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University of Pennsylvania: Department of Physics
Lab Manual 2006
IntroLab Manual
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University of Pennsylvania: Department of Physics
Lab Manual 2006
Lab Manual Table of Contents
Section 1: Introduction
1.1 Introduction to the Labs..........................................................................................................................................3
1.2 Writing Physics Lab Reports ...................................................................................................................................5
1.3 Reporting Results in the Laboratory.......................................................................................................................6
1.4 Making Graphs .......................................................................................................................................................8
General Guidelines for Graphs ................................................................................................................................8
Graph Presentation...................................................................................................................................................8
Least Squares Fitting ..............................................................................................................................................11
1.5 Linearizing Data ...................................................................................................................................................13
The Dial Caliper ..........................................................................................................................................................15
The Micrometer ...........................................................................................................................................................16
Photogates and what they are good for........................................................................................................................18
The Sonic Ranger ........................................................................................................................................................21
Introduction to the Oscilloscope..................................................................................................................................26
Error Table of Contents/Definitions ............................................................................................................................35
2.1 Types of Error........................................................................................................................................................36
2.2 Precision vs. Accuracy...........................................................................................................................................38
2.3 Writing Experimental Numbers: Significant Figures and Scientific Notation.......................................................39
2.4 Uncertainty of Measurements................................................................................................................................40
2.5 Averaging ..............................................................................................................................................................41
2.6 Averaging and Uncertainty for Large Data Sets....................................................................................................42
2.7 Standard Deviation ................................................................................................................................................44
2.8 Standard Deviation of the Mean ............................................................................................................................44
Example: Calculating the mean, the standard deviation, and the standard deviation of the mean ........................45
Formulae for calculating standard deviation and uncertainty ...............................................................................46
Summary Tables for Error Propagation Rules .......................................................................................................47
2.9 Addition and Subtraction.......................................................................................................................................48
2.10 Multiplication and Division.................................................................................................................................48
2.11 Percent Uncertainties ...........................................................................................................................................49
2.12 Multiplication by a known constant.....................................................................................................................49
2.13 Functional Error: Propagation of errors with functions .......................................................................................50
2.14 An Example of Propagation of Errors .................................................................................................................51
2.15 Another Error Propagation Example ...................................................................................................................52
2.16 Propagation of Error Using the Values of Uncertainty........................................................................................54
2.17 Estimation of Uncertainty by Multiplying the High and Low Values of the Factors ..........................................54
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University of Pennsylvania: Department of Physics
Lab Manual 2006
1.1 Introduction to the Labs
Where…..Labs are located on the third floor of DRL, in the north wing and west wing. You may
meet in a different room each week. The specific lab rooms are listed on two bulletin boards
located next to 3N18 and at the intersection of the west wing and the center corridor next to
3W5. For the first week only, sections that have more than 14 students enrolled will have one
room listed. When everyone arrives, the TAs will divide the class and half will go to another
room.
Lab Website... The website for the Undergraduate Labs is www.physics.upenn.edu/~uglabs It is
listed on the Physics homepage (under Academics/UNDERGRADUATE/Physics and
Astronomy Courses/ Undergraduate Labs). Your lab manual, lab schedule, pre-labs, video
demos, registration information and other announcements are all here.
When…The lab "week" is different for different courses. For that reason, the date given for each
lab in the Master Experiment Schedule on-line is the first day in the lab week cycle, and the
other sections all do the same lab until the last day of the cycle. There will be a copy of the
master schedule on the bulletin boards next to 3W5 and 3N18.
The Lab Manual….The lab manual is a pdf document with about 50 pages. It is intended to be
your reference on how to prepare for lab, how to write a lab report, do error analysis and graphs
and use various instruments. When you view this document, look on the ACROBAT toolbar for a
button called "Navigation Pane" that splits the window into two frames- this will display the
bookmarks in the left frame. There is also a "Bookmarks" tab on the left edge of the window.
The bookmarks will help you navigate through this document. There is a table of contents in the
manual. The page number is shown in the center of the navigation bar at the bottom of the page
and on the bottom of each page.
The lab manual will be available in lab on your computer and in hard copy. The manual cannot
be printed in lab. You can print sections or single pages by specifying their page numbers in the
Print window.
The Lab Write-ups… Each individual lab has a pdf document that is linked to the on-line
schedule and also to the EXPERIMENTS page on the web-site. The write-ups will contain PreLab questions. It is advisable to print only what you need for the next lab or two. Changes may
be made two or more weeks before lab. The changes will be listed on the lab homepage in the
NEWS table. The instructions will become less detailed with each subsequent lab. Details will
increasingly be left to your discretion after the initial few labs.
The write-ups contain figures and photographs of the apparatus. The figures are illustrations
only, and are not to be taken too literally. Since no one can learn to drive by studying a picture
of a Chevrolet, your lab instructor may demonstrate the apparatus.
The Lab Notebook… You must buy a carbonless copy notebook to write your data and
calculations in. This is available in the bookstore. You will turn in a copy of your data at the end
of lab (whether you hand in a report or not) and keep a copy for your reference.
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University of Pennsylvania: Department of Physics
Lab Manual 2006
Videos…. Videos are available that demonstrate and explain the equipment. View the VIDEOS
before doing your labs. They are streaming video that requires a fast internet connection. They
are not available in lab. If it's more convenient, you will be able to see the videos in the MathPhysics library or the Van Pelt library and in MMETS in the basement of DRL.
Pre-Labs…For each lab that you do, you will be expected to turn in a pre-lab (built into the labwrite-up) to your TA before you start your lab. The pre-labs are designed to direct your reading
and give you practice in doing some of the calculations that you will need to do in lab.
About the lab sessions….Physics labs are scheduled for a two hour period, which means that
you must be out of the room (there's usually another class coming in after you) in one hour and
fifty minutes. That's not much time-- other science labs are three or four hours long. So you need
to be well prepared and efficient. It is advisable to outline a data sheet layout beforehand so that
you'll be ready to record data.
Your TA….Your lab instructor will help you with the hardware and software used to carry out
the experiment, and grade the lab reports and pre-labs. At the beginning of the period the
instructor will first collect your PreLab and then give a brief introduction, going over key points.
He or she will assume that you have read the lab write-up. After that you will be on your own.
Your instructor will give you hints. You must expect to analyze sources of experimental errors.
About the computers….
A file called readMeFirst is available on the desktop to give you software tips.
You will be using a networked laser printer. There is a printer icon on your computer desktop
telling you where your printer is located. Your printouts will have a light watermark
corresponding to the letter or number marked on the CPU of the computer you are using. Give
your printouts a title with your name and your partner’s name and a short description.
Keep only the minimum software running or the computer may run out of memory and freeze
without warning! Our lab software, LoggerPro, stores a huge amount of data. It is a good idea to
restart LoggerPro after collecting data for a few minutes to avoid using up all the memory.
We have both Macintosh and Windows computers. If you are using a Macintosh- one big
difference is that closing a window DOES NOT QUIT the software.
Lab Reports…The lab report will either be done in the lab or afterwards and then handed in by
a deadline stipulated by your instructor. Staple all pages together. You must keep all your graded
lab reports until you receive your final grade for the course or lab (if taken separately).
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University of Pennsylvania: Department of Physics
Lab Manual 2006
1.2 Writing Physics Lab Reports This is a brief discussion of what is expected in a lab report
for the Introductory Physics courses. You will receive additional information in class. The report,
other than actual data and graphs should only be a few pages.
Cover Sheet
 Must include: name of experiment and date performed; group partners’ individual names
and roles; TA’s name; Professor’s name; course and lab section numbers.
 Cover sheet template can be found on lab computers (but do not edit the file).
Intro/Abstract
 Brief statement (1 paragraph) describing the goals of the lab and methods/procedure used
(do not duplicate the laboratory manual description)
 Written in a way that a person not familiar with the lab can quickly understand what the
goals and methods used are.
Data
 A carbonless copy of all handwritten data must be submitted to TA at end of each lab
(any acceptable lab notebook will do this). Show units and uncertainties.
 Should be well organized and arranged in tables for clarity.
 Sample printouts of computer generated graphs or tables are appropriate.
 Show uncertainties in the data points and the uncertainty in the slope of the best fit line.
Analysis
 Outline major calculations and provide an example calculation for each one. Calculate
best values and uncertainties (should be done in lab to make sure that the data is correct
and sufficient).
 Include graphs and tables with proper titles and units in sequence or at the end of the
report.
 State the equations used in calculations and define all variables.
Summary/ Error
 A short summary (no more than a few paragraphs) considering what could have caused
error in your measurement. Include all sources of error (the more insightful the better).
 Determine the largest source of error and describe how it could be improved.
Conclusion
 Brief (a few paragraphs). Compare your results to accepted values. Discuss what the data
shows, why, and what you’ve learned.
Questions
 Attach questions and answers on a separate sheet. If they are answered in the report put a
reference next to question.
Grading
 Grades are 7-10 with 7 considered passing, 8 considered average, and 9 considered
insightful and outstanding. A 10 is for the rare report that is perfect--good enough to
publish. Grades are based on overall clarity, brevity, completeness, and level of insight
into the physics of the experiment.
 Comments will suggest how to improve or reward excellence. A check by a section
means OK.
HS 9/11/2006 labIntro.doc
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University of Pennsylvania: Department of Physics
Lab Manual 2006
1.3 Reporting Results in the Laboratory
Why do we need to express uncertainties in measurements? Two reasons: to have a way to
compare experimental numbers with theoretical numbers, and to be able to make absolute
judgments based on those comparisons.
Reason 1: There are right ways, wrong ways, and really, really terrible ways to compare your
experimental value to the value you expect from a theoretical calculation. Let’s look at a few
examples in which we have measured acceleration in the lab and calculated what we thought the
value should be:
Experimental a (m/s2)
0.406
Theoretical a (m/s2)
0.415
In this case we can’t make any comparison. Without an associated uncertainty, these numbers
are meaningless. If the “theoretical” value was calculated from a measured quantity, it will have
an uncertainty too. Always report an uncertainty!
Experimental a (m/s2)
0.406  0.010
Theoretical a (m/s2)
0.415  0.008
experimental acceleration
falls into the range
0.396 < aexp < 0.416
experimental acceleration
falls into the range
0.407 < atheo < 0.423
These two intervals overlap, so the values must agree within uncertainty. As far as we are able
to determine, they are the same!
Experimental a (m/s2)
0.401  0.005
Theoretical a (m/s2)
0.415  0.005
experimental acceleration
falls into the range
0.396 < aexp < 0.406
experimental acceleration
falls into the range
0.410 < atheo < 0.420
The two intervals do not overlap, so these values do not agree within uncertainty. It doesn’t
matter that they are “close” or “similar” or “almost the same.” They do not agree within
uncertainty. The values could differ because they described fundamentally different phenomena,
or you could have underestimated the associated uncertainties and any difference can be
attributed to additional sources of error. This distinction is not always obvious.
To evaluate our results, we should find the percent deviation from the calculated or theoretical
value. This will tell us how close the two values are. The percent deviation is:
 a exp  atheo 
(100)
% deviation  


atheo


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University of Pennsylvania: Department of Physics
Lab Manual 2006
The percent deviation is not the uncertainty. It is the amount by which your values deviate from
the norm. Using the above example, we find the percentage deviation to be 3.4%.
Experimental values may not always compare with the theoretical values, as in these examples:
Experimental a (m/s2)
0.406  0.006
0.402  0.010
0.408  0.006
Theoretical a (m/s2)
0.425  0.01
All of the experimental values shown above lie within error of each other, but none lie within the
range of theoretical values. A valid conclusion here is that there is some systematic error that
our calculations have not taken into account.
Experimental a (m/s2)
0.38  0.01
0.405  0.012
0.435  0.006
Theoretical a (m/s2)
0.425  0.005
In this example none of the experimental values lie within error of each other, and no value is
within error of the theoretical value. The conclusion for this case is that your experiment
requires major improvement. It is not necessarily your fault, but that must be your conclusion.
Reason 2: Ambiguous, vague, approximate statements are not acceptable in a formal lab report.
Some examples:
“Since our values are relatively close, we determined that mass does not play a role …”
“The acceleration of the sphere is approximately constant …”
“These two values are very similar …”
“Our recorded values are moderately off from our expected values …”
“Energy was conserved somewhat …”
Some examples of acceptable statements:
“Since our results agree within uncertainty, we conclude that mass does not play a role …”
“The acceleration of the sphere is constant within our accepted level of error …”
“These two values do not agree within their associated uncertainties; they differ by 3.45% …”
“Our recorded values do not agree with the theory within uncertainty, however, they seem to
follow the same general trend …”
“Energy was/was not conserved …”
5/28/2009 reporting results.doc
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University of Pennsylvania: Department of Physics
Lab Manual 2006
1.4 Making Graphs
General Guidelines for Graphs:
Whether your graphs are made by computer or by hand- follow the same basic guidelines.
Do you have enough points to characterize your curve shape?
For example, if you have 2 points, you can always draw a line between them, similarly 3 points
define a parabola, 4 a cubic function, and so forth. Therefore, to make a meaningful fit, you need
the number of data points to exceed the minimum number of points necessary to define the
shape. In general, the more data points you collect, the better the fit becomes.
Does the best fit curve fall within the error bars of most or all of the data points?
The error bars are markers that visually show 1σ around the data points. Therefore, you would
expect your best fit line to pass through at least 70% of the error bars.
Does the data randomly appear on both sides of the best fit curve?
The best fit curve minimizes the sum of the distances from the data points. It may not fit the data
set equally well everywhere in the data set. (Consider what a linear fit would look like when
applied to sinusoidal data.) If your curve is not characterizing your data correctly, try another fit.
Does your curve shape have a physical meaning?
Always take a moment to see if you can explain the curve shape by a physical process in your
experiment. If you can’t, it can be useful to double check the equipment, take more data, or
tinker with the experiment until you can.
Real Data: Real life data sets often display several types of behavior that may make your data
not look “perfect”. For example, if you measure the period of a simple pendulum over a long
l
period of time, you will not observe a perfect fit to T  2
. For small oscillations over a
g
short period of time you can recover the equation. Over long periods of time the pendulum’s
oscillations will decrease in amplitude, and you may observe an exponential decay envelope on
top of the sinusoidal behavior.
Graph Presentation:
o Titles should tell the reader exactly what is graphed.
o Remove any stray lines, legends, points, and any other unintended additions by the
computer that does not add to your graph. (For example, many programs will add
meaningless piecewise lines connecting your data points.)
o Remember the error bars! They give your fit meaning. Give them a good display.
o Label axes to clearly show the units and the scale
o Choose scales so that the points can be easily plotted and read. Usually one division
should represent 1, 2, 5, or 10 units. The scales used on the two axes need not be the
same.
o Arrange the scales so the data covers most of the area of the graph. The origin (0,0)
is usually included, but not always.
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University of Pennsylvania: Department of Physics
Lab Manual 2006
Figure 1:Example Graph using Excel
A linear fit of a data set with error in both x and y. The error in x is constant, the error in y
varies (see data table below)
Y versus X
12
10
8
Y axis label (units)
y = 9.8352x + 0.4782
2
R = 0.995
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
X axis label (units)
Table 1: Data Set for Figure1 Example Graph using Excel
X (units)
X error (±) (units)
Y (units)
Y error (±) (units)
0.00
0.20
0.70
0.50
0.10
0.20
1.30
0.40
0.20
0.20
2.66
0.35
0.30
0.20
3.40
0.12
0.40
0.20
4.26
0.16
0.50
0.20
5.20
0.20
0.60
0.20
6.34
0.10
0.70
0.20
7.04
0.36
0.80
0.20
8.62
0.30
0.90
0.20
9.52
0.24
9
0.9
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University of Pennsylvania: Department of Physics
Lab Manual 2006
Finding the “best line” and the uncertainty in the slope
Figure 2 shows the "best" (heavy line) or most
representative straight line that fits the data points as
well as two other (red) lines . Approximately the same
number of points lie above and below the best line. The
best line is used to find the slope and the intercept.
The two red lines might represent the data nearly as well
as the best line. One red line has the largest plausible
slope and one has the smallest. The largest slope line
can be constructed by drawing a line which goes
between a point below the best fit line on the left side of
the graph and a point above the best fit line on the right
side of the graph. The smallest slope line can be
Figure 2: Finding the “best fit”
constructed by drawing a line which goes between a
point above the best fit line on the left side of the graph and between a point below the best fit
line on the right side of the graph. The differences between the slopes and intercepts of these
lines yields the uncertainties in the slope and intercept.
Finding the Uncertainty in the Slope
Figure 3 illustrates the method used for finding the
uncertainty in the slope of the best line.
The dashed lines define the slope triangle of the best fit
line. The vertical height of the slope triangle is called the
"rise" and the horizontal width is called the "run". The
slope is found from "rise/run" --- please note that the
slope triangle is for the best fit line and not for individual
data points.
The uncertainty in the slope, expressed as a fraction of
the slope, is  slope slope  and is found as follows:


2

 

2 run
2
 slope
  rise

rise
run
slope 

Figure 3: Finding the uncertainty
in the slope.
In this example, rise should be approximately the same as the uncertainty in the y
measurements and similarly run should be approximately the same as the uncertainty in the x
measurements. Since the uncertainty in the rise does not affect the uncertainty in the run, the
uncertainties are added in quadrature. (See the Propagation of Errors section for a complete
explanation.)
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University of Pennsylvania: Department of Physics
Lab Manual 2006
Data that do not fit a straight line
When the distances between the data points and the "best" line are much larger than the error
bars, the data do not fit a straight line. An experimenter faced with data of this type should
conclude one or more of the following:

The phenomenon under study cannot be described by a linear relationship between the
variables

the uncertainties have been grossly underestimated

the data are as precise as indicated but are inaccurate due to mistakes made reading a
measuring instrument (e.g., interpreting 1.23 cm as 2.23 cm.)

the data are plotted incorrectly.
Very Precise Data and a Good Fit to a Straight Line
The graph in Figure 4 shows very precise data (error
bars too small to plot).
If the errors are too small to draw minimum and
maximum slope lines, you might want to consider
using the method of least squares.
Least Squares Fitting
Figure 4: A very good fit to a
straight line.
The points in Figure 5 were generated from measurements of the velocity of a moving object at
different times. Due to experimental "noise" the data are scattered about a line but do not
necessarily lie exactly on it. The best lines have been drawn "by eye".
In the graphs in Figure 5, let yn be the
measured value of the velocity at time
xn (mentally replace the standard y
and x variables with v and t).
According to theory the data points
are expected to form a straight line.
Since the points (yn, xn ) do not lie
exactly on a single line, we cannot
describe them with the equation for a
straight line with slope m and
intercept b, x = m t + b. Indeed, the
Figure 5: Two examples of a best fit line fit by eye.
equation will "predict" that at time xn
the velocity has some value which will be different from the measured value yn.
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University of Pennsylvania: Department of Physics
Lab Manual 2006
Calculating the Least Squares Fit from data
In Figure 5 the left data set has only a few data points, the right has a lot more. Numerical
values are omitted for clarity. The reason the data points do not all lie exactly on a single line is
because of random deviations in the motion and/or measuring process. The line drawn to
represent the data points is just one of many plausible lines that could be drawn.
While many lines could be drawn through the data, some lines are better than others. What is the
"best" line --- the single line that represents the data as well as possible? If the points would all
lie on a single line, except for random errors, then we can find the "best" line (the most likely
one) from a Least Squares Fit on the data.
Example 2: Calculating the LSQ fit line
It is possible to calculate the slope and intercept of the
"best fit" line by hand, although it takes a lot of time.
Table 2: Example 2 Calculating the
The details of this calculation are in the Least Squares
LSQ fit line
Fitting with Excel document in the USEFUL
x
y
yfit
y - yfit (y - yfit)2
DOCUMENTS section of the webpage. It is much faster
0.000 0.500
0.000
0.500
0.250
to let a program, such as LoggerPro or Excel, calculate
1.000 1.000
1.000
0.000
0.000
the fit for you, but it is important to understand how
the calculation is done.
2.000 1.500
2.000
-0.500
0.250
Consider the data in the Table 2. Units have been left
out here for simplicity (you must always include
them). The values of y in the best fit line satisfy the
equation y fit  y'n  mx 'n  b . The method of least
squares adjusts the values of the slope and intercept so
that on average yn and yn' are as close to each other as
possible. The idea is to find the line that minimizes
the sum of all the squares of the differences between
the actual (yn) and predicted (yn') values. In other
words, the method minimizes the sum of
(yn –yn')2 in the last column to the right in the table.
JB 5/28/2009 making graphs_lsq.doc
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3.000
3.000
3.000
0.000
0.000
4.000
4.500
4.000
0.500
0.250
5.000
5.000
5.000
0.000
0.000
6.000
5.500
6.000
-0.500
0.250
7.000
7.000
7.000
0.000
0.000
8.000
8.500
8.000
0.500
0.250
9.000
9.000
9.000
0.000
0.000
10.000
9.500
10.000
-0.500
0.250
University of Pennsylvania: Department of Physics
Lab Manual 2006
1.5 Linearizing Data
A linear relation has the form y  mx  b , which is useful for showing direct relationships such
as F=ma and V=IR. A graph of the force of gravity on the y-axis and mass on the x-axis would
yield a line with a slope equal to the acceleration due to gravity. A graph of voltage vs. current
would give a value for resistance. This is very good stuff!
What about equations which are non-linear? How could a least squares fit help with that? The
trick is to linearize your data and then to apply a least squares fit.
Consider the two graphs in Figure 6, which both deal with the equation y  ax 2 . In the graph on
the left we plotted y vs x and got a parabola. In the graph on the right we have linearized the
function and plotted y vs. x2 and got a line. The difference between the two graphs is that in
Figure 6A x is treated as the independent variable and in Figure 6B x2 is treated as the
independent variable. x is not always the independent variable; think of plotting y vs. a new
variable u , where u  x 2 .
(A) shows a graph of y vs. x
of the quadratic function ax2.
(B) shows a graph of y vs. x2,
a linear relation with slope a.
(A)
(B)
Figure 6
Examples of linearizing equations

Consider the magnetic force on electrons and the circular motion the electrons undergo in
m 2eV
. We want a graph whose slope
a uniform magnetic field. The equation is eB 
r
m
gives the charge-to-mass ratio of the electron, e/m, just like Sir J. J. Thompson did in
1897. We want the charge as a function of the mass --- this means that the charge should
be the dependent variable and the mass should be the independent variable, and both of
them should be present to the first power only. The first step is to get the mass out from
under the square root, which we can do by squaring both sides of th equation. After that,
we just rearrange the terms to get the charge on the left-hand side and the mass on the
right-hand side.
eBr
2eV
e 2 B2 r 2 2eV
eB 2 r 2
2V




 2V  e  2 2 m
2
m
m
m
m
m
Br
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University of Pennsylvania: Department of Physics
Lab Manual 2006
This rearrangement involved only simple algebra! The final equation gives the charge as
a linear function of mass. In lab we only measure the voltage, V, the magnetic field, B,
and the radius, r. Calculating 2V B 2 r2 would give the charge-to-mass ratio.

The period of a pendulum, given by T  2 l g . In lab we only measure the period and
the length of the pendulum, but we want a linear graph. What to do? Squaring both sides
4 2
of the equation gives T 2 
l . This yields a linear graph of the square of the period
g
vs. the length of the pendulum with a slope of

4 2
and an intercept of zero.
g
Torricelli’s law says that a tall column of liquid with a small hole at the bottom will have
water coming out of the hole with a velocity v  2gh . In lab we will measure v, the
velocity of the water, and h, the height of the column. If we graph v vs. h, we won’t get a
straight line. However, if we graph v vs. h , we’ll get a line with a slope of 2g and an
intercept of zero.
In these examples we took a non-linear relationship and found a way to rearrange it as a linear
relationship. The key to manipulating an equation to get a linear relationship is to understand
what you must treat as the independent and dependent variables. This is a skill that improves
with practice, but it is certainly worth the time it takes to learn!
JB 5/28/2009 linearizing data.doc
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University of Pennsylvania: Department of Physics
Lab Manual 2006
Laboratory Equipment
The Dial Caliper
An ordinary ruler can be accurately read to the nearest millimeter (0.1~centimeter) and visually
interpolated to a few tenths of a millimeter. A dial caliper is an instrument that allows a length
measurement accurate to a few hundredths of a millimeter.
Examine the dial caliper image below.
Figure 1: The Dial Caliper
The primary markings on the ruled portion of the scale between the jaws of the caliper
correspond to tens of millimeters, the secondary markings to fives of millimeters and the tertiary
markings to millimeters. Intermediate values are read off the dial. One complete revolution of the
dial is equivalent to five millimeters. You have to figure out from the position of the jaw whether
to use the numbers from 1 to 5 or 6 to 10 on the dial.
Since each millimeter on the dial is subdivided into twenty divisions, the instrument is graduated
in intervals of 0.05 mm as shown on the dial face. Readings can be visually interpolated to a
fraction of this interval.
In this example, the reading in Figure 1 is 12.10  .02 mm .
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The Micrometer
Figure 2: The Micrometer
Note that a scale is engraved on the micrometer barrel with primary divisions at one millimeter
intervals (above the horizontal line) and half-millimeter divisions (below the horizontal line) at 1
mm intervals. Rotation of the thimble advances the head of the micrometer 50 divisions, which
equals 0.5 millimeters (the smallest division is 0.01 mm).
The reading in Figure 2 is 7.17  .002mm .
To take a reading:
Gently close the micrometer and note that the zero marking of the thimble is aligned exactly with
the horizontal line on the barrel. (If this is not true, record the actual zero reading and use it to
correct subsequent measurement.) Slowly rotate the thimble so as to open the micrometer; note
that two full rotations of the thimble are required to advance the moving part one mm. The
calibrations below the horizontal line on the barrel between the millimeter marks are necessary
for determining if the thimble has been rotated a complete revolution beyond the primary
millimeter marks.
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Figure 3: Close up of the scale (horizontal scale is enlarged).
In the example above, the thimble rests at the 12 mm mark between 2.5 and 3.0 mm on the
main scale. The thimble indicates an additional reading of 0.12 mm; thus the proper reading is
2.62 mm.
For another view of the micrometer, visit our Photo Gallery,
http://www.physics.upenn.edu/~uglabs/Photo_gallery/101_150photoThumbs.html#misc101
JB 5/28/2009 dialCalMicrometer.doc
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Photogates and what they are good for
Mechanics is mostly about motion -- things moving from here to there-- and to study motion we
need to be able to measure distance (displacement) and time.
When we are observing suitable and large enough objects, we will use a Sonic Ranger, which
can measure both position and time simultaneously. However, in some experiments we will use
photogates, which are photoelectric devices that measure time intervals but not distances. They
are like electronic “eyes“.
The figure below shows a drawing and a photograph of the photogate we use.
One arm contains a tiny infrared transmitter and the other arm an infrared receiver. When the
beam is interrupted by an opaque object between the arms, the device sends a signal to the
computer saying that the beam is blocked. The LED indicator on the side of the photogate lights
up when the beam is blocked.
What the computer does with that information depends entirely on the way the software is set.
We use a single photogate in several experiments, and two photogates working together in the
projectile motion experiment. Different LoggerPro files are used for each experiment to handle
the photogate data in the appropriate way.
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A single photogate can be used in one of two ways:
1) ON-OFF mode (Falling Body, Rotational Motion)
The photogate goes ON when the beam is blocked and OFF when the beam is unblocked. The
time interval between switching ON and OFF is recorded.
2) (ON)- (OFF/CLEAR/ON) mode (Picket Fence)
The photogate goes ON when the beam is blocked and stays ON when unblocked. The next time
the beam is blocked it does three things in immediate succession: it goes OFF, CLEARS the
recorded time interval and goes ON.
The two photogates used in Projectile Motion are set so that when the first photogate is blocked
it goes ON and when the second photogate is blocked it goes OFF and the time interval between
the two events is recorded.
JB 5/28/2009 photogates.doc
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Analyzing Motion Data with a Photogate: Interpolating
As the picket fence falls through the
photogate, the displacement intervals are
constant.and therefore the time intervals
cannot be. Since the time intervals are not
constant, the velocity measured during each
time interval cannot be plotted directly
against the time recorded at the end of each
time interval. The measured velocity is the
average for each time interval, and is the
same as the instantaneous velocity at the midpoint of each time interval. One either has to
calculate the mid-point times, and plot the
velocities against them, or use interpolation
to find the instantaneous velocity as follows:
x12
and  t12  t 2  t1
t12
Since the acceleration is constant, the slope can be found by interpolation:
The average velocity for the time interval  t12 is v12 
v23  v12 v 2  v12

t13
t12
where v2 is the instantaneous velocity at time t2 .
v  v
v 
This leads to  v2   23 12  12 t12
 t13
t12 
v2 
which can be rewritten as
v23t12  v12 t 23
which is calculated from distance and time intervals only!
t13
interpolationEQ.doc 01/23/03 HS
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The Sonic Ranger
The sonic ranger is a device for measuring the position of an object with respect to the position
of the ranger. It sends out short ultrasonic pulses (bats can probably hear them, but we can't)
which are reflected by an object a distance, x, away from the ranger. You may hear a series of
clicks as the ranger sends pulses --- but you are not hearing the pulses! Those clicks are the gold
foil rattling in place. The data produced by a sonic ranger is much more detailed than that
produced by the photogates. Accordingly, the software running the ranger comes with a package
of analytical tools, LoggerPro, which is described in the Analyzing Sonic Ranger Data section
of the lab manual.
The software measures the distance x by measuring the time required for a pulse to make a round
trip from the active surface (the gold foil) of the ranger to the reflecting object and then back to
the ranger. The ultrasonic pulse bounces off almost any hard object. The software is
programmed with the speed of sound and the air temperature and therefore is able to find the
distance (half the round trip time of the pulse times the speed of sound) of the reflecting object.
Figure 1: Sonic Ranger
In a typical situation a plastic “flag” attached to the cart on an air track reflects the sound pulses.
If the equipment is not properly aligned, the sound will be reflected from something else (e.g.,
your hand or the closest wall) and a spurious signal will result. The sections Analyzing Sonic
Ranger Data and Least Squares Fitting in the lab manual provide additional information,
including the use of automated least squares fitting to exact information from the graphs. Figure
2 shows a cart on an inclined plane. The position of the cart (“x”) is measured by the sonic
ranger.
Figure 2: Cart on an inclined plane.
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If the cart is released it will
accelerate “down hill”
strike the spring bumper at
the bottom, rebound and
move uphill, come to a
momentary stop, and then
repeat the motion over
again. The data will
resemble Figure 3.
When the computer starts
to record the data the cart is
approximately one meter
from the ranger and moving
away from it (the distance
is initially increasing with
time). About two seconds
later the cart reaches its
maximum distance from the Figure 3: Sonic ranger data from a cart on an inclined plane.
ranger and begins to slide down the track, coming closer to the ranger. Finally, the cart strikes
and rebounds off of a bumper at the bottom of the track. Then the motion begins all over again.
Each section of the graph is a parabola corresponding to motion with constant acceleration.
Analyzing the data --- for example, finding the acceleration of the cart --- requires tools that are
discussed in the Analyzing Sonic Ranger Data section of the lab manual. The three cusp-like
minima show the cart rebounding from the spring bumper and the rounded maxima show the cart
reaching its greatest height up the hill.
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Analyzing Sonic Ranger Data
The graphs below show the position and velocity of a cart as a function of time as it bounces off
the bottom of a tilted air track. Here's is how we analyze this data.
Figure 1: Position vs. Time
The three cusp-like minima show the cart rebounding from the spring bumper and the rounded
maxima show the cart reaching its greatest height up the hill.
Each section of the graph is a parabola corresponding to motion with constant acceleration.
We would like to measure the acceleration but that's not easy with this graph. A better (more
useful) graph would show the velocity as a function of the time since the acceleration is the slope
of the velocity vs. time graph.
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Figure 2: Velocity Vs. Time
To make this plot the software has been reset to display the velocity.
Each straight line segment represents one complete up and down motion: Positive velocity shows
the cart moving up hill; zero velocity is the cart at the very top; and negative velocity shows the
cart moving downhill.
Note the experimental noise--a fact of life.
The slope of the straight-line segments is the acceleration of the cart. But to find the slope we
must first isolate one segment of the motion.
Another plot of velocity as a function of time is shown below. Two changes have been made to
the graph.
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Figure 3: Velocity Vs. Time
1. A fraction of the graph has been high lighted, which means chosen for particular analysis.
2. A (linear) least squares fit has been done to find the straight line that best represents that
the high-lighted data. The software always called the vertical axis "Y" and the horizontal
"X", but here they are velocity and time. Thus the acceleration is -0.0892 m/s2
PS 8/21/2000 analyzingSonicRangerData
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Instrument Notes: Introduction to the Oscilloscope
The cathode ray oscilloscope is a device that displays graphs of time-dependent voltages that
vary too rapidly to be analyzed with a conventional voltmeter.
As you will see, an oscilloscope is a complicated device, but if you take reasonable precautions
not to physically abuse it you probably will not do any damage. The most important precaution
is: Never use an oscilloscope to study high voltages. In particular do not connect it to an AC
outlet.
At all times, but especially when the spot of the screen is stationary, keep the brightness
(intensity) low to avoid damage to the fluorescent screen. In general use the minimum intensity
that provides reasonable visibility.
The Cathode Ray Tube
The heart of the oscilloscope is a cathode ray tube, the essential features of which are shown in
Figure 1.
Figure 1: Oscilloscope diagram.
Electrons are produces by the heated cathode, a process known as thermionic emission.
They are accelerated by an electric field between two electrodes, the heated cathode and the plate
(with the hole) held at a potential difference of several thousand volts and emerge as a narrow
beam. When the electron beam strikes the fluorescent screen on the face of the tube it produces a
luminous spot.
The trajectory of the electron beam and hence, the position of the spot on the screen is controlled
by the horizontal and vertical deflecting plates. When a potential difference (a voltage) is applied
to a pair of plates, the electrons are accelerated transversely and the spot made on the screen
moves accordingly. The design is such that the vertical and the horizontal positions of the spot
are directly proportional to the voltage on the corresponding plate. It is this property which
makes the oscilloscope useful as a voltage-measuring device.
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In the most common mode of use, a time-dependent voltage applied to the vertical plates is
displayed by the deflecting the beam horizontally with an internally generated saw-tooth voltage
applied to the horizontal plates. This sweep voltage moves the beam across the screen at a
constant rate, returns it to its initial position, and then repeats the sweep. The result looks like a
plot of input voltage versus time. The saw-tooth waveform of the sweep voltage is illustrated in
Figure 2.
Figure 2: Sweep waveform (sawtooth)
The sweep time is adjustable over a large range, and thus the scope can be used to display
voltages whose characteristic frequencies vary by many orders of magnitude. Due to the
persistence of vision (around 0.05 seconds) and the decay time of the fluorescence, a plot of
voltage on the vertical plates is displayed as a function of time.
Preliminary use of the scope
Your instructor may ask you to experiment with the oscilloscope before actually using it as a
measuring device. If so, gain familiarity with the oscilloscope and its many controls. All of the
controls are described at the end of the write-up. The voltage to be displayed in the scope is
provided by a signal generator, as shown in Figure 3 on the next page. The notations to the right
of the figure are recommended settings of the scope controls.
Figure 3: Basic Setup
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Study the trace produced when time-varying voltages are produced by the generator are applied
to the plates. The function generator can produce sine, square and triangular waves at a range of
frequencies. You aim here is to compare the voltage characteristics as defined by the generator
with the same characteristics read off the scope screen.
You must first decide about the source of the sweep voltage. The best thing to do is to
synchronize the horizontal sweep with the output of the signal generator. Set SOURCE S2 to
either CH1 or CH2 which instructs the scope to use the corresponding input voltage to trigger the
sweep. Vary the magnitude and frequency of the input voltage and use the scope to measure
these quantities. The TIME VARIABLE H2 should be set to CAL’D, the POSITION H3 to a
setting that provides fill view of the waveform, and the TIME/DIV (H1) to whatever you need to
display the waveform. Experiment with this control and be sure you understand how to read time
intervals off the face of the screen and how to convert these readings to a frequency.
The front panel of the Leader Model LBO-522 oscilloscope that you will use is shown in
Figure 4. There are many controls but the instrument is not as complicated as it appears at first
sight. It is useful to divide the controls into sets according to their function.
Vertical Beam Control
Horizontal Beam Control
Formation of the Electron Beam
Figure 4: Front Panel of the LBO-552
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Sweep (Trigger) controls
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Lab Manual 2006
The Probe
Voltages are measured by attaching the tip of the probe to one point in the circuit and grounding
the alligator clip. The only tricky thing about the probe is the setting “X1” and “X10”, which
appear as the switch markings on a slide switch. When set to x10 the probe has high input
resistance (10 Mega-ohms) and low capacity which is what one wants to avoid loading the
circuit. However, all input voltages are attenuated by a factor of 10. Therefore when the probe is
set this way, all vertical readings must be multiplied by 10.
Formation of the Electron Beam
A1: POWER. Press “on” to turn the instrument on.
A2: INTEN: Controls the brightness of the beam spot which should be kept as low as
possible (consistent with seeing some kind of trace or signal)
A3: FOCUS: Controls the sharpness of the beam spot on the screen
A4: ILLUM: This controls the lighting of the square grid on the screen, which is called the
graticule
A2
A3
A4
Figure 5: Power and beam controls
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Trace Selection
The dual trace oscilloscope is capable of displaying two voltages at the same time, both relative
to a single point in the circuit (which is not always ground). There are two probes (or sometimes,
connecting wires), one for each trace, which are connected to B1 and B2. The common ground
connection is made with an alligator clip or a third wire. The connections shown in the figure
will display V1-V2 on channel 1 (CH-1) and V3-V2 on channel 2 (CH-2) respectively.
CH1: only channel 1. Probe connected to B1
CH2: Only channel 2. Probe connected to B2
CHOP: Both channels are displayed with the beam switching from one to the other
rapidly.
ALT: Both channels are displayed but the entire traces are drawn in succession.
ADD: The input voltages are added and the result displayed on the screen.
B4 CH-2 POL: When pushed in, this button inverts (changes the sign) of the display for channel
2 with respect to channel 1 when channel 1 is the source.
B5: This connects to the common ground for both traces. The lead with the alligator clip should
be connected here.
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Vertical Deflection of the Beam
The vertical deflection of the two traces are controlled by identical set of controls. The controls
for channel 1 are located at the top left side of the scope and those for channel 2 are located on
the lower left side.
V1: This controls the way the scope
responds to the signal applied to the probe.
AC: The scope responds to a time
dependent voltage only. A steady
state or direct current component, if
any is suppressed and does not
affect the display.
GND: The input voltage is
grounded (i.e. exactly zero voltage
is applied to the plates) and the
input terminal is opened so that the
probe is electrically disconnected
from whatever it is attached to.
DC: The input voltage is directly
connected to the deflecting plates
so that both alternating and steady
state components of the input
voltage affect the position of the
trace.
V2 Position: Controls the position of the
trace on the screen
V3 Scale: This is a dual knob switch. The
outer control sets the sensitivity, i.e. the
voltage corresponding to a beam deflection
of one division on the graticule. The inner
knob can be both rotated and pulled out. It
should be rotated completely clockwise to
the CAL'D position (otherwise the
sensitivity settings on the outer knob is
meaningless). When the knob is pulled out
the sensitivity is magnified by a factor of
10, i.e. a sensitivity of 0.1V/div on the
outer knob really means 0.01V/div.
Figure 7: Channels 1 and vertical position
controls
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Horizontal Deflection of the Electron Beam
These control the horizontal position of the traces on the face of the tube. Note that a single set of
controls is used for both traces.
H1 Scale: This sets the sweep rate of the beams, i.e. the rate at which the spot moves across the
face of the tube. For example, if set at 1ms, then it takes the beam 1ms to move one major
horizontal division and a sine wave with a period of 2ms would occupy 2 full divisions. The
calibration is correct only if H2 is set fully clockwise to CAL’D. (The switch settings XY refers
to another mode of use in which one input voltage is applied to the vertical plates and the other
to the horizontal plates).
H2 Time Variable: As noted above, the sweep rate setting is meaningful only if this is set to
CAL’D. The knob has an in and out position as well. When pulled out, the time scale is
expanded by a factor of five. For example if H1 is set to 1ms and the knob is pulled out, one
division on the graticule corresponds to 0.2ms. In normal use, the knob should be pushed in and
set to CAL’D.
H3 Position: This sets the horizontal position of both traces on the face of the screen. It is
convenient because it allows one to place the traces at a convenient point on the graticule.
H1
H3
H2
Figure 8: Horizontal scale and position controls
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Sweep (Triggering) Controls
The oscilloscope is used to display a repetitive signal but it can do this only if the trace starts its
horizontal motion at a fixed point of the waveform. If the horizontal motion started at an
arbitrary time of the input waveform nothing stationary would be seen. The controls in this block
set the point on the input waveform that triggers (initiates) the horizontal motion.
S1 Level: This knob can be pushed in and out, and rotated. This difference between the in and
out settings is the disposition of the trace when zero voltage is applied to the plates. In the AUTO
position (pushed in) zero voltage produces a horizontal line; in the NORM position (pulled out) a
zero voltage signal is suppressed and the trace does not appear. The angular position of the
controls determines whether or not the scope triggers on a positive or negative (voltage) segment
of the waveform. It should generally be set at PRE-SET, which causes the scope to choose a
point near the center of the waveform.
Figure 9: Sweep (Trigger) Controls
S2 Source. The scope can receive triggering information either from the signal itself (as implied
above) or from an external source.
CH1: Triggers on the input voltage to the channel-2 whether or not this is actually
displayed on the scope.
CH2: Triggers on the input voltage to channel 2, whether or not this is actually
displayed on the scope.
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ALT: Triggers on the alternate sweeps on the input voltage to channels 1 and 2.
LINE: Triggers at the ac line frequency of 60Hz.
EXT: Triggers according to an external voltage applied to the input ‘EXT
TRIGGER’ marked as S6 in the figure.
S3 COUPLING: The normal setting of this is AC which means that the scope
triggers on the AC component of the incident voltage. The other settings are used
for television repair work or to reject high frequency noise and it will normally not
concern us.
S4 SLOPE: Used to control whether triggering occurs on a part of the waveform
with a positive or negative slope.
S5 HOLDOFF: This is used to delay or advance triggering. It should be set to
NORMAL for all practical purposes.
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Error Table of Contents/Definitions
Term
Instrumental
Limitation
Systematic Errors
Random Error
Precision
Accuracy
Range
Uncertainty
Normal Distribution
Propagation of Error
Standard Deviation
(SD)
Standard Deviation
of the Mean (SDM)
Correlated Errors
Uncorrelated Errors
Quadrature Rule
Definition
A limitation due to the ability of a measuring device to
measure to a particular degree of fineness
Error due to a mistake which does not change during
the measurements
Variations in the result of repeated measurements of
the same quantity due to unnoticed variations in
technique or changes in the environment
Independent measurements of the same quantity
closely cluster about a single value
Independent measurements cluster about the true
value of the measured quantity
The difference between the largest and smallest value
Error of measurement- but not a mistake
If a measurement is subject to random error and
negligible systematic error, the measured values will
be distributed on a symmetric bell-shaped curve
centered on the true value of x
Errors in independent measurements combine when
performing mathematical calculations
A measure of the spread, or average uncertainty of a
set of measurements. SD is defined so that 67% of the
data lies within ± SD or one standard deviation from
the mean.
A measure of the uncertainty in the mean itself;
decreases with increasing number of measurements N
An error in one measurement will affect a correlated
measurement
An error in one measurement that has no effect on
another measurement
Uncorrelated (independent) errors are added in
quadrature
35
Section
2.1
2.1
2.1
2.2
2.2
2.5
2.4
2.6
2.9-2.15
2.7
2.8
2.9
2.9
2.10
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Errors and Uncertainty
2.1
Types of Error
There are three types of limitations to measurements:

Instrumental limitations Any measuring device can be used to measure to a given degree
of fineness. Our measurements are no better than the instruments we use to make them.

Blunders and systematic errors These are caused by some kind of mistake which does not
change during the measurements. For example, if the platform balance you used to weigh
something was not correctly set to zero with no weight on the pan, all your subsequent
measurements of mass would be too large. Systematic errors never enter into a quoted
uncertainty. They've either been eliminated if identified or lurk in the background
producing a shift from the true value.

Random errors These arise from unnoticed variations in measurement technique, tiny
changes in the experimental environment, etc. Random variations affect precision. Truly
random effects average out if the results of a large number of trials are combined.
Instrumental Limitations
The terms resolution and sensitivity are used to describe a measuring device. What do they mean
and what is the difference between them?
The resolution is the smallest increment that can be measured.
Example: A meter stick is divided into centimeters and millimeters. We can surely measure to
the nearest millimeter, and visually interpolate perhaps to one-quarter of a millimeter. The
resolution of a meter stick is then 0.25 millimeter or 250 m. A measured length could be any
place within a 0.5 mm wide region centered around the reading on the meter stick.
We will assume that the accuracy of any single reading with a meter stick is plus or minus 1/4 of
the smallest scale division.
The sensitivity of a measuring device is the smallest measured quantity that can actually be
resolved.
Example: A balance may have a scale with gram divisions, but if the pan will move only when
the mass changes by ten grams, then the one-gram divisions are meaningless. The sensitivity of
this balance would be ten grams.
Note: The limitation of a measuring instrument is the LARGER of the resolution of the
instrument and the sensitivity of the instrument.
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Blunders and Systematic Errors
Blunders are facts of life. The word itself is not very scientific and is used here just to make the
point that intelligent people occasionally make very stupid mistakes that can ruin an experiment.
In September 1999 NASA lost a lander on Mars worth $150,000,000 because the engineers used
pounds for the forces used to adjust the orbit instead of Newtons (1 lb force = 4.45 Newtons).
The force was low by a factor of 4 and as a result the lowest point of orbit became 57 km instead
of the planned 226 km --- too low for spacecraft survival. That's a blunder!
"Outlier" Data: Sometimes you will record a value for a measurement that is clearly "out of line"
with your other measurements. Almost certainly such a value represents a mistake --- either an
incorrect reading, or an incorrectly recorded reading. The best thing to do is to ignore a data
point that seems to be inconsistent with the rest of the data set. Don't just erase the number.
Rather, make a note to the grader that the point is omitted from subsequent calculations. If
possible, explain why the point is obviously a mistake.
Systematic Errors: Systematic errors have nothing to do with uncertainty. They are errors
arising from defects in the measuring devices or faulty procedure. If the one centimeter divisions
on a ruler are actually 0.95 centimeters apart, all measurements made with the ruler will be
systematically too large. Sometimes systematic errors creep into an experiment from extraneous
effects that the experimenter is unaware of or ignores.
A common systematic error arises from the assumption that the end of a ruler can be used as a
reference. The ruler marking that reads one centimeter may not be one centimeter from the end
of the meter-stick. The end of the meter stick might be worn away from frequent use. To avoid
this, never use the end of the meter stick to measure a length; use one of the marked points
instead. (See figure 2 in section 1.4)
Systematic errors can arise from the way an experimenter reads a measuring instrument. For
example, if an experiment is designed so that the object must start its motion at a particular point,
but because of your angle of view the object always starts slightly below that point, then all
measurements of the fall distance will be systematically in error. This is called parallax.
Random Errors
No matter how many measurements are made the results will be spread over a range of values.
Variations in the result of repeated measurements of the same physical quantity come from
random errors due to unnoticed variations in techniques, or unknown changes in the
environment, etc.
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2.2
Lab Manual 2006
Precision vs. Accuracy
Everyone wants to do accurate measurements … or is it precise measurements that we want?
Precise does not necessarily mean accurate. What is the difference?


A precise measurement is one where independent measurements of the same quantity closely
cluster about a single value (possibly wrong)
An accurate measurement is one where independent measurements cluster about the true
value of the measured quantity.
Precision of a measurement is defined as the spread or range of measured values, regardless of
whether those measured values are in fact spread out around the correct value. If random
fluctuations in the environment, the measuring technique, etc. are excessive or if the measuring
instruments are very poor, the resulting measurements will exhibit large fluctuations compared to
the size of the quantity measured. One can still find the average but the error bars will be quite
big. Such a measurement exhibits low precision. Conversely, if random fluctuations are
relatively small and the instruments good, the measurement will exhibit high precision. Thus
precision refers only to the size of the error bars compared to the size of whatever is measured.
One can make perfectly precise measurements with a faulty ruler and get an inaccurate result.
Accuracy, on the other hand, refers to the relationship between the true value of a quantity and
the measured value. A measurement will be accurate to the extent that systematic errors produce
shifts small compared to the true value of the quantity.
Systematic errors are not random and therefore can never cancel out.
They affect the accuracy but not the precision of a measurement.
Figure 1 illustrates the difference between precision and accuracy. The center of the bullseye
represents the "true" value of a measured quantity and the dots individual measurements. The
scatter in the points represents the precision.
A:
B:
C:
D:
Low-precision, Low-accuracy: the average (the X) is not close to the center
Low-precision, High-accuracy: the average is close to the true value
High-precision, Low-accuracy: the average is not close to the true value
High-precision, High-accuracy: and the average is close to the true value
Figure 1: Precision
and Accuracy. This
picture was taken
from "Precision and
the Terminology of
Measurement" by
Volker Thomsen.
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University of Pennsylvania: Department of Physics
2.3
Lab Manual 2006
Writing Experimental Numbers: Significant Figures and Scientific Notation
Every measured number must be accompanied by
an estimate of its uncertainty and the appropriate
units. For example, say that the best estimate of
a length L is 2.59 cm, but the length might be as
small as 2.57 cm or as large as 2.61 cm. A
proper way to report the measurement is
L = 2.59 ± 0.02 cm.
Significant Figures
Experimental numbers must be written in a way
consistent with the precision to which they are
known. In this context one speaks of significant
figures or digits that have physical meaning.
As a general rule, all definite digits and the first
doubtful digit are considered significant. The
length measurement in the last example was
quoted to three places because we were certain
of the first two digits but dubious about the third.
Recording the length as 2.590 cm would be
incorrect because it would imply that the nine
was definite and the zero in doubt (in fact the 9
is in doubt).
The Dinosaur in the Natural
History Museum --- from Mike
Cohen
A man walked into the Natural
History Museum. He saw a
dinosaur skeleton on display. He
walked up to the guard and asked,
“How old is that dinosaur?”
The guard replied, “65 million and
30 years old.”
The man said, “How do you know
its 65 million and 30 years old?”
The guard said, “Oh, because it
was 65 million years old when I
started working here 30 years ago.”
Example: Writing L=2.593  0.03 cm would be wrong. It is impossible for the third decimal
place to be meaningful when the second is uncertain by 3.
NOTE: One significant figure should be used for the error or occasionally two, especially if the
second figure is a five. The number of significant figures must be consistent with the
uncertainty.
Scientific notation is useful for reporting results because only significant digits are written down
and the decimal place indicated by the power of ten. The numbers 1.2 km and 1.2 x 103 m
manifestly have the same number of significant figures.
Rounding Off Numbers
To keep the correct number of significant figures, numbers must be rounded off. The discarded
digit is called the remainder. There are three basic rules for rounding:

Rule 1: If the remainder beyond the last digit is less than 5, drop the last digit. Rounding to
one decimal place, the number 5.346 would become 5.3.

Rule 2: If the remainder is greater than 5, increase the final digit by 1. The number 5.798
becomes 5.8 if rounding to 1 digit.
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
Lab Manual 2006
Rule 3: If the remainder is exactly 5 then round the last digit to the closest even number.
The number 3.55 would be rounded to 3.6 and 3.65 rounds to 3.6 also.
From: Hurlburt, R. (1994) Comprehending Behavioral Statistics, Brooks/Cole, Pacific Grove, CA.
Most people and most calculators round up to the next digit and because of that carry an extra
digit. However, to prevent rounding bias you must round off to the closest even digit so that
numbers from 1 to 5 are rounded down and numbers from 6 to 10 are rounded up.
Zeros as a significant digit
Zeros that precede the first non-zero digit of a result are not significant figures. As an example, a
measurement of 2.31 cm has 3 significant figures. When written as 0.0231 m, the first two zeros
serve to force the decimal point to the correct position. They are not significant figures.
Zeros at the end of a measurement are taken to be significant figures. For example, a racetrack
might have a length of 1.200 km (or 1.200 x 103 meters). This implies the distance is known to
the nearest meter. Writing this as 1200 m is ambiguous, since it unclear whether the distance is
known to the nearest meter or perhaps just to within 100 meters. Scientific notation is a good
way to avoid this confusion.
2.4
Uncertainty of Measurements
Physics is an "exact science" that is based on experiment. All experimenting is based on
measurement which is by nature inexact; thus the term uncertainty. A directly-measured
quantity is something we do not compute from other measured quantities. The uncertainty of a
directly measured quantity is the larger of the uncertainty due to instrumental limitations and the
uncertainty due to random errors.
Sometimes great advances are made through data with relatively large errors. Recent data from
the Hubble Space Telescope have enabled scientific groups using different approaches to
measure the Hubble Constant (a quantity whose value determines the age and fate of the
universe) to get results that agreed within roughly 10%. This is great news, because only a few
years ago different measurements disagreed by more than a factor of two. The age of the
universe, which can be computed from this constant, has finally settled down to values consistent
with other physical estimates.
Consider how we measure the
length of the gray object using
the ruler shown with it in
Figure 2. We will assume that
the instrumental error in the
position of any point is one
half(1/2) of the smallest scale
division or 0.05 cm .
Determining the length of the
gray rectangle actually requires
length  8.7  0.05 cm  1.0  0.05 cm  7.7  0.07 cm
0.07
length  7.7cm  (
 100%)  7.7cm  1%
7.7
Figure 2: Measuring the length of an object with a ruler
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University of Pennsylvania: Department of Physics
Lab Manual 2006
subtracting two measurements. The left end of the object is at 1.00 ± 0.05 cm, and its right end is
at 8.70 ± 0.05 cm. The length is therefore 8.70 ± 0.05 cm – (1.00 ± 0.05 cm) = 7.70 ± 0.07 cm.
You might wonder about why the uncertainty is 0.07 cm. We found it using the rule of "adding
uncertainty in quadrature", which is explained in detail in section 2.9, Propagation of Errors.
The uncertainty may be expressed in two different ways. One is the absolute uncertainty, that
is, the uncertainty expressed in the units of the measured quantity, as in L=7.7  0.07 cm.
Alternately, one can express the uncertainty as a percentage uncertainty, which is independent
of the units used to express the result. In this case, since 0.07/7.7 ~ 1%, we would write L = 7.7
cm ± 1% .
Now that we know how to present one measurement with its uncertainty, we need to understand
how to combine multiple measurements of the same quantity to form a best estimate and an
uncertainty for the true value of this quantity. Multiple measurements contain more information
than any single measurement, so you shouldn’t be surprised that the uncertainty in the measured
quantity decreases when multiple measurements are combined. Exactly how this works is the
subject of the next section.
2.5
Averaging
Why take an average? How does it change the uncertainty? Measuring something once tells us
next to nothing about the uncertainty. We must repeat our measurements, often many times, to
gain a sense of the uncertainty due to random errors. It is important to realize that one can repeat
a measurement endlessly and the resulting numbers will continue to be spread over the range of
values; measuring many additional times will not decrease the spread in values. But by
averaging the set of measurements we obtain a result which is more precise than any single
measurement and is more repeatable.
Example: (Data Set 1 below)You must measure the length of an object. You know that a single
measurement is not enough, so you make five measurements. To find the best value, you
calculate the mean, or average value, which we will write as x :
72  77  82  86  88
x
 81 cm
5
Data Set 2 is another set of five measurements that has the same mean but a different spread or
range. Here is a comparison of the two data sets:
x1
x2
x3
x4
x5
Mean
Range
Data Set 1 (cm)
72
77
82
85
88
81
16
Data Set 2 (cm)
80.
81
81
81
82
81
2
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University of Pennsylvania: Department of Physics
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The values of the Data Set 1 are scattered widely with a range R  xmax  xmin given by
88  72  16 cm. For the purposes of this course, where we frequently work with small data sets,
we will assume that the uncertainty of any single measurement is called x and is given by half
the range. Then the uncertainty associated with Data Set 1 is x1  R / 2  8 cm (regardless of its
instrumental uncertainty which might be much smaller). The uncertainty associated with Data
Set 2 is smaller: x2  1 cm.
The spread in the data will increase as more measurements are made when the number of
measurements is small. Beyond a certain number of measurements, no matter how many more
measurements are made, the results will continue to be spread out over roughly the same
interval. This spread is inherent in the measuring process.
The average value becomes more and more precise as the number of measurements N increases.
Although the uncertainty of any single measurement is always x , the uncertainty in the mean
eavg becomes smaller as more measurements are made. In fact, for the case of small data sets, the
uncertainty in the mean is eavg  x /
definition of the standard deviation.
N . This relationship is derived from the statistical
The best estimate of the true value of x is the mean value, x .
x
For a small data set, the mean value with uncertainty is x 
N
For Data Set 1 the uncertainty in the mean is e avg  8 /
5  4 cm ( eavg is rounded to one
significant figure because it corrects the last significant digit of x  81 cm).
For Data Set 1 we report the length as: 81  3 cm
For Data Set 2 we report the length as: 81.0  0.4 cm
For Data Set 2 the uncertainty in the mean is so small that we must add another significant figure
to the value for the mean.
2.6
Averaging and Uncertainty for Large Data Sets
Suppose we make twenty measurements of the period (time for one swing) of a pendulum, as
shown in Table 1 on the next page. The uncertainty in each single measurement is 0.1 sec . A
convenient way to analyze these data is to make a histogram (Figure 3) where the horizontal axis
is values of the period x , and the height of each bar represents the number of times the value of
the period occurs. For example, since three of the measurements of the period were 5.7 sec, the
bar at 5.7 sec is 3 units high. By eye we would say the average value is about 5.8 sec, with an
uncertainty near 0.05 s, but we will make this more precise using formulas given below.
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University of Pennsylvania: Department of Physics
6
5
Count (frequency)
Table 1:Sample data for
the pendulum period
Trial Period Trial Period
(sec)
(sec)
#
#
1
5.8
11
5.4
2
6.1
12
5.7
3
5.9
13
5.8
4
5.5
14
5.7
5
5.7
15
6.0
6
5.8
16
5.9
7
6.0
17
5.8
8
5.9
18
5.5
9
5.9
19
6.2
10
5.6
20
5.8
Lab Manual 2006
4
Frequency
3
2
1
0
5.4 5.5 5.6 5.7 5.8 5.9
6
6.1 6.2 6.3
Period (sec)
Figure 3:Histogram of the sample data from Table 1, showing
the number of occurrences (Count ) of various periods.
The study of random errors is a science in itself; here we desire only a general understanding.
Imagine that there is a "true" value of a quantity (the period of the pendulum, for example), xT .
If a large number of measurements are made, and if the variation from measurement to
measurement is due to the combined effect of numerous different sources of randomness, then
one can show mathematically that in the limit of an infinite number of measurements, the
histogram of values follows a "normal distribution" commonly known as "the bell curve"
(Figure 4), which has its peak at xT and has a width  (also known as the standard deviation).
Any single measurement might be quite different from the true value, and any finite number of
measurements (like our data set) might have its peak at a value different from the true value.
However, as more and more measurements are taken, the peak of the histogram tends to lie
closer and closer to the true value.
The theory of statistics tells us how to use a large data set to generate estimates of the standard
deviation  , the "true" value xT , and the uncertainty in our estimate of the true value. Statistics
help us quantify the fact that while repeated independent measurements of the same physical
quantity always yield results that differ from each other, they will tend to scatter about some
average value. Additional measurements do not reduce the spread of the data set but they help
better define the interval in which most of the numbers are likely to be.
The theory of statistics tells us how to handle this data set. You won’t be surprised to discover
that the best estimate of the true value xT is given by the average (also called the mean) of the
data, known as x .
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University of Pennsylvania: Department of Physics
x
x1  x2  ....  x N 1

N
N
Lab Manual 2006
N
x
n 1
(1)
n
Figure 4: Illustration of normal distribution and standard deviation.
x
x1  x2      x N
1 N
  xi
N
N i 1
2.7 Standard Deviation
From the large data set we can also estimate the uncertainty associated with any single
measurement. You probably are already familiar with the standard deviation (SD) of a set of
measurements. It is exactly this quantity which is the best estimate of the actual standard
deviation  . We call the estimate of the SD s to distinguish it from  , which is the actual SD –
it can only be determined by doing an infinite number of measurements. We calculate s as
follows:
1/2
 1 N

s
xi  x 2 

 N  1 i 1

You may wonder why the equation for s has (N-1) in the denominator instead of N. This reflects
the fact that if you make 1 measurement, you have no information about how large  really is.
So, it is reasonable that the formula for s is indeterminate when there is only 1 measurement. If
you use a calculator to determine the SD, you should be sure to know whether it uses (N-1) in the
denominator or N. If it uses the latter, you will have to correct your result for this discrepancy.
2.8 Standard Deviation of the Mean
Although the uncertainty associated with any single measurement is s , the uncertainty in x , the
estimate of the true value, is much less than this. Roughly speaking, making more measurements
gives us more information about the true value xT . To be precise, the uncertainty in x , also
known as the standard deviation of the mean (SDM), is called eavg and is given by:
eavg 
44
s
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University of Pennsylvania: Department of Physics
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As expected, eavg decreases as the number of measurements increases. This is why we take
multiple measurements: the spread in individual measured values does not decrease but the
probable difference between the average and the true value becomes smaller.
Example: Calculating the mean, the standard deviation, and the standard deviation of the
mean from the data in Table 1
Note: Twenty data points are just barely enough data to get a good
value for the SD and SDM.
The data is listed in the table to the right. You can use your
calculator or Excel to find that the average is x  5.80
The SD from equation (2) is
1/ 2
 1 20
2
s    x  x  
 19 i 1

 0.202 sec
The SDM from equation (3) is eavg 
s
0.202

 0.045 sec .
N
20
x
x  x 2
x
x  x 2
5.8
6.1
5.9
5.5
5.7
5.8
6.0
5.9
5.9
5.6
0
0.09
0.01
0.09
0.01
0
0.04
0.01
0.01
0.04
5.4
5.7
5.8
5.7
6.0
5.9
5.8
5.5
6.2
5.8
0.16
0.01
0
0.01
0.04
0.01
0
0.09
0.16
0
The estimate of the uncertainty of any single measurement of the
pendulum period is s  0.209 sec . The best estimate of the true period is x  eavg = 5.80
0.045 sec.
We can compare this exact calculation with our earlier method of estimating using the
uncertainty calculated from the range of the data. The results are as follows.

The range or spread in these measurements is R  6.2  5.4  0.8 sec

The uncertainty is x  R / 2  0.4 sec

The SDM is approximately eavg 

The result of the measurement of the period is reported then as
x
0.4

N
20
 0.09 sec
x  eavg  5.80  0.09 sec
The data set is large enough that the equations for a small data set are no longer correct. They
give a value for the SDM that is about a factor of two too large. As a rule of thumb, if the data
set has 5 or fewer measurements, use the formulas appropriate for a small data set. If it has 15 or
more measurements, use the formulas for a large data set. If it is in between, you may choose to
use either set of equations but should state clearly what procedure you are following.
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University of Pennsylvania: Department of Physics
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Summary of terms used in averaging
x
x
R
x
s

e avg
The value of a particular measurement
Average (mean) of all measurements of x (the best value of x )
Spread or Range of the data (largest - smallest value of x )
Uncertainty in a single measurement x or approximate Sample Standard Deviation for
small data sets
Uncertainty in a single measurement x or Sample Standard Deviation for large data sets
Uncertainty in a single measurement x or Sample Standard Deviation for infinitely large
data sets
Uncertainty in x or approximate Standard Deviation of the Mean for small data sets OR
large data sets
Formulae for calculating standard deviation and uncertainty
R  x max  x min
x 
eavg 
s
R
2
x
R

N 2 N
1 N
xi  x 2

N  1 i 1
eavg 
s
N
Range or Spread of the data x1 , x 2  x N
Uncertainty of a single measurement or Sample Standard
Deviation for a small number of data points
Uncertainty in the mean or Standard Deviation of the Mean for a
small number of data points.
Uncertainty of a single measurement or Sample Standard
deviation for data points x1 , x 2  x N . Used for large data sets
Uncertainty in the mean or standard deviation of the mean. Used
for large data sets.
CJ 5/28/2009 errors.doc
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University of Pennsylvania: Department of Physics
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Summary Tables for Error Propagation Rules
Estimating the Uncertainty
Small data sample
Uncertainty in x
1
Range
2
x

N
x 
Uncertainty in the mean
eavg
Large data sample
Standard Deviation
Standard Error / Standard
Deviation of the Mean
s
eavg
1 N
2
(x  x n )

N  1 n1
s

N
Uncertainty Propagation Rules
When calculating error with an angle, always have your numbers in radians!
Uncorrelated Errors
Addition / Subtraction
Multiplication
Division
Correlated Errors
2
2
(x  y)  (x)  (y)
2
2
x   y 
(xy)  xy     
 y 
x
 x  x  x 2 y 2
 
   
 
 
 

 y  y  x   y 
Power
x n 
x
 n
n
x
x
Multiplication by Constant
(3x)  3x
Functional Uncertainty
f 2
f 2
2
2
 f    x   
 
 y 
x 
y 
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University of Pennsylvania: Department of Physics
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Propagation of Errors in Addition, Subtraction, Multiplication and Division
2.9 Addition and Subtraction
Addition and subtraction are treated the same way. As far as propagation of errors is concerned,
subtraction is just addition with a sign change. If we want to add two numbers, x and y with
absolute uncertainties  x and  y , then the simplest estimate of the uncertainty in the sum is
x  y . This value is actually the LARGEST that the uncertainty could be, and it would only
occur in the rare instance that the errors in x and y were both at their maximum positive value at
the moment of observation. The sum x  y overestimates the uncertainty in the sum.
If x and y are truly independent measurements, it is likely that the combined uncertainty will be
larger than the difference between  x and y but smaller than sum of  x and y . In vector
terminology, the combined value is between that of added parallel vectors (+) and opposite
parallel vectors (-). The combined value corresponds to vectors that are perpendicular, which are
combined with the Pythagorean sum or quadrature.
Using statistical theory it is possible to prove that if the errors are independent of each other or
uncorrelated, the uncertainty in the sum should be calculated by adding them in quadrature:
( x  y )  ( x ) 2  ( y ) 2
(2)
The term quadrature originated from geometry. The diagonal of a quadrilateral with dimensions
x and y has a length (x)2  (y)2 . This rule holds for the sum or differences of more than
two quantities.
When errors are not independent of each other we say they are correlated. If the two
uncertainties are dependent on each other, then the assumption of identical signs and values is
correct and we use the maximum uncertainty found by simple addition.
Example:
If we take the sum of these two experimentally determined numbers:
( 7 .15  0 .03)  ( 9 .00  0.02 )  16 .15  0 .03 2  0 .02 2 16 .15  0 .04
2.10 Multiplication and Division
Suppose we need to find the product of two experimentally determined numbers, x and y with
absolute uncertainties  x and  y .
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University of Pennsylvania: Department of Physics
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We can find the uncertainty in the product xy by computing
 ( xy )  ( x  x )( y  y )  xy (3)
since this is the difference between the largest value of the product and the most likely value.
Since the term from the product of the two uncertainties (xy) is numerically very small, it can
be dropped giving (xy)  xy  yx
(3a)
x
 e x (4)
x
Dividing equation (3a) by the product xy finally gives
A fractional uncertainty in x can be written as
and similarly for e y
 ( xy ) x y


(5)
xy
x
y
So the FRACTIONAL uncertainty of the product is the sum of the FRACTIONAL uncertainties
of the two factors if the errors are correlated. In general this is an overestimate of error.
Addition of the uncertainties in quadrature gives a better estimate of the uncertainty because the
uncertainties in x and y are usually independent.
2
2
y 
 xy
 x  
The correct uncertainty in xy is
    
 y 
xy
x
(6)
2.11 Percent Uncertainties
It's usually more convenient to use percent uncertainties. This becomes absolutely necessary
when dealing with quantities having different units from each other.
The % uncertainty is then
%ex 
The % uncertainty in the product xy is
%exy 
x
 100%
x
(7)
%e x   %e y 
2
2
(8)
Example (with units omitted) :
2
2
 20.52  1%  5%  20.52  5%
(5.13  0.05)  ( 4.0  0.2)  (5.13  1%)  ( 4.0  5%)
2.12 Multiplication by a known constant
What happens to the uncertainty if one of the factors is a known constant and not a measured
quantity? For example, if 3 is a known constant:
(9)
3(y  y )  3y  3y
In other words, the absolute error is multiplied by the same constant.
Example:
2  (3.1 0.1)  6.2  0.2
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University of Pennsylvania: Department of Physics
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Powers of x : (correlated errors)
If q  x n
Then
q
x
n
x
q
(10)
Examples:
(4.0  1.0) 2  (4.0  25%)2  16.0  50%  16.0  8.0
(4.0  1.0)1/ 2  (4.0  25%)1/ 2  2.0  12.5%  2.0  0.2
2.13 Functional Error: Propagation of errors with functions
If you need to propagate the error in x for an expression which is not a power of x, you will need
to calculate the functional error to find the uncertainty. If f (x) is a function of x
f ( x  x )  f ( x )  f ( x  x )
Then
There are two ways to find the functional error. The lower and upper limits of f (x) can be
calculated directly, by substituting x  x and x  x into the function and calculating the
difference.
Another way to find the functional error applies more generally to all situations and can lead to
simplifications. It is done in the same way that we take the derivative with respect to x of the
function. The absolute value is used because the functional uncertainty is a positive number.
f (x) 
df (x)
 x
dx
(11)
If the function has more than one variable,
Then f (x  x, y  y)  f (x, y)  f (x  x, y  y)
Using partial derivatives,
f (x,y) 
f
f
x 
y
x
y
(12) Approximately OR with correlated errors
The correct way (uncorrelated errors) to calculate the functional error is
2
2
f 
f 
2
2
f 
x   

y 
x 
y 
(13)
Note: Quadrature eliminates the need to use absolute values as in equation (12)
Example:
We have measured an angle  and need the error on a trigonometric function such as cos  :
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University of Pennsylvania: Department of Physics
  30  5 or  
quantity e .

6


36
Lab Manual 2006
 0.524  0.087 radians. It is necessary to use radians for the
The x-component of the force vector F with magnitude 25N is given by
 3
  21.65 N
Fx  F cos   25

2


To find the error on Fx , where the uncertainty in F is eF  F   2 N
We can find the fractional errors and add the two terms in quadrature:
ecos  
d cos
e  sin  e where e is in radians
d 
2

0.50.087
e F  sin( )e 
 2 
    
    

3
Fx
F
cos 
25


2
eFx
2
2
2
The final answer is Fx  21.65  2.17 N
2.14 An Example of Propagation of Errors
The words “error” or “uncertainty” as used here refer to the actual quantity by which the measure
is uncertain.
Ex: 12  3 cm
If the expression “fractional error” or “fractional uncertainty” is used, the error value is divided
by the measured value. This can be expressed as a fraction or a decimal.
Ex: 12 cm  0.25
Note the subtle but critical difference in the location of the units in each expression. Since a
fraction or a decimal is the ratio of two values, the units cancel and do not apply to the
uncertainty. It is not hard to overlook this small difference in the statement of a measurement.
To make the distinction more obvious, the fraction or decimal is often converted to a “percentage
error” or “percentage uncertainty”, with the accompanying % symbol to emphasize the
difference.
Ex: 12 cm  25%
The absence of units in fractional and percentage uncertainties makes them the only way to
combine the uncertainties in measures of different units, such as when calculating density or
velocity.
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University of Pennsylvania: Department of Physics
Lab Manual 2006
Combining Uncertain Measurements: Addition and Subtraction
A
+1cm
-1cm
-1cm
+1cm
15 cm
B
10 cm
Figure 1
Figure 1 illustrates the problem of adding the measured lengths of rods A and B. In a perfect
world the combined length would be 25, but in each case the measurement is uncertain by 1 cm.
A = 15  1 cm
B  10  1 cm
Since the two measurements do not depend on each other, one could have a positive variation
while the other was negative. So the outcome could be as large as 16+11=27 or as small as
14+9=23. The error of  2 in the addition is just the sum of the errors of the individual measures
. But each of these cases assume the two uncertainties are both positive or both negative. It is
just as likely that they have opposite signs, so the combinations could also be 16+9=25 or
14+11=25. In other words, half of the possible outcomes have little or no error. So the error of
 2 is an upper limit, not a frequent expectation. As a rough estimate the error will fall
somewhere between 0 and  2. An investigation of statistics will show that the center value of
 1 is a bit too small. The best uncertainty value for two measures that are added or subtracted is
found using quadrature.
A2  B 2
Note that the values used in this case are the actual uncertainties with appropriate units included.
2.15 Another Error Propagation Example: Multiplication and Division
If you are uncertain of the dimensions of your room by 1 foot, what is the uncertainty in your
calculation of how much carpet the room needs?
L  15  1 ft W  10  1 ft
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University of Pennsylvania: Department of Physics
Lab Manual 2006
+1ft
+1ft
-1ft
-1ft
10 ft
15 ft
Figure 2
The sizable zones of uncertainty on the top and right side of the figure immediately indicate that
the uncertainty of the product (area) is a lot bigger than the sum of the factor uncertainties
(1+1=2). The correct way to propagate error in multiplication and division is to convert to
percent errors, and then add them in quadrature. Finally, the percentage answer is re-converted
to a number and quoted with the correct number of significant figures.
Propagation of Error in Multiplication Using Percentages
15  1 ft  15 ft  6.7%
10  1 ft  10 ft  10%
Addition
Area  150 ft  16.7%
 16.7% of 150 =  25
Area = 150  25 ft
To paraphrase Goldilocks,
“Oh that uncertainty is much too large!”
2
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University of Pennsylvania: Department of Physics
Lab Manual 2006
Quadrature
(6.7%) 2  (10%) 2  145 %  12%
Goldilocks, again,
 12% of 150 =  18
“Oh, that uncertainty’s just right!”
Area = 150  18 ft2
2.16 Propagation of Error Using the Values of Uncertainty
Addition of Uncertainties
(15  1 ft )  (10  1 ft )  150  2 ft 2
If Goldilocks were to compare this uncertainty of  2 to the size of the dotted uncertain regions
in Figure 2 above, she would certainly say, “Oh that’s much too small!”
2.17 Estimation of Uncertainty by Multiplying the High and Low Values of the Factors
If there were no uncertainty, the area would be 15 x 10 = 150 ft2 .
Because the length and width are independent, random variation permits four possible
combinations of signs for the two uncertainty values:
(+,+), (-,-), (+,-), (-,+)
The first two cases give the greatest variations:
(+,+): 16 x 11 = 176 (error = +26) and
(-,-): 14 x 9 = 126
(error = -24)
Rounded off and expressed as percentages, this would convert to
150  25 ft = 150ft  16.7%
Of course these are the biggest possible errors, not the most probable. Half of the possible
outcomes have almost no error:
(-,+): 14 x 11 = 154 (error = +4)
and
(+,-): 16 x 9 = 144
(error = -6)
Obviously, these values are too small, but a realistic estimate of positive uncertainty might be
made by taking the average of the max of +26 and the min of +4 (+15) and then doing the same
for the negative uncertainties (-15). This produces an area value of
150  15 ft = 150 ft  10%
Although this is not the correct approach, the outcome is close to the result from quadrature, and
it may shed some light on why the correct calculation does not report the maximum value for
uncertainty.
JB 5/28/2009 error propagation.doc
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University of Pennsylvania: Department of Physics
Lab Manual 2006
Index
Micrometer ..............................................................16
Accuracy..................................................................38
Averaging ..........................................................41, 46
Oscilloscope ......................................................26–34
horizontal deflection ...........................................32
sweep (triggering) controls .................................33
vertical deflection ...............................................31
waveform of sweep voltage ................................27
Best fit line ....................................................8, 10, 12
Cathode Ray Tube ...................................................26
Computers in lab........................................................4
Percent deviation .......................................................7
Photogate ...........................................................18, 19
Precision ..................................................................38
Pre-Labs.....................................................................4
Probe........................................................................29
Propagation........................................................47–54
addition and subtraction......................................52
addition and subtraction......................................48
example of multiplication and division...............52
example of propagation of errors ........................51
multiplication and division .................................48
multiplication by a known constant ....................49
multiplication using percentages.........................53
summary tables for..............................................47
uncertainty propagation rules..............................47
using values of uncertainty .................................54
with functions .....................................................50
Dial Caliper .............................................................15
Dual Trace Oscilloscope................. See Oscilloscope
Electron Beam .........................................................29
Equipment................................................................15
Errors
functional ............................................................50
propagation of.....................................................48
quadrature ...................................10, 35, 41, 48–54
random errors......................................................37
resolution ............................................................36
sensitivity............................................................36
systematic errors .................................................37
table of contents ..................................................35
types of................................................................36
Function Generator ..................................................28
Range.......................................................................45
Graph using Excel .....................................................9
Graphs .......................................................................8
Scientific Notation...................................................39
Significant Figures...................................................39
Sonic Ranger ...........................................................21
Analyzing Sonic Ranger Data.............................23
Standard Deviation ............................................43, 44
formulae for calculating......................................46
of the Mean (SDM).............................................44
Instrumental Limitations..........................................36
Lab Manual................................................................3
Lab Notebook ............................................................3
Lab Reports ...........................................................4, 5
Lab Write-ups............................................................3
Least Squares Fitting .........................................11, 12
Linearizing Data ......................................................13
Linearizing equations ..............................................13
LoggerPro ................................................4, 12, 18, 21
Uncertainty ....................................................6, 44, 45
addition of...........................................................54
estimating............................................................47
estimation of by multiplying the high and low
values of the factors .......................................54
formulae for calculating......................................46
in the Slope .........................................................10
percent uncertainties ...........................................49
rules ....................................................................47
values used for propogation of error ...................54
Mean, calculating the...............................................45
Measurements..........................................................38
accurate measurement.........................................38
precise measurement...........................................38
uncertainty of ......................................................40
55