonstrator will check
their place of work to see that it is left in satisfactory condition. A mark deduction
will be applied to your in-lab performance grade if the work space is left in a
unsatisfactory condition.
0.2.7
Lab Safety
Scientists very commonly live to a grand old age in spite of their daily encounters with
many hazards. The main reason for this is that a scientist doing an experiment is paying
very close attention to everything that happens, is expecting the unknown and can react
quickly to it. Your best protection against accidents in the lab is a constant thoughtful
alertness which never permits your actions to become mechanical and reflex.
Specific hazards which exist in particular experiments will be stressed in the respective lab
outline. Please pay very careful attention to these warnings and act accordingly.
Notify the demonstrator or lab technologist of any accident or injury no matter how insignificant it may seem.
In the case of a fire, at the sound of the fire alarm in the building, the university stipulates
that everyone must leave the building. In the case of a fire in the lab students and demonstrators must leave the building immediately. The demonstrator is responsible for taking
the appropriate action to scale the lab marks, if needed.
A 24-hour Emergency Services Telephone Centre operates on York Campus and
can be alerted by calling 33333 on all campus telephones or 416-736-5333 on
public or off-campus telephones.
Health services are located in York Lanes.
0.3
Measurement and Uncertainties
0.3.1
Measurements
There are several requirements that must be met if a measurement is to be useful in a
scientific experiment:
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Introductory Information
The Number of Determinations It is a fundamental law of laboratory work that a
single measurement is of little value because of the possibility of gross mistakes or smaller
random errors. Accordingly, it is customary to repeat all measurements as many times as
possible. The laws of statistics lead to the conclusion that the value having the highest
possibility of being correct is the arithmetic mean or average, obtained by dividing the sum
of the individual readings by the total number of observations. Because of time limitations,
we often suggest you do a small number of repetitive measurements but remember this
reduces the reliability and respectability of your results.
Zero Reading Every measurement is really a difference between two readings, although
for convenience, most instruments are calibrated so that one of these readings will be zero.
In many instruments, this zero is not exact and may shift slightly over time due to wear
or usage. Thus, it is essential that the zero reading be checked before every measurement
where it is one of the two readings. In some cases the zero can be reset manually, while in
others it is necessary to record the exact zero reading and correct all subsequent readings
accordingly. E.g., when measuring the length AB shown in Fig. 1, a ruler could be placed
(1) with 1.2 cm at A, then length AB = (4.0 - 1.2) cm = 2.8 cm. The more usual ruler
position (2) allows the length AB to be read as 2.8 cm directly, but remember this is still
the difference between two readings: 2.8 cm and 0.0 cm.
Figure 1: Example of measuring length
Accuracy Quantitative work requires that each measurement be made as accurately as
possible. The main units of a scale are usually divided, and the eye can easily subdivide
a small a distance of 1 mm into five parts reasonably accurately. Thus, if a linear scale
is divided into millimeters on a high quality ruler, a reading could be expressed to 0.2 of
a millimeter (examples would be: 4.6 mm and 27.42 cm, where 3/5 and 1/5 of a mm are
estimated by eye). In cases where the reading falls exactly on a scale division, the estimated
digit would be 0 (e.g., 48.50 cm) indicating that you know the reading more accurately than
48.5 cm. It would not be possible to take a reading with greater accuracy then 0.2 mm
with this equipment. If the scale is not finely engraved (the lab meter sticks, for example)
a lower accuracy of 0.5 mm would be possible.
The accuracy desired from a measurement dictates the choice of instrument. For example,
a distance of 4 m should not be measured by a car’s odometer, nor a distance of 2 km with
a micrometer. The student learns to decide which instrument is most appropriate for a
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Introductory Information
specific measurement. Ideally, all measurements for any one experiment should have about
the same percentage accuracy.
Significant Figures A significant figure is a digit which is reasonably trustworthy. One
and only one estimated or doubtful figure can be retained and regarded as significant in
any measurement, or in any calculation involving physical measurements. In the examples
presented in the Accuracy paragraph above, 4.6 mm has two significant figures, 27.42 cm
has four significant figures. Including extra digits is not acceptable as it incorrectly implies
a higher accuracy for the quantity being determined.
The location of the decimal point has no relation to the number of significant figures. The
reading 6.54 cm could be written as 65.4 mm or as 0.0654 m without changing the number
of significant figures - three in each case.
The presence of a zero is sometimes troublesome. If it is used merely to indicate the location
of the decimal point (as in 0.0654 m), it is not called a significant figure, but if it is between
two significant digits (as in a temperature reading of 20.5° C) it is always significant. A zero
digit at the end of a number tends to be ambiguous. In the absence of specific information
we cannot tell whether it is there because it is the best estimate or merely to locate the
decimal point. In such cases the true situation should be expressed by writing the correct
number of significant figures multiplied by the power of 10. Thus a measurement of the
speed of a subatomic particle of 128,000 m/s is best written as 1.28 × 105 m/s to indicate
that there are only three significant figures. The latter form is called standard notation,
and involves a number between 1 and 10 multiplied by the appropriate power of 10. It is
equally important to include the zero at the end of a number if it is significant. If reading
a meter-stick, one estimates to a fraction of a millimeter, then a reading of 20.00 cm is
written quite correctly. In such a case, valuable information would be thrown away if the
reading were recorded as 20 cm. The recorded number should always express the degree of
accuracy of the reading.
In computations involving measured quantities, carry only those digits which are significant.
Consider a rectangle whose sides are measured as 10.77 and 3.55 cm (the doubtful digits
have been highlighted here in bold). When these lengths are added to find the perimeter
the last digit in the answer will also be doubtful.
10.77
10.77
3.55
+3.55
28.64
When the lengths are multiplied to obtain the area, any operation by a doubtful digit results
in a doubtful digit.
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Introductory Information
10.77
×3.55
0.5385
5.385
+32.31
38.2335
In the result only one doubtful digit is retained and the area is 38.2 cm2 . When rounding
off a number with too many insignificant figures, use the usual rule of retaining the last
digit unchanged if the first dropped digit < 5, and increase it by 1 if the first dropped digit
≥ 5.
In the lab this year, most of your calculations will probably be limited to three significant
figures by the measuring devices. A calculator quickly produces about 8 mathematically significant figures, but, in your final answer, record only those figures that have physical
significance.
0.3.2
Uncertainties (Errors)
A measurement is only of value if it has attached to it quantitative limits within which
it is accurate - that is, its uncertainty. An uncertainty of 50% or even 100% is a vast
improvement over no knowledge at all: an accuracy of ±10% is a great improvement over
±50% and so on. In fact, much of science is directed toward reducing the uncertainties in
specific quantities of scientific interest.
The uncertainty in a reading or calculated value is technically called an error. The word
has this precise meaning in science and carries no implication of mistake or misjudgement.
This manual uses the words error and uncertainty interchangeably.
A Systematic Error is one which always produces an error of the same sign. That is, all
data points are systematically high or low. Systematic errors may be sub-divided into three
groups: instrumental, personal, and external. Some systematic errors can be corrected for
if they are due to known instrumental malfunction (e.g., a timer is known to not start at
0.0 s but at 0.2 s) ... or personal malfunction (e.g., you need glasses!).
An Instrumental Error is caused by faulty or inaccurate apparatus; for example, an
undetected zero error in a scale or an incorrectly adjusted clock. If 0.2 mm has been worn
off the end of a ruler, all readings will be 0.02 cm too high.
Personal Errors are due to some peculiarity or bias of the observer. Probably the most
common source of personal error is the tendency to assume that the first reading taken is
correct. A scientist must constantly be on guard against any bias of this nature and make
each measurement as if it were completely isolated from all previous experience. Other
personal errors may be due to fatigue, the position of the eye relative to a scale, etc.
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Introductory Information
External Errors are caused by external conditions such as wind, temperature, humidity,
vibration, etc. Examples include the expansion of a scale as the temperature rises or the
swelling of a meter stick as humidity increases.
Random Errors (often referred to as statistical errors) are a reflection of the randomness
of the measurement process and so observations above and below the “true” value are
equally probable. Because random errors are subject to the laws of chance, their effect in
the experiment may be lessened by taking a large number of observations. A simplified
statistical treatment of random errors is described in Appendix B of this manual.
The Error Interval If it is not practical or possible to repeat a measurement many
times, the errors in measurement must be estimated differently.
Since the last digit recorded for a reading is only an estimation, there is some possibility of
error in this digit due to the instrument itself and the judgement of the observer. Hence,
the best that can be done is to assign some limits within which the observer believes the
reading to be accurate.
A reading of 6.540 cm might imply that it lay between 6.538 and 6.542 cm. The reading
would then be recorded as (6.540 ± 0.002 cm). The scales on most instruments are as
finely divided by the manufacturer as it is practical to read. Hence, the error interval will
probably be some fraction of the smallest readable division on the instrument; it might
be 0.5 of a division, or perhaps 0.2 of a division. The error interval is a property of the
instrument and the user, and will remain the same for all readings taken provided the scale
is linear.
Remember that measurement of a quantity (such as length) also involves a zero reading, so
the error in the quantity will be twice the reading error. Note that it is essential to
quote an error with every set of measurements.
Absolute Uncertainty The estimation of an error interval gives what is called an “absolute” uncertainty. It has the same units as the measurement itself; e.g., (6.540 ± 0.002) cm.
The absolute uncertainty 0.002 cm is recorded to one significant figure, which in turn,
defines the last significant digit in a measurement.
When using a ruler, micrometer screw, or Vernier caliper, you will be required to interpolate
between scale divisions. For most of these types of instruments it is reasonable to estimate
the absolute uncertainty as ± half of the smallest scale division. For example, if the smallest
division on a metre stick in 1 mm, the absolute uncertainty is ± 0.5 mm.
When using weights, with mass written on them, the absolute uncertainty should be taken
as half of the last significant digit. For example, for the mass m = 200 g, the absolute
uncertainty is δm = 0.5 g. When using digital instruments, such as digital multimeters, the
absolute uncertainty is the sum of a reading error and an instrument error. The reading
error is ± the last stable digit displayed. For example, if the digital voltmeter reading is
11.6 V, the reading error is ± 0.1 V. The instrument error is specified by the manufacturer,
which typically is 1.5% of the reading value. For example, if the digital voltmeter reading
8
Introductory Information
is 11.6 V, the instrument error is (0.015)(11.6 V) = 0.2 V. The total uncertainty is 0.1 V +
0.2 V = 0.3 V. The voltmeter reading should be recorded in the form: (11.6 ± 0.3) V.
Relative and Percentage Uncertainties Frequently, a statement of the absolute uncertainty δx, is not as meaningful as a comparison of the size of the uncertainty with the
size of the measurement itself, x. This comparison is expressed by a relative (or fractional)
uncertainty:
0.3 V
δx
in the case above would be
≃ 0.03
|x|
11.6 V
or expressed as a percentage uncertainty:
0.3 V × 100%
≃ 3%
11.6 V
An uncertainty δx = ± 0.2 cm, for example, is much more important in a measurement of
2 cm (δx/|x| = 0.1 or 10%) than in a measurement of 2 m (δx/|x| = 0.2/200 = 0.001 or
0.1%). Relative and percentage uncertainties have no units.
Uncertainties in Calculated Quantities The measurements themselves are often not
the desired end-products of an experiment. That is, the measurements are used to calculate
something. A simple example is the measurement of speed. To get this you need to measure
a distance divided by a time interval, both of which have associated errors. So the error
in the speed will depend on both the error on the distance and the error on the time
interval.
Until you have studied differentials in math, you can calculate errors compounded in computation using the rules which follow. These are derived by a differential method (Appendix
D).
RULE 1: For Addition and Subtraction
When addition and/or subtraction occur in a calculation, the resultant maximum absolute uncertainty in the answer is the sum of
the absolute uncertainties of all the measured quantities occurring
in the calculation.
Example: Let x = 2.66±0.02 and y = 1.79±0.02. Find the magnitudes of the uncertainties
of (x + y) and (x − y).
Solution:
(x + y)
2.66 ± 0.02
+1.79 ± 0.02
4.45 ± 0.04
9
Introductory Information
(x − y)
2.66 ± 0.02
−1.79 ± 0.02
0.87 ± 0.04
This is reasonable since
x = 2.66 ± 0.02
→
2.64 ≤ x ≤ 2.68
y = 1.79 ± 0.02
→
1.77 ≤ y ≤ 1.81
Adding and subtracting in the most unfavourable ways to obtain the maximum possible
uncertainty gives
4.41 ≤ (x + y) ≤ 4.49
0.83 ≤ (x − y) ≤ 0.91
which can be expressed as 4.45 ± 0.04 and 0.87 ± 0.04 as above.
RULE 2: For Multiplication and Division
When multiplication and/or division occur, the maximum relative
uncertainty of the product or quotient is equal to the sum of the
relative uncertainties of each factor in the function.
Example: Let x = 2.66 ± 0.02 and y = 1.79 ± 0.02. Find the magnitudes and the uncertainties of z = (x × y) and w = (x/y).
Solution:
z = (x × y) = 2.66 × 1.79 = 4.7614
δz
δx
δy
=
+
|z|
|x| |y|
δz
0.02 0.02
=
+
|z|
2.66 1.79
δz
= 0.0075 + 0.011
|z|
δz
= 0.0185
|z|
where z = 4.7614 so,
δz = 0.0185 × 4.7614 = 0.088 = 0.09
Therefore, z = (4.76 ± 0.09)
Please observe that the uncertainty δz = 0.09 was rounded to one significant figure.
10
Introductory Information
Similarly,
w = (x/y) = 2.66/1.79 = 1.486
δw
δx
δy
=
+
|w|
|x| |y|
δw
0.02 0.02
=
+
|w|
2.66 1.79
δw
= 0.0185
|w|
We have w = 1.486 so δw = 0.0185 × 1.486 = 0.02749 = 0.03 and therefore w = 1.48 ± 0.03.
Where addition and/or subtraction and multiplication and/or divisions are all involved in
one formula, the calculation is more complicated. The rules, above, should then be applied
to one part of the function at a time and then combined.
Example: Find the uncertainty of the quantity given by the expression Z = AB 2 +
C/D where A, B, C, D are measured quantities, and δA, δB, δC, δD are the corresponding
absolute uncertainties.
Solution:
Let I1 = AB 2 = A × B × B and I2 = C/D
δI1
δA
δB
δB
=
+
+
|I1 |
|A| |B| |B|
,
δI2
δC
δD
=
+
I2
C
D
δI1
δA 2δB
=
+
|I1 |
|A|
|B|
,
δI2
δC
δD
=
+
|I2 |
|C| |D|
δA 2δB +
I1
|A|
|B|
,
δI2 =
δI1 =
δI = δI1 + δI2 =
δC
δD +
|I2 |
|C| |D|
δA 2δB δC
δD +
I1 +
+
|I2 |
|A|
|B|
|C| |D|
δA 2δB δC
δD C
+
+
|AB 2 | +
|A|
|B|
|C| |D| |D|
δI =
2
δI = δA(B ) + δB(2|A||B|) + δC
11
1
|D|
+ δD
C
|D2 |
Introductory Information
In the calculation of uncertainties, it is generally assumed that whole numbers occurring in
formulas have no uncertainty. Similarly, when using physical constants in formulas (such
as g), one usually includes more significant figures than the measured quantities in the
experiment, so that uncertainty of such physical constants is negligible.
Since uncertainties are only estimated, they should never be quoted to more than one
or two significant figures. Also, because their accuracy is limited, approximations and
simplifications can often be made which make their actual calculation much easier.
RULE 3: For Multiplication/Division with constant
Whenever multiplication with an exact/known number with no uncertainty occurs, the uncertainty is the product of the absolute value
of known number times the uncertainty.
Example: Find the uncertainty of the quantity given by the expression y = Bx, where
x = 2.66 ± 0.02 and B is exactly 4 with no uncertainty.
Solution:
y = Bx = 4 × 2.66 = 10.64
δy = |B| × 0.02 = 0.08
Therefore, y = 10.64 ± 0.08
Rules for Stating Uncertainties and Answers Uncertainties are only estimated and
as such they should be rounded to one significant figure. The only exception to this rule
is when the leading digit in the uncertainty is 1. In this case two significant digits might
be justified. For example, the uncertainty 0.14 rounded to one significant digit would be
reduced very significantly.
The final answer of the measured or calculated quantity should have the last significant
digit in the same decimal position as the uncertainty.
Example:
(26.3 ± 0.5) s
(48 ± 2) m
(36.82 ± 0.06) N
(15.34 ± 0.14) × 102 kg
Comparison
Occasionally in the lab you will be asked to compare either:
(1) several values for the same quantity which you have measured using different methods,
(2) a value which you have measured or calculated with a standard value in a table.
12
Introductory Information
For experimentally determined values, the word compare means more than a comparison
by eye alone – it means a mathematical comparison.
In case (1) the best way to compare the value is to calculate what percentage the average
deviation is from the mean (Appendix B).
In case (2), if your measurement has an uncertainty associated with it, then you should see
whether the range of your measured values including your uncertainty is consistent with
the standard (accepted) value. You should ask the question does:
|your value − standard value| ≤ your uncertainty
If the above it true, then measurement agrees with the standard value.
In case (2) if your measurement does not contain an uncertainty, it may be instructive to
calculate the percentage difference.
|your value − standard value|
× 100%
standard value
For more information on how errors and uncertainties are determined, please see Talyor, J.R.
An Introduction to Error Analysis: The study of uncertainties in physical measurements,
University Science Books, 1997. Additionally, the introductory chapter of your physics
textbook contains important information on significant figures and units.
0.4
Graphs
In this course it will frequently be necessary to plot a series of results (i.e., make a graph).
A graph is often the most concise and meaningful way to display data. In this course,
plotting will be done using Logger Pro. Plotting experimental data and deriving significant
information from the resulting graph is rather different from the process of plotting the
graph of a known analytic function. An experimental measurement is not exact, but rather
is represented by a small range of possible values (e.g., 2.38 ≤ x ≤ 2.42, which we usually
express as x = 2.40 ± 0.02, or y = 1.42 ± 0.03).
On a graph we would represent the uncertainty by plotting the point as shown in Fig 2,
where the two error bars cross at (2.40, 1.42) and their lengths are 2 × 0.02 along the x
direction and 2 × 0.03 along the y direction.
Once all the data points are plotted, you will often want to draw the best smooth curve
(usually this will be a straight line) through the set of points. Due to the uncertainties the
experimental points will never fall exactly on the smooth curve. The ”best fit” to the points
is the curve which comes closest to the points and goes through the error bars of most of
the points.
Generally, one variable is under your control and is known as the independent variable.
By convention this is plotted on the horizontal, x axis. The second measured quantity will
vary in some dependent way as you vary the first quantity and is called the dependent
variable. This is plotted on the vertical, y axis.
13
Introductory Information
Figure 2: Plotted data point with Error Bar
The following general guidelines should be used when preparing graphs:
0.4.1
Labeling
Graphs should have some kind of title to give meaning to the data displayed. For example,
Variation of length of rubber band with load is a meaningful title, where as Length vs load
is not good enough. Axes should always be labelled with the name of the quantity being
displayed and the units in which it is measured (e.g., Spring extension (mm)).
0.4.2
Scales
Aim to spread your data out across as much of the graph as possible. If the value for one
variable extends from 90 to 212 (units), you can use a scale that ranges from 90 to 220
(rather than from 0 to 220) –unless, of course, you do know, or want to know, something
about the situation at the zero value.
0.5
Fitting a Straight Line to the Data
If the relationship between two variables is in the standard straight line form, y = mx + b,
the constants m and b can be determined from the graph of y vs x where m is the slope of
the line and b is the y-intercept.
0.5.1
Estimate “by eye”
To get an estimate of the line parameters m and b, first draw a straight line which comes
as close as possible to all of the data points. Avoid the tendency to force the extrapolation
of the line through the origin even if intuition tells you it should go there. Limitations of
apparatus and other side effects of which you are not aware can cause distortions close to the
origin. Draw your line considering only the region in which measurements were taken.
Often it will not be convenient to start the x-axis at 0 and hence the y-intercept will not
occur on the graph, but b can easily be determined, once m is known, by substituting the
co-ordinates of one point on the line into the equation of the line (recall that each and every
point on a line must satisfy the equation of the line).
14
Introductory Information
Figure 3: Determining the Slope from Plotted Data
Once the line has been drawn (Fig.3) you can calculate its slope to determine some information pertinent to the experiment.
From analytical geometry
m=
rise
y2 − y1
=
x2 − x1
run
where (x1 , y1 ) and (x2 , y2 ) are points which lie on the line. Since your experimental points,
in most cases, do not lie exactly on the line, do not use them for determining the slope.
Instead, choose two arbitrary points lying (exactly) on the line as far apart as possible
and determine the slope from them.
Since the variables you plot will be physical quantities with units, the rise and run should
be given units, and thus the slope will also have units.
0.5.2
The Method of Least Squares
While the method above is useful for getting an approximate feel for the slope and intercept,
there are a number of analytical approaches which give precise results. One of these methods
is the Method of Least Squares. The Method of Least Squares is used when one wishes to
fit a given equation to a set of data. We shall illustrate the method here for the simplest of
cases, the straight line, but the method can be used for any functional form.
Consider Fig.4. The mathematically best straight line through the set of data points is
shown, together with the data. (Notice, this data set not have errors associated with each
data point.) Not every data point falls on the line. There is a difference between the fitted
y value and the actual y value. The purpose of least squares is to minimize these deviations
over all the N data points.
The equation of a straight line may be given by:
y = mx + b
15
Introductory Information
Figure 4: Data with a Linear Fit
The difference between the fitted y, on the straight line, and the actual yi may be given
by:
b + mxi − yi
Summing the squares of these differences gives us the total square uncertainty, or (if divided
by N ) the variance.
X
σ=
(b + mxi − yi )2
P
where
means “the sum over all data points”. It is the sum that we wish to minimize.
We accomplish this by taking the derivative of σ with respect to each of the individual
parameters b and m and set these simultaneously equal to zero. The details of this approach
are given in Appendix C.
16
Chapter 1
Data Analysis using Logger Pro
Objective
1. To install Logger Pro Software.
2. To learn how to enter data and uncertainties in Logger Pro.
3. To learn how to graph data with error bars.
4. To learn how to use the fitting tool to fit curves to experimental data.
5. Qualitatively identify the best fit curve to a given data set.
6. To learn how to use the Word template file to complete your lab report.
Apparatus Logger Pro Software, Word Document Lab Report Template, Personal Computer/Laptop
1.1
Getting Started
Welcome to the laboratory component of your 1st year physics course. Everything you
need for the in-person laboratories is available on our eClass site and in the Introductory
Information section of the lab manual. To conduct your laboratory report you will be
provided with a Word template via our eClass site https://eclass.yorku.ca. Please
download this file from our eClass site now! This Word file template will be used to complete
your laboratory report and is available for download on eClass. Refer to the sample lab
report provided in Appendix A of the lab manual to help you complete all the sections of
your lab report. Read and sign the Assertion of Compliance with Academic Honesty
Regulations (ACAHR). Failure to sign this document will result in a grade of
zero on the lab report. This ACAHR portion of the template is on all lab report
templates and must be completed for every lab report submitted during this course.
All active York University students have access to Microsoft Office 365 software for FREE.
If you don’t already have Word on your personal device visit YorkU MS Office365 and follow
17
Chapter 1. Data Analysis using Logger Pro
the instructions.
1.2
Part I: Installing Logger Pro on your computer
During your on-campus laboratories, you will be provided with a laptop to use to acquire,
display, analyze your experimental data, and to complete your lab report. Along with Word,
one of the main software packages you will be using is Logger Pro. This introductory
laboratory has been designed to give you a chance to work with the software on your
personal device prior to attending the on-campus labs. Below are links to download the
Logger Pro software from Vernier Software & Technology. The software is available to
all students registered in a first year physics course at York University. After the file has
downloaded to your computer, double-click the file to install the software.
1.2.1
Logger Pro 3.16.2 Downloads
Windows 11, 10 or 7 Link: https://www.vernier.com/d/gnny5
macOS 12, 11, 10.15, 10.14, 10.13 Link: https://www.vernier.com/d/hpfeu
For more detailed instructions on how to download and install LoggerPro, see:
https://www.vernier.com/til/2069/
Logger Pro For Older Computers: For Windows and Mac computers that are no longer
receiving updates, you will need an older version of Logger Pro.
Windows 8.1 Link: https://www.vernier.com/d/otayn
Windows 8, Windows Vista, Windows XP This installer is no longer available due to driver
licensing requirements.
Mac OS X 10.12, 10.11, 10.10Link: https://www.vernier.com/d/bstnw
Mac OS X 10.9 Link: https://www.vernier.com/d/oewti Password: experiment
Mac OS X 10.8 Link: https://www.vernier.com/d/pdwat Password: exploration
Mac OS X 10.7 Link: https://www.vernier.com/d/dpen3 Password: experiment
1.3
Part II: Entering Data and Creating a Graph with Error
Bars
Once you have downloaded and installed Logger Pro, double click on the desktop shortcut
icon to start the program. Take a few minutes and note the toolbar across the top of the
window and the blank table and graph below the toolbar. Hover the mouse across the
toolbar icons and note the names of the different icons.
To populate the table simply enter data in each cell. Let’s create a table with 2 columns
and 5 rows of data. Enter the following numbers in the X column: 1, 2, 3, 4, 5. Enter the
18
Chapter 1. Data Analysis using Logger Pro
following numbers in the Y column: 162, 97, 72, 45, 38. Your table should like the one
shown below:
Table 1.1: Table of Data
X
1
2
3
4
5
Y
162
97
72
45
38
Notice as you added the numbers in the table data points also appear in the graph. There
should be 5 data points displayed on the graph. The axes of the graph can be re-scaled.
Change the scale of the horizontal axis (x-axis) by clicking on the largest value shown on
the axis (bottom right corner of the graph) and change the value to 6. Your Logger Pro
window should look like Fig.1.1 once you have entered all data values provided in Table 1.1
and re-scaled the x-axis.
Figure 1.1: Entering Data in Logger Pro. As you enter data, the graph is automatically
populated.
The data provided represent time in seconds (s) on the X column and the Y column data
are the number of decays observed as a function of time from a radioactive element. For
example the number of decays observed at 2 seconds was 97 events (or counts). If you look
at your graph you can see a red marker at position X = 2 and Y = 97.
Let’s rename the columns to match our data information. Double click on the X in the table.
19
Chapter 1. Data Analysis using Logger Pro
A new window, “Manual Column Options” will appear. Select the “Column Definition”
tab. In the “Labels and Units:” section change the name from ‘X’ to ‘time’, ‘t’ in the “Short
Nm:” box and ‘s’ in the “Units” box. Click “Done”. Notice the new name in the table for
the 1st column. Repeat for the Y column entering the following in the “Labels and Units:”
section: Change ‘Y’ to ‘Decays’ for the name, ‘N’ for the “Short Nm:” and ‘N’ for the
“Units”. Notice how the x-axis and y-axis labels on the graph are also updated when you
change the column options.
In Logger Pro the data table and the graph sizes can be adjusted. Simply click to highlight
the table or the graph and adjust the size by clicking and dragging the small squares in
the outer highlighted region. Try it! We are going to be adding more columns to the data
table. Make the table wider and decrease the size of the graph so they don’t overlap.
1.3.1
Adding a Calculated Column
In many experiments, collected data is used in calculations to obtain a final value. Let’s
create a calculated column to subtract a background (Bckg or B) value associated with our
“Decays” column. Click on Data at the top of the window and select ‘New Calculated
Column...’. A ‘New Calculated Column’ window will appear. In the “Column Definitions”
tab, in the “Labels and Units:” section change the name from ‘Calculated Column’ to
‘Decays-Bckg’, ‘Decays-B’ in the “Short Nm:” box and ‘N’ in the “Units” box.
In the “Expression” section, equations can be entered to calculate the value to appear in
the new calculated column. For our Decays data set we are informed there is a background
of 7 decays associated with our measurements presented in the “Decays” column. We are
going to use the “Expression” section to subtract the background decays from our Decays
column. First, click on the “Functions >” box and select “integer” from the drop down
menu. Next, click on the “Variables(Columns)>” and select “Decays” column. To complete
the expression, click outside the brackets and enter “ - 7”. Your expression in the box should
look exactly as shown in Fig.1.2.
Once you have entered the expression click “Done” and the calculated column, now named
“Decays-B”, will be populated with the values as shown in the table below:
Table 1.2: Calculated Column Data
Decays-B
155
90
65
38
31
To add this data set to your graph, right-click on the y-axis and select “Graph Options”
(or click anywhere on the graph area and select “Graph Options” from the “Options” menu
above the toolbar). Select the “Axes Options” tab and place a check mark in the box
beside “Decays-Bckg (N)” and remove the check mark from the box beside “Decays (N)”.
20
Chapter 1. Data Analysis using Logger Pro
Figure 1.2: Creating a new calculated column in Logger Pro. When you enter an equation
in the “expression” section, the data in this column will be automatically populated when
you click “Done”.
Click “Done”. This will display the calculated column data on the graph and remove the
“Decays” data from the graph. Notice how the y-axis label is also updated to “Decays-Bckg
(N)”.
Your Logger Pro window should now look like Fig.1.3
1.3.2
Adding and Graphically Displaying Uncertainties
We are going to add uncertainties to our data table and display them on the graph. You are
informed the error associated with the time column is a constant ± 0.3 seconds for each value
in the table. Double-click on the “time” column. The “manual column options” window
will appear. Select the “options” tab and click on the box beside “Error Bar Calculations”
This will place a check mark in the box and activate the section. This is a fixed value
uncertainty and you can enter the value “0.3” in the box beside the “Error Constant +/-:”.
Click “Done”. You should now see horizontal error bars appear on every data point in your
graph.
There are also uncertainties associated with the Decays column. The uncertainty associated
with the “Decays” data is not a constant and thus we will have to create and assign a new
column to express the uncertainty associated with these values. Click on Data at the top
of the window and select ‘New Manual Column...’. A ‘New Manual Column’ window will
appear. Name the column, ‘delta N’, in the ‘Short Nm:’ enter δN. You can access the Greek
letters by clicking on the drop-down menu to the right of the ‘Short Nm:’ box. Enter ‘N’
for in the ‘units’ box. Press “Done”.
21
Chapter 1. Data Analysis using Logger Pro
Figure 1.3: After creating a new calculated column in Logger Pro and changing the graph
options the graph now displays time on the horizontal axis and the decays minus the background on the vertical axis.
In this new column enter the uncertainty values associated with the “Decays-B” values as
shown in Table 1.3 below:
Table 1.3: Uncertainty associated with the “Decays-B” data
Decays-B
155
90
65
38
31
δN
23
20
19
18
16
This column, δN will be the error associated with the “Decays-B” column. Now let’s assign
the “δN” error column to the “Decays-B” column. Double click on the ‘Decays-B’ column.
Select the ‘options’ tab in the ‘Manual Column Options’ window and activate the ‘Error
Bar Calculations’ section by clicking on the box to the left of ‘Error Bar Calculations’.
Select ‘Use Column:’ and use the drop down menu to select the ‘Data Set delta N’ column
from the listed options. Click ‘Done’. You should see vertical error bars appear on your
graph.
Note the error bars for the decays are vertical as they are related to the error in the “DecaysB” data on the y-axis on your graph. Take a screen capture of your Logger Pro window
using Ctrl+Alt+PrtScr (or use the “Snipping Tool” app) and paste in the provided Word
template file in the Data section.
22
Chapter 1. Data Analysis using Logger Pro
1.4
Part III: Using the fitting tool in Logger Pro
In experiments once data is collected, one looks for a relationship that best describes the
observed data. The curve fit option in Logger Pro allows one to fit a set of experimental
data to a variety of equations. Let’s explore the curve fitting options available in Logger
Pro. Click on the “curve fit” icon on the toolbar (or click on Analyze and select ‘Curve
Fit...’) to activate the “Curve Fit” window. In this window you will see a graph of the data
to be analyzed on the top left with a ‘General Equation’ box below the graph. Logger Pro
has a wide selection of available equations to choose from. You can also use the “Define
Function...” option if the equation of choice is not provided in the general equations. Select
the Linear equation from the listed General Equations and click ‘Try Fit’. You will see
a solid black line appear on the small graph. Click ‘Ok’ to close the window and see the
fit on the main graph. You will notice along with the fitted curve there is also a box
with information regarding the equation of fit. The window displays the equation of the
line along with the best fit values with uncertainties. You can right click on this box and
select ‘Linear Fit Options...’ to select different available options to be displayed. Click on
‘Appearance...’ and change the font size to 12 and click ‘OK’ to exit the ‘Linear fit options’
window. (The information box can also be moved anywhere on the graph, simply click and
hold anywhere on the information box and drag it.)
Notice how the line of best fit is within the error bars on the graph. Let’s see if we can find
another expression that also fits this set of data. Click on the ‘curve fit’ icon on the toolbar.
Scroll down the list of ‘General Equation:’ and select ‘Natural Exponent’. Click ‘Try Fit’
and then ‘Ok’. Right click on the new information box and select ‘Curve Fit Options..’.
Click ‘Appearance...’ and change the color to blue and the font size to 12. Now you can
easily differentiate the two curves.
Take a screen shot of your Logger Pro Window displaying the two curve fits and paste in
your lab report under the Data Analysis section.
Both of these fits are within the error bars. How does one qualitatively decide which fit best
represents this data set? With this limited data set, one option would be to collect more
experimental points. Another option is to try and improve on your systematic uncertainties
to decrease the size of the error bars. Let’s explore the second option. After thinking
over your experimental setup you realize you can improve on the uncertainty for both the
time and decays measurements. You repeat the experiment with these improvements and
decrease the uncertainties to the values listed in Table 1.4.
Table 1.4: Improved Uncertainty associated with time and Decays
δt
0.1
0.1
0.1
0.1
0.1
δN
12
9
8
6
5
23
Chapter 1. Data Analysis using Logger Pro
Change the constant error value in your time column options to 0.1 and update your δN
column with the new values presented in Table 1.4. Notice how the error bars in your graph
are changing. Do you have more confidence on which curve fit better represents this data
set now? Take a screen shot of your Logger Pro window with these new error bar values
and paste in your report also in the Data Analysis section.
Along with the best fit parameters presented in the Curve Fit box, Correlation and RMSE
values associated with the fit are also displayed. RMSE stands for the Root-Mean-SquareError. This value is a measure of how close the curve fit is to the data points. The lower
the value the better the fit!
1.4.1
Questions:
Answer these questions in the Discussion section of your laboratory report. Include the
subsection number, use full sentences, refer to your data and provide as much detail as
possible.
1. What is the equation for the linear fit?
(Express your answer in the form of y = (m ± δm)x + (b ± δb), where m and b (and
their uncertainties) are your best fit parameters determined by the Logger Pro curve
fitting, and y and x represent “Decays-Bckg” and “time”, respectively.)
2. What is the equation of the natural exponent fit? (Express your answer using the
natural fit equation in the format as described for the linear fit above.)
3. What are the two RMSE values from the two equation fits (linear and natural exponent)? Which fit better represents the experimental data?
1.5
Explore Logger Pro Capabilities
Now that you have created a table and a graph, explore the options available in Logger
Pro. Try right and left clicking on the axes of the graph and on the graph and explore the
different options.
For extra practise in Logger Pro, there are tutorials available within Logger Pro. A tutorial
about curve fitting is available along with many other tutorials on using Logger Pro.
To access the tutorials, choose “Open” from the “File” menu. Within the “Experiments”
folder, open the “Tutorials” folder to see all the available tutorials.
24
Chapter 2
Uniformly Accelerated Linear
Motion
Motion and changes in motion lie at the heart of physics. The genius of Galileo, Descartes,
and Newton in the 17th Century started us on our way. Their legacy has been refined and
re-refined in the ensuing centuries but the basic concepts have remained the same.
Objective
1. To learn how to analyze experimental data and how to prepare a laboratory report
using the WORD lab report template.
2. To calculate the instantaneous velocity of a moving cart on an inclined track.
3. To use the linear fit option in Logger Pro and determine the acceleration of the cart
travelling down an inclined frictionless track.
4. Calculate the value of g and compare to the known value of g = 9.81 m/s2 .
Apparatus Dynamics cart and track system, lab stand, data acquisition system (LabQuest
Mini), motion detector, reflecting plate, digital level, lab laptop.
Suggested Reading Depending on which course you are in, you will have an assigned
physics text book. For additional reading for this lab refer to the relevant linear motion
section in your physics textbook.
2.1
Introduction
The experience of several centuries shows three quantities – displacement, velocity, and
acceleration (all functions of time) – are sufficient to describe the motion of an object.
The position in three dimensions is ⃗r = xî + y ĵ + z k̂. The displacement is defined as the
change in the position of the object: ∆⃗r = ⃗r(tf ) − ⃗r(ti ) ≡ r⃗f − r⃗i , where r⃗f is the final
25
Chapter 2. Uniformly Accelerated Linear Motion
position (i.e., the position at tf ) and r⃗i is the initial position of the object. The velocity ⃗v is
the rate at which the position changes. For infinitesimal changes in position we have:
⃗v (t) =
d⃗r(t)
dt
The acceleration ⃗a is defined as the rate of change of velocity:
⃗a(t) =
d⃗v (t)
dt
(2.1)
In this experiment you will consider motion with a constant acceleration. This occurs
whenever a constant net force acts. For this kind of motion ⃗a(t) = ⃗a (i.e., the acceleration
is not a function of time) and equation (2.1) becomes:
d⃗v (t)
= ⃗a
dt
which has the solution ⃗v (t) = ⃗v0 + ⃗a[t − t0 ] where ⃗v0 is the velocity at the instant t = t0 . If
the motion is only in one direction (and setting t0 = 0) then this can be written
v(t) = v0 + at
(2.2)
where v ≡ |⃗v |, i.e., v is the magnitude of ⃗v , and similarly a ≡ |⃗a|.
Since we want to examine uniformly accelerated motion we must control the design of the
experimental procedure to ensure that we are indeed studying such motion. The basic
requirement is a constant net force. As you read the instructions be conscious of how the
procedure and the equipment fulfill this basic requirement.
In this experiment you will study the motion of a cart moving down an inclined frictionless
track. The track permits a nearly constant net force from gravity because friction forces
are small enough that they can be ignored. The cart, released from rest, starts to move
with constant acceleration (refer to Figure 2.1 where g, the downward acceleration due to
gravity, has the known value 9.81 m/s2 ).
Figure 2.1 shows the decomposition of the force due to gravity into the component forces
parallel and perpendicular to an inclined plane. Note, only the force parallel to the incline
plane, mg sin θ is responsible for the motion along the inclined plane, and m is the mass
of the object (in this experiment, m is the mass of the cart). Thus, according to Newton’s
second law:
ma = mg sin θ
a = g sin θ
Experimentally the acceleration of the cart can be determined using equation 2.2. This
equation represents a straight line with the slope equal to acceleration a, when the magnitude of the velocity v is plotted on the vertical axis (y-axis) vs time t on the horizontal axis
(x-axis).
26
Chapter 2. Uniformly Accelerated Linear Motion
Figure 2.1: Forces on an object on an inclined plane.
2.2
Experimental Setup and Measurements
The experimental setup is shown in Fig.2.2. The provided lab laptop can be interfaced
with the motion detector via the LabQuest Mini to measure the position of the cart. The
reflecting plate is attached to the low-friction cart, which is capable of nearly frictionless
motion because it is on a low-friction track. The laptop serves to collect, analyze and
graphically display the experimental data.
The motion detector emits short pulses of sound which are reflected from the reflecting
plate mounted on the low-friction cart. The reflected pulses are detected by the same
motion detector that emits them. The reflected signal is detected by the motion detector,
amplified and digitized by the electronic unit, and then transmitted to the Logger Pro
software on the laptop. The time, t, that elapses between the emission and detection of
sound pulses is measured precisely by the software, and is used to compute the distance of
the reflecting plate from the detector using the formula x = vt/2, where v = 343 m/s is the
speed of sound in air. The data from the experiment is also graphically displayed on the
laptop screen while it is collected.
Getting to know the equipment: As a group take a few minutes to get to know the
equipment. Pick up the cart with the attached reflecting plate and notice the wheel mechanism. (Some setups do not require an attached reflecting plate, the edge of the cart acts
as the reflecting plate.) Place the cart gently back on the track and ensure all wheels are
on the track grooves. Move the cart up and down. When the cart is securely in place it
slides smoothly along the track. Practise releasing the cart from the top of the track and
ensuring it has a smooth path down the track.
Each student must collect their own data set using their assigned laptop. The
data acquisition system that controls the experiment is Logger Pro. Working as a group
repeat the experimental procedure taking turns to collect data on your assigned laptop. For
each student in the group follow steps 1-5 below to collect data:
27
Chapter 2. Uniformly Accelerated Linear Motion
Figure 2.2: Experimental Setup.
2.2.1
Experimental Procedure
1. Connect your Laptop to the LabQuest mini using the USB cable.
2. To begin the experiment, double-click on the Logger Pro icon on the desktop. The
window shown in Fig.2.3 will appear. Note the position icon on the bottom left below
the table. This indicates the connection to the motion detector is active and will
display the position in real time during your experiment.
3. Setup of data collection in Logger Pro: Hover the mouse across the toolbar icons
and note the names of the different icons. Click on the clock icon Data Collection.
28
Chapter 2. Uniformly Accelerated Linear Motion
A new window will appear. Enter the following information: Duration: 3 seconds,
Sampling Rate: 8 samples/second. Click Done.
4. Collect the data:
(a) Move the cart with the fitted sound-reflecting plate facing the motion detector
near the top end of the inclined track.
(b) While holding the cart motionless, click Experiment on the menu and then
click on Zero so that the motion detector sets the distance between it and the
reflecting plate to zero. (Shortcut to zero detector, press Ctrl+0.)
(c) When you are ready to begin the experiment, click on the collect button located
in the menu toolbar and release the cart shortly after the button turns red so that
the computer starts collecting data just before the cart starts moving. (Shortcut
to start button, press the “spacebar” on the keyboard.)
(d) You may have to repeat the above steps several times to get comfortable with
the timing of the cart release.
5. Once you obtain your data, disconnect the labquest mini from your lab
laptop and the next student in your group can now collect their data set
using their lab laptop by repeating the above steps. Once all group members
have collected their data set you may proceed to the next step.
Figure 2.3: Logger Pro Window.
6. Your experimental data will be plotted on the screen in near real time. The data
table should be in the form of columns of time, t, position, x and velocity, v. The
two graphs should display position vs time on the top, and velocity vs. time on the
29
Chapter 2. Uniformly Accelerated Linear Motion
bottom, as shown in Fig.2.3. Click on the “Autoscale Graph” icon in the toolbar to
automatically resize your graph to fit the data.
7. Take a screen shot of the Logger Pro window of your successful trial result and paste
in the Data section of your Word lab report template file. You will be using Word
to complete all your lab reports and submitting through eClass at the end of your
lab section. You can use Ctrl+Alt+PrtScr while Logger Pro is active on your screen
to only capture the Logger Pro Window, or the “Snipping Tool” app to capture and
copy the screen selection and then paste in Word.)
8. Observe your data and note the time corresponding to the largest position (m). This
value corresponds to the furthest distance from the motion sensor (or closest to the
bottom of the track). This value is the largest position data point displayed on your
position-time graph. Create a table in Word in the Data Analysis section of your
lab report to record these two values. (In Word, select Insert from the menu, click
Table and create a 2 x 2 table). Label the table as shown below:
Table 2.1: Time at largest position.
time, t (s) at largest displacement
largest displacement position (m)
2.2.2
(your value)
(your value)
Questions
Answer these questions in the Discussion section of your laboratory report before moving
on to the next section. Include the subsection number, use full sentences, refer to your data
and provide as much detail as possible.
1. For the measurement of the constant acceleration we are only interested in the data
from the start of motion until just prior to the cart hitting the end stop at the bottom
of the track. Why is that?
2. This set of data corresponds to the uniformly accelerated linear motion of the cart
moving down an inclined plane and will be the focus of your data analysis. From
your graph and data table what is the time range corresponding to the uniformly
accelerated linear motion of the cart moving down the inclined plane?
3. Observe the data on the graphs corresponding to the cart motion after it hits the
bumper at the bottom. Describe the motion (position and velocity) of the cart at
the bumper, and as it continues after contact with the bumper. How does the graph
description compare to your observation of the cart during the experiment?
2.3
Calculating the Acceleration
1. Use your position x vs time t data to find the dependence of velocity |⃗v | vs time t.
The instantaneous velocity is equal to the slope of the graph of x vs t at a particular
instant of time. It can be approximated by the average velocity |⃗v | = ∆x/∆t, where
30
Chapter 2. Uniformly Accelerated Linear Motion
∆x = xi+1 − xi is the change of the position in time ∆t = ti+1 − ti . Select two
adjacent points within the region of uniformly accelerated linear motion of your cart
and calculate the instantaneous velocity using the approximation |⃗v | = ∆x/∆t. Show
your work in the Data Analysis section of your Word lab report file.
2. Click on the tangent icon on the toolbar in Logger Pro. This function gives you the
tangent at any point along your data on the selected graph. What are the tangent
values on your position vs time graph for the two time points you used to calculate
the instantaneous velocity above? How do they compare with your calculated instantaneous velocity? Report your findings in the Data Analysis section of your lab
report.
3. Assume that the absolute uncertainty for position is ±1.0 mm and for time is ±0.005
s. As ∆x = xi+1 − xi (difference between two positions), then δ(∆x) = δxi+1 + δxi ,
where δxi+1 and δxi are uncertainties of positions xi+1 and xi , respectively. Thus,
δ(∆x) = δxi+1 + δxi = 1.0 + 1.0 = 2.0 mm.
The uncertainty δ(∆t) = δti+1 + δti = 0.005 + 0.005 = 0.01 s and is determined by
the electronic unit controlling the Logger Pro acquisition system.
To find the uncertainty of velocity δv you will use the procedure described in section 0.3 Measurement and Uncertainties of the lab manual (steps shown below).
Calculate δv for your calculated instantaneous velocity in the first step above using
equation (2.3) .
v =
δv
v
=
δv =
x
t
δ(∆x) δ(∆t)
+
∆x
∆t
δ(∆x) δ(∆t)
+
∆x
∆t
v
(2.3)
4. Fit a straight line representing the velocity vs time graph using the Linear Fit icon
on the tool bar: Click on the velocity vs time graph to select it then select the Linear
Fit icon on the toolbar. Note the two [ ] brackets, solid black line and Linear Fit Box
with fit values that appear on the graph. Logger Pro creates a line of best fit using
the method of least squares as described in section 0.5.2. The [ ] brackets represent
the data set included in the fit that is calculated by Logger Pro. Since we are only
interested in the motion of the cart down the incline, move the brackets
using the mouse to only encompass the linear motion of the cart down the
incline. (This should correspond to your selected data set you defined in
Question 2 in section 2.2.2.) Notice the values within the Linear Fit Box change
as you adjust the [ ].
5. The acceleration of the cart down the incline plane is equal to the slope of the line of
best fit. What is the acceleration (with uncertainty) of your cart? (Double click on the
31
Chapter 2. Uniformly Accelerated Linear Motion
Linear Fit Box and ensure Show Uncertainty is checked.) Present your acceleration
value in the form of a = af it value ± δa. Remember to use proper significant figures in
your reported answer. (Calculations conducted in Logger Pro work like a calculator
and provides many numerical figures in the calculated values. The user needs to
correct number of significant figures from the provided data.)
6. Present your acceleration value and the graph with the line of best fit in the Data
Analysis section of your lab report. To copy the velocity vs time graph: Left click
to select the graph, than right click and select copy and paste in your lab report.
2.3.1
Questions:
Answer these questions in the Discussion section of your laboratory report. Include the
subsection number, use full sentences, refer to your data and provide as much detail as
possible.
1. How does your error value compare with the tangent points provided by Logger Pro?
In other words, are the tangent points within the error margin of your calculated
value? Show your work.
2.4
Measurement of g
Measure the incline angle θ of the track using the digital level. Determine the value of the
gravitational acceleration g from your data using the equation a = g sin θ. From the rules
given in the Introductory Information chapter we have that the uncertainty in g is:
δg
g
δa δ sin θ
+
a
sin θ
δa cos θδθ
=
+
a
sin θ
δθ
δa
+
=
a tan θ
δa
δθ
δg = g
+
a
tan θ
=
Note: the uncertainty in sin θ given the uncertainty in θ is δθ is given by: δ sin θ = cos θδθ.
Estimate the uncertainty in the angle θ and calculate the uncertainty in g. Note, the value
of θ must be in radians to calculate the uncertainty. Show your work in the Data Analysis
section of your lab report.
2.4.1
Questions:
Answer these questions in the Discussion section of your laboratory report. Include the
subsection number, use full sentences, refer to your data and provide as much detail as
possible.
1. Is your result for g consistent with the standard value? (Be sure to define what you
mean by “consistent with.” See section Compare in 0.3.2.)
32
Chapter 2. Uniformly Accelerated Linear Motion
2. Discuss with your group members what is the largest source of uncertainty in your
determination of g in this experimental setup. Can you think of ways of changing
how the experiment is done in order to reduce the uncertainty in g? (List at least two
changes.)
end of lab
33
Chapter 3
Thrust and Friction
In the investigation of Forces and Newton’s Laws, the concepts of Thrust and Friction
appear time and time again. Thrust is a force which acts on an object (for example an
airplane or a boat) in reaction to a force being applied on something else (like the air or
water). It is a consequence of Newton’s Third Law. For example, in a motor boat, the
propeller of the motor forces water backwards, thus forcing the boat forwards – the force
on the boat forwards is in reaction to the force the propeller exerts on the water backwards.
Friction is the force which results from the interaction of the surface of a moving object
with another surface. The force of friction is always directly opposite to the velocity, and
in some systems is a source of unwanted resistance to motion. However, in many cases,
friction is highly desirable (imagine walking to class without friction).
Objective
1. To apply the Laws of Motion to a system and to predict the motion.
2. To observe the differences between static and kinetic friction.
3. To measure the acceleration of a fan cart due to an applied thrust force.
4. Create tables in Logger Pro and in Word to record and organize data.
Apparatus Dynamics cart and track system, cart with fan, friction blocks, computer,
data acquisition system (LabQuest Mini), weights, stopwatch, DC power supply, digital
level, motor with shaft and string.
Suggested Reading Depending on which course you are in, you will have an assigned
physics text book. For additional reading for this lab refer to the relevant laws of motion
section in your physics textbook.
35
Chapter 3. Thrust and Friction
3.1
Thrust
3.1.1
Introduction
In this lab activity you will study the acceleration of a cart when acted upon by a thrust
force generated by a fan mounted onto the cart. Note that in contrast to the Linear Motion
experiment, the track on which the cart moves will be horizontal, so the force of gravity
will always be perfectly balanced by the Normal force. You will investigate the acceleration
of the cart for forces of varying magnitude, carts of varying weights, and forces applied at
various angles with respect to the direction of motion. Note that when the fan is turned on
at a particular setting, it very quickly spins up to a steady speed and achieves a condition
of providing a constant thrust (force).
3.1.2
Setup and Explore
In this exercise, you will have a chance to make qualitative observations regarding the
motion of the cart, but first the track must be setup. The experimental setup is shown in
figure 3.1.
Figure 3.1: Experimental setup of Thrust experiment. The speed of the fan is controlled
using the On-Off switch located on the fan cart.
36
Chapter 3. Thrust and Friction
1. It is critical that the track is perfectly level so that the force of gravity can be perfectly
balanced by the Normal force. Use the digital level to check the level of the track. If
an adjustment is needed contact the TA.
2. Gently place the fan cart on the track, and ensure the wheels of the cart are properly
in the grooves of the track. The cart slides easily and smoothly along the track when
all the wheels are in the track grooves. Notice the removable weights which can be
placed along the sides of the cart.
Include your free body diagrams and your observations for steps 3 - 8 below in the
Discussion section of your laboratory report. Include the subsection number, use
full sentences, and provide as much detail as possible.
3. Draw a free-body diagram of the cart with the fan turned off. (You can create shapes
in Word, located on the Insert tab to create arrows for the free body diagram.)
4. Turn on the fan to the lowest setting and observe the motion of the cart.
5. Draw a free-body diagram of the cart with the fan turned on. Question: Is this
constant velocity? Question: Is this constant acceleration?
6. Observe and comment on the motion of the cart as a function of fan speed.
7. Observe and comment on the motion of the cart as a function of fan angle.
8. Observe and comment on the motion of the cart with and without the weights.
3.1.3
Quantify
Now, we will investigate more closely the effect of the direction of the thrust (fan angle).
To do this, we want to determine the acceleration of the cart. We could use a computerbased data-collection system like was used in the Linear Motion lab. Here, however, we will
apply our physics knowledge to determine the acceleration. For constant acceleration, ⃗a,
we have
1
⃗r(t) = ⃗r0 + v⃗0 ∆t + ⃗a∆t2
(3.1)
2
To determine the acceleration, ⃗a, using equation (3.1), you should measure the time ∆t it
takes the cart, starting from rest, ⃗v (t0 ) = v⃗0 = ⃗0, to travel to position ⃗r(t). The distance
travelled is ∆⃗r = ⃗r(t) − ⃗r0 . When collecting data, hold the cart fixed at a selected starting
position on the track, turn on the fan, then start the timer as you release the cart and stop
the time when the card has travelled the selected distance, ∆⃗r. For this exercise, set the
fan speed to 2, and include the weights on the cart.
Procedure:
You will use Logger Pro to create a table and record your values using the instructions
provided below. Disconnect the USB connection to the Lab Quest Mini from your laptop
if it is connected.
37
Chapter 3. Thrust and Friction
1. Click on Logger Pro to start the program. Note: you can adjust the size of the table
by dragging your cursor when placed at the corners of the window. Try It: Reduce
the size of the graph and increase the size of the table window.
2. Double-click on the name of the column located at the top of the table for column
options. Change the name to Fan Angle and select θ for the short name from the
drop down menu and the degrees symbol for the units. Repeat for the second column
and change the name to Delta Time 1 and enter Time 1 and s for the short name
and the units respectively.
3. Next click on Data and select New Manual Column, name it Delta Time 2 and
enter Time 2 and s for the short name and units, respectively.
4. Repeat the previous step and add one more manual column for Delta Time 3. (You
will be conducting the time measurement three times therefore we need three time
columns, Time 1, Time 2, and Time 3).
5. Click on Data and select New Calculated Column for the mean time. Choose an
appropriate name, short name and units for mean time. For the expression part, type
in the proper mathematical expression for calculating the mean time by using the
three time columns as your variables and the appropriate mathematical operations
for the mean time. The expressions for the mean can be found in Appendix B. Your
expression should look like:
(“Delta Time 1” + “Delta Time 2” + “Delta Time 3”)/3
6. Create one additional New Calculated Column, for the fan cart acceleration, ⃗a,
and enter the correct mathematical expression, by rearranging equation (3.1), where
∆⃗r = ⃗r(t) −⃗r0 is your chosen travel distance along the track and ∆t is the mean time.
Now that your table is setup in Logger Pro you are ready to take and record data. Work
in pairs to run the experiment and collect the data.
1. Measure the time it takes for the cart to travel your chosen travel distance on the
track, ∆⃗r, down the track for fan angles, 0, 15, 30, 45, 60 and 75 degrees.
Note: At each angle repeat the time measurement 3 times. As you enter the time
data in the table, the mean and acceleration should automatically be populated using
the expressions you entered in each calculated column when you created your table.
(Notice the graph is also populated with data.) It’s a good idea to use your calculator
to manually check if the calculation in Logger Pro is as expected.
2. Include a screen capture of the Logger Pro table in the Data section of your Word
lab report.
3. Plot your data points for Acceleration vs Fan Angle in the graph on Logger Pro.
The x-axis should be Fan Angle and the y-axis the acceleration column. Use the
“Autoscale Graph” button located on the toolbar to automatically scale your graph.
(To change the y-axis data left-click on the name of the y-axis and select the desired
data.)
38
Chapter 3. Thrust and Friction
4. Data Analysis: With the help of geometry, let’s investigate if the sine (or cosine)
equation for the component of Thrust in the direction of motion as a function of fan
angle make sense. Add a fit to your Logger Pro graph using the Curve Fit icon on the
toolbar. Click on the Curve Fit icon in the toolbar and a new window will appear.
In the General Equation box, scroll down and select the sine equation and click on
“Try Fit”. A line of best fit will appear on your data along with a curve fit box with
the best fit values. What is the RMSE? (The RMSE indicates how well the chosen
equation fits to the data. The lower the value the better the fit!) Report your findings
and include your graph in the Data Analysis section of your lab report. (Select the
graph and right click and select copy (or press Ctrl+C) to copy the graph (or use the
“Snipping Tool” App) and use Ctrl+V to paste in your Word laboratory file.)
3.1.4
Questions:
Answer these questions in the Discussion section of your laboratory report. Include the
subsection number, use full sentences, refer to your data and provide as much detail as
possible.
1. Does your data follow the expected trend? (Hint: Think about the RMSE value of
the curve fit.)
2. What force balances the component of Thrust perpendicular to the track?
3. What assumption was made in order to use equation 3.1?
4. Calculate the average deviation of the mean time for one of your fan angle measurements. Indicate which fan angle data your are using and shown your work! (See
Appendix B)
5. Explore Some More: Set the fan angle to 90 degrees. Comment on the motion
of the cart when the fan is turned on. Draw a free body diagram for the fan cart.
3.2
Friction
3.2.1
Introduction
Forces of friction are of immense importance because up to 1/3 of all energy produced in
the world is “wasted” because of them. In this lab you will be investigating sliding and
static, dry forces of friction, as opposed to fluid friction (viscous drag). The static force of
friction fs is given by the following inequality:
fs ≤ µs N
(3.2)
where µs is the coefficient of static friction and N is the normal force. (Here “normal”
means that the force is perpendicular to the direction of fs .) The inequality signifies that
the static force of friction may have a range of values, from zero to a maximum value of
fs,max = µs N .
39
Chapter 3. Thrust and Friction
The kinetic force of sliding friction fk is given by the following equation:
fk = µk N
(3.3)
where µk is the coefficient of kinetic friction and N is the normal force (directed perpendicular to the direction of motion).
The kinetic force of sliding friction for the most part does not depend on the speed of the
object (at very low and very high speeds this may not be true). Static and kinetic forces of
friction do not depend on the apparent surface area although they depend on the actual or
real contact area between the two surfaces. The difference between the apparent and actual
contact areas is explained in the figure below.
Figure 3.2: Microscopic surfaces are very uneven are never in complete contact.
Surfaces, at a microscopic level, are very uneven and so two surfaces are never in total contact. The object that slides is supported by the surface at the top of roughness irregularities,
called asperities. These asperities deform until the weight of the object is fully supported.
Approximately just 1 part in 10,000 of the apparent area is usually in contact, which is
the reason why friction is effectively independent of the apparent area of contact. For very
smooth surfaces the actual contact area may be a significant fraction of the apparent contact
area and in this case friction is dependent on the size of the surface.
The actual contact area depends on the magnitude of the normal force. The larger the
normal force the more the two surfaces push against each other, increasing the actual
contact area and the force of friction. Because friction, from the microscopic point of view,
occurs due to interactions between electrical charges, the larger the actual contact area
between two surfaces, the larger the force of friction. The electrical interaction that causes
friction is quite complex and it is still a current topic of research.
3.2.2
Measuring Static and Kinetic Coefficients of Friction
You’ve likely had the experience of pushing or pulling a heavy object on the floor – it is
harder to get the object moving initially than it is to keep it moving. That is because the
coefficient of static friction is greater than the coefficient of kinetic friction, meaning the
static friction force is greater. In this experiment, we will apply a slowly increasing force to
a stationary object, and measure the forces which oppose the motion.
3.2.3
Setup
Setup your track as shown in figure 3.3. Place the cart with the dual-force sensor already
connected on the track and connect the cart to the string attached to the DC motor shaft.
40
Chapter 3. Thrust and Friction
Figure 3.3: Experimental setup for measuring static and kinetic coefficients of friction.
A separate string is used to connect the force sensor to the friction block being pulled.
Additional weights can be mounted on the friction block in order to increase the magnitude
of the normal force. The cart with the force sensor can be pulled either by hand or by a DC
motor. You should try both ways but it is very likely that you will conclude that pulling the
friction block by hand does not give as good results as using the motor. A variable power
supply is used to control the speed of the motor. For voltages of about 3 V, the speed of the
motor is very low, which allows collecting many experimental points during the transition
of the block from being stationary to being in motion.
The force sensor is connected to the LabQuest Mini interface. As in previous labs, you will
use the Logger Pro software to analyze the experimental data.
3.2.4
Procedure
Create a simple table in the Data section of your Word lab report to record your data.
Click on the Insert tab and click on the Table Icon. Create a 6 x 4 table (6 columns, 4
rows). Click on the first row of the first column and type Material. Similarly name the
other columns with Additional Mass Added (kg), Total Mass (kg), Normal Force (N), Max
Static Force, fs (N), Kinetic Friction Force, fk (N). Including the units with the name of
41
Chapter 3. Thrust and Friction
the column is good practice. Each row can be used to enter your measured and calculated
data for each measurement. Include a title above your table with the material (duct tape,
cork or plastic) of your friction block.
For this experiment you will work with a partner to run the experiment and collect the
data. As a pair you will follow the data collection procedure, section (3.2.5), twice.
The first time, one student will connect the labquest mini to their laptop and
collect data for one of the friction blocks. The second time, the second student
will connect the labquest mini to their laptop and collect data for the other
friction block.
3.2.5
Data Collection Procedure
1. Plug in the Labquest Mini to the laptop using the USB cable.
2. set the voltage on the power supply to 3 V and turn the main power switch off.
3. Ensure both strings are not under tension at this point. Start Logger Pro and then
zero the force detector by clicking on the Zero icon in the toolbar.
4. Set the duration of the data collection is 10 s. (If not already set to 10s, click on the
Data Collection icon in the toolbar and changing “Duration” to 10 s.)
to begin to acquire data just before you turn on the power switch
5. Click on
on the power supply for the motor.
6. Turn the power switch off after collection is complete. A typical dependence of the
force as a function of time is shown in figure 3.4.
Figure 3.4: Example of the force as a function of time for a friction block starting from rest
and continues in motion after the breakaway point.
7. The maximum magnitude of the recorded force is equal to the maximum value of the
static force of friction fs , max. Click on the Examine tool
determine fs,max . Record your value in your table.
42
in the toolbar and
Chapter 3. Thrust and Friction
8. Determine the kinetic force of friction fk using the approximately horizontal part after
the breakaway point of the graph. Small fluctuation of the force should be averaged
out by using the Statistics tool (STAT icon in the toolbar
) and finding the
mean value of fk . Once the
icon is initiated, move the two small black brackets
activated on the graph to highlight only the region of interest after the breakaway
point. Record your value in your table.
9. Repeat steps 1 to 8 for the same block material, two more times with different masses.
Unwind the string from the motor shaft before each data collection. You
should have 3 data points for the same friction block. Include a screen shot of your
data table and one of your graph measurements with the statistics box obtained in
Logger Pro in the Data section of your lab report.
3.2.6
Data Analysis
Use the equation fs ≤ µs N to determine µs for your three data points. The normal force is
N = (Mblock + Madded )g, where Mblock is the mass of the block and Madded is the additional
mass placed on the block, and g is the acceleration of gravity. N = (Mblock + Madded )g.
(This is true only if the string connecting the force sensor to the block is horizontal.) The
mass of the different blocks are: Mblock−duct−tape = (88.0 ± 0.1)g, Mblock−cork = (92.9 ± 0.1)g
and Mblock−plastic = (95.5 ± 0.1)g.
Similarly, use the equationfk = µk N to determine µk . As for the static friction, N =
(Mblock + Madded )g.
In the Data Analysis section of your lab report, list your calculated values of µs and µk
for your block material using a table (use the “Insert Table” option in Word to create a
table). Include the mean and average deviation of your results for µs and µk . (Use the
equations provided in Appendix B.)
3.2.7
Questions
Answer these questions in the Discussion section of your laboratory report. Include the
subsection number, use full sentences, refer to your data and provide as much detail as
possible.
1. List 3 possible sources of uncertainty in your measurements.
2. Is the force of kinetic friction independent of velocity? What evidence do you have?
3. What block material did your lab partner use and how do their µs and µk values compare with your block material values? What can you conclude about the coefficients
of static and kinetic friction of these two materials?
end of lab
43
Chapter 3. Thrust and Friction
44
Chapter 4
Oscillatory Motion and
Conservation of Energy
Objective
1. To study oscillations of a spring-mass system.
2. To measure the spring constant of a spring.
3. To investigate the angular frequency and phase shift of an oscillating system.
4. To study conservation of energy in a spring-mass system.
Apparatus
masses, springs, stopwatch, data acquisition system (LabQuest Mini)
Reading Depending on which course you are in, you will have an assigned physics text
book. For additional reading for this lab refer to the relevant oscillatory motion section in
your physics textbook.
4.1
Spring-Mass System
4.1.1
Introduction
In this part of the lab you will analyze a special kind of motion, called oscillatory motion.
A good illustration of oscillatory motion is the motion of a mass attached to a spring. If
the spring is displaced from the equilibrium position and then released, the mass oscillates
up and down about the equilibrium position. If the frictional forces, which are always
present, could be neglected the spring would oscillate for an indefinite period of time. That
idealized, frictionless oscillatory motion of the mass on the spring is called Simple Harmonic
Motion (abbreviated SHM). There is a long list of systems to which the ideal of SHM
applies including: pendula, vibration of parts of musical instruments, vibration of atoms
and molecules in solids, oscillation of an electric and a magnetic field in electromagnetic
45
Chapter 4. Oscillatory Motion and Conservation of Energy
Figure 4.1: Motion of a mass attached to a spring.
wave, and time variation of voltage and current in alternating-current circuits. Simple
harmonic motion in one dimension (call it x) can be represented by
2πt
x = A cos
+ ϕ0
(4.1)
T
where:
• x = the displacement from the equilibrium position (x = 0 is taken as the equilibrium
position)
• A = the amplitude (maximum displacement from the equilibrium position)
• T = the period (time of one complete oscillation).
• ϕ0 = the phase constant (set by the displacement at t = 0).
The frequency f of the oscillatory motion is defined as the inverse of the period T
f=
1
T
It is equal to the number of oscillations in 1 second. Recall that the unit of frequency is
Hertz [Hz]. Using the angular frequency ω = 2πf (unit is radians/second) allows equation
(4.1) to be rewritten in the more commonly used form
x = A cos (ωt + ϕ0 )
(4.2)
x
= −Aω sin (ωt + ϕ0 )
dt
(4.3)
Note that this also implies:
v=
46
Chapter 4. Oscillatory Motion and Conservation of Energy
and so x and v are out of phase by π/2 (since x ∝ cos (ωt + ϕ0 ) while v ∝ sin (ωt + ϕ0 ))
So far we have described what SHM is but we have not explained what causes that kind of
motion. When the mass connected to the spring is displaced, the spring exerts a force on
the mass in the opposite direction to the displacement. The bigger the displacement, the
bigger the force. Since this is just in one dimension |F⃗ | ≡ F and we then have F = −kx
where k is the so-called “spring constant.” The negative sign expresses the fact that the
force F has a direction opposite to the displacement. Using Newton’s second law (where in
one dimension |⃗a| ≡ a) we have
−kx = ma
Rearranging gives
a=−
k
x
m
but a = d2 x/dt2 so we have
k
d2 x
=− x
2
dt
m
The solution of the above differential equation is (be sceptical, check by differentiating
twice!)
!
r
k
x = A cos
t + ϕ0
(4.4)
m
Comparing equations (4.2) and (4.4) leads us to conclude
r
ω=
k
m
(4.5)
That is, the angular frequency of oscillation depends on the mass m and the spring constant
k, and not on the amplitude A.
The experimental setup is shown in Fig. 4.2.
4.1.2
Measurements and Calculations
1. Displace the mass m = 500 g suspended at the end of the spring about 10 cm from
the equilibrium position and measure the time t for N = 10 oscillations. Repeat
the measurement of the time t at least eight times (n = 8). The uncertainty of
your measurement is not given by the accuracy of the stopwatch but rather by your
reaction time. The best estimate of the time t is the average of your eight, or more,
measurements and the uncertainty δt is given by the standard deviation. Use Logger
Pro (ensure the LabQuest mini is not connected to your laptop) to create a table
to record your data. Label the first two columns n and t, and create an additional
calculated column, labelled T for the Period, and enter the appropriate equation to
calculate the Period from your measured time.
2. Include a screen capture of your table in the Data section of your lab report.
47
Chapter 4. Oscillatory Motion and Conservation of Energy
Figure 4.2: A mass is suspended at the end of a spring. When it is displaced downwards
and released it will oscillate up and down.
3. Create a graph of n vs. T and use the STAT tool to obtain the mean and standard
deviation of the period. Include a screen capture of your graph in the Data Analysis
section of your lab report. Present your result for the period in the form:
(T ± δT ) s
4. The spring constant k can be determined using equation (4.6 by utilizing the relation
ω = 2πf and substituting T = 1/f in equation (4.5) and solving for k.
r
ω = 2πf =
r
m
T = 2π
k
k = 4π 2
k
m
m
T2
(4.6)
Based on the two rules described in Section 0.3.2 the uncertainty is calculated as:
δk
δm 2δT
=
+
k
m
T
(4.7)
The uncertainty δm is specified by the last significant digit written on the mass. For
example, 10 g indicates δm = 0.5 g. If you use two masses 10 g + 5 g, then the total
48
Chapter 4. Oscillatory Motion and Conservation of Energy
uncertainty is δm = 1.0 g. Determine the spring constant k and its uncertainty δk
using equations (4.6) and (4.7). Record the result in the form: (k ± δk) N/m in the
Data Analysis section of your lab report.
5. A different way to determine the spring constant is to measure the elongation of the
spring when acted on by a known force, in this case, the force of gravity on a mass
suspended from the spring Fg = mg. If the elongation of the spring is ∆x when acted
on by force mg, then mg = k∆x or
k=
mg
∆x
(4.8)
Its uncertainty is
δk
δm δ∆x
=
+
k
m
∆x
where we assumed that g is known relatively precisely.
With the 500 g mass suspended from the spring, measure the location of the bottom
of this mass – this is your reference location x1 . Attached a second mass m to the
500 g mass and measure the new location of the bottom of the 500 g mass – this is
reference location x2 . ∆x is calculated the usual way ∆x = x2 − x1 . The mass m to
use in equation 4.8 is the added mass only. Measure the spring constant using this
technique and present the result in the form: (k ± δk) N/m.
6. Verify if the spring constants determined using two different methods are equal within
the experimental uncertainty, that is, if the uncertainty intervals overlap. The uncertainty intervals overlap if the difference between the two spring constants, |k1 − k2 |,
is smaller than the sum of their two uncertainties δk1 + δk2 .
4.1.3
Questions:
Answer these questions in the Discussion section of your laboratory report. Include the
subsection number, use full sentences, refer to your data and provide as much detail as
possible.
1. Does the period T depend on the amplitude of the oscillation?
2. The mass of the spring should be taken into account when analyzing oscillations of a
spring-mass system. It can be shown that one third of the mass of the spring must
be considered as a portion of the total mass. Designating the mass of the spring by
ms and the suspended mass by m
k = 4π 2
(m + ms /3)
T2
Using a mass value of 12 g for the spring, was it justified to ignore the mass of the
spring? Explain.
3. Researchers try to design their experiments in such a way that the contribution of
various measured quantities to the overall uncertainty are comparable. Which of the
49
Chapter 4. Oscillatory Motion and Conservation of Energy
quantities: mass m, period T , and elongation of the spring x, contributed most to the
overall uncertainty δk? Propose a more precise way to measure this quantity.
4.2
Conservation of Energy
4.2.1
Introduction
The Law of Conservation of Energy states that energy can be converted into different forms,
but it cannot be created or destroyed. In an oscillating spring/mass system, energy can be
transferred between: spring potential energy, gravitational potential energy, kinetic energy,
energy lost as heat in the spring, and energy lost due to drag (viscous friction). In this
experiment you will investigate this energy transfer.
The spring potential energy Uspring is given by the expression:
1
Uspring = kx2
2
where k is the spring constant of the spring and x is the displacement from the equilibrium
position.
Gravitational potential energy Ug is given by:
Ug = mgh = mgx
where m is the mass of the oscillating object, g is acceleration due to Earth’s gravity, and
h is the height with respect to some reference level. Since we have a vertically mounted
spring, and we already called the displacement from the springs equilibrium x, this h is
equal to x.
Kinetic Energy K is given by
1
K = mv 2
2
where m is the mass of of the oscillating object, and v is the speed of the object.
The energy lost as heat in the spring and energy lost due to drag (viscous friction) occur
in all systems, however, in the system under investigation these losses are expected to be
small, and for now, will be ignored (we have no easy way to measure them). Therefore, the
total energy of the system is:
1
1
Etot = Uspring + Ug + K = kx2 + mgx + mv 2
2
2
50
(4.9)
Chapter 4. Oscillatory Motion and Conservation of Energy
4.2.2
Setup and Data Collection
To collect data in this experiment, a sound probe (as was used in the Linear Motion Experiment) is used to measure the distance from the probe to the hanging mass and graphically
display the experimental data using Logger Pro.
Ideally, we would have a spring which hangs without its coils compressed together. To
achieve this situation, we will hang an initial mass onto the spring using the mass holder
before we start. The mass holder weighs 50 g, and you should place a disk of 200 g onto the
holder. This is the initial mass, mi . This mass is needed simply to allow the spring to act
more like an ideal spring. The level of the bottom of the mass hook will be the reference
level for the gravitational potential energy and spring potential energy.
1. We need to assign the present position of the spring as Zero. With the initial mass,
mi steady, select Experiment ⇒ Zero from the top menu.
2. Now we will apply our oscillating mass to the spring. Choose a value which doesn’t
overly extend the spring, and place it on the mass holder. This is the added mass,
madded . The spring will now hang at a lower position (do not re-zero the probe).
3. To get the system to oscillate, pull the mass holder down by a few centimeters and
release. Click on
to begin data collection. You will see two graphs being
plotted, one of the position vs time and one of the velocity vs time. The two data sets
will be fit with a sinusoidal function (simple harmonic motion) while the table on the
left hand side of the screen of time, position, and velocity will be used to determine
the energy. Take a screen capture of your visible data table (∼ 20 rows) and paste it
in the Data section of your lab report.
4.2.3
Data Analysis
If this is truly SHM then the position and velocity as a function of time should be described
by equations (4.2) and (4.3), respectively.
1. One of the fit functions available to you in Logger Pro is A*sin(Bx+C)+D. Use the
Curve Fit tool to fit the two data sets using this function. Include a screen capture
of the Logger Pro graphs with the fits.
2. There are three measurements of ω derived from the two fits. The parameter B is
directly ω in both fits while the ratio of the amplitude of the velocity function (ωA see eq 4.3) to that of the position function (A - see eq. 4.2) also yields a value of ω.
Write the three values of ω that you measured in your report.
3. Take the average of the three ω values. Qualitatively, how does this value compare
with the ω you get using equation 4.5? The value of k for this spring is 18.0 ± 0.5
N/m.
4. The parameter C in the fit function is the phase. Are the position function and velocity
function out of phase by π/2 according to the fits?
51
Chapter 4. Oscillatory Motion and Conservation of Energy
Table 4.1: Conservation of Energy of a Spring-Mass System
Time(s)
Position (m)
Velocity (m/s)
USpring (J)
Ug (J)
K (J)
Etot (J)
5. Based on the Law of Conservation of Energy, the total energy of the system should
remain constant over time (i.e. for all rows in your data table). What energies do
we need to consider? Using the total energy of the system equation, eq. (4.9), spring
potential energy, gravitational potential energy and kinetic energy can be computed
from the position and Velocity values provided in the table. Energy lost as heat in the
spring, and energy lost due to drag (viscous friction) are variables we cannot measure,
but are likely quite small in this experiment, so we’ll ignore them for now.
To find if energy was conserved in this experiment, pick 3 times during the oscillations
from your data table, preferably when the mass is at a variety of different positions,
and compute the spring potential energy, gravitational potential energy and kinetic
energy for each. Create a table in word or Logger Pro to display your data similar to
the one shown in Table 4.1. Include a sample of your energy calculations in your lab
report.
Special Note When computing the Spring Potential Energy and the Gravitational
Potential Energy, we only need to consider madded , not the initial mass mi required to
extend the spring. (At all times, the extra Spring potential energy of mi cancels the
extra Gravitational Potential Energy of mi .) However, when computing the Kinetic
Energy, we need to consider the total mass of mi + madded , since there is actually a
larger mass moving than just madded . Use the value of k = 18.0 ± 0.5 N/m.
4.2.4
Questions:
Answer these questions in the Discussion section of your laboratory report. Include the
subsection number, use full sentences, refer to your data and provide as much detail as
possible.
1. Find the mean and average deviation of the total energy. Was energy conserved in
this experiment?
2. Did we need to include the heat lost in the spring and work done by viscous drag in
our analysis? Explain.
end of lab
52
Chapter 5
Angular Motion
Circular motion, which is a special form of angular motion was initially thought to be natural
motion (Galileo and Descartes in the early 17th century). We have learned otherwise since
then thanks to Newton (latter part of 17th century). Our common experience with angular
motion is much less than with linear motion, so our intuition about angular motion is not
developed well. The first thing we need to tackle is how to describe it. As a starter we
can try the ideas we used for linear motion. After all they may ′ work ′ , and we know
the mathematics for them. So we shall try displacement, velocity, acceleration and force.
However, we need to be cautious and to be prepared to adapt the ideas transplanted from
linear motion. Shortly, we shall do a theoretical analysis of the motion of a mass moving in
a circle. The analysis has two parts: 1) motion at a constant speed (NOT constant velocity)
and 2) motion at a constant magnitude of acceleration (NOT constant acceleration). A
sound understanding of the difference between a constant scalar and a constant vector
is essential to grasp the relationship between the concepts and the mathematics in this
experiment. In this lab we will examine two concepts: Centripetal Force and Moment
of Inertia.
Objective
1. To determine the mass of an object moving in a circle through measurement of the
centripetal force.
2. To determine the moment of inertia of a mechanical model of a diatomic molecule.
Apparatus
Centripetal force apparatus, Vernier caliper, stop watch, measuring tape.
Suggested Reading Depending on which course you are in, you will have an assigned
physics text book. For additional reading for this lab refer to the relevant centripetal
acceleration and moment of inertia sections in your physics textbook.
53
Chapter 5. Angular Motion
5.1
Centripetal Force
5.1.1
Introduction
This experiment may seem simple but it is not. A good grasp of the principles and how
the experiment is designed requires substantial thought. Please pay careful attention to:
(i) How we ensure the motion is in a circle. (ii) How we ensure there is a centripetal force
of constant magnitude. And (iii) How we acquire the required data.
Figure 5.1: When a particle moves in a circle its velocity is always tangent to the circle.
As the particle moves around the circle the velocity ⃗v changes, hence it is accelerating. If
the speed v = |⃗v | is constant, then the velocity change is only in the direction. That is, ∆⃗v
is directed towards the centre of the circle. So the acceleration defined as ∆⃗v /∆t has the
same direction as vector ∆⃗v . It can be shown that the size of this acceleration in circular
motion, called centripetal acceleration aR , is
aR =
v2
r
(5.1)
where v, the magnitude of the velocity ⃗v , is the speed and r is the radius of the circle. If
the particle makes one revolution in time T (period) then its speed is
v=
2πr
distance
circumference
=
=
T
time
period
(5.2)
The period, T, is related to the frequency, f (number of revolutions per second) by
f=
54
1
T
(5.3)
Chapter 5. Angular Motion
Equation (5.2) can then be written as
v = 2πf r
(5.4)
v = ωr
(5.5)
or
where ω = 2πf is the angular speed. Using the relation of equation (5.5), the size of the
radial acceleration (equation (5.1)) can be written as
aR = ω 2 r
(5.6)
According to Newton’s second law a particle can accelerate only when an unbalanced force
acts upon it. So a force is needed to have circular motion. The required force is called a
centripetal force and its magnitude is:
FR = maR = m
v2
= mω 2 r
r
(5.7)
We want to check that there is a centripetal acceleration equal to ω 2 r pointing radially
inward when a mass moves around a circular path. How can we make this check? We shall
use the relation F⃗ = m⃗a, measure F⃗ and ⃗a, then calculate m. This is to be compared to
the known value of m.
5.1.2
About the Experimental Setup
You will use the apparatus shown below, in figure 5.2. (Note, the underlined words tell you
about key factors in the design of this experiment.) A bob of mass Mb is suspended by a
cord from one end of a horizontal bar. (What is the purpose of the counter mass shown in
5.2 a and b?).
The bar is supported at the top of a vertical shaft. When the shaft is rotated by a motor
the mass Mb has a tendency to swing outward, i.e. to move along a tangent (Fig. 5.2a)
while in motion. This is opposed by a spring joining the mass with the shaft, i.e. the
stretched spring supplies the required centripetal force. What is the direction of the force
exerted by the spring on the mass? An adjustable pointer mounted on the base serves as a
radius indicator. The force exerted by the spring on a moving mass can be measured when
the system is at rest. This can be accomplished by attaching a string with a hook over a
pulley and adding masses until the spring extension is the same as during the motion (Fig.
5.2b).
5.1.3
Measurements
1. Assemble the equipment as shown in Fig. 5.2a. Place the counter-mass (labelled M1)
on the end of the horizontal bar, attach the bob to the string and spring, and find the
balance between the bob and counter-mass. To balance the horizontal bar, one can
tip the apparatus so that the shaft is horizontal and free to rotate. When the system
55
Chapter 5. Angular Motion
Figure 5.2: Angular Momentum Apparatus. On the left, (a) shows the rotating setup and
on the right, (b) is a schematic of the stationary setup.
is aligned the crossbar with the masses does not rotate. Once the masses are balanced
secure the horizontal bar firmly to the shaft. Fix the screw joining the horizontal
bar and the shaft.
2. Switch on the motor, give a small push to the rotating mass and wait at least 30s
until the speed is constant. The speed can be changed by selecting different grooves
on the motor shaft. The highest groove corresponds to the lowest speed. Select the
groove which gives a speed such that the rotating bob hangs vertically with respect
to the horizontal bar.
3. Adjust the pointer so that the bob is directly over the pointer.
4. Measure the time t for about n = 20 revolutions. Repeat the time measurement N =
4 times. Create a table in Logger Pro to record your data of N and time, t.
5. Create a graph (N on the x-axis and time on the y-axis) and use the STAT icon on
the tool bar to find the mean and the standard deviation. For your data, the standard
deviation will serve as the uncertainty δt. Take a screen capture (Ctrl+Alt+PrtScr)
of Logger Pro displaying your table and graph and include it the Data section in your
lab report.
56
Chapter 5. Angular Motion
6. The angular speed ω can be calculated using the formula:
ω=
2πn
t
(5.8)
7. Measure the radius, r, of rotation from the tip of the pointer to the centre of the shaft.
Don’t forget to include the error, δr, associated with your measurement of r.
8. Assemble the equipment as shown in Fig. 5.2b. Add some masses to pull the bob
over the pointer. Record the total mass, m (including the mass of the hook which is
50g), and its uncertainty, δm . Determine the force (weight) F = mg.
9. Find the ratio F/(rω 2 ). According to equation (5.7) the ratio represents the mass of
the bob Mb . Using (5.8) we have
Mb =
mgt2
F
=
rω 2
4π 2 n2 r
Calculate the uncertainty δMb
δm 2δt δr
δMb
=
+
+
Mb
m
t
r
take δt as the standard deviation calculated using Logger Pro in step 5 above, δm is
the uncertainty in the mass, m, and δr is the error associated with the measurement
of r. Note that quantities n and g do not contribute to the uncertainty.
Record your result in the form (Mb ± δMb ).
10. Verify if measurements of Mb and Mb,balance are consistent, that is, if |Mb −Mb,balance | ≤
δMb and given Mb,balance is (0.449 ± 0.005)kg.
5.1.4
Questions:
Answer these questions in the Discussion section of your laboratory report. Include the
subsection number, use full sentences, refer to your data and provide as much detail as
possible.
1. If you were to increase the frequency of rotation, how would the radius r and the
centripetal force F change? Would both quantities change; only one or none? Justify
your answer!
2. Identify the centripetal forces in the following two situations:
(a) The Moon moving around the Earth
(b) An electron moving around the nucleus.
57
Chapter 5. Angular Motion
5.2
Rotational Inertia (Moment of Inertia)
5.2.1
Introduction
Again, this part of the experiment has a deceptively simple appearance. You need to pay
careful attention to how the ideas of displacement, velocity, acceleration, mass, and force
are adapted to describe changes in this special case of angular motion. Pay careful attention
to:
• the rotational variables analogous to displacement, velocity, and acceleration that are
used to describe angular motion
• the difference between a constant magnitude for acceleration or force versus a constant
acceleration or force vector
In the first part of this lab we dealt with the rotation of a single mass. We neglected the
dimensions of the rotating mass and treated it as a point particle. In the second part of the
lab you will analyze the rotation of an extended rigid body (i.e., one consisting of many
particles). Each element of a rigid body rotates with the same angular speed ω since each
point of the body makes the same number of rotations in a given time. The definition of
angular speed ω is:
dθ
ω=
dt
where θ is the angular distance measured in radians. This is similar to the definition for
linear speed v:
dx
v=
dt
where x = linear distance. Similar to the magnitude of the linear acceleration a = dv/dt,
we can define the magnitude of the angular acceleration α:
α=
dω
dt
(5.9)
Both the angular speed and the magnitude of the angular acceleration are the same for all
particles of a rigid body. Usually there is a simple relation between the magnitudes of the
angular and linear accelerations
a = rα
(5.10)
where r is the distance from the body to the axis of rotation. We will delay discussion of the
circumstances for which a ̸= rα. The angular acceleration of the rigid body is proportional
to the quantity called torque. Torque is a measure of the ability of a force to rotate the
body about a certain axis. The magnitude of torque τ is defined as
τ = rF sin β = F d
(5.11)
where the symbols are as shown in figure 5.3 below.
F⃗ is the force acting on the body, and d is the perpendicular distance from the line of action
of force F⃗ to the axis of rotation. The force F⃗ can rotate the body more effectively if d is
58
Chapter 5. Angular Motion
Figure 5.3: Torque measures the ability of a force to rotate a body about a certain axis.
bigger, and hence τ is bigger. Torque is related to the angular acceleration.
⃗τ = I α
⃗
(5.12)
where I is what is called the moment of inertia. This relation is a rotational analogue of
Newton’s second law for linear motion (F⃗ = m⃗a) with F , m, and a replaced by τ , I, and
α. The moment of inertia of a particle of mass M a distance R from the axis of rotation
is:
I = M R2
(5.13)
The moment of inertia of an extended body is found by dividing the body into small elements
and finding the sum of the moment of inertia of each element
X
I=
Mi Ri2
(5.14)
i
where Ri is the perpendicular distance from Mi to the axis of rotation. Thus the moment of
inertia of an extended body depends upon the size and shape of the body, and the location
of the axis of rotation.
5.2.2
Experimental Setup
We have a model for angular motion based on analogies from linear motion. Now we need
to check the predictions with a carefully controlled experiment. In particular we want to
examine torque, angular acceleration, moment of inertia and how they are related. Again,
pay close attention to the way in which the ideas discussed are built into the design of the
experiment.
Make changes to the equipment used in part A of the lab (Fig. 5.4 ). Remove the spring,
the crossbar with the hanging mass and the belt from the motor shaft. Use the long thin
crossbar. With one aluminum and one brass mass, set the masses as far apart as possible on
the crossbar (securely fastened into a depression on the bar). The rotation is only about
59
Chapter 5. Angular Motion
one point. To make this centre-of-mass coincide with the shaft, one can tip the apparatus
so that the shaft is horizontal and free to rotate. When the system is aligned the crossbar
with the masses does not rotate. Once the masses are balanced secure the crossbar firmly
to the shaft.
A nylon string with a mass on one end is attached to the shaft of the rotational apparatus.
The shaft is rotated so that the string winds around it until m is close to the pulley. The stop
watch should be started simultaneous to releasing the mass m. The stopwatch is stopped
at the instant that the system completes n rotations.
In the first part, a uniform angular velocity was maintained in each run. In this experiment,
the angular velocity increases during a run because m is accelerating. This is uniform
acceleration. [Think about how do you decide that ⃗a is constant.]
To simplify the analysis of the situation described above, friction and the mass of the pulley,
which are small, will be ignored. [This about how does this simplify the analysis!]
Suppose m falls with a constant acceleration ⃗a. The net force acting on m is m⃗a; if the size
of the tension in the string is Ts , then
ma = mg − Ts
(5.15)
Be sure you know why this is true. Since the pulley requires a negligible force to turn it,
Figure 5.4: Schematic of Experiment
the tension in the string is the same throughout. The taut string applies a constant torque
⃗τ to the central shaft of radius r, causing the system to rotate.
τ = Ts r is the magnitude of the torque
60
(5.16)
Chapter 5. Angular Motion
Note that mg is not explicitly involved. Recall that the analogue of Newton’s second law,
F⃗ = m⃗a, for rotational motion is ⃗τ = I α
⃗ . In this particular case ⃗τ = I α
⃗ are constant.
Substituting τ = Iα into (5.15), along with (5.16) gives the relationship
Ts =
Iα
= mg − ma
r
(5.17)
The magnitude of the linear acceleration can be related to the magnitude of the angular
acceleration α by considering the point where the string just leaves the shaft. In this case
a = rα where r is the radius of the shaft. If the system starts from rest and completes n
revolutions (= 2πn radians) in t seconds, the magnitude of the angular acceleration is given
by
1
1
2πn = αt2 (Compare x = at2 )
(5.18)
2
2
Substituting for a and α in (5.17) gives an expression for I which only involves easily
measured quantities.
2
t g
−r
(5.19)
I = mr
4πn
In the above formula, m is the hanging mass in Fig.5.4, and r is the radius of the shaft.
Note that this I is the moment of inertia of the entire rotating system, i.e. the sum of the
moments of inertia of all the component parts.
5.2.3
Measurement and Calculations
You are now going to attempt to determine the rotational inertia of two masses orbiting
about an axis. However, our system is comprised of more than just the masses. The support
rod for the masses, the rotating shaft and some bearings also contribute to the rotational
inertia of the two masses. Described below is a method of cancelling out the inertia of the
unwanted components, allowing us to measure the inertia of just the masses.
1. Ensure the drive belt connecting the motor to the rotating shaft used in Part I is
removed.
2. Note the mass of M1 = (540 ± 5)g and M2 = (271 ± 5)g
3. Measure the radius r of the shaft using the Vernier caliper.
4. Set up the experiment as shown in Fig. 5.4 with the balance-point of the cross bar
attached to the rotating shaft. Be sure that the screw securing each mass is tightened
into an indent on the cross bar.
5. Measure the position (R1 and R2 ) of the masses with respect to the axis of rotation.
How well do you know where the centre of each mass is? Briefly describe your method
of measuring R1 and R2 in the Data section of your lab report.
6. Have the TA check your setup to ensure that everything is secured properly.
61
Chapter 5. Angular Motion
7. To apply a known torque to the shaft (to cause rotation) we will use a third mass
m = 500g at the end of the nylon string. Gravity acting on this third mass supplies
a torque ⃗τ to the system about the shaft.
8. Release the mass and measure the time t for n = 8 rotations of the system. Repeat
2 more times. Create a table in Word to record your measurements and calculate
the mean and average deviation (refer to Appendix B for equations). You need to
rewind the hanging mass before each measurement.
9. In order to determine the rotational inertia of the support system, Isupport , remove
masses M1 and M2 , but NOT the cross bar. This time, use a smaller third mass,
m = 50g at the end of the nylon thread so it does not spin too fast. Repeat the
measurements performed in step 8 on the remaining part of the apparatus and record
your data in a table.
5.2.4
Data Analysis
Using equation (5.19) one can calculate I for the system and I for the support using the
measurements from above. In order to extract the Moment of Inertia for the two mass
system (I2M ) we need to subtract the moment of inertia of the support from the system. In
other words,
I2M = Isystem − Isupport
(5.20)
1. Determine the rotational inertia Isystem for the whole system and the rotational inertia
Isupport for the support system using equation (5.19). You can then calculate I2M using
equation (5.20). The difference I2M is the rotational inertia of the two masses M1 and
M2 alone without the support rod. This is a useful model for any diatomic molecule
or binary star system.
2. Calculate the expected rotational inertia of two masses using the equation below (see
Fig. 5.4).
I2MExpected = M1 R12 + M2 R22
(5.21)
3. Calculate the percentage difference between I2M and I2MExpected . If the two different
calculations agree then we can feel pretty confident about the validity of the relationship ⃗τ = I α
⃗.
5.2.5
Questions:
Answer these questions in the Discussion section of your laboratory report. Include the
subsection number, use full sentences, refer to your data and provide as much detail as
possible.
1. Did you neglect any frictional forces when calculating the Moment of Inertia? If so,
what kind?
2. In Fig.5.4, the cord wrapped around the shaft should form only one layer. Which
of the following quantities: torque applied to the shaft, angular acceleration of the
62
Chapter 5. Angular Motion
system, Moment of Inertia of the system, acceleration of the falling mass, are changed
if more than one layer is formed by the cord? Explain why.
end of lab
63
Chapter 6
Standing Waves
Objective
1. To study transverse and longitudinal standing waves.
2. Calculate the speed of standing transverse waves in a rope.
3. Measure the linear mass density of a stretched string.
4. Calculate the speed of standing longitudinal standing waves in a spring.
5. To study sound standing waves in a tube and measure the speed of sound.
Apparatus Mechanical wave driver, sine wave signal generator, orange string, spring,
weights, measuring tape and stands, standing speed of sound apparatus, function generator,
oscilloscope.
Suggested Reading Depending on which course you are in, you will have an assigned
physics text book. For additional reading for this lab refer to the relevant wave motion
section in your physics textbook.
6.1
Introduction
Interference is a fundamental property of waves. It occurs when two or more waves coexist
in the same medium and produce a resultant wave. A particularly interesting interference
occurs when two identical waves (of the same amplitude and frequency/wavelength) travel
in the same medium in opposite directions. The superposition of these two waves under
certain conditions leads to standing waves. Standing waves can be created only in a medium
of finite size, for example, in a rope or spring fixed at both ends or in a solid rod of a finite
length. A simple harmonic wave traveling in the positive x-direction can be described by
the following wave function:
y = A sin(kx − ωt)
65
Chapter 6. Standing Waves
In this wave equation A is amplitude, k = 2π/λ is the wave vector (in rad/m), and ω = 2πf
is angular frequency (in rad/s). It is important to distinguish angular frequency ω = 2πf
and frequency f = 1/T , where T is the period. The wavelength λ, frequency f , and speed
v of the wave are inter-related through the equation
v = λf
The speed of mechanical waves depends on the properties
p of the medium. For example,
the speed of a transverse wave in a rope is given by v = T /µ , where T is the tension in
the rope and µ = m/L is called linear mass density; m is the mass of the rope and L its
length.
p
The speed of the longitudinal wave moving in the spring is v = kL/µ, where k is the
spring constant of the spring (in units of N/m), L is the length of the spring, and µ = m/L
is called linear mass density of the spring with m being the mass of the spring.
When two identical waves travel in opposite directions, they interfere and a new resultant
wave is created. According to the superposition principle, the net displacement of the
resultant wave is equal to the algebraic sum of the displacements due to the individual waves.
If the displacements of the two waves are given by equations: y1 = A sin (kx − ωt) and
y2 = A sin (kx + ωt) then the resultant wave is y = y1 +y2 = A([sin (kx − ωt)+sin (kx + ωt)]
Using the identity
(a + b)
(a − b)
sin a + sin b = 2 sin
cos
2
2
we obtain
y = 2A sin(kx) cos(ωt)
The frequency of the resultant wave is the same as the frequency of the individual waves,
but the amplitude is 2A sin(kx) and so depends on x.
If the crests or troughs of the two waves traveling in opposite directions occur at the same
positions, the waves interfere ”constructively” and the resultant wave is bigger than the
individual ones. When the crest of the first wave coincides with the trough of the second
wave, then we have ”destructive” interference between the waves.
The maximum amplitude occurs for sin(kx) = 1, which happens when
kx =
nπ
,
2
n = 1, 3, 5, ...
Since k = 2π/λ, the maxima occur for when x = nλ/4. The positions at which the
amplitude is a maximum are called antinodes and are separated by λ/2.
The minimum amplitude occurs when sin(kx) = 0, which happens when
x=
nλ
,
2
n = 1, 3, 5, ...
Points corresponding to zero displacement are called nodes.
66
Chapter 6. Standing Waves
6.2
Standing Waves in a Rope and a Spring
6.2.1
Transverse Standing waves in a rope
The experimental setup for this section is shown in Figure 6.1. The suspended mass creates
Figure 6.1: The experimental setup to produce standing waves in a rope
tension T in the rope, T = Mhanging g. A mechanical wave driver, whose frequency is
controlled by the sine wave signal generator, creates a harmonic wave at one end. The wave
is reflected and inverted at the fixed end and returns to the source where it is reflected again.
If the mechanical wave driver sends out a new crest just as the reflected crest reaches it,
the new and reflected waves will reinforce each other. The two identical waves traveling in
opposite directions interfere and for certain discrete frequencies form a standing wave.
You will record your data in a table created in Logger Pro. To setup your table start Logger
Pro.
1. Double click on the column name to change it. A new window will appear. Enter the
column name: Standing Wave, enter N for the Short Nm and click Done.
2. Similarly, double click on the second column name and change the name to Wavelength, the Short Nm to λ (use the drop down menu to select the symbol) and enter
m for metres in the units box.
3. Next create a new column: Click on Data and select New Manual Column.... A
new window will appear. Name the column frequency with a Short Nm: f and units
Hz.
67
Chapter 6. Standing Waves
4. The last column will be created to automatically calculate the speed, v using the
formula v = λf . Click on Data and select New Calculated Column.... In the
new window that appears enter Speed for the name, ν for the Short Nm and m/s
for the units. In the expressions box, select columns λ and f by clicking on the
variables(columns) box and create the expression “wavelength” * “frequency”. This
will automatically multiply the data in the λ and f columns and populate the speed
column along each row.
Procedure: Setup the apparatus as shown in Fig. 6.1. Ensure the hanging mass, Mhanging ,
is not touching the table. (Adjust the orange string and distance between the mechanical
vibrator and lab stand as needed.) Do not adjust this distance (Length, Ls in Fig.
6.1) during the experiment.
Create at least four standing waves and measure their wavelength (remember that the
distance between two adjacent nodes is λ/2) using the steps below.
1. Turn on the sine wave signal generator and adjust the frequency to create a standing
wave. The frequency of the standing wave is displayed on the signal generator.
2. Measure the wavelength of the standing wave using the measuring tape.
3. Enter the frequency and wavelength values in your Logger Pro table. Notice the graph
and speed column are automatically populated as you enter your data. To display
speed on the y-axis of the graph, left click on the y-axis name and select speed from
the menu.
4. Determine the mean speed of the wave and the standard deviation from the mean
using the statistics icon on the toolbar. Include a screen capture of the table and
graph in Logger Pro (Ctr + Alt + PrtScn), or use the “Snipping Tool” app, and
include (paste) it in the Data section of your report.
p
Data Analysis: Next, calculate the speed of the wave using the formula v = T /µs , where
tension T = Mhanging g and linear mass density µs = mstring /Ls . In the experiment the
string is under tension (stretched). The linear mass density for the unstretched string,
µu = 0.0011 ± 0.0001 kg/m.
To get the linear mass density for the stretched string, µs , perform the following procedure
and record your measurements in the Data Analysis section of your lab report:
1. Measure the distance between the two pivot points of the stretched string (Ls ± δLs ).
2. To obtain the mass of the stretched string use the following procedure:
(a) Make a mark on the string pivot point near the end with the hanging mass, using
a piece of tape. Remove the mass while holding the string.
(b) Gently remove the string from the hook of the mechanical wave driver and place
it on the table.
(c) Measure the actual (unstretched) length of string corresponding to the stretched
length used on the apparatus. Call this Lu ± δLu .
68
Chapter 6. Standing Waves
(d) The mass of string used in the experiment is then mstring = µu Lu . (Don’t forget
to calculate the uncertainty in the mass!)
3. Calculate µs and its uncertainty. The µs to use in the expression for the velocity is
equal to mstring /Ls .
p
4. Calculate the velocity using v = T /µs , where tension T = Mhanging g. The expression for the uncertainty in v is:
δv
1
=
v
2
6.2.2
δT
δµs
+
T
µs
Questions
Answer these questions in the Discussion section of your laboratory report. Include the
subsection number, use full sentences, refer to your data and provide as much detail as
possible.
1. How does your calculated wave speed compare to the measured wave speed of the
transverse standing waves? Is the calculated speed within the uncertainty provided
by the statistics fit in Logger Pro?
2. Imagine that you create a second harmonic in a short and long rope (both of the same
mass density and with the same mass suspended at the end). Which of the three
quantities: λ, f and v, would be larger, smaller or the same for both rope?
3. How should the amplitude of the standing wave depend on the frequency of the wave?
Does it match with you observations? Can you think of some reasons why or why
not?
6.2.3
Longitudinal Standing Waves in a Spring
Procedure: Rearrange the standing wave apparatus as shown in Figure 6.2. Attach one end
of the spring to the hook of the mechanical wave driver and the other end to the upper
clamp of the lab stand. Ensure the spring is vertically aligned between the mechanical wave
driver and clamp, and perpendicular to the table surface. Create at least four standing
waves in the spring using the same procedure outlined in section 6.2.1. It is easier to see
longitudinal standing waves in the spring with 5 or more nodes. Tip: You can hold a blank
piece of paper behind the spring to help you see the wave nodes more distinctly. Measure
their wavelengths and record frequencies for which they occur. Use the formula v = λf to
determine speed v. You can reuse the table created in Logger Pro for the transerve wave
experiment. Clear the data in the cells (or create a copy) and enter your new measurements
for the longitudinal waves.
Determine the mean value of the speed and the standard deviation of the mean using the
statistics tool. Include a screen capture of Logger Pro displaying your table and graph in
your report.
69
Chapter 6. Standing Waves
Figure 6.2: Apparatus for studying longitudinal waves in a spring
p
Data Analysis: The formula for the speed is v = kL/µ where L is the length of the spring
used to obtain standing waves, k is the spring constant, and µ is the linear mass density of
the spring. Using the fact that µ = mspring /L we can rewrite this as
s
v=
kL2
=
mspring
s
k
mspring
L
The values for the spring constant k is (1.9 ± 0.1) N/m and mspring is (0.0274 ± 0.0001) kg.
Measure the spring length, L, (see Fig. 6.2) using a ruler or a measuring tape while it is
installed in the apparatus and calculate the value for the speed (uncertainty calculation
is optional).
70
Chapter 6. Standing Waves
6.2.4
Questions
Answer these questions in the Discussion section of your laboratory report. Include the
subsection number, use full sentences, refer to your data and provide as much detail as
possible.
1. How does your calculated wave speed compare to the measured wave speed of the longitudinal standing waves? Calculate the percentage difference as described in section
0.3.2.
2. Is the calculated speed within the uncertainty provided by the statistics tool in Logger
Pro?
3. Imagine that you create a third harmonic in two springs of the same length, and of
the same linear mass density, but one soft and the other stiff. Which of the three
quantities: λ, f and v, would be larger, smaller or the same for both springs?
6.3
Sound standing waves in the air tube
Sound waves are longitudinal waves. When a sound wave travels through the air, the air
particles vibrate along the direction of motion of the wave. This results in series of highand low-density regions of the air. The simplest kind of sound wave is called a ”simple
harmonic wave”, which corresponds to a sinusoidal variation in the density, as shown in
the figure below. Consider a simple harmonic wave generated by a loudspeaker, travelling
in a glass tube closed at one end. The wave will be reflected and inverted at the closed
end of the tube. The reflected wave returns to the source and there it is reflected again.
If the speaker is sending out a new crest just as the reflected crest reaches it, the new and
reflected waves will reinforce each other and a standing wave will be produced.
The closed end of the tube is always a node, as the air particles at the closed end do not
have the freedom to vibrate along the axis of the tube (sound is a longitudinal wave!). The
open end of the tube is an antinode, as particles have complete freedom of motion. The
wave which is created in the tube is called a standing wave and can occur only if the
length of the tube is equal to an odd number of the quarter-wavelengths:
L=
nλ
4
n = 1, 3, 5, ...
(6.1)
The wavelength λ, frequency f and the speed of sound v are inter-related via v = λf . Note
that the n in this equation is not the number of nodes, rather it is the number of quarter
wavelengths. You will experimentally determine λ and since you will know f can then find
the speed of sound v.
6.3.1
Experimental Setup
A schematic diagram of the experimental set up is shown in Figure 6.5. The length of the
air column in the glass tube may be varied by changing the position of the piston. A simple
harmonic wave is produced by the loudspeaker attached at the open end of the glass tube.
The loudspeaker is driven by the signal generator. A small microphone attached at the top
71
Chapter 6. Standing Waves
Figure 6.3: Illustration of transverse wave. Density is represented by variation of the shading
lines (b). The higher density corresponds to more closely spaced lines. Note that density
changes with distance (a) and time at a fixed position (c).
of the tube is connected to an oscilloscope. The signal from the microphone is amplified and
displayed on the screen of the oscilloscope. Resonance occurs when the maximum amplitude
signal is seen on the oscilloscope. Create a table in Logger Pro similar to the one shown in
table 6.1 to record your data and calculations.
Table 6.1: Acoustic standing waves in a hollow cylinder data and calculations
N
Frequency
(Hz)
Difference in Resonance
Lengths = λ/2 (m)
700
Resonance
Lengths
(m)
L1 =
1
2
700
L2 =
L3 − L2 =
3
700
L3 =
L4 − L3 =
4
700
L4 =
L5 − L4 =
5
700
L5 =
L2 − L1 =
72
Wavelength
λ (m)
Velocity
v = λf
(m/s)
Chapter 6. Standing Waves
Figure 6.4: Acoustic standing wave in a tube. This corresponds to n = 3 as there are 3
quarter wavelengths present within the tube.
6.3.2
Measurements and Calculations
1. Power on the oscilloscope, function generator, and microphone.
2. Adjust the frequency of the signal generator to about 700 Hz, and the amplitude to
low level sound, about half of the maximum level.
3. Move the piston in the tube starting at the end closest to the speaker until you hear a
substantial increase in the sound intensity (maximum amplitude on the screen of the
oscilloscope). That corresponds to the first resonance (formation of a standing wave).
Find other resonances in the tube.
4. Use the measuring tape to measure each resonance length, from the piston to the
open end of the tube. Record the resonance lengths in your table and complete the
calculations as shown in Table 6.1.
5. Adjust the frequency of the signal generator now to 1400 Hz and record all the positions of the piston for which a high intensity sound is detected at the open end of the
tube. Enter your positions in the table and repeat the same calculations as before.
6. Use the STAT tool in Logger Pro to calculate the average value of the velocity v and
the average deviation σ. In your calculations use all the velocities you obtained for
both frequencies since the speed of sound does not depend on the frequency. Include
a screen capture of your Logger Pro window with the table, graph and statistics box.
7. Compare you measured value of the speed of sound you determined experimentally
and the standard value for the speed of sound, 343 m/s (at room temperature and
zero altitude).
73
Standing Waves
Figure 6.5: Experimental setup for studying acoustic standing waves.
6.3.3
Questions:
Answer these questions in the Discussion section of your laboratory report. Include the
subsection number, use full sentences, refer to your data and provide as much detail as
possible.
1. Explain why speed of sound does not depend on the frequency of sound wave?
2. Explain how the speed of sound can be found keeping the piston at the same position
and changing the frequency of the sound wave with the signal generator?
3. With your group discuss what the wave envelopes for the first three standing waves
look like at the 700 Hz frequency. Use your measurements to help you explain.
end of lab
74
Appendix A
Sample Lab Report
Please read and sign the Assertion of Compliance with Academic Honesty Regulations
below. Failure to sign this document will result in a grade of zero on the lab report.
Assertion of Compliance with Academic Honesty Regulations. I assert that I have read
and understood York University’s Senate policy on Academic Honesty (Academic Honesty
Link) and will abide by that policy in the completion of the lab exercises and preparation
of this lab report. Furthermore, specific to these labs I assert that:
• I will not use data, answers to questions, use as a reference, or include any information
from another student’s lab report either from this year or a previous year.
• • I will not use web-based tutoring sites (ex. Chegg) or AI tools (such as chat GPT,
Grammarly) to complete my lab report.
• I will not use a cell phone/personal computer/tablet in the lab unless explicitly provided for by an accommodation or by authorization from the TA or Lab Coordinator.
• Although often working with a partner(s), I am fully aware of how all data was
collected, and I was directly involved in the experimental process of each lab exercise.
• Although often working with a partner(s) and naturally may have discussed answers
to question posed in the lab manual, I composed my own individual responses.
• The data presented in this lab report was the data I actually observed in the lab, and
not altered in any way for any reason.
By entering your name and date you agree to the terms presented above.
Name: Physics Rocks
Date: Sept. 18, 2022.
Experiment Title: Lab N - Heating
Laptop No.: 21
Lab Partner Name(s): Partner’s Name(s)
75
Appendix A. Sample Lab Report
Objective: In this section ensure you right the objectives of the lab in your own words.
DO NOT COPY THE LAB MANUAL. To determine how the temperature of an
object changes with time.
Equipment: In this section list all the equipment you used to conduct the experiments.
• Digital Thermometer
• Digital timer
Data: Use this section to report your data collected during an experiment.
T (◦ )
0.5
1.3
2.4
3.8
5.6
7.7
10.2
13.0
16.2
19.6
23.5
27.6
t (s)
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
δt = 0.01 s
,
76
δT = 0.5◦ C
Appendix A. Sample Lab Report
Figure A.1: Temperature as a function of Time
Data Analysis: Include any curve fitting results and calculations in this section.
Curve Fitting
Shown in Figure A.2 are the results of trying to fit the data to a straight line, T = mt + b,
and to a quadratic function, T = At2 + Bt + C. Clearly the quadratic function describes
the data better since ...
Uncertainty Calculations
The variable Q is related to the rate of change of the temperature. The uncertainty in the
average value of Q is:
∆Ti
, where ∆Ti = Ti+1 − Ti , ∆ti = ti+1 − ti
∆ti
δT = δTi+1 + δTi = 0.5 ◦ C + 0.5 ◦ C = 1.0 ◦ C
Qi =
δt = δti+1 + δti = 0.01 s + 0.01 s = 0.02 s
1 ◦C
0.02 s ◦
+
(4 C/s) = 1.2 ◦ C/s
0.8 ◦ C
0.2 s
1 ◦C
0.02 s δQ6 =
+
(12.5 ◦ C/s) = 6.25 ◦ C/s
◦
2.5 C
0.2 s
δQ1 =
Discussion and Answers to Questions: Answer any questions in this section and include any discussion related to your experiments.
77
Appendix A. Sample Lab Report
Figure A.2: Linear and Quadratic fits to the data
Conclusions: The conclusions should include your findings during the lab with a short
summary of the experiment. The temperature was found to vary .... The uncertainty on
the value of the fit parameter was ... This agrees/disagrees with expectations ... The
experiment could possibly be improved by ...
By the way, just to show that there are devices which have characteristics that vary with
temperature as in this sample lab report, here is the variation of breakdown voltage with
temperature for a device called a Silicion Photomultiplier (SiPM). Silicon Photomultipliers
are used extensively for reading out light signals in the ALPHA antihydrogen experiment.
78
Appendix A. Sample Lab Report
Figure A.3: SiPM Breakdown voltage versus temperature
79
Appendix B
Statistical Treatment of Random
Error
When it is practical to make many readings of a measurement, the scatter of the readings
can be used to indicate the uncertainty.
For example, the width d of a piece of wood could be measured at many different points
and a quantitative estimate of the scatter of the readings can be obtained by calculating the
difference between each reading and the average value. The average of these values gives
the average deviation.
Width (cm)
Sum
12.32
12.35
12.34
12.38
12.32
12.36
12.34
12.38
98.79
Sum of Absolute deviation from mean
Devitation
from Mean
(cm)
0.03
0.00
0.01
0.03
0.03
0.01
0.03
0.03
0.17
Mean d = 98.97/8 = 12.35. Average Deviation σ = 0.17/8 = 0.02.
You would report the measured width as (d ± σ) = (12.35 ± 0.02)cm.
In general, if a quantity x is measured N times, then the mean of x is
P
x=
i xi
N
81
Appendix B. Statistical Treatment of Random Error
and the average deviation as
P
σ=
i |xi − x|
N
There is a lot more to statistics than this, starting from the design and sampling techniques
of experiments through the calculation of results, but this is sufficient for our elementary
purposes.
82
Appendix C
Method of Least Squares
The method of least squares is used when one wishes to fit a set of data to a given equation.
We shall deal with the simplest of cases, the straight line
Figure C.1
Consider Fig. C.1. The mathematically best straight line through the set of data points is
shown, together with the data. Not every data point falls on the line. There is a difference
between the fitted y value and the actual y value. The purpose of least squares is to minimize
these deviations over all the data points (N points).
The equation of a straight line may be given by:
y = mx + b
The difference between the fitted y, on the straight line, and the actual yi may be given
by:
b + mxi − yi
83
Appendix C. Method of Least Squares
Summing the squares of these differences gives us the total square uncertainty, or (if divided
by N ) the variance.
X
σ=
(b + mxi − yi )2
P
where
means ”the sum over all data points”. It is the sum that we wish to minimize.
We accomplish this by taking the derivative of σ with respect to each of the individual
parameters b and m and set these simultaneously equal to zero.
The derivatives with respect to b only may be expressed as:
X
∂σ
=2
(b + mxi − yi ) = 0
∂b
follows from simple calculus
The operator ∂/∂b means ”take the derivative with respect to b only and treat all other
numbers as constant”.
Expanding the summation, dropping the factor of 2, and moving the term with yi to the
right of the equals sign gives:
X
X
X
b+
mxi =
yi
the parameters b and m can be moved from inside the summation, as they do not change
during the summations, i.e.:
X
X
bN + m
xi =
yi
The parameter b is multiplied by N , the number of data points, as summing the constant,
b, N times is bN .
Similarly:
X
∂σ
=2
(bxi + mx2i − xi yi = 0
∂m
and this equation may be manipulated as the one for b:
X
X
X
bxi =
mx2i =
xi yi
to yeild
b
X
xi + m
X
x2i =
X
xi yi
solving equations gives:
P
Intercept : b =
Slope : m =
N
P
P
P
x2i
y − xi yi xi
P i2
P 2
N xi − ( xi )
P
P P
xi yi − xi yi
P
P
N x2i − ( xi )2
84
Appendix C. Method of Least Squares
In order to find the best straightPline through
P 2 Pa given set
P of data points, one needs to know
the number of points, the sums
xi ,
xi ,
yi and
xi yi then substitute these into the
equations above. Note that the denominators in the equations are the same and, therefore,
need only be calculated once.
It can be shown that the uncertainty of the slope is given by the following expression
sP
[(yi − y) − m(xi − x)]2
P
Uncertainty in Slope : δm =
(N − 2) (xi − x)2
P
where :
xi
x = mean(x) =
N
P
and y = mean(y) =
yi
N
The method of least squares requires the evaluation of a few complicated expressions. To
evaluate these expressions precisely and quickly, it is worth setting up a computer program.
85
Appendix D
Uncertainty Calculations using
Differentials
If x, y etc., are measured quantities how does one calculate the resultant uncertainty in p
where
(i) p = x + y
(ii) p = x − y
(iii) p = xy
(iv) p = x/y
and combinations of all the above?
Let the absolute uncertainties be interpreted as differentials dx, dy, etc
Consider (i)
p=x+y
Taking differentials dp = dx + dy
and considering (ii)
p=x−y
Taking differentials dp = dx − dy
The resultant uncertainty which we estimate is to be a maximum (upper) and minimum
(lower) bound. Since the uncertainties are always ±dx; ±dy, etc., we must, therefore,
choose the sign which will give the greatest uncertainty. This, in effect, means that after
the differentials have been taken, as a last step before substituting numbers, we should
change all - signs to + signs.
Therefore, for addition and subtraction dp = dx + dy i.e. the final uncertainty is
the sum of absolute uncertainties.
87
Appendix D. Uncertainty Calculations using Differentials
Consider
(iii) p = xy
Taking differentials : dp = xdy + ydx
This would be complicated to compute, but may be simplified by dividing (2) by (1).
dp
xdy + ydx
dy dx
=
=
+
p
xy
y
x
(D.1)
Note that dp/p, dx/x, dy/y are just the relative uncertainties in p, x, y.
Multiply through equation (3) by 100 to convert to a percentage uncertainty
dp
dx
dy
× 100 =
× 100 +
× 100
p
x
y
For division: Consider
(iv) p = x/y
ydx − xdy
Taking differentials : dp =
y2
This also seems complicated to compute but can be simplified by dividing equation (5) by
equation (4):
dp
=
p
ydx−xdy
y2
x
y
=
ydx − xdy
y
y2x
dp
dx dy
=
−
p
x
y
By the same reasoning as in (ii) above, we will now change the -sign to + sign.
dp
dx dy
=
+
p
x
y
which is the same as (3)above.
Therefore, for multiplication and division, the final uncertainty, expressed as a
percentage, is the sum of percentage uncertainties of the components.
Note that if one of the factors is taken to a power e.g. p = y a then by differentials
dp = ay a−1 dy
dp
ay a−1 dy
ady
=
=
a
p
y
y
88
Appendix D. Uncertainty Calculations using Differentials
However, if p = 2xy then dp = 2(xdy + ydx) so
dp
2(xdy + ydx)
dy dx
=
=
+
p
2xy
y
x
When addition, subtraction, multiplication and division are all involved in one formula,
things become more complicated. These may be treated by the differential process or by
using the rules on one part of the function at a time.
Examples
The dimensions of a room are:
length (l) :
(3.04 ± .02) m
width (w) :
(3.66 ± .02) m
height (h) :
(2.74 ± .02) m
What are the resultant uncertainties in
(a) the perimeter of the floor?
Perimeter P = 2× length + 2 × width = 2l + 2w = 2(3.04) + 2(3.66) = 13.40 m.
Uncertainty in perimeter is δP = 2δl + 2δw = 2(0.02) + 2(0.02) = 0.08 m.
So the perimeter of the floor is therefore (13.40 ± 0.08) m.
(b) the floor area?
Area Af loor = length × width = l × w = (3.04 m)×(3.66 m) = 11.13 m2
Uncertainty in Area is :
δAfloor
Afloor
δAfloor
dl dw
+
l
w
dl dw
+
Afloor
=
l
w
0.02 0.02
=
+
11.13
3.04 3.66
=
= 0.13 m2
So the area of the floor is (11.13 ± 0.13) m2
(c) the wall area?
Awalls = 2(length × height) + 2(width × height)
= 2(3.04)(2.74) + 2(3.66)(2.74)
= 36.71 m2
and
89
Appendix D. Uncertainty Calculations using Differentials
δAwalls = δ(2l × h) + δ(2w × h)
δl δh
δ(2l × h) =
2lh
+
l
h
0.02 0.02
δ(2l × h) =
2(3.04)(2.74)
+
3.04 2.74
2
= 0.23
m
δw δh
2lh
δ(2w × h) =
+
w
h
0.02 0.02
δ(2l × h) =
2(3.04)(2.74)
+
3.04 2.74
= 0.26 m2
δAwalls = (0.23 + 0.26) m2
= 0.49 m2 = 0.5 m2
So the area of the walls is (36.7 ± 0.5) m2 .
90