1. No category

Brock University
Physics Department
St. Catharines, Ontario, Canada L2S 3A1
PHYS 1P91 Laboratory Manual
Physics Department
c Brock University, 2014-2015
Copyright Contents
Laboratory rules and procedures
1
Introduction to Physica Online
5
1 The pendulum
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2 Check your schedule!
19
3 Angular Motion
21
4 Collisions and conservation laws
29
5 The ballistic pendulum
35
6 Harmonic motion
41
A Review of math basics
47
B Error propagation rules
49
C Graphing techniques
51
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Laboratory rules and procedures
Physics Department lab instructors
Frank Benko, office B210a, ext.3417, [email protected]
Phil Boseglav, office B211, ext.4109, [email protected]
This information is important to YOU, please read and remember it!
Laboratory schedule
To determine your lab schedule, click on the marks link in your course homepage; your lab dates are
shown in place of the lab marks and correpond to the section number that you selected when you registered
for the course. You cannot change these dates unless you have a conflict with another course and make
the request in writing.
The schedule consists of five Experiments to be performed every second week on the same
weekday. On any given day there are four different Experiments taking place, with up to three groups
of no more than two students in a group performing the same experiment. You need to:
• prepare for your scheduled experiment. Out of schedule Experiments cannot be accomodated;
• be on time. The laboratory sessions begin at 2:00 pm and end no later than 4:45 pm and
you will not be allowed entry once the experiments are under way.
Lab report format and submission
You are required to submit the Discussion component of your Lab Report to www.turnitin.com
prior to the lab submission deadline. Instructions for registering and submitting your work are found in
your course web page. Be sure to have a working account before you need to use it.
After you submit your Discussion to the Turnitin webpage, Turnitin will email to your
Turnitin login email address (i.e. [email protected]) a receipt tagged with the submission date and a
unique ID number. Include this email as part of your lab report.
Submit your report in the clearly marked wooden box across the hall from room MC B210a. Reports
are due by midnight one week after the experiment is performed. For example, the report for
an experiment performed on a Tuesday is due by midnight on the Tuesday following.
Compile your Lab Report as follows:
• submit the complete Lab Report in a clear-front document folder.
Do not use three-hole Duo-tang folders, envelopes or submit a stapled set of pages;
• insert the first lab worksheet so that your name is visible through the folder front cover.
Do not include a title page as the first experiment page is the title page;
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2
• add the other lab worksheets in the proper sequence, followed by printouts and pages of calculations.
• At the end of the Lab Report include a complete copy of the email sent by Turnitin,
followed by a complete copy of the Discussion that you submitted to Turnitin. This can be a printout
from your Word Processor or a copy downloaded from the Turnitin web site.
• Note: you should anticipate and be prepared for the likelyhood that Turnitin may not provide an
immediate email response following your Discussion submission; this response may take several hours.
Submit your work well ahead of the submission deadline.
• Note: Late Lab Reports will receive a zero grade, no exceptions.
• Note: Lab Reports not formatted as outlined will receive a 20 % grade deduction.
• Note: Marked Lab Reports will be returned to you during your next Lab session. Be sure to pick-up
your marked labs and review the comments made by the marker.
The lab manual
Your lab manual is available as a .PDF document in your course webpage. This allows you to
print a copy of the experiment that you need for the current lab. It also allows the department to make
quick edits to the manual to fix typographical errors, etc.
This lab manual contains five experiments. Each experiment consists of three components, and
completing the lab means reaching all three of the milestones described below.
1. Pre-lab review questions, to be completed before entry into the lab, are intended to ensure
that the student is familiar with the experiment to be performed. A Lab Instructor will initial
and date the review page if the questions are answered correctly. The review questions contribute
to your lab grade.
• You will be required to leave the lab if the review questions are not completed as
instructed. Missing your assigned lab date could result in a grade of zero for that Experiment.
• Be sure to have a TA check and initial the completed review questions before you begin the
lab. Lab reports missing the initials will be subject to a 20 % grade deduction.
• In case of difficulties with any of the review questions, a student is expected to seek help from
a lab instructor well before the day of the lab.
2. A lab component is the actual performing of the experiment. Marks are deducted for failing
to complete all of the required procedures, follow written instructions, answer questions, provide
derivations, the improper use of rounding and incorrect calculations. The lab report markers use a
standard marking scheme to grade the lab reports.
At the end of the lab session, if the lab procedures have been completed as required, a Lab Instructor
will also initial and date the front page of your Experiment.
• An incomplete lab component will not be initialled; you will need to finish the work on
your time and have it signed before submission. A report missing this signature will be subject
to a 20 % grade deduction.
3. The final component is the compilation of the experimental data, its analysis, and a
critical assessment of the results into a lab Discussion. This component is worth 40 % of the
lab mark.
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The Discussion should consist of a series of paragraphs rather than an itemized list of one-line answers.
You do not need to review the theory or reproduce formulas or tables of experimental data contained
in the workbook as part of the discussion. You should:
• begin the discussion with a tabulated summary of your data, properly rounded according
to the associated margin of error;
• thoughtfully answer and expand on the given guide questions, outline your observations, summarize the results of the experiment and support your conclusions with data or reasoned arguments;
• assess the validity of your results by comparing your values and their associated errors with
values estimated from the theory or cited in your textbook or other literature.
Suggestions for improving the experimental procedure and a summary of the implications of the
obtained results will make the discussion complete.
A guide to team collaboration
To ensure that the collaborative nature of the experimental team is expressed in a fair and mutually
advantageous way for every member of the team:
• Come prepared and ready to participate constructively as part of your lab team!
• Do not sit idle and expect others to provide you with their data. The data gathering procedures
should be undertaken by all the members of the team. While it may not be practical to have every
student perform the same reading every time, each member of the team must become familiar with
the equipment and perform some of the readings. The lab instructor will ask procedural questions
during the lab and you will be expected to know what is going on in the experiment.
All measurements are to be made by more than one student. This is a very effective way to verify
a measurement; the use of an incorrect value in a lengthy calculation can waste a lot of your team’s
lab time and result in an incomplete lab. All labs finish by 4:45pm sharp.
• Do your own calculations!. There is sufficient time during the lab for this to be accomplished.
As above, this is also a good strategy; comparing the results of several independent calculations can
expose numerical errors and lead to the correct result or give you the confidence that your result is
indeed correct. To access a calculator on your workstation, type xcalc in a terminal window.
• Submit your own set of graphs. Enter your own data, include your name and a description of
the plotted data as part of the title. This approach will also expose any errors in the data entry or
the computer analysis of the data. Needless to say, the Discussion section of the lab report is not to
be a collaborative effort.
• Warning: Do not copy someone else’s review questions, calculations or results. This is an insult
to the other students, negates the benefits of having an experimental team and will not be tolerated.
Any such situations will be treated as plagiarism. You should review in your student guide Brock
University’s definition and description of plagiarism and the possible academic penalties.
• Warning: Do not allow others to copy the content and results of your calculations or review
questions; doing so makes you equally responsible under the definition of plagiarism. Do not feel
pressured to allow another student to copy your work; inform the lab instructor.
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Academic misconduct
The following information can be found in your Brock University undergraduate calendar:
”Plagiarism means presenting work done (in whole or in part) by someone else as if it were one’s
own. Associate dishonest practices include faking or falsification of data, cheating or the uttering of false
statements by a student in order to obtain unjustified concessions.
Plagiarism should be distinguished from co-operation and collaboration. Often, students may be permitted or expected to work on assignments collectively, and to present the results either collectively or
separately. This is not a problem so long as it is clearly unbderstood whose work is being preented, for
example, by way of formal acknowledgement or by footnoting.”
Academic misconduct may take many forms and is not limited to the following:
• Copying from another student or making information available to other students knowing that this
is to be submitted as the borrower’s own work.
• Copying a laboratory report or allowing someone else to copy one’s report.
• Allowing someone else to do the laboratory work, copying calculations or derivations or another
student’s data unless specifically allowed by the instructor.
• Using direct quotations or large sections of paraphrased material in a lab report without acknowledgement. (This includes content from web pages)
Specific to the Physics laboratory environment, you will be cited for plagiarism if:
• you cannot satisfactorily explain to the lab instructor how you arrived at some numerical answer
entered in your laboratory workbook;
• you cannot satisfactorily describe to the lab instructor how you derived a particular equation in the
lab procedure or as part of the review questions;
• your data is identical to that of some other student when the lab procedure stated that each student
should obtain their own data.
The above points are based on the conclusion that if you cannot explain the content of your workbook,
you must have copied these results from someone else.
In summary, you are allowed to share experimental data (unless otherwise instructed) and compare
the results of calculations and derivations for correctness with other members of your group, but the
derivation of results must be your own work.
Academic penalties
A first offence of plagiarism in the lab will result in the expulsion of all parties concerned from that
lab session and the assignment of a zero grade for that particular lab. A record will be made of the event
and placed in your student file.
A subsequent offence will initiate academic misconduct procedures as outlined in the Brock University
undergraduate calendar.
Introduction to Physica Online
Overview
Figure 1: Physica Online opening screen
Physica Online is a web-based data acquisition and plotting tool developed at Brock University for
the first-year undergraduate students taking introductory Physics courses with labs. It is accessible on the
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INTRODUCTION TO PHYSICA ONLINE
Web at
http://www.physics.brocku.ca/physica/
When you first point your browser to this page, you should see approximately what is shown in Fig. 1.
The “engine” behind the Web interface is the program physica written at the Tri-University Meson
Facility (TRIUMF) in Vancouver. It is also available in a stand-alone menu-driven Windo$se version from
http://www.extremasoftware.com/.
First and foremost, Physica Online is a plotting tool. It allows you to produce high-quality graphs
of your data. You enter the data into the appropriate field of the Web interface, select the type of graph
you want, and make some simple choices from the self-explanatory menus on the page. After that, a single
button generates the graph. You can view it, print it, save it as a PostScript file for later inclusion into
your lab report.
Physica Online is a fitting and data analysis tool. The physica engine has extensive and powerful
fitting capabilities. Only a small subset is used in the “easy” default web mode, but it is sufficient for all
of the experiments that Brock students encounter in their first-year labs. The full “expert” mode is also
available for those needing more advanced capabilities of full physica; some learning of commands may
be required.
Physica Online is a data acquisition tool. The web interface connects to a LabProTM by Vernier
Software or a similar interface device, typically attached to a serial port of one of the thin clients1 (or of
some serial-port server). You can then copy-and-paste the returned data into the data field of Physica
Online, ready to be plotted and/or analysed.
Below, we will follow the approximate sequence of a typical lab experiment. A demo mock-up of one
such experiment (RC time constant determination) is available online, even from outside of the lab. It
may be useful to open a browser, and point it to the demo RC lab while reading this manual.
Acquiring a data set
The data acquisition hardware consists of a variety of interchangeable sensors connected to a programmable
interface device called a LabPro. This unit samples the sensors and transmits the data to a serial port of
a thin client (or a personal computer).
To acquire a set of data from a sensor press Get data in the control panel of Physica Online. In the
main plot frame to the right, a LabPro frame will open up, similar to the one seen in Fig. 2. In this frame
you have to set several options by hand.
Begin by identifying the IP address of the thin client to which the LabPro hardware is attached. The
thin client is identified as ncdXX, where XX should be set as indicated by the label on the terminal. Usually,
it is the one you are sitting at, but sometimes you may need to use the LabPro attached to another thin
client. Several groups of students can use the same hardware device to collect data, but not at the same
time! In the example shown in Fig. 2, the IP is set to ncd36.
Next, specify one or more channels from which the data will be read. There are four available analog
channels, Ch1–Ch4, used to attach probes of voltage, temperature and light intensity. The two digital
channels, Dig1 and Dig2, are used to connect probes such as photogate timers and ultrasonic rangefinders.
More than one channel can be selected; in this case, more than two columns of data will be returned by
the LabPro. In the example of Fig. 2, a single voltage probe is attached to Ch1.
Select the number of data points to be collected, the delay between successive data points and then
initiate the data acquisition by pressing the Go button. Once the data collection begins, a progress
message appears in this window indicating the time required to complete the data collection. Be patient
1
Thin clients are desktop devices that provide a display, a keyboard, a mouse, etc., but that do not have a disk or an
operating system software. Instead, they connect to one of several possible servers, running whatever operating system that
is required. The files and applications run on these servers, and the thin clients, or Xterminals, take care of the interactions
with the user.
INTRODUCTION TO PHYSICA ONLINE
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Figure 2: LabPro configuration frame
and let the LabPro process complete. If something is wrong and the browser is unable to communicate to
the LabPro, it will time out after a few extra seconds of waiting. This may happen if the network is busy
or if more than one browser is trying to obtain the data from the same LabPro; check that your IP field
is set correctly.
Once the data acquisition is complete, you will have two columns of data in front of you. The next
step is to select the data using your mouse, and copy-and-paste it to the data entry field below, as shown
in Fig. 3. You can also do this by pressing the Copy from above button. You are now ready to plot
and fit the data.
Graphing your data
The data entry field of Physica Online is usually filled through a copy-and-paste operation from the LabPro
frame, as described in the previous section. You can also manually enter into this field any other data2
that you wish to graph and analyse. You can press Ctrl+A to select all of the contents in the data window
and Ctrl+X to delete those contents. On some machines, you need to use Alt+A and Alt+X.
The default settings are appropriate for generating a scatter plot of the data; all you need to do after
entering or pasting in the data is to press Draw .
You have the option of associating error bars with each data point by entering one or two extra columns
into the data field; the third column, if present, would be interpreted as ∆y, and the fourth one, if present,
as ∆x. If the error is the same for all data points, you may use the the dx: and dy: fields below the data
field. To omit the error bars, set these values to zero (this is the default).
There are several graphing options available. A scatter plot graphs a set of coordinate points using a
chosen data symbol of a specific size. Check the line between points box to connect the data points
with line segments or the smooth curve box to interpolate a smooth curve through the data points. After
you made all your selections and pressed Draw , you may see the plot that looks similar to that shown
in Fig. 4.
2
Feel free to use Physica Online for preparing graphs for your other lab courses!
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INTRODUCTION TO PHYSICA ONLINE
Figure 3: LabPro data has been copy-and-pasted into the data entry field
Figure 4: Physica Online data scatter plot
INTRODUCTION TO PHYSICA ONLINE
9
The fit to: y= option allows you to fit an equation of your choice to the data points. A second-order
polynomial is pre-entered as the default, but in each experiment this will need to be changed to reflect the
expected form of the y(x) dependence. The simple web interface allows up to four fitting parameters A, B,
C, D, which is enought for most of the first-year lab data. Much more elaborate fitting is possible from the
“Expert Mode” of Physica Online. One essential point about fitting, especially when the fitting equation
contains non-linear functions such as sin(x) or exp(x), is to have a good set of approximate initial guesses
for all fitting parameters. Examine the scatter plot of your data carefully, estimate the approximate values
of all parameters your use in the fitting equation and enter your initial estimates in the fields provided. The
default values of A = B = C = D = 1 are almost never going to work. If the fit fails to converge, Physica
Online will return a text error message when you press Draw , you should then re-examine whether the
fitting equation and the initial guesses for all parameters have been entered correctly.
You can fit more than one function to your data set, such as for example, a steady-state straight line
followed by an exponential decay. These fits are explained further in the experiment in which they occur.
You can also constrain the fit to two separate regions of your data set. In this case, you must copy-andpaste the fitting equation used in the fit to: y= box into the constrain X to: box or the error values
will be incorrect.
Additional settings allow you to display the plot with the axes scaled in linear or logarithmic units,
and the scale limits and increments can be manually set. A grid can be optionally included. The font and
size of the text used to label the axes and in the title is also user selectable. Fig. 5 shows a set of values
Figure 5: Physica Online fit and plot parameters
and settings that Ms. Jane Doe may have used for her data set. When she presses Draw , Physica Online
returns the plot shown in Fig. 6.
An Expert mode button is available if other, more advanced features of physica are desired. Selecting this mode passes on all the settings from the “Easy Mode” and allows further changes to be made
directly to the physica macro script. There is on-line help and tutorials, as well as hardcopy reference
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INTRODUCTION TO PHYSICA ONLINE
Figure 6: Experimental data and the fit, with settings from Fig. 5
manuals if you want to learn how to use these advanced features. For example, you may wish to plot two
data sets on the same graph and add a legend. Feel free to change the macro; if you run into difficulties,
press Easy mode and start again.
You can press Print to redirect the output to a PostScript printer. Only a few printer names are
accepted as valid by the web script; your TA will tell you which one to use. If you leave the Print to:
field blank, a PostScript file will be sent to your browser; if the browser knows how to display PostScript
(though GhostView, or Adobe Acrobat, or a similar external application) it will do so. Otherwise, it should
offer you an option to save it as a file; this is useful for later including your plot in a lab report or attaching
it to an email.
Your browser’s Back button will come in handy on occasion. If you find yourself hopelessly lost, you
can also use Reload to bring you back to the starting point, although this will reset the graph settings
such as title and axis labels to their default values.
first name (print)
last name (print)
Brock ID (ab13cd)
TA initials
grade
Experiment 1
The pendulum
A simple pendulum consists of a compact mass m suspended from a fixed point by a string of length L, as
shown in Fig. 1.1.
Figure 1.1: Pendulum experimental arrangement
The equilibrium position of m is O. Here, the tension P~ exerted by the string on m is exactly equal and
opposite to the weight m~g of the mass. This is not true however if m is anywhere else along the arc, say
at an angle α (in radians) with respect to O. Then the weight will have a component tangent to the arc,
whose magnitude is m~g sin α. If α is small, we may approximate sin α ≈ α, so that the force on m is equal
to mgα. This force is a restoring force, as it will drive m back to its equilibrium position O. When a mass
is acted on by a restoring force, whose magnitude mgα is proportional to the deviation α from O, then
the resulting motion is an oscillation around the equilibrium position. In our case, m will swing back and
forth around O. The time for one complete swing is defined as the period T . An equation for acceleration
due to gravity g is given by:
g=
4π 2 L
T2
(1.1)
This equation predicts that the period T is independent of the mass m of the ball, and that T is also
independent of the angle α through which the ball swings, as long as the approximation sin α ≈ α is valid.
For α = 15◦ ≈ 0.2618 radians, there is a difference of approximately 1.2% between α and sin α.
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EXPERIMENT 1. THE PENDULUM
Introduction to error analysis
The result of a measurement of a physical quantity must contain not only a numerical value expressed in
the appropriate units; it must also indicate the reliability of the result. Every measurement is somewhat
uncertain. Error analysis is a procedure which estimates quantitatively the uncertainty in a result. This
quantitative estimate is called the error of the result. Please note that error in this sense is not the same as
mistake. Also, it is not the difference between a value measured by you and the value given in a textbook.
Error is a measure of the quality of the data that your experiment was able to produce. In this lab, error
will be considered a number, in the same units as the result, which tells us the precision, or reliability,
of that experimental result. Note that error value, represented by the Greek letter σ (sigma), is always
rounded to one significant digit; the result is always rounded to the same decimal place as σ (see below).
Error of a single measurement
Consider the measurement of the length L of a bar
using a metre stick, as shown in Figure 1. One can
see that L is slightly greater than 2.1 cm, but because the smallest unit on the metre stick is 1 mm,
it is not possible to state the exact value. We can,
however, safely say that L lies between 2.1 cm and
2.2 cm. The proper way to express this information
is:
L ± σ(L) = 2.15 ± 0.05 cm
This expression states that L must be between,
(2.15−0.05) = 2.10 cm and, (2.15+0.05) = 2.20 cm,
which is our observation. The quantity σ(L) =
Figure 1.2: Measurement with a metre stick
±0.05 cm is referred to as the maximum error. This
number gives the maximum range over which the correct value for a measurement might vary from that
recorded, and represents the precision of the measuring instrument.
Propagation of errors
In many experiments the desired quantity, call it Z, is not measured directly, but is computed from one
or more directly-measured quantities A, B, C, . . . with a mathematical formula. In this experiment, the
directly-measured quantities are T , y and b, and the desired quantity g is calculated from g = 4π 2 L/T 2 ,
with L = y + 12 b. The following rules give a quick (but not exact) estimate of σ(Z) if σ(A), σ(B) etc.
are known Always use the absolute value of an error in a calculation . Error rules are tabulated in the
Appendix.
1. If Z = cA, where c is a constant, then σ(Z) = |c|σ(A). This is used only if A is a single term. For
example, it can be used for Z = 3y, so that σ(Z) = 3σ(y), but not for Z = 3xy.
2. If Z = A + B + C + · · ·, then σ(Z) = σ(A) + σ(B) + σ(C) + · · · . For example, if
1
b
2
1
σ(L) = σ(y) + σ
b (See 2. above.)
2
1
σ(L) = σ(y) + σ(b) (See 1. above.)
2
L = y+
then
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3. To derive an error equation for any relation, rewrite that relation as a series of multiplications, then
apply the change of variables method as shown in the Appendix to evaluate the error terms:
g=
4π 2 L
−→ g = 4π 2 LT −2 −→ g = ABCD, ( letting A = 4, B = B = π 2 , C = L, D = T −2 )
T2
Then
and
σ(A) = σ(4),
σ(g)
σ(A) σ(B) σ(C) σ(D)
=
+
+
+
,
|g|
|A|
|B|
|C|
|D|
(Appendix, Rule 4)
σ(π)
σ(T )
σ(D)
σ(B)
= |2|
= | − 2|
, σ(C) = σ(L),
.
|B|
|π|
|D|
|T |
(Rules 1,6)
The quantities 4 and π are constants and have no error (strictly speaking, σ(4) = σ(π) = 0), therefore
these terms do not contribute to the overall error. The error equation for g then simplifies to
σ(π)
σ(g)
σ(4)
=
+2
|g|
|4|
|π|
σ(T )
σ(L)
+ |−2|
+
|L|
|T |
σ(T )
σ(L)
σ(g)
=
+2
.
|g|
|L|
|T |
−→
The right hand side of the above equation, called the “relative error” of g, results in a fraction that
describes how large σ(g) is with respect to g. The desired quantity, σ(g), is obtained by multiplying
both sides of the equation by g:
σ(T )
σ(L)
+2
σ(g) = |g|
|L|
|T |
.
Standard deviation of a series of measurements
When the same measurement or calculation is repeated several times, an error estimate can be calculated
form the variation in this set of values. This is convenient because on other error values, measured or
calculated, need to be known. For example, the five results for g can be used to estimate an error σ(g).
To calculate the standard deviation σ(g) for a sample of N values of g:
1. determine the average hgi of N values gi , where i = 1, 2, · · · , N by summing the values and then
dividing by the number of values:
hgi =
1
(g1 + g2 + · · · + gN );
N
2. for each gi , calculate the deviation ∆gi = gi − hgi from the average hgi;
3. for each gi , calculate the square of the deviation (∆gi )2 to make all the values positive;
4. calculate the variance σ 2 (g) of the sample by summing the squares of the deviations (∆gi )2 and
dividing by the number of values minus one, N − 1
σ 2 (g) =
1
2
(∆g12 + ∆g22 + · · · + ∆gN
);
N −1
5. undo the previous squaring operation by taking the square root of the variance. This is the root
average squared deviation, or standard deviation σ(g), of b:
σ(g) =
q
σ 2 (g)
A large σ value indicates a result which is not very precise. The theory of statistics shows that the
probability
that a further measurement of g falls within the range of σ(g) is 68% and that σ is proportional
√
to 1/ N , so σ will decrease as the number of samples N is increased.
14
EXPERIMENT 1. THE PENDULUM
Rounding a final result: X ± σ(X)
The value of σ(x) is rounded to one significant digit whether it represents a maximum error estimate,
calculated error, or standard deviation of a sample. The result corresponding to this error must be rounded
off and expressed to the same decimal place as the error. For example, hxi = 25.344 mm and σ(x) =
0.0427 mm. Rounded to one digit, σ(x) = 0.04 mm. Rounded to the same decimal place, hxi = 25.34 mm.
The final result is expresses as hxi ± σ(x) = (25.34 ± 0.04) mm.
Do not use a rounded off value in further calculations. Use the original unrounded value. Use of a
truncated value will decrease the quality of your result.
Powers of 10
Express both the result and its error to the same power of 10. This allows the reader to immediately judge
how large the error is relative to the result:
1. 2.68 × 10−2 ± 5 × 10−4 should be written as 0.0268 ± 0.0005 or (2.68 ± 0.05) × 10−2 . Note the
parentheses, indicating that both the result and the error are to be multiplied by 10−2 .
2. 1.634 ± 3 × 10−3 m should be written as 1.634 ± 0.003 m
Format of calculations
Record all calculations, in the appropriate space if provided or on a separate sheet of paper. A calculation
is performed in three lines. The first line displays the formula used. In the second line, the variables in the
formula are replaced with the actual values used in the calculation. The third line shows the final answer
formatted according to the section on Rounding above and if any, the units associated with the result.
Review questions
Using a textbook, the Physics Handbook, the Internet or some other source, find the accepted value of g.
Cite your reference so that a reader can find this information. Be specific; do not cite ’my physics book’,
’my buddy’ or ’i remember it from high school’ as your source. Include specific info such as the uncertainty,
or error, in the value, and the condition (i.e. sea level) under which it was measured.
g = ..............................................................................
An error value is only meaningful when expressed with ..... significant digits.
The measurement error σ of a scale graduated in minutes and seconds is ....................
My Lab dates: Exp.1:...... Exp.3:...... Exp.4:...... Exp.5:...... Exp.6......
I have read and understand the contents of the Lab Outline (sign) ................
CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT!
Procedure and analysis
You will determine the period T of a pendulum of length L by monitoring the distance of the pendulum bob
from a computer controlled rangefinder. This device determines the distance to an object by transmitting
an untrasonic ping and measuring the time required to receive the echo of the original signal after it is
reflected from the object.
15
The transmitted signal propagates in a narrow conical beam and as it is the echo from the nearest
object that is accepted, you will need to be sure that you are pinging the pendulum bob and not the
pendulum arm or some other structure in the cone of the beam. As the pendulum bob swings toward and
away from the rangefinder, the variation in distance over time can be approximated by a sine wave. By
analysing the properties of this sine wave, we can calculate the period T of the pendulum.
The pendulum is an aluminum ball suspended by a light nylon string from a sliding arm. Assume the
mass of the string is negligibly small compared to the mass of the ball. To calibrate the pendulum:
1. Release the string to set the ball on the table. Align the bottom of the sliding arm with the zero
mark on the scale. Adjust the string length so that the top of the ball lightly contacts the bottom of
the arm, ensuring that the string is not stretched. Secure the string with the clamping nut.
2. Raise the arm away from the ball and carefully reposition the arm until it once again just contacts
the top of the ball. If the bottom of the arm is no longer at the zero mark, repeat the sequence of
adjustments until the bottom of the arm is in line with the zero mark of the scale and the arm lightly
contacts the top of the pendulum ball. All members of the group must check the calibration.
The scale is now calibrated to display the length of the string y from the bottom of the sliding arm,
the pivot point of the pendulum, to the top of the ball, to a precision of one millimetre (mm). This
number is not exactly equal to L since the ball is not a point-mass. The length of the pendulum L is the
sum of the length of the string y and half the ball’s diameter b given in Table 1.1:
1
L = y + b.
(1.2)
2
Mount the larger ball m1 and calibrate the pendulum. Adjust the sliding arm so that the string length
y is approximately 0.3 m. Record the actual length to a precision of 1 mm.
• Set the pendulum swinging in a straight line, keeping α less than approximately 15◦ . Wait several
seconds to allow for any stray oscillations present in the bob to dissipate.
• Shift focus to the Physicalab software. Check the Dig1 box and choose to collect 50 points at 0.1
s/point. Click Get data to acquire a data set.
• Click Draw to generate a graph of your data. Your points should display a nice smooth sine
wave, without peaks, stray points, or flat spots. If any of these are noted, adjust the position of the
rangefinder and acquire a new data set.
• Select fit to: y= and enter A*sin(B*x+C)+D in the fitting equation box. Click Draw . If
you get an error message the initial guesses for the fitting parameters may be too distant from the
required values for the fitting program to properly converge. Look at your graph and enter some
reasonable initial values for the amplitude A of the wave and the average distance D of the wave
from the detector. The value of B (in radians/s) is given by B = 2π/T since the period of a sine
wave is 2π = B ∗ x radians, where x = T is the period in seconds. Estimate the time T between two
adjacent minima of the sine wave. Estimate and enter an initial guess for B. (See Appendix A)
• Label the axes and identify the graph with your name and the string length y used. Click Print
to print your graph. Every student should print a copy of each graph.
• Record in Table 1.1 the trial length y and the computer calculated value of the fitting parameter B
(radians/second). Do a quick calculation for g to make sure that the value is reasonable.
• Repeat the above steps for m1 with y = 0.45, 0.60, 0.75 and 0.90 m.
• Mount the second ball m2 , recalibrate the pendulum and verify the calibration, set the string
length to approximately y = 0.5 m, and record this one measurement in Table 1.1.
16
EXPERIMENT 1. THE PENDULUM
Run, i
m (kg)
b (m)
1,m1
0.0225
0.02540
2,m1
0.0225
0.02540
3,m1
0.0225
0.02540
4,m1
0.0225
0.02540
5,m1
0.0225
0.02540
1,m2
0.0095
0.01904
y (m)
L (m)
B (rad/s)
T (s)
T 2 (s2 )
gi (m/s2 )
Table 1.1: Table of experimental results
Part I: Determining g from a single measurement
• Use Equation 1.2 and the single measurement of m2 to calculate L and g. Show the calculation
below, in three steps as previously outlined and recalling the proper application of significant figures
and physical units.
L = ...............................
g = ...............................
...............................
...............................
...............................
...............................
• The measurement errors in y, b, and T , represented by σ(y), σ(b), and σ(T ), respectively, are
determined from the scales of the measuring instruments. The micrometer used to measure the ball
has a resolution, or scale increment of 0.00001 m. The data logger can measure time with a precision
of 0.00002 s. Summarize these measurement errors below:
σ(y) = ±..................
σ(b) = ±.................. σ(T ) = ±..................
• Calculate the error σ(L) in the length L and the error σ(g) in g. Express the final values as L ± σ(L)
and as g ± σ(g), with the correct units.
σ(L) = ...............................
σ(g) = ...............................
...............................
...............................
...............................
...............................
L = .............. ± ...............
g = .............. ± ...............
17
Part II: Determining g from a series of measurements
• Use Equation 1.1 to calculate the five measurements of g for m1 . Record the results in Table 1.1 and
Table 1.2. Include all the calculations for m1 as part of your Lab Report.
• Use Table 1.2 to calculate the average and standard deviation of the sample of g values, then summarize the result in the proper format below:
i
gi
gi − hgi
(gi − hgi)2
1
2
3
4
5
hgi =
variance =
σ(g) =
Table 1.2: Calculation template for the standard deviation of g.
g = ............. ± .............
Part III: Determining g from the slope of a graph
Equation 1.1 can be written as:
g
T2
(1.3)
4π 2
This is the equation of a straight line Y = mX + b, where in this case Y = L, X = T 2 , b = 0, and the
slope m = rise/run = g/(4π 2 ).
L=
• Using the Physica Online software, enter the five coordinates (T 2 , L) in the data window. Select
scatter plot. Click Draw to generate a graph of your data. Select fit to: y= and enter A*x+B
in the fitting equation box. Click Draw . The computer will generate a line of best fit through the
data points. Print the graph, then record below the slope A and error σ(A)
A = ............. ± .............
• Calculate below values for g and σ(g).
g = ....................... = ....................... = .......................
σ(g) = ....................... = ....................... = .......................
g = ............. ± .............
18
EXPERIMENT 1. THE PENDULUM
Summary of results
Now that you have quantitatively estimated the reliability of your various results for g by application of error
analysis, you can make a meaningful comparison of these results with the accepted value of g = 9.80 m/s2 .
• In the space below, plot and label your three results for g and the error bars representing the
uncertainty in each of these results. Begin by drawing on the given line a scale that is representative
of the range in your results. The range of agreement in two results is determined by the region of
overlap of their error bars. For clarity, plot your data and error bars above one another.
• Which of the error bars overlap the accepted value of g? ...............................
• Is there a region where all the error bars overlap? .....................................
• Do any of your error bars overlap the accepted value of g? ..............................
IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK!
Discussion
Your Discussion should summarize the essence of this experiment in a clear and organized way, so that
someone reading (or marking) this report will obtain a good understanding of what was done, how it was
done, and of the results that were obtained. Here are some ideas on the issues you should address in your
discussion:
Begin by tabulating your three experimental values and the accepted value of g. Identify the source of
the data and include appropriate units. Give the table a descriptive title.
Compare your three results for g. Do your values agree with the accepted value of g. Explain your
reasoning.
Consider which method is likely to provide the most reliable value for g. Which method is the least
reliable? Explain your conclusions.
From your graph, determine whether T varies with L. Does T vary with the mass of the ball? Explain.
Do your conclusions agree with the predictions of Equation 1.1?
Both the averaging method and the line of best fit use the same five experimental values for L and T .
What similarities do you think there may be in these two approaches to determining g?
If drawing a line through the plotted experimental points, how would you choose where to place the
line? How might this choice affect your resulting value of g?
What is the resolution, or scale, of the time measurement and how did you determine this? How might
the resolution be improved?
A Final Note: Have you submitted your Discussion to Turnitin and printed and included the receipt
that Turnitin sent you? If not, you will lose 40% of your grade. Also, be sure to submit your Lab
Report only in a clear-front folder (no duo-tangs, envelopes or staples) or you will lose 20% of your
grade.
first name (print)
last name (print)
Brock ID (ab13cd)
TA initials
grade
Experiment 2
Check your schedule!
This is a reminder that there is no Experiment 2 and that you need to check your lab schedule by following
the Marks link in your course homepage to determine the experiment rotation that you are to follow. The
lab dates are shown in place of lab grades until an experiment is done and the mark is entered.
My Lab dates:
Exp.3:.........
Exp.4:.........
Exp.5:.........
Exp.6.........
Note: The Lab Instructor will verify that you are attending on the correct date and have prepared for the
scheduled Experiment; if the lab date or Experiment number do not match your schedule, or the review
questions are not completed, you will be required to leave the lab and you will miss the opportunity to
perform the experiment. This could result in a grade of Zero for the missed Experiment.
To summarize:
• There are five Experiments to be performed during this course, Experiment 1, 3, 4, 5, 6.
• Everyone does the first experiment on the first scheduled lab session.
• The next four experiments are scheduled concurrently on any given lab date.
• To distribute the students evenly among the scheduled experiments, each student is assigned to one
of four groups, by the Physics Department. The schedule is entered as part of your lab marks.
Lab make-up dates: You may perform one missed lab on December 1, 2, 3, 4, 5 from 2-5 pm.
N otes : ..........................................................................
................................................................................
................................................................................
................................................................................
19
20
EXPERIMENT 2. CHECK YOUR SCHEDULE!
first name (print)
last name (print)
Brock ID (ab13cd)
TA initials
grade
Experiment 3
Angular Motion
Imagine placing a drop of paint near the edge of a
rotating wheel. As the wheel rotates with an angular speed ω, the marked point moves along a circular trajectory, its path a certain arc length s. When
a wheel turns through a full circle, θ = 2π, this arc
length is s = 2πr; for half a circle, θ = π, the arc
length is half the circumference of a circle, sπr, etc.
For an arc of radius r that spans an angle θ, s = θr.
In this way, the radius of the circle acts as a natural
unit of angle: one radian is the angle that spans the
arc of length s = r. It is natural to express angles
in radians: the circumference of a circle, 2πr, corresponds to the angle of 2π radians. Another way
of seeing this is to express all lengths in units of r
or, equivalently, to choose a unit circle, r = 1. The
circumference of a full unit circle is 2π, for the angle
Figure 3.1: Rotational coordinates
of θ = 2π radians. Note that since the radian is a
unit that represents a ratio of two lengths, θ = s/r,
it is a dimensionless quantity.
To convert between radians and other units of measuring angles, conversion coefficients such as π/180◦
or 180◦ /π are introduced as appropriate.
If the particle moves a distance s and sweeps an angle θ in a time t, it has a linear speed v = s/t
along the circular path and an angular speed ω = θ/t = (s/r)/t = v/r. The tangential acceleration of the
particle is a = v/t and the corresponding angular acceleration is α = ω/t = a/r. The inverse relations are,
of course, v = rω and a = rα, so all translational quantities scale with r.
The kinetic energy of the particle is then
K=
mv 2
mr 2 ω 2
Iω 2
=
=
.
2
2
2
The quantity I = mr 2 represents the rotational moment of inertia of a point mass m placed a distance r
from the center of rotation; for extended bodies, adding up contributions from many small point masses
that make up the body yields a result that is always proportional to M R2 , where M is the total mass and
R is some measure of the size of the body. However, there will be a numerical factor in front of M R2 that
will vary with the shape of the body. It has been calculated and tabulated for many common shapes: disks,
rods, spheres, etc. The physical meaning of I is that it’s a measure of the body’s resistance to a change in
its angular speed, in other words to an applied torque attempting to cause an angular acceleration α. This
21
22
EXPERIMENT 3. ANGULAR MOTION
is in analogy to an applied force trying to cause a linear acceleration, where the measure of the body’s
resistance is the familiar inertial mass m. If Newton’s Second Law for translational motion is F = ma, or
m = F/a, for rotational motion the analogous relation is τ = Iα, or I = τ /α, where τ denotes the applied
torque.
Torque acting on a rotating body
To understand what torque is, let us consider a rigid body with its centre of mass at a point r = 0,
constrained to rotate about that point. The radius vector ~r of a point on the body has the origin at r = 0
and ends at the point, a distance r from the origin, as shown in Figure 3.1. If a force F~ is applied at that
point, in the plane of rotation of the disc, at a distance r from the centre, and at an angle φ to the radius
vector ~r, the magnitude of the turning torque τ produced by this force is:
τ = rF sin φ
The torque τ depends both on the distance from the centre of rotation and on the direction of the applied
~ into a radial component F~r and a tangential component F
~t so that F
~ = F~r + F~t ,
force F~ . Decomposing F
~
it can be seen that only the tangential component of F causes torque and affects the magnitude of τ . A
force applied through the centre of rotation has a zero tangential component (φ = 0, Ft = 0) and the radial
component alone produces no torque and will not cause angular acceleration. As the angle of the force
changes, so does the torque experienced by the body; for a given magnitude of the force, F , the maximum
torque is produced when F~ = F~t is perpendicular to ~r and τ = rFt = rF .
Resisting the force F~ is the total mass
M of the rotating body. Suppose that this mass consists of
P
many particles of mass mi so that M = i mi . In a translational motion, the force acts equally on all the
P
component particles of the body at once, according to Newton’s second Law F~ = i (mi~a) = M~a. In a
rotation, the rotational effect of F~ on a particle is proportional to the distance r from the centre of rotation.
While the entire rigid body experiences a single angular accelaration α, common to all its particles, each
of them will experience a tangential acceleration ai that is proportional to the distance ri from the centre
of rotation.
The rotational moment of inertia I of a rigid body composed of many particles is simply the sum of
the individual rotational moments of inertia of all particles:
I=
X
mi ri2
(3.1)
i
For a thin hoop, where all the particles are located at the same common radius away from the centre,
ri = R, Eq. 3.1 reduces to I = M R2 ; for other shapes, the calculation may not be so simple.
The rotational form of the Newton’s Second Law is τ = Iα:
F =
X
i
(mi a) = M a
−→
τ=
X
(mi ri2 α) = Iα.
i
The torque τ plays the same role for rotational motion that the force F plays for translational motion.
Determining the rotational inertia of a disc
Consider a homogeneous disc of radius R and mass M constrained to rotate without friction around the
centre of mass. A massless string is wrapped around the outer edge of the disc and connected to a mass
m that is subjected to the force of gravity Fg , as shown in Figure 3.2. The string experiences a tension T
due to the weight of m; at the other end of the string the same tension T acts on the edge of the disc. The
displacement of the falling mass is given by Equation 3.2:
23
y = y0 + v0 t + at2 /2
(3.2)
When a force is applied, the disc, initially at rest,
begins to spin as m falls with linear acceleration
a=g−
T
.
m
(3.3)
The rotational acceleration of the disk is
α=
τ
TR
a
= =
.
R
I
I
(3.4)
By combining Eqs. 3.3 and 3.4, the rotational inertia I of the disc can be expressed in terms of a as
follows:
g
Figure 3.2: A falling block causes the disc to rotate
−1 .
(3.5)
I = mR2
a
Measuring the acceleration of the falling mass, and comparing it to the acceleration of the free fall, yields
an operational measurement of the moment of inertia of the disk.
Review questions
Determine from your textbook or the Internet the equation for the rotational inertia I of a solid uniform
disc of radius R, thickness h and mass M . You will use this expression to calculate the theoretical I for
the disc used in this experiment.
......................................................................
......................................................................
Derive equation 3.5. Begin by evaluating the force F on the cradle due to the tension T of the string acting
on the edge of the disc. Show a complete, step-by-step solution.
......................................................................
......................................................................
......................................................................
......................................................................
......................................................................
......................................................................
CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT!
24
EXPERIMENT 3. ANGULAR MOTION
Procedure and analysis
Figure 3.3: Experimental setup
The equipment consists of a plastic disc bolted to a metal drum (this combination is called the cradle),
that can rotate nearly friction free, around a vertical axis. A torque τ can be applied to the cradle with a
string-and-pulley combination, at the end of which a mass m is attached. The pulley is arranged so that
the string leaving the drum is perpendicular to the axis of rotation and to the radius vector.
Pulley A is part of a photogate system. The C-shaped photogate has an infrared transmitter at one end
and an infrared detector at the other end. As the pulley rotates, the spokes interrupt the beam, turning
on and off the signal at the detector, thus allowing LabPro to count a series of pulses. A light emitting
diode (LED) on the photogate shows the current on/off state at the detector.
The pulley has ten spokes and a circumference of 155 mm, so there are twenty pulses transmitted every
rotation of the pulley. Each pulse represents a radial distance of 7.75 mm travelled by the string and hence
the same distance y moved by the mass m.
After the data acquisition is initiated, LabPro waits for the first transition and marks this event with
the elapsed time and assigns it an initial distance y = 0. After all subsequent transitions, the new time
and total distance travelled are sent. Note that the time t = 0 set by LabPro may not coincide with the
start of the motion, therefore:
• For the most reliable results, adjust the cradle by monitoring the LED so that a transition will occur
as soon as the cradle begins to rotate. Wait for the first data point to be sent, then release the cradle.
You should delete any of these (0,0) data points at the start of your data set.
Part 1: Determining the rotational inertia of the cradle
• Wrap the string, trying not to overlap the strands, around the second smallest pulley (r = 0.02282 m)
on the cradle and arrange the string path as shown in Figure 3.3. Place a 10 g weight on the mass
holder and note how the weight accelerates as it falls. It should not shake sideways. If it makes
sudden vertical jerks, then the string was binding as it unravelled from the pulley and should be
wound less tightly. Rotate the cradle to raise the weight to the top of the assembly.
25
• Change focus to the graphing software. Select Dig2, the channel that the photogate should be connected to, then choose to collect 20 points with 0.5 seconds between points. Calibrate the photogate
by monitoring the LED as previously described, then release the weight. Press Get data to acquire
the data set. As the weights change, you may need to adjust the number of points collected and/or
the sample time. Try to stop the weight before it reaches the floor; it will save you the trouble of
having to re-spool the string on the pulleys.
At the end of the run, review your graphed data; it should be a smooth curve that represents the
falling of a mass m under constant acceleration a. To determine a, you will now fit Equation 3.2 to
your data set.
• Delete any (0, 0) points. Select fit to: y= and enter A+B*x+C*x**2/2 in the fitting equation
box. Click Draw to perform a fit of your data. Click Print to generate a representative graph of
your data. Fill in Table 3.1, then calculate an average acceleration hai for the various masses. Make
a printout of one trial, properly labelled, and include it in your report.
m (kg)
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
a1 (m/s2 )
a2 (m/s2 )
hai (m/s2 )
Table 3.1: Acceleration data for unloaded cradle assembly
• You may now wish to evaluate the rotational inertia Ic for the unloaded cradle, using Equation 3.5
and the values of a for a given m. This would not likely lead to the correct Ic value because you
do not know the actual mass m0 required to overcome the friction of the pulley system, representing
the least mass required to keep the cradle rotating at a constant speed, once started by hand, to
overcome the static friction of the system. Note that the cradle will likely begin to rotate with only
the mass holder attached (m = 0). In this case, to prevent the system from rotating, m0 would need
to be negative (i.e. the mass holder would need to be lighter).
There is however a method often used that does not require m0 to be known, and m0 is actually
obtained from the follwong procedure: begin by rearranging Equation 3.5 so that m is expressed as
a function of a:
Ic = m T r
2
g
g
− 1 ≈ mT r 2
a
a
gr 2
= (m − m0 )
a
!
→ m = m0 +
Ic
a.
gr 2
(3.6)
Here, the equation can be simplified only if a ≪ g so that (g/a − 1) ≈ (g/a). The total mass
mT = m − m0 that causes the tension T on the string is the sum of the mass m that you used and
the mass m0 of the mass holder. The resulting equation is that of a straight line with slope Ic /(gr 2 )
and y=m intercept at x=a=0 equal to m0 .
• Enter into an empty data window your set of (hai, m) coordinates. The scatter plot of your data
should show a linear behaviour. Select fit to: y= and enter A+B*x in the fitting equation box.
Click Draw and record below the fit parameters A and B along with their appropriate units.
A = ............... ± ...............
B = ............... ± ...............
26
EXPERIMENT 3. ANGULAR MOTION
• Calculate Ic using Equation 3.6 and use the appropriate error propagation rules to evaluate the
corresponding uncertainty σ(Ic ):
Ic = ..............................
σ(Ic ) = ..............................
..............................
..............................
..............................
..............................
Ic = ................. ± .................
Part 2: Determining the rotational inertia of cradle and disc
• Use the peg on the steel disc to center it on the cradle, adding a small piece of paper under the disc
to prevent it from slipping. As before, determine the rotational inertia It of cradle-plus-disc.
m (kg)
0.030
0.040
0.050
0.060
0.070
0.080
0.090
0.100
a1 (m/s2 )
a2 (m/s2 )
hai (m/s2 )
Table 3.2: Acceleration data for cradle assembly with disc
It = ..............................
σ(It ) = ..............................
..............................
..............................
..............................
..............................
It = ................. ± .................
• Subtract Ic from It to obtain Id , the rotational inertia of the disc alone.
Id = ..............................
σ(Id ) = ..............................
..............................
..............................
..............................
..............................
Id = ................. ± .................
27
• Use the equation from the review question to calculate the theoretical rotational inertia for the disc
that you used, as well an estimate of the error.
Id(theoretical) = ..............................
σ(Id ) = ..............................
..............................
..............................
..............................
..............................
Id(theoretical) = ................. ± .................
IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK!
Discussion
Begin by presenting a tabulated summary of your results. Address the following issues as part of your
discussion; as always, you should expand this list with your own observations, points and ideas.
Do your theoretical end experimental results for Id agree with one another? Explain.
Use the largest value obtained for the acceleration a to estimate the maximum error introduced in the
rotational inertia I by the approximation used in Equation 3.6.
Are the results for m0 for the cradle and the cradle-plus-disc systems consistent with your observations?
Describe how the two systems behaved when m = 0, then analyse the results from your fits to reach your
conclusions.
If you spun the cradle using the smallest pulley on the drum (r = 1.539 cm), should the rotational
inertia IC of the system change? How, if at all, should the acceleration a change as a function of the pulley
radius r? Explain your reasoning by referring to the relevant equation(s).
Perform a trial using the smallest pulley with pull mass m = 30 g on the unloaded cradle.
Compare this a with the previous result for a obtained using the second smallest pulley and same pull
mass. Do these two a values obtained by varying the pulley radius r agree with the results predicted by
the theory? Show your data, equations, calculations and results below.
......................................................................
......................................................................
......................................................................
......................................................................
......................................................................
......................................................................
28
EXPERIMENT 3. ANGULAR MOTION
first name (print)
last name (print)
Brock ID (ab13cd)
TA initials
grade
Experiment 4
Collisions and conservation laws
The velocity ~v of a body of mass m is determined by its speed, a scalar quantity, and its direction of motion,
represented by a unit vector. If we consider a collision of two such masses m1 and m2 with velocities ~v1b and
~v2b before the collision, and velocities ~v1a and ~v2a after the collision, we note that it is generally difficult,
if not impossible, to predict these resulting velocities ~v1a and ~v2a . To accomplish this, one would need to
have a complete knowledge of the physical characteristics of the objects (size, shape, et cetera) and of the
geometry of the interaction.
The linear momentum P~ of a body is a vector equal to the product of its mass m (a scalar) and its
velocity ~v (a vector). The law of conservation of linear momentum states that:
If there are no net external forces acting on the masses, then the total momentum P~b before the
collision is equal to the total momentum P~a after the collision.
A mathematical formulation of this law for a collision between two masses m1 and m2 is a vector equation
and can be expressed as follows:
P~1b + P~2b = P~1a + P~2a
m1~v1b + m2~v2b = m1~v1a + m2~v2a .
(4.1)
Rearranging Equation 4.1 in terms of the change in the momentum ∆P~1 of m1 and ∆P~2 of m2 reveals
that the net change will be zero and that the vectors will be oriented anti-parallel to one another:
P~1b − P~1a + P~2b − P~2a = 0 → (P~1b − P~1a ) = −(P~2b − P~2a ) → ∆P~1 = −∆P~2
Similarly, the change in the velocity for the two masses will result in two anti-parallel vectors:
m1 (~v1b − ~v1a ) = −m2 (~v2b − ~v2a ) → m1 ∆~v1 = −m2 ∆~v2
(4.2)
Since the velocity vectors are collinear, the vector equation can be simplified to a scalar equation and
expressed in terms of the magnitudes of the velocities |~v2a − ~v2b | and |~v1b − ~v1a |. Rearranging Equation 4.2
as a ratio of masses m1 and m2 :
m1
|∆~v2 |
|~v2b − ~v2a |
|~v2a − ~v2b |
=−
=−
=
.
m2
|∆~v1 |
|~v1b − ~v1a |
|~v1b − ~v1a |
(4.3)
The mass ratio equation predicts that for equal masses m1 and m2 , the change in the velocity of the
two masses should be the same. It also predicts that the two resulting velocity vectors will point in the
same direction since the the components of vector 1 have been reversed.
It is also of interest to know whether kinetic energy K = mv 2 /2 is conserved during a collision. The
parameter Q, defined as the ratio of the total kinetic energy Ka after the collision to the total kinetic
29
30
EXPERIMENT 4. COLLISIONS AND CONSERVATION LAWS
energy Kb before, is used as a measure of the energy lost during a collision. Note that Q depends on the
square of the velocity and hence will be very sensitive to variations in v.
Q=
Ka
=
Kb
1
2
2 m1 v1a
1
2
2 m1 v1b
2
2 + m v2
+ 12 m2 v2a
m1 v1a
2 2a
=
1
2
2
2
m1 v1b + m2 v2b
+ 2 m2 v2b
(4.4)
The kinetic energy, and therefore Q, are scalar quantities. Q can range in value from 0 to 1. If Q = 1,
the kinetic energy of the system is conserved and the collision is said to be elastic. A collision is said to
be inelastic when Q < 1. This is the expected result of our experiment since some of the kinetic energy
is changed to heat and sound energy during the collision, and there will be frictional forces acting on the
masses throughout the interaction. On the other hand, rotational kinetic energy may be imparted onto
the pucks as they are pushed.
In this experiment the mass ratio of two colliding pucks will be calculated to determine whether or
not linear momentum was conserved during the collision. The ratio Q will be used to estimate the kinetic
energy lost during the interaction.
Review questions
• Sketch the collision shown in Figure 4.2 then add the vectors ~v1 = ~v1b −~v1a and ~v2 = ~v2a −~v2b . What
can you conclude about the magnitude and direction of these two vectors? Should the momentum
be conserved in the collision? And the velocity? When is the velocity conserved? Explain.
...................................................
...................................................
...................................................
• Show below that, if a certain condition is met, Equation 4.3 can be rewritten in terms of distance
vectors rather than velocity vectors, to analyse your collision record as in Figure 4.2. Then you can
then measure the magnitude of the velocity vectors which have units of distance/time with a ruler
that measures distance. What is this necessary condition?
......................................................................
......................................................................
......................................................................
• Derive the error σ(m1 /m2 ) in the theoretical mass ratio m1 /m2 using the appropriate error propagation rules. The answer is given by Equation 4.5 in the analysis section of this lab. Do not simply
copy Equation 4.5 as your answer. Show a complete step by step solution (see Appendix).
CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT!
31
Procedure and analysis
The equipment consists of a flat glass plate in a metal frame which can be levelled by varying the height of
four adjustable legs. The glass plate is covered by a conductive black pad. A sheet of regular white paper
is placed over this conducting layer. The sheet must be flat and free of any kinks or other deposits.
The collision components are two heavy metal pucks. Each puck is tethered to a plastic hose that
provides the puck with compressed air and carries within it an electrode wire. The hose should hang freely,
without any twists or interference from the rest of the equipment.
When turned on, the air exits from a hole at the bottom of the puck, creating a high pressure layer
between the puck and the paper and causing the puck to levitate and move with negligible friction. When
the frame is properly levelled, the puck will float in place, without any lateral movement.
A spark timer is used to provide the high voltage
pulses required in the experiment. One terminal of the
timer is connected through the electrode wire in the hose
to an insulated needle at the bottom of the puck.
The other terminal of the spark timer is connected
to the conductive sheet, completing the electric circuit.
The firing rate of the timer can be adjusted with a control
knob on the unit. To turn on the timer, depress and hold
the pushbutton switch at the front of the spark timer
unit. Sparks are generated from the needle through the
white paper to the conducting paper beneath. These
pulses produce black spots on the bottom of the white
paper, marking the positions of the pucks at equal time
Figure 4.1: Trails left by moving pucks
intervals.
Consequently, it is not necessary to know the actual time interval between firings of the spark timer.
In your calculations refer to this unknown time unit as ”u” (unknown time unit). Puck speeds would then
be expressed in units of cm/u or mm/u.
• Place a new sheet of white paper over the conducting layer, making sure that the paper is flat and
clean. Very slowly turn on the air until the pucks begin to float. Excessive air pressure will burst
the air hose. Level the frame by adjusting the height of the four legs until the pucks float in place.
• To record the collision, have an assistant depress the switch of the spark timer just before you release
the pucks, and release the switch just after the pucks rebound from the edges of the frame. Every
member of the team needs to make their own collision record to analyse.
• Place the pucks against the launchers in adjacent corners of the frame, and release them toward the
centre of the paper, where they will collide. Do several trial runs. While any collision is theoretically
valid, for ease of analysis your collision should be fairly symmetric as in Figure 4.1 and take about
3 seconds to complete. Small initial velocities will cause the pucks to slow down due to friction and
yield incorrect results. The pucks should not rotate as this would give them rotational kinetic energy.
We are concerned only with translational kinetic energy.
• The mass of each puck has been measured and is displayed on the puck. Assign this mass value to
the corresponding trace for future reference, keeping in mind that the traces appear on the bottom
of the paper. Be sure to match each mass with the corresponding trace, otherwise your Q value
will not make sense.
• A good collision record will show for each of the for trails a series of 4-6 collinear dots, equidistant
and longer than 100 mm in total. If your collision did not turn out, flip the sheet of paper paper and
try again.
32
EXPERIMENT 4. COLLISIONS AND CONSERVATION LAWS
• Evaluate the quality of your collision record. Measure the distance between the adjacent dots of a
trail. Did the distance between the dots in each trail remain the same and did these dots form a
straight line? Is the trail a valid representation of a vector? Note your observations below:
................................................................................
................................................................................
................................................................................
It can be seen from Equation 4.3 that the generation of the vectors ~v1b , ~v2b , ~v1a , and ~v2a is required.
For example, the vector ~v1b can be obtained from
the collision record by joining a series of dots along
the Puck 1 trail (before the collision) spanning n
time intervals.
• Draw four proper vectors on your sheet, as
shown in Figure 4.2. It is important to use
this same number n of time intervals for all
the vectors so that you are, in effect, applying
the same time scale n ∗ u to each vector.
The vectors ~v2a − ~v2b and ~v1b − ~v1a depend on the
magnitude and direction of the component vectors, hence a graphical vector addition must be
performed. A reasonably long vector or line segment
with well defined endpoints can be transposed very
accurately by visually estimating the final placement of this vector as follows:
Figure 4.2: Join dots to obtain vectors
• Using a long ruler, extend the line segment
AB beyond the region of point D.
• Place the ruler so that the edge rests on point
C and is parallel to the previously drawn line.
• Draw a line through C beyond points A and
D. The measurement error for all vectors is
σ(v) = ± ..............
• Measure near A and D the perpendicular distance between the two lines to verify
that they are indeed parallel. Determine the
length AB with a ruler or the span of a compass and draw a line segment of length AB
from C to D to define the vector CD.
Figure 4.3: Transposition of a vector
This method of extending the length of a line segment can also be used on the resultant vectors to see
if they are parallel. By the law of conservation of momentum, the change in momentum of one puck is
equal and opposite to that of the other puck. The two resultant vectors should then be nearly equal in
magnitude since m1 ≈ m2 and parallel since the a − b components in the vectors are reversed.
33
m1
|v1b |
|v1a |
|~v1b − ~v1a |
Time units
m2
|v2b |
|v2a |
|~v2a − ~v2b |
(g)
(mm)
(mm)
(mm)
(u)
(g)
(mm)
(mm)
(mm)
Table 4.1: Experimental data: masses and vector magnitudes
• Compare the magnitudes of directions of your resulting vectors. To quantify the test for parallelism,
calculate the angle θ between the two extended line segments. It is given by θ = arctan(dy/dx),
where dx is the length of the line and dy is the difference in the distance of the two lines at their
opposite ends. Show your calculations below.
......................................................................
......................................................................
......................................................................
• Determine a theoretical value (m1 /m2 ) and error σ(m1 /m2 ) for the mass ratio from the given values
of m1 and m2 . The measurement error for these masses is ±0.1 g. Show your calculatios below
m1
=
m2
.............................
m1
σ
m2
=
..................................
.............................
(Theoretical)
.................................
m1
= ............... ± ...............
m2
• Use Equation 4.3 to calculate a value and error for the experimental mass ratio of the pucks.
m1
=
m2
σ
.............................
.............................
(Experimental)
m1
m2
=
..................................
..................................
m1
= ............... ± ...............
m2
34
EXPERIMENT 4. COLLISIONS AND CONSERVATION LAWS
• Use Equation 4.4 and the theoretical m1 and m2 values determine Q and σ(Q). You can approximate σ(Q) by letting Q = (A + B)/(C + D). Show the complete solution on a separate sheet
of paper.
Hint: solving and calculating a result for the σ(Q) error equation can be a tedious error-prone venture.
To simplify the task, note the symmetry in the equation, apply a divide-and-conquer strategy and avoid
repeating calculations.
For example, there are four identical mv 2 terms, each of which will appear several times in the Q and
σ(Q) equations. Evaluate these four terms only once, record the result and then compare the four values.
You would expect them to be similar since the m and v values are also similar. Then you can confidently
evaluate Q and apply the values to σ(Q).
Likewise, the error equation for each term will be identical in form and these error terms should also
evaluate to similar results.
Q=
σ(Q) =
.............................
..................................
.............................
..................................
Q = ............... ± ...............
IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK!
Discussion
Include your collision record, your worksheets and a tabulated summary of your results.
Discuss the results in terms of momentum changes. Was momentum conserved? How do you establish
this? What was the quality of your vector addition? Did the experimental and theoretical mass ratios
agree within their respective margin of error? Explain.
Discuss your result for the parameter Q. Was energy conserved? Was there a significant loss of energy
during the interaction? Was the collision elastic or inelastic?
Was your Q value greater than unity? Can Q be greater than unity? Consider how someone performing
this experiment could arrive at a value of Q > 1.
Review the test you performed on the two resultant vectors and use your findings to support your
conclusions regarding the quality of the data obtained from the analysis of the collision record.
The conservation of energy requires that no energy may be lost during an interaction. However, it
can be changed to other forms, which may not be monitored during the interaction. In this collision
experiment, what other forms of energy would the initial kinetic energy be transformed into?
Consider how the apparatus used in this experiment may have introduced random and systematic errors
into the experimental procedure. Which of these errors do you think would have a significant impact on
the experiment? Which are likely to be insignificant?
first name (print)
last name (print)
Brock ID (ab13cd)
TA initials
grade
Experiment 5
The ballistic pendulum
In this experiment we explore the transfer and conservation of energy and momentum in a collision of
two objects. One of these objects is a small projectile of mass m that is given a certain velocity v
by a launcher. The second object is a stationary
pendulum. As the projectile hits the pendulum,
a re-distribution of energy and momentum takes
place.
In a certain class of collisions, the projectile is
captured by the pendulum. For such inelastic collisions, there is only one combined object that is
moving at the end, and carries all of the kinetic energy K and momentum P . If the mass of the pendulum is M then the total mass of the combined
object at the end of the collision is MT = M + m.
If the bob is stationary when the projectile hits, it
contributes nothing to the total kinetic energy and
momentum of the system before the collision. Thus
the Law of Conservation of Momentum in this case yields:
Figure 5.1: Ballistic Pendulum
Pbef ore = mv = Paf ter = (M + m)vT = MT vT
(5.1)
where vT is now the velocity of the combined object immediately after the collision. Knowing v and MT ,
we can use Equation 5.1 to determine vT .
As the pendulum begins to swing after the collision, another physical process takes place; a conversion
of the kinetic energy of the moving object into into gravitational potential energy as it swings up, losing
kinetic energy and gaining potential energy. At the bottom of the swing, the pendulum of mass MT has
all of its energy in the form of kinetic energy. At the top of the swing, all of the pendulum’s energy is
converted into gravitational potential energy, and as MT momentarily pauses and reverses its motion, the
kinetic energy falls to zero. Thus we can write:
1
Ebottom = K = MT vT2 = Etop = MT gh.
2
(5.2)
Combining Equations 5.1 and 5.2 to eliminate vT gives us an expression that relates the initial velocity v
of the projectile to the final elevation h of the combined object.
v=
MT p
2gh,
m
35
(5.3)
36
EXPERIMENT 5. THE BALLISTIC PENDULUM
It is more convenient to express h in terms of the angle θ of the swing
v=
MT q
2gRcm (1 − cos θ)
m
(5.4)
where Rcm is the distance from the pivot point to the centre of mass of the combined rod, block, and block
contents.
There is another energy conversion taking place, even before the collision. In the experiment, the
launcher gives the projectile its initial kinetic energy and momentum by releasing the potential energy
and momentum of a compressed spring an converting it into the kinetic energy of the moving projectile.
Conservation of energy in this case yields
1
1
Einitial = P E = kx2 = Ef inal = K = mv 2
2
2
(5.5)
where x is the spring displacement from its equilibrium (relaxed) length and k is the spring constant. As
the projectile is released from rest, the projectile has no initial kinetic energy. As the projectile begins to
accelerate, a conversion of energy takes place where the potential energy stored in the spring is converted
into the kinetic energy of the projectile. At the point where the projectile releases from the spring, all of
the spring’s stored potential energy has been converted into the projectile’s kinetic energy. We can then
determine a value for the spring constant k of the launcher by equating the initial potential energy of the
spring to the final kinetic energy of the projectile:
k=
mv 2
x2
(5.6)
Review question
• Derive an equation that expresses the error σ(v) in the velocity v given by Equation 5.4, expressed
below as a product of terms. Perform a change of variables and apply the appropriate error propagation rules (Appendix B) to each term to arrive at a final answer. Show a complete step by
step solution.
1/2
v = MT ∗ m−1 ∗ 21/2 ∗ g1/2 ∗ Rcm ∗ (1 − cos θ)1/2
......................................................................
......................................................................
......................................................................
......................................................................
......................................................................
σ(v) = ................................................................
CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT!
37
Procedure and analysis
CAUTION: Always wear safety glasses while using the launcher.
The launcher component of the ballistic pendulum consists of a precision spring encased in an aluminum
barrel. One end of the spring is secured to the closed end of the barrel. The other end is attached to a
piston that slides along the inside of the barrel. The projectile rests on the face of the piston. A trigger
mechanism allows the piston to be locked in one of three force settings: short, medium or long range.
Note: When the launcher is in the discharged position, the spring is subjected to a small compression,
or preload, so that it will not rattle when released. This preload is in the order of 1 mm. For this
experiment, we will assume that when the launcher is discharged the potential energy stored in the spring
is approximately zero.
• Load the ballistic pendulum by placing the projectile in the barrel and pushing it with the plunger
rod until the trigger locks. This is the short range setting. Further compression selects the medium
range and finally, the long range setting.
The work done to compress the spring is now stored as the spring’s potential energy V = kx2 /2, where k
is the spring constant and x is the compression distance. The launcher mechanism is designed so that the
elastic limit of the spring is not exceeded during compression.
• Once the launcher is loaded, slowly withdraw the plunger, making sure that the trigger is indeed
locked and that the projectile has not rolled away from the face of the piston, as this would cause an
innacurate firing of the launcher.
• To fire the launcher, gently pull the string upward to release the trigger.
The projectile of mass m will be accelerated by the expansion of the spring to achieve a final velocity v
at the moment that it finally loses contact with the face of the piston. At this point, the potential energy
V stored in the spring will have been totally converted into the kinetic energy K = mv 2 /2 of the projectile.
The pendulum consists of a rod and bob of combined mass M attached to a pivot point. When an
impact takes place, the pendulum catches the impacting mass m, changing the total mass of the pendulum
to MT = M + m, and is caused to swing about the pivot point to a maximum angle of deviation θ, relative
to the initial vertical position of θ = 0◦ . The pendulum drags with it a pointer that stops at the limit
of the swing and identifies the value of θ on a degree scale concentric with the pivot. The friction of the
pointer is negligible.
All of the kinetic energy of the projectile is transferred to the pendulum which, being initially at rest,
gains a kinetic energy K = MT v 2 /2. After rising against the force of gravity g a maximum height h from
the vertical position, the pendulum stops as all of the pendulum’s kinetic energy has been converted to
gravitational potential energy V = MT gh. This type of collision is classfied as totally inelastic since the
bodies involved stick together, moving as one after the collision.
• Check that two brass weights are attached to the pendulum and that the bob is properly oriented in
order to catch the projectile.
• Swing the pendulum to 90◦ and lock it in position with a slight push. Load the steel ball projectile
into the launcher, and latch the trigger at the short range setting. Gently lower the pendulum to the
vertical position, and move the angle indicator to the 0◦ mark. If the indicator does not reach zero,
you will need to subtract the offset from your angle reading.
38
EXPERIMENT 5. THE BALLISTIC PENDULUM
range
θ1
θ2
θ3
θ4
θ5
hθi
σ(θ)
short
medium
long
Table 5.1: Experimental angle values at three force settings
• Perform five short range launches, recording the angle θi reached in trial i = 1 . . . 5 in the appropriate
spaces of Table 5.1. These values should be within ± 0.5◦ of one another, otherwise redo the set of
measurements. What is the resolution of the protractor scale? ..................
• Repeat the five above trials using the medium and then the long range settings.
• Calculate an average value hθi for the five angles θi obtained in each of the three sets of trials and
enter these in Tables 5.1 and 5.2. To avoid some lengthy standard deviation calculations, let the error
in the angle θi be the measurement error of the angle scale, σ(θ) = ± 0.25◦ . Discuss the validity
of this assumption in your conclusions.
• With the digital scale, measure the masses m and MT .
m = .................. ± .................. kg
MT
= .................. ± .................. kg
• Remove the pendulum arm and determine the centre of mass point of the pendulum/ball combination
by balancing the unit on the edge of the measuring apparatus, as shown in Figure 5. Make sure that
the arm of the pendulum is parallel with the length of the scale. The centre-of-mass distance Rcm is
the distance from the edge to the centre of the pivot hole on the arm of the pendulum of mass MT .
Be sure to replace the pendulum arm when you are done.
Rcm = .................... ± .................. m
Figure 5.2: Experimental setup for determining the centre-of-mass of the pendulum
39
• With the launcher discharged and the ball removed, use the scale on the plunger to determine the
offset depth of the face of the piston from the front end of the barrel. You will need to subtract this
offset from all of the following depth measurements.
offset = .................... ± .................. m
• With the ball removed, compress the piston until it latches at the short range setting. Measure the
depth from the face of the piston to the front end of the barrel. Subtract from this length the offset
depth of the piston and record the result as x in Table 5.2.
• Repeat the above step for the medium and long range settings and complete Table 5.2.
• Use Equation 5.4 and the error equation derived in the review section of this lab to calculate the
initial velocity v and the error σ(v) of the steel ball projectile at the three force settings. Enter the
results in Table 5.2. Show your calculations below for the short range force setting and include all
other calculations as part of your Discussion.
v=
MT q
2gRcm (1 − cos θ)
m
σ(v) = |v|
σ(MT ) σ(m) σ(Rcm ) | cos[θ + σ(θ)] − cos[θ − σ(θ)] |
+
+
+
|MT |
|m|
2|Rcm |
|4(1 − cos θ)|
...................
............................................................
...................
............................................................
v = .................... ± .................. m/s
• Calculate the maximum kinetic energy of the steel ball at the moment that it lost contact with the
piston of the launcher and enter the value in Table 5.2. Show a complete calculation for the short
range setting and include all other calculations as part of your Discussion.
K=
1
mv 2
2
σ(K) =
............................
..............................................
............................
..............................................
K = .................... ± .................. J
If no energy is lost (or gained) during the interaction, this kinetic energy K is equal to the potential
energy V = kx2 /2 of the spring before the ball was discharged, K = kx2 /2. This equation is that of a
straight line Y = M X with Y = K, X = x2 and slope M = k/2. By plotting K as a function of x2 we
can extract from the slope a value for the spring constant k of the launcher spring. To do this, you will
measure x relative to a fixed reference point, the end of the barrel, at the three force settings.
40
EXPERIMENT 5. THE BALLISTIC PENDULUM
v (m/s)
x (m)
x2 (m2 )
K (J)
±
±
±
±
±
medium
±
±
±
±
±
long
±
±
±
±
±
range
hθi
short
◦
Table 5.2: Parameters for the calculation of the kinetic energy K and the force constant k
• Shift focus to the Physicalab software and enter in the data window the three data pairs and corresponding errors in the format (x2 , K, σ(K), σ(x2 )). Select scatter plot. Click Draw to generate
a graph of your data. Your graphed points should well approximate a straight line. Select fit to:
y= and enter A*x+B in the fitting equation box. Click Draw to perform a linear fit of the data.
Label the axes and include a descriptive title. Click Print to generate a hard copy of your graph.
Every student requires their own graph.
• Summarize the values for the slope, the Y-intercept Y(X=0) and their associated errors, then calculate
a value for k and σ(k):
slope
= .................. ± .................. J/m2
Y(X=0) = .................. ± .................. J
k = .................. ± .................. J/m2
IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK!
Discussion
Tabulate your final data and attach your computer printout and all calculation worksheets.
Discuss whether your results show a linear relationship between K and x2 and if three data pairs are
sufficient to define such a relationship. Does the spring force constant k change as the spring compression
x is increased? Explain.
Consider the value for the Y-intercept from the Least Squares Fit of your data. What does this value
represent? Is your result consistent with your expectations? Explain.
Refer to your three sets of angle data. Discuss the validity of the assumption made regarding the
magnitude of the error σ(θ) in the angle θ.
You calculated the force constant k of the launcher spring by observing the interaction of the projectile
with the ballistic pendulum apparatus. What needs to be true for this method to be valid? Can we use this
experimental setup to determine the actual energy lost by the projectile between the time that it leaves
the piston and the time that it collides with the pendulum bob? Explain.
first name (print)
last name (print)
Brock ID (ab13cd)
TA initials
grade
Experiment 6
Harmonic motion
A simple harmonic oscillator (SHO) is a model system that is widely used, and not just in the elementary
physics exercises.The reason for this is profound: the same fundamental equations describe the motion of
a mass-on-a-spring, and also of the interatomic forces that hold all matter together. Literally, everything
we touch, at the atomic level, is held together by springs connecting pairs of atoms. The details of these
interactions are studied in Quantum Mechanics and Solid-State Physics, but two masses connected by a
coiled elastic wire represent an excellent model system,from which much can be learned.
With no external forces applied to the material, the interatomic springs are at their equilibrium length,
neither stretched nor compressed. The application of an external stretching force to the material will cause
these springs to extend, thereby increasing the bulk length of the material. When the applied external force
is removed, the springs return to their equilibrium lengths, restoring the material to its original dimensions.
This restoring force may be overcome by
A large enough applied external force will cause the object to deform permanently or to break. The
maximum force applicable without permanent distortion is called the elastic limit of the material.
Hooke’s Law states that the amount of stretch or compression x exhibited by a material is directly
proportional to the applied force F . The proportionality constant is called the spring constant k. The
spring constant, k, has the units of Newtons per metre (N/m) and is a measure of the stiffness of the
material being considered. This proportionality between force and elongation has been found to hold true
for any body as long as the elastic limit of the material is not exceeded.
Perhaps the most convenient way of expressing a relationship of the Hooke’s Law is to write;
F = −kx,
(6.1)
The minus sign expresses the fact that the force exerted by the spring, Fs , is always opposing the deformation. It is often described as a restoring force
If Fs is the only force acting on the system, the system is called a simple harmonic oscillator, and it
undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant
amplitude and a constant frequency, which does not depend on the amplitude. Harmonic motion is
equivalent to an object moving around the circumference of a circle at constant speed.
In the absence of friction, the oscillations will conrinue forever.Friction robs the oscillator of it’s energy,
the oscillations decay, and eventually stop altogether. Frequently, this friction term is proportional to the
velocity of the moving mass (think of the air resistance), Fd = Rv.
If a frictional force Fd = −Rv, proportional and opposed to the velocity is also present, the harmonic
oscillator is described as a damped oscillator. Depending on the strength of the friction coefficient R, the
system can either oscillate with a frequency slightly smaller than in the non-damped case, and an amplitude
decreasing with time (underdamped oscillator); or decay exponentially to the equilibrium position, without
oscillations (overdamped oscillator).
41
42
EXPERIMENT 6. HARMONIC MOTION
Mass on a vertical spring
When a mass m is attached to a hanging spring, the spring is subjected to a force Fg = mg and will be
stretched to a new equilibrium position y0 where
Fg = −Fs
⇒
mg = ky0
⇒
m=
k
y0 .
g
(6.2)
If the mass is displaced from y0 and released, it will begin to oscillate about y0 according to
y = A0 cos(ω0 t + φ),
ω0 =
q
k/m = 2πf0 =
2π
T0
(6.3)
where A0 and φ are the initial amplitude and phase angle of the oscillation, T0 is the period in seconds,
and ω0 is the angular speed in radians/second. If a damping force Fd is present, the oscillation decays
exponentially at a rate determined by the damping coefficient γ:
q
R
(6.4)
,
ωd = ω02 − γ 2
2m
Note two interesting details; first, the damped frequency of oscillations, ωd , is made smaller than ω0 by
the subtraction of γ 2 under the square root. However, if R is not very large, this reduction is not all
that noticeable, even though the decrease in the amplitude due to the e−γt term may be readily observed.
Second, in the case of the air resistance, Fd = −Rv is an approximate relation, valid only for slow speeds.
As the speed increases, the turbulent air flow offers resistance that is proportional to the square of the
velocity, Fd = −Cv 2 . Here C is the so-called drag coefficient. Unfortunately, this expression cannot be
solved analytically and would have to use numerical methods (computer) to evaluate. Note also that
the hanging spring is stretched by its own weight and may exhibit twisting as well as lateral oscillations
when stretching. One other simplifying assumption: this experiment assumes an ideal massless spring
connected to a point mass m. Even with all these approximations, Equation 6.4 lends itself very nicely to
an experimental investigation.
y = A0 cos(ωd t + φ)e(−γt) ,
γ=
Review questions
Two springs with spring constants k1 and k2 are connected together
as shown in figure 6.1. With mass m1 attached, the springs have
lengths d1 and d2 . Increasing the pull mass to m2 stretches the
springs a distance x1 and x2 for a total stretch of x. Derive an
equation for the effective force constant ke of the two spring in
series. (Hint: The stretching force F is the same for both springs.)
What is ke when k1 = k2 ? Does the k of a spring depend on the
length of the coiled spring?
.................................................
.................................................
Figure 6.1: Effective spring constant
.................................................
.................................................
CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT!
43
Procedure and analysis
The experimental setup consists of a vertical stand from which
hangs a spring. A platform hanging from the bottom of the spring accepts various mass loads. The
spring mass system is free to move in the vertical direction. A collinear rangefinder measures the distance
to the bottom of the platform by sending a stream of ultrasonic pulses and measuring the time for each
echo to return.
Part 1: Static properties of springs
In the following exercises you will determine the force constants of two different springs, labelled #1 and
#2, and verify that their combined force constant satisfies the equation derived in the review section of
the experiment. Note: Use positive k values in all the following steps.
Equation 6.2 represents a straight line with slope k/g = ∆m/∆y so that k = |∆m/∆y| ∗ g. By varying
m, measuring y0 and plotting the resulting data, the force constant k for the spring can be determined.
Note that the absolute distance y0 does not matter, however the change in distance ∆y with mass ∆m is
critical.
• Suspend spring #1 from the holder and load it with the 50 g platform. Use a ruler to carefully
measure the distance y from the top of the table to the bottom of the platform. Record your result
in Table 6.1. Determine the mass required to bring the platform near the top of the table and record
this new distance, then select several intermediate masses and repeat the measurements.
mass (kg)
y (m)
Table 6.1: Data for determining the spring force constant k1 of spring #1
• Use the Physicalab software to enter the data pairs (y, m) in the data window. Select scatter plot.
Click Draw to generate a graph of your data. Select fit to: y= and enter A+B*x in the fitting
equation box. Click Draw to perform a linear fit on your data. Label the axes and enter your
name and a description of the data as part of the graph title. Click Print to generate a hard copy
of your graph. Show below the steps used to evaluate k1 and the associated error σ(k1 ). (Recall that
the error is always rounded to one significant digit while the value is rounded to the same decimal
place as the error. Appendix B includes a review of error propagation rules).
B = .............
± .............
A = .............
± .............
k1 = ...............................
σ(k1 ) = ...............................
................................
................................
................................
................................
k1 = ...............
± ...............
44
EXPERIMENT 6. HARMONIC MOTION
• Repeat the above procedure using spring #2, recording your results in Table 6.2, then determine the
spring constant k2 for the spring. Show all calculations on a separate sheet and include these as part
of your lab report.
mass (kg)
y (m)
Table 6.2: Data for determining the spring force constant k2 of spring #2
B = .............
± .............
k2 = ...............
A = .............
± .............
± ...............
• Connect together springs #1 and #2, and repeat the procedure, entering your data in Table 6.3, to
determine the experimental effective spring constant ke of the two springs:
mass (kg)
y (m)
Table 6.3: Data for determining the effective spring force constant ke of springs #1 and #2
B = .............
± .............
ke = ...............
A = .............
± .............
± ...............
• Calculate the theoretical effective spring constant ke and σ(ke ) for the two springs:
ke =
k1 k2
k1 + k2
σ(ke ) = ...............................
................................
................................
................................
................................
ke = ...............
± ...............
45
Part 2: Damped harmonic oscillator
You will now explore the behaviour in time of an oscillating mass. By accumulating a series of coordinate
points (ω, m) and fitting your data to Equation 6.4, you can use several methods to determine the spring
constant for the oscillating mass system. Begin by selecting one of the two springs to use:
• Starting with a 100 g, raise the platform a small distance, then release it to start the mass swinging
vertically. Wait until the spring/mass system no longer exhibits any erratic oscillations.
• Shift focus to the Physicalab software. Check the Dig1 box and choose to collect 200 points at 0.05
s/point. Click Get data to acquire a data set.
• Select scatter plot, then Click Draw . Your points should display a smooth slowly decaying sine
wave, without peaks, stray points, or flat spots. If any of these are noted, adjust the position of the
rangefinder and acquire a new data set.
• Select fit to: y= and enter A*cos(B*x+C)*exp(-D*x)+E in the fitting equation box. Click
Draw . If you get an error message the initial guesses for the fitting parameters may be too distant
from the required values for the fitting program to properly converge. Look at your graph and enter
some reasonable initial values for the amplitude A of the wave and the average (equilibrium) distance
E of the wave from the detector. C corresponds to the initial phase angle of the sine wave at x=0.
• The angular speed B (in radians/s) is given by B= 2π/T . Estimate the time x= T between two
adjacent minima of the sine wave then estimate and enter an initial guess for B.
• The damping coefficient D determines the exponential decay rate of the wave amplitude to the
equilibrium distance E. When D = 1/x, the envelope will have decreased from A0 to A0 e−1 =
A0 /e = A0 /2.718 ≈ A0 /3. Make an initial guess for D by estimating the time t=x required for the
envelope to decrease by 2/3.
• Check that the fitted waveform overlaps well your data points, then label the axes and include as
part of the title the value of mass m used. Click Print button to generate a printout of your graph.
Every student should print a copy of ONE sample graph.
• Record the results of the fit in Table 6.4, then complete the table as before, being careful to not have
any of the hanging masses fall on the rangefinder:
mass (kg)
y0 (m)
ω0 (rad/s)
γ (s−1 )
Table 6.4: Experimental results for damped harmonic oscillator
• The spring constant k can be determined as before from the slope of a line fitted through the (y0 , m)
data in Table 6.4. Generate and print the graph, then record the result below:
k(y0 ) = ...............
± ...............
46
EXPERIMENT 6. HARMONIC MOTION
• The spring constant k can also be determined fron the period of oscillation of the mass. Rearranging
the terms in Equation 6.3 yields m = k/ω02 . Enter the coordinate pairs from Table 6.4 in the form
(ω0 , m) and view the scatter plot of your data. Select fit to: y= and enter A+B/x**2 in the
fitting equation box. This is more convenient than squaring all the ω0 values and fitting to A+B/x.
Click Draw to perform a quadratic fit on your data. Print the graph, then enter below the value
for the spring constant:
k(ω0 ) = ...............
± ...............
• The damped harmonic oscillator equation 6.4 predicts that the frequency of oscillation depends on the
damping coefficient γ so that your experiment actually yields values for ωd rather than ω0 . Calculate
ω0 from ωd using an average value for γ. Which ωd value should be used? Why? Is the difference
between ωd and ω0 significant?
................................................................................
................................................................................
IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK!
Discussion
Begin by tabulating your spring constant results.
In the linear fit of your (y, m) data, what are the units and meaning of the fit parameter A?
Do the experimental and theoretical results for ke agree within experimental error? Explain.
Do the results for k for the spring used in parts 1 and 2 agree within experimental error? Explain.
It is assumed that the rangefinder is properly calibrated, so that it measures length correctly. Based
on your results, is this assumption valid? Explain your conclusion.
How would you determine if the rangefinder is not properly calibrated?
Check the calibration of your rangefinder against the ruler. Show your results and calculations
below. Why would you want to select two calibration points to be a maximum distance apart. Are two
calibration points sufficient? Why?
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
first name (print)
last name (print)
Brock ID (ab13cd)
TA initials
grade
Appendix A
Review of math basics
Fractions
a b
ad + bc
+ =
;
c d
cd
a
b
= ,
c
d
If
then
ad = cb
and
ad
= 1.
bc
Quadratic equations
Squaring a binomial:
Difference of squares:
(a + b)2 = a2 + 2ab + b2
a2 − b2 = (a + b)(a − b)
2
The two roots of a quadratic equation ax + bx + c = 0 are given by x =
−b ±
√
b2 − 4ac
.
2a
Exponentiation
(ax )(ay ) = a(x+y) ,
ax
= ax−y ,
ay
a1/x =
√
x
a,
a−x =
1
,
ax
(ax )y = a(xy)
Logarithms
Given that ax = N , then the logarithm to the base a of a number N is given by loga N = x.
For the decimal number system where the base of 10 applies, log10 N ≡ log N and
log 1 = 0 (100 = 1)
log 10 = 1 (101 = 10)
log 1000 = 3 (103 = 1000)
Addition and subtraction of logarithms
Given a and b where a, b > 0: The log of the product of two numbers is equal to the sum of the individual
logarithms, and the log of the quotient of two numbers is equal to the difference between the individual
logarithms .
log(ab) = log a + log b
a
log
= log a − log b
b
The following relation holds true for all logarithms:
log an = n log a
47
48
APPENDIX A. REVIEW OF MATH BASICS
Natural logarithms
It is not necessary to use a whole number for the logarithmic base. A system based on “e” is often used.
Logarithms using this base loge are written as “ln”, pronounced “lawn”, and are referred to as natural
logarithms. This particular base is used because many natural processes are readily expressed as functions
of natural logarithms, i.e. as powers of e. The number e is the sum of the infinite series (with 0! ≡ 1):
e=
∞
X
1
=
n!
n=0
1
1
1
1
+ + + + · · · = 2.71828 . . .
0! 1! 2! 3!
Trigonometry
Pythagoras’ Theorem states that for a right-angled triangle c2 = a2 + b2 .
Defining a trigonometric identity as the ratio of two sides of the triangle,
there will be six possible combinations:
b
c
c
csc θ =
b
sin θ =
a
c
c
sec θ =
a
cos θ =
sin(θ ± φ) = sin θ cos φ ± cos θ sin φ
cos(θ ± φ) = cos θ cos φ ∓ sin θ sin φ
tan θ ± tan φ
tan(θ ± φ) =
1 ∓ tan θ tan φ
b
sin θ
=
a
cos θ
a
cos θ
cot θ = =
b
sin θ
tan θ =
sin 2θ = 2 sin θ cos θ
cos 2θ = 1 − 2 sin2 θ
2 tan θ
tan 2θ =
1 − tan2 θ
180◦ = π radians = 3.15159 . . .
1 radian = 57.296 . . . ◦
sin2 θ + cos2 θ = 1
To determine what angle a ratio of sides represents, calculate the inverse of the trig identity:
if
b
sin θ = ,
c
then
θ = arcsin
b
c
For any triangle with angles A, B, C respectively opposite the sides a, b, c:
a
b
c
=
=
, (sine law)
sin A
sin B
sin C
The sine waveform
If we increase θ at a constant rate from
0 to 2π radians and plot the magnitude
of the line segment b = c sin θ as a function of θ, a sine wave of amplitude c and
period of 2π radians is generated.
Relative to some arbitrary coordinate system, the origin of this sine wave
is located at a offset distance y0 from the
horizontal axis and at a phase angle of θ0
from the vertical axis. The sine wave referenced from this (θ, y) coordinate system is given by the equation
y = y0 + c sin(θ + θ0 )
c2 = a2 + b2 − 2ac cos C. (cosine law)
first name (print)
last name (print)
Brock ID (ab13cd)
TA initials
grade
Appendix B
Error propagation rules
• The Absolute Error of a quantity Z is given by σ(Z), always ≥ 0.
• The Relative Error of a quantity Z is given by
σ(Z)
|Z| ,
always ≥ 0.
• To determine the error in a quantity Z that is the sum of other quantities, you add the absolute errors
of those quantities (Rules 2,3 below). To determine the error in a quantity Z that is the product of
other quantities, you add the relative errors of those quantities (Rules 4,5 below).
Relation
Error
1.
Z = cA
σ(Z) = |c|σ(A)
2.
Z = A + B + C + ···
σ(Z) = σ(A) + σ(B) + σ(C) + · · ·
3.
Z = A − B − C − ···
σ(Z) = σ(A) + σ(B) + σ(C) + · · · (Error terms are always added.)
4.
Z = A × B × C × ···
(Use only if A is a single term, i.e. Z = 3x.)
5.
Z=
6.
Z = Ab
σ(Z)
|Z|
σ(Z)
|Z|
σ(Z)
|Z|
7.
Z = sin A
σ(Z) = σ(sin A) =
8.
Z = log A
σ(Z) = σ(log A) =
A
B
=
=
=
σ(A)
σ(B)
|A| + |B|
σ(A)
σ(B)
|A| + |B|
|b| σ(A)
|A|
+
σ(C)
|C|
+ ···
(Note the absolute value of the power.)
| sin[A+σ(A)]−sin[A−σ(A)] |
2
| log[A+σ(A)]−log[A−σ(A)] |
2
(Similar for cos A)
• a, b, c, . . . , z represent constants
• A, B, C, . . . , Z represent measured or calculated quantities
• σ(A), σ(B), σ(C), . . . , σ(Z) represent the errors in A, B, C, . . . , Z, respectively.
How to derive an error equation
Let’s use the change of variable method to determine the error equation for the following expression:
y=
Mq
0.5 kx (1 − sin θ)
m
• Begin by rewriting Equation B.1 as a product of terms:
49
(B.1)
50
APPENDIX B. ERROR PROPAGATION RULES
y = M ∗ m−1 ∗ [ 0.5 ∗ k ∗ x ∗ (1 − sin θ)]
= M ∗ m
−1
1/2
∗ 0.5
∗ k
1/2
∗ x
1/2
1/2
(B.2)
1/2
∗ (1 − sin θ)
(B.3)
• Assign to each term in Equation B.3 a new variable name A, B, C, . . . , then express v in terms of
these new variables,
y=A ∗ B ∗ C ∗ D ∗ E ∗ F
(B.4)
• With σ(y) representing the error or uncertainty in the magnitude of y, the error expression for y is
easily obtained by applying Rule 4 to the product of terms Equation B.4:
σ(y)
σ(A)
σ(B)
σ(C)
σ(D)
σ(E)
σ(F )
=
+
+
+
+
+
|y|
|A|
|B|
|C|
|D|
|E|
|F |
(B.5)
• Select from the table of error rules an appropriate error expression for each of these new variables as
shown below. Note that F requires further simplification since there are two terms under the square
root, so we equate these to a variable G:
A = M,
B=
m−1 ,
C = 0.51/2 ,
D = k1/2 ,
E = x1/2 ,
F = G1/2 ,
G = 1 − sin θ,
σ(A) = σ(M )
Rule 1
σ(B)
|B|
σ(C)
|C|
σ(D)
|D|
σ(E)
|E|
σ(F )
|F |
Rule 6
σ(m)
= | − 1| σ(m)
|m| = |m|
= 21 σ(0.5)
=0
|0.5|
= 21 σ(k)
= σ(k)
2|k|
|k|
1 σ(x)
σ(x)
= 2 |x| = 2|x|
σ(G)
= 21 σ(G)
|G| = 2|G|
σ(G) = σ(1) + σ(sin θ) = 0 +
since σ(0.5) = 0
Rule 6
Rule 6
Rule 6
| sin[θ+σ(θ)]−sin[θ−σ(θ)] |
2
Rules 3,6
• Finally, replace the error terms into the original error Equation B.5, simplify and solve for σ(y) by
multiplying both sides of the equation with y:
σ(y) = |y|
σ(M ) σ(m) σ(k) σ(x) | sin[θ + σ(θ)] − sin[θ − σ(θ)] |
+
+
+
+
|M |
|m|
2|k|
2|x|
|4(1 − sin θ)|
(B.6)
first name (print)
last name (print)
Brock ID (ab13cd)
TA initials
grade
Appendix C
Graphing techniques
A mathematical function y = f (x) describes the one to one
relationship between the value of an independent variable x
and a dependent variable y. During an experiment, we analyse
some relationship between two quantities by performing a series of measurements. To perform a measurement, we set some
quantity x to a chosen value and measure the corresponding
value of the quantity y. A measurement is thus represented
by a coordinate pair of values (x, y) that defines a point on a
two dimensional grid.
The technique of graphing provides a very effective method
of visually displaying the relationship between two variables.
By convention, the independent variable x is plotted along the
horizontal axis (x-axis) and the dependent variable y is plotted
along the vertical axis (y-axis) of the graph. The graph axes
should be scaled so that the coordinate points (x, y) are well
distributed across the graph, taking advantage of the maximum display area available. This point is especially important when results are to be extracted directly form the data
presented in the graph. The graph axes do not have to start
at zero.
Scale each axis with numbers that represent the range of
values being plotted. Label each axis with the name and unit
of the variable being plotted. Include a title above the graphing area that clearly describes the contents of the graph being
plotted. Refer to Figure C.1 and Figure C.2.
Figure C.1: Proper scaling of axes
Figure C.2: Improper scaling of axes
The line of best fit
Suppose there is a linear relationship between x and y, so that
y = f (x) is the equation of a straight line y = mx + b where m is the slope of the line and b is the value
of y at x = 0. Having plotted the set of coordinate points (x, y) on the graph, we can now extract a value
for m and b from the data presented in the graph.
Draw a line of ’best fit’ through the data points. This line should approximate as well as possible the
trend in your data. If there is a data point that does not fit in with the trend in the rest of the data, you
should ignore it.
51
52
APPENDIX C. GRAPHING TECHNIQUES
The slope of a straight line
The slope m of a straight-line graph is determined by
choosing two points, P1 = (x1 , y1 ) and P2 = (x2 , y2 ),
on the line of best fit, not from the original data, and
evaluating Equation C.1. Note that these two points
should be as far apart as possible.
m =
m =
rise
run
∆y
y2 − y1
=
∆x
x2 − x1
(C.1)
Figure C.3: Slope of a line
Error bars
All experimental values are uncertain to some degree due
to the limited precision in the scales of the instruments
used to set the value of x and to measure the resulting value of y. This uncertainty σ of a measurement is
generally determined from the physical characteristics of
the measuring instrument, i.e. the graduations of a scale.
When plotting a point (x, y) on a graph, these uncertainties σ(x) and σ(y) in the values of x and y are indicated
using error bars.
Figure C.4: Error Bars for Point (x, y)
For any experimental point (x ± σ(x), y ± σ(y)), the
error bars will consist of a pair of line segments of length
2 σ(x) and 2 σ(y), parallel to the x and y axes respectively and centered on the point (x, y). The true value lies within the rectangle formed by using the error
bars as sides. The rectangle is indicated by the dotted lines in Figure C.3. Note that only the error bars,
and not the rectangle are drawn on the graph.
The uncertainty in the slope
Figure C.5 shows a set of data points for a linear relationship. The slope is that of line 2, the line of best fit
through these points. The uncertainty in this slope is
taken to be one half the difference between the line of
maximum slope line 1 and the line of miinimum slope,
line 3:
σ(slope) =
slopemax − slopemin
2
(C.2)
The lines of maximum and minimum slope should go
through the diagonally opposed vertices of the rectangles
defined by the error bars of the two endpoints of the
graph, as in Figure C.5.
Figure C.5: Determining slope error
53
Logarithmic graphs
In science courses you will encounter a great number of functions and relationships, both linear and nonlinear. Linear functions are distinguished by a proportional change in the value of the function with a change
in value of one of the variables, and can be analyzed by plotting a graph of y versus x to obtain the slope m
and vertical intercept b. Non-linear functions do not exhibit this behaviour, but can be analyzed in a similar
manner with some modification. For example, a commonly occuring function is the exponential function,
y = aebx ,
(C.3)
where e = 2.71828 . . ., and a and b are constants.
Plotted on linear (i.e. regular) graph paper, the
function y = aebx appears as in Figure C.6. Taking the natural logarithm of both sides of equation (C.3) gives
ln y = ln aebx
ln y = ln a + ln ebx
ln y = ln a + bx ln e
ln y = ln a + bx (since ln e = 1)
Figure C.6: The exponential function y = aebx .
Equation (C.4) is the equation of a straight line
for a graph of ln y versus x, with ln a the vertical
intercept, and b the slope. Plotting a graph of ln y
versus x (semilogarithmic, i.e. logarithmic on the vertical axis only) should result in a straight line, which
can be analyzed.
There are two ways to plot semilogarithmic data for analysis:
1. Calculate the natural logarithms of all the y values, and plot ln y versus x on linear scales. The slope
and vertical intercept can then be determined after plotting the line of best fit.
2. Use semilogarithmic graph paper. On this type of paper, the divisions on the horizontal axis are
proportional to the number plotted (linear), and the divisions on the vertical axis are proportional to
the logarithm of the number plotted (logarithmic). This method is preferable since only the natural
logarithms of the vertical coordinates used to determine the slope of the lines best fit, minimum and
maximum slope need to be calculated.
Semilogarithmic graph paper
The horizontal axis is linear and the vertical axis is logarithmic. The vertical axis is divided into a series
of bands called decades or cycles.
• Each decade spans one order of magnitude, and is labelled with numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 1.
• the second “1” represents 10× what the first “1” does, the third 10× the second, et cetera, and
• there is no zero on the logarithmic axis since the logarithm of zero does not exist.
A logarithmic axis often has more than one decade, each representing higher powers of 10. In Figure C.7,
the axis has 3 decades representing three consecutive orders of magnitude. For instance, if the data to be
plotted covered the range 1 → 1000, the lowest decade would represent 1 → 10 (divisions 1, 2, 3, . . . , 9),
54
APPENDIX C. GRAPHING TECHNIQUES
Figure C.7: A 3-Decade Logarithmic Scale.
the second decade 10 → 100 (divisions 10, 20, 30, . . . , 90) and the third decade 100 → 1000 (divisions 100,
200, 300, . . . 900, 1000).
Another advantage of using a logarithmic scale is that it allows large ranges of data to be plotted. For
instance, plotting 1 → 1000 on a linear scale would result in the data in the lower range (e.g. 1 → 100)
being compressed into a very small space, possibly to the point of being unreadable. On a logarithmic
scale this does not occur.
Calculating the slope on semilogarithmic paper
The slope of a semilogarithmic graph is calculated in the usual manner:
m = slope
rise
=
run
∆(vertical)
=
.
∆(horizontal)
For ∆(vertical) it is necessary to calculate the change in the logarithm of the coordinates, not the change
in the coordinates themselves. Using points (x1 , y1 ) and (x2 , y2 ) from a line on a semilogarithmic graph of
y versus x and Equation C.4, the slope of the line is obtained.
m =
m =
m =
rise
run
ln y2 − ln y1
x2 − x1
ln (y2 /y1 )
x2 − x1
(C.4)
Note that the units for m will be (units of x)−1 since ln y results in a pure number.
Analytical determination of slope
There are analytical methods of determining the slope m and intercept b of a straight line. The advantage
of using an analytical method is that the analysis of the same data by anyone using the same analytical
method will always yield the same results. Linear Regression determines the equation of a line of best fit
by minimizing the total distance between the data points and the line of best fit.
To perform “Linear Regression” (LR), one can use the preprogrammed function of a scientific calculator
or program a simple routine using a spreadsheet program. Based on the x and y coordinates given to it, a
LR routine will return the slope m and vertical intercept b of the line of best fit as well as the uncertainties
σ(m) and σ(b) in these values. Be aware that performing a LR analysis on non-linear data will produce
meaningless results. You should first plot the data points and determine visually if a LR analysis is indeed
valid.