University Physics - A Laboratory Manual
Bruce W. Zellar
Fall 2015
ii
c Copyright 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012 by the
author(s), Bruce W. Zellar except where noted. Printed in the United States
of America. All rights reserved. This book, or parts thereof, may not be reproduced in any form without the express written permission of the author(s).
Contents
ix
Preface
xi
1 P1-1: Uncertainty in Measurements
1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Significant Figures . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Use of Significant Figures . . . . . . . . . . . . . . . .
1.2.2 Rules for Using Significant Figures in Calculations . .
1.3 Uncertainty or Error . . . . . . . . . . . . . . . . . . . . . . .
1.4 Precision and Accuracy . . . . . . . . . . . . . . . . . . . . .
1.5 Propagation of Error . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Absolute and Percent Error . . . . . . . . . . . . . . .
1.5.2 Rules Used To Calculate Uncertainty . . . . . . . . . .
1.5.3 Propagation of Error Examples . . . . . . . . . . . . .
1.6 Experimental Procedure . . . . . . . . . . . . . . . . . . . . .
1.6.1 Volume and Density of a Cylinder . . . . . . . . . . .
1.6.2 Equipment Usage . . . . . . . . . . . . . . . . . . . . .
1.6.3 Experimental Procedure . . . . . . . . . . . . . . . . .
1.6.4 Calculations . . . . . . . . . . . . . . . . . . . . . . . .
1.6.5 Questions . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.6 P111/112 Data Sheet - Experiment 1: Error Analysis
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1
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2 Statistical Analysis
13
2.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Statistical Distributions . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Some Important Parameters of the Gaussian Distribution 13
2.2.2 An Example of Standard Deviation of the Mean . . . . . 15
2.2.3 Some Important Parameters of the Poisson Distribution . 16
2.3 t-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Chi-Squared Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
iii
iv
3 Simple Harmonic Motion
3.1 Objectives . . . . . . . . . . . . . . . . . . . . . .
3.2 Introduction . . . . . . . . . . . . . . . . . . . .
3.2.1 Simple Harmonic Motion . . . . . . . . . .
3.2.2 Hooke’s Law . . . . . . . . . . . . . . . . .
3.2.3 Period Of An Oscillating Mass On A Spring
3.2.4 An Alternate Derivation Of The Period . .
3.3 Experimental Concept . . . . . . . . . . . . . . . .
3.4 Experimental Procedure . . . . . . . . . . . . . . .
3.4.1 Materials . . . . . . . . . . . . . . . . . . .
3.4.2 Period Measurement . . . . . . . . . . . . .
3.5 Analysis and Calculations . . . . . . . . . . . . . .
3.6 Report Requirements . . . . . . . . . . . . . . . . .
CONTENTS
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21
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4 Conservation of Angular Momentum
4.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Relating Torque to Moment of Inertia . . . . . . . . . . .
4.2.2 Angular Momentum . . . . . . . . . . . . . . . . . . . . .
4.2.3 Conservation of Angular Momentum . . . . . . . . . . . .
4.2.4 Conservation of Angular Momentum - Alternate Derivation
4.2.5 Experimental Concept . . . . . . . . . . . . . . . . . . . .
4.3 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Data Collection - Part A . . . . . . . . . . . . . . . . . .
4.3.3 Data Collection - Part B . . . . . . . . . . . . . . . . . . .
4.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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37
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5 Coupled Oscillations and Normal Modes
5.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Normal Mode Vibrations . . . . . . . . . . . .
5.2.2 Fast Fourier Transform . . . . . . . . . . . . .
5.3 Experimental Procedure . . . . . . . . . . . . . . . . .
5.3.1 Materials . . . . . . . . . . . . . . . . . . . . .
5.3.2 Part A - Determining Spring Constant k. . . .
5.3.3 Part B - Determining the Angular Frequency
Fourier Transform (FFT). . . . . . . . . . . . .
5.3.4 Part C - Determining the Angular Frequency
Fourier Transform (FFT) Continued. . . . . . .
5.3.5 Part D - Determining the Angular Frequency ω
citation. . . . . . . . . . . . . . . . . . . . . . .
5.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
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ω - Fast
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ω - Fast
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- By Ex. . . . . .
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44
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CONTENTS
v
6 Simple Pendulum
6.1 Objectives . . . . . . . . . . . . . . . . . . . . . .
6.2 Introduction . . . . . . . . . . . . . . . . . . . .
6.2.1 Period for the Small Angle Approximation .
6.2.2 The Average Acceleration Approximation .
6.2.3 Period When Angles Are Not Small . . . .
6.3 Evaluation of the Constant C . . . . . . . . . . . .
6.4 Experimental Procedure . . . . . . . . . . . . . . .
6.4.1 Materials . . . . . . . . . . . . . . . . . . .
6.4.2 Data Collection . . . . . . . . . . . . . . . .
6.5 Calculations . . . . . . . . . . . . . . . . . . . . . .
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47
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7 Physical Pendulum
7.1 Objectives . . . . . . . .
7.2 Introduction . . . . . .
7.3 Experimental Procedure .
7.3.1 Materials . . . . .
7.4 Calculations and Analysis
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57
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8 Coupled Pendulums
8.1 Objectives . . . . . . .
8.2 Introduction . . . . .
8.3 Experimental Procedure
8.3.1 Materials . . . .
8.3.2 Procedure . . . .
8.4 Calculations . . . . . . .
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61
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9 The Oscilloscope
9.1 Objective . . . . . . . . .
9.2 Introduction . . . . . . . . .
9.2.1 Horizontal Amplifier
9.2.2 Vertical Amplifier .
9.2.3 Trigger Circuit . . .
9.3 Procedure . . . . . . . . . .
9.4 Report Requirements . . . .
9.5 Questions . . . . . . . . . .
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63
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69
10 AC and the Series Circuit
10.1 Objective . . . . . . . . . . . . . . . . . . .
10.2 Introduction . . . . . . . . . . . . . . . . . . .
10.2.1 Impedance and the LCR Series Circuit
10.2.2 More on Resonance . . . . . . . . . . .
10.3 Procedure . . . . . . . . . . . . . . . . . . . .
10.3.1 Part A - Determination of L and r . .
10.3.2 Part B - Resonance Curves and Q . .
10.4 Calculations . . . . . . . . . . . . . . . . . . .
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71
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78
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vi
CONTENTS
10.5 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 Double Pendulum
11.1 Objectives . . . . . . . . . . .
11.2 Introduction . . . . . . . . .
11.2.1 Generalized Coordinates
11.2.2 The Double Pendulum .
11.3 Experimental Concept . . . . .
11.4 Experimental Procedure . . . .
11.4.1 Materials . . . . . . . .
11.4.2 Frequency Measurement
11.5 Analysis . . . . . . . . . . . . .
11.6 Report Requirements . . . . . .
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and Lagrange’s
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Equations
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79
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Appendix A - Reports
89
11.7 Report Writing Tips . . . . . . . . . . . . . . . . . . . . . . . 89
Appendix C - Sample Report
95
List of Figures
1.1
1.2
Vernier Caliper . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Micrometer Caliper . . . . . . . . . . . . . . . . . . . . . . . . . .
8
9
2.1
2.2
Gaussian Distribution (Bell Curve) . . . . . . . . . . . . . . . . .
Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . .
14
14
3.1
Mass on a spring. . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.1
4.2
4.3
4.4
4.5
4.6
28
29
30
30
33
4.8
Slice of rigid body rotating about z-axis . . . . . . . . . . . . . .
Edge view of off-axis particles rotating about z-axis. . . . . . . .
c 1994 PASCO Scientific) . . .
Apparatus with rotating disk. (
c 1994 PASCO Scientific) . . .
Apparatus with rotating disk. (
c
Rotating disk with ring. ( 1994 PASCO Scientific) . . . . . . .
c 1994
Smart pulley with pulley removed and in new position. (
PASCO Scientific) . . . . . . . . . . . . . . . . . . . . . . . . . .
c 1994 PASCO SciLocation of modified smart pulley mount. (
entific) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Display of Logger Pro table and graph. . . . . . . . . . . . . . . .
5.1
5.2
Coupled Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coupled Masses Set-Up . . . . . . . . . . . . . . . . . . . . . . .
40
44
6.1
Simple Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . .
48
7.1
Simple Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . .
58
8.1
Two Coupled Pendulums . . . . . . . . . . . . . . . . . . . . . .
62
9.1
9.2
9.3
Chopped Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Series RC Circuit. . . . . . . . . . . . . . . . . . . . . . . . . . .
Lissajous Pattern. . . . . . . . . . . . . . . . . . . . . . . . . . .
64
67
68
10.1 Series LCR circuit. . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Circuit components vector (phasor) relationships. . . . . . . . . .
10.3 Circuit component vector addition. . . . . . . . . . . . . . . . . .
72
72
73
4.7
vii
33
34
36
viii
LIST OF FIGURES
10.4 Response curve - Lorentzian. . . . . . . . . . . . . . . . . . . . .
10.5 (a) LCR series circuit (b) same circuit with DSA and DSG/A
attached. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
11.1 A double pendulum, one mass suspended below the other. . . . .
11.2 A double pendulum - heights displaced shown. . . . . . . . . . .
82
82
77
11.3 Illustration of marks made in spark recording paper by spark timer. 96
11.4 Left: free-fall aparatus with falling body. Right - schematic of
free-fall aparatus, showing electric circuit. . . . . . . . . . . . . . 96
11.5 * Determined from least squares fit. ** Cannot be determined
without an additional data point. . . . . . . . . . . . . . . . . . . 98
11.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
11.7 * Determined from the least squares fit. ** Cannot be calculated
(by definition). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
”I had no need of that hypothesis.” - Pierre-Simon LaPlace
”Back off man, I’m a scientist.” - Peter Venkman
”Extraordinary claims require extraordinary evidence. Since the beginning of time there have been literally hundreds of thousands of predictions for the end of the world, and we’re still here.” - Carl Sagan
”When you can measure what you are speaking about, and express it in
numbers, you know something about it, when you cannot express it in
numbers, your knowledge is of a meager and unsatisfactory kind; it may
be the beginning of knowledge, but you have scarely, in your thoughts
advanced to the stage of science.” - William Thomson - Lord Kelvin,
1891
”Thus far I have explained the phenomena of the heavens and of our sea
by the force of gravity, but I have not yet assigned a cause to gravity....I
have not as yet been able to deduce from phenomena the reasons for these
properties of gravity, and I do not ’feign hypotheses.’ ” - Isaac Newton,
Book III of Principia
ix
x
Preface
A CONCEPTS-BASED LABORATORY COURSE
The underlying ideas for understanding uncertainty can be broadly categorized
as follows:
• All measurements have an associated uncertainty, which should be quantified
and reported.
• A calculated result has an associated uncertainty based on its dependent
values.
• The design of an experiment and skill of conducting the experiment affects
the uncertainty in the measurement.
• Uncertainty is used to compare results and draw conclusions.
Sequence of measurement concepts
• Predictive versus descriptive question. Students should understand the
difference between a question asking for a prediction (often involving probability) of what might happen and a question asking for a description of what
did happen. Typically a laboratory question asks a predictive question.
• Purposes of multiple measurements. Many students do not appreciate
the purpose of multiple trials and might expect to obtain the same number
repeatedly. Students should understand the idea that multiple trials can
produce a range of numbers that provide more information than just one
number.
• Using range overlap. When comparing two sets of data, one should look
at the degree of overlap between the data in the two sets. The decision of how
much overlap is necessary to show agreement is a difficult one and depends
on specific circumstances
• Stacking. This technique often is used on repeatable measurements to increase precision. For example, the thickness of one piece of paper can be
determined by measuring the height of a stack of 500 pieces of paper and
dividing by 500. Students should be able to appraise the benefits and disadvantages of stacking.
xi
xii
PREFACE
• Systematic or random mechanism. Students should think about what
causes the variation in their data. Whether the mechanism is causing a systematic or random variation should affect how they attempt to reduce its
effects and how they interpret their result.
• Internal versus external variation. Students must be able to distinguish
between variation coming from the measurement process, which is external to
the system, and variation caused by something internal to the system being
measured. They should understand that external variation detracts from their
conclusions, but internal variation is neither harmful nor beneficial.
• Displaying data with representations. Introductory students tend to
report their data in tables, but frequently a graph or chart is more convincing.
Different types of representations convey different information and are useful
for different amounts and types of data. For example, a histogram is useless
for displaying the results of five readings.
• Low probability data. Students should consider which characteristics are
necessary to allow an outlying data point to be discarded.
• Minimize external variation. There are typically several ways to reduce
the uncertainty of an experiment’s results, which may or may not require
more resources. Students should consider how to improve every aspect of
their data collection.
• Range propagation. Students should know that uncertainty in experimental data will imply uncertainty in a calculated result, and should be able to
find this uncertainty. Predict uncertainty. Given an experimental method, we
can estimate from experience what the uncertainty in the data would likely
be and can estimate the uncertainty in the results. Students should be able
to carry out this estimation and use it to compare experimental methods
without actual implementation.
• Generalize theory. Typically the theory taught to students is true for a
limited set of circumstances (for example, when friction effects are minimal).
Often, invalidating effects emerge in the student laboratory. These are not
errors, but show a need to generalize the model to include such effects.
Experiment 1
P1-1: Uncertainty in
Measurements
1.1
Objectives
What are Significant Figures? What Rules Govern the Use of Significant Figures? What is Uncertainty or Error? What is Absolute Uncertainty or Absolute
Error? What is Percent Uncertainty or Percent Error? What is Propagation of
Error and the Rules Governing It?? How is Uncertainty calculated?
1.2
Significant Figures
The first step in the acquisition of experimental data is the direct observation
of the relative position and motion of an object. Such an observation, or measurement, is possible only through the detection of sensations produced in the
environment surrounding the object in question. The value of such an observation depends on the accuracy of the measurement. The accuracy is always
limited by the refinement of the apparatus in use and the skill of the observer.
The accuracy can be improved with refinements in the measuring equipment and
the observer’s skill in using it, but no matter how ’improved’ the measurement is
made, there is always a point when the observation becomes uncertain. At this
point, the measurement must be estimated. However, it gives information that
is meaningful about the measurement in question and is therefore significant.
We define a Significant Figure as one that is know to be reasonably trustworthy.
1
2
EXPERIMENT 1. P1-1: UNCERTAINTY IN MEASUREMENTS
1.2.1
Use of Significant Figures
This definition does not really seem very helpful, so here are a few rules that
will clarify things.
The decimal place has nothing to do with the number of significant figures.
If a zero is used to merely indicate the location of the decimal place, it is
not significant.
If a zero is located between two figures that are significant, it is always
significant.
A zero located at the end of a number tends to be ambiguous.
Since the last digit in a measurement is uncertain and must be estimated,
only one doubtful digit is kept and treated as significant.
For example, a certain length measurement is made using a meter-stick and
is recorded as 7.36cm. The final digit, 6, is an estimate on the scale using
the millimeter division. This reading has three significant figures. This same
number can be written as 73.6mm and it also has three significant figures.
Expressing this in meters gives 0.0736m or as 7.36 × 10−2 m, once again, both
expressions have three significant figures.
1.2.2
Rules for Using Significant Figures in Calculations
We now know how to specify the number of significant figures but we also
need to know how to use them in calculations. Numbers that do not represent
quantities that are measured have an unlimited number of significant figures.
For example, an inch is defined to be 2.54cm exactly. The 2.54 has an unlimited
number of significant figures. If it did not, then unit conversions would suffer
unnecessary round-off and truncation errors. Likewise, pure numbers, such as
Pi (π), also have unlimited significant figures.
When disregarding digits that are not significant, the digit to be retained
often has to be rounded. For example, if a quantity is calculated to be 13.468
(which has 5 significant figures) and should only have 4 significant figures, then
we round the 6 to a 7, so we would have 13.47, because we are dropping the 8.
If the 8 were instead a 2 (ie. 13.462), then we would round it to 13.46. When
the digit to be dropped is 5, it is a bit more tricky. If the quantity were 13.465,
we would round it to 13.46. If it were 13.475, we would round it to 13.48.
When adding or subtracting, it is convenient to arrange the numbers in
columns to determine how many decimal places should be kept. For example,
Summarizing, we have five rules:
Any number that represents a numerical count in an exact definition has
an unlimited number of significant figures.
1.3. UNCERTAINTY OR ERROR
3
When a digit to be dropped is less than 5, the preceding digit is retained
without change. When it is greater than 5, the last digit to be kept is
rounded up by 1. When digit is exactly 5, the digit to be kept is rounded
so that the last digit kept is an even number.
When Adding or Subtracting numbers, arrange them in columns and keep
no column that is to the right of a column containing a doubtful figure.
When Multiplying or Dividing, the result should have no more significant
figures than the factor having the least number of significant figures.
The roots and powers of a number should have as many significant figures
as the number itself.
1.3
Uncertainty or Error
In a physical experiment, one is faced with two types of determination or measurement; direct and indirect. A direct determination establishes a value for
the quantity under consideration by the use of some kind of measuring device.
An indirect determination involves one or more separate direct determinations
which are used to find another physical quantity by means of a functional relation (ie. a formula). The merit or reliability of such a determination is called
the Uncertainty or Error. The word Error, does not mean mistake. A
mistake can be corrected, but the Uncertainty can never be eliminated, only
reduced. The Uncertainty or Error is an inherent ’fuzziness’ in the quantity.
This fuzziness is random in nature.
At this point, it is convenient to discuss types of error, specifically the difference between Systematic and Random Errors. When a given measurement
is repeated several times, they generally do not agree exactly; this is the result
of a random error, which often comes about from a number of factors. Some of
these are:
Errors of Judgment: Estimates of a fraction of the smallest division of a a
scale on an instrument may vary in a series of measurements.
Fluctuation Conditions: Important factors in a given experiment such as
temperature, pressure, or line voltage may fluctuate during the measurements, affecting the results.
Small Disturbances: Small mechanical vibrations, and the pickup of spurious electrical signals will contribute random errors to some types measurement.
Lack of Definition in the Quantity Measured: For example, measurements
with a micrometer of the thickness of a steel plate having non-uniform
surfaces will in general not be reproducible.
4
EXPERIMENT 1. P1-1: UNCERTAINTY IN MEASUREMENTS
Randomness in the Quantity Measured: Repeated measurements of the number of disintegrations per second in a radioactive source will give different
values because radioactive disintegrations occur randomly in time.
In contrast, Systematic Errors present in the measurement process will produce
a constant offset from the ’true’ value, in a series of repeated measurements.
For example, a systematic error will be present if the measuring device is used
improperly or if it is not calibrated correctly.
1.4
Precision and Accuracy
If the measured values cluster around a ’true’ value closely, the measurement
is regarded as accurate. And, if an experiment has a small systematic error it
is regarded as having high accuracy. A precise measurement is one in which
the spread of measured values is small, and, naturally, the source of the error is
random. One should note that a measurement can be very precise but inaccurate
if the systematic errors are large. One should keep this in mind when one applies
the following error analysis technique.
1.5
Propagation of Error
1.5.1
Absolute and Percent Error
The most natural way in which to judge the merit of an experimental value is
to compare it to an actual value, when it is know. One way to do this is to find
the Percent Difference. The Percent Difference is defined as:
(accepted value − measured value)/(accepted value) ∗ 100%
(1.1)
Close agreement would presume an ’accurate’ determination, but, in a majority
of cases, an actual value is not known. Furthermore, it is difficult to know
when or if such a difference is significant. In that case, one can judge the
merit by examining the range of uncertainty of the determination. This range
in uncertainty is referred to as Absolute Uncertainty or Absolute Error.
For a direct measurement, the Absolute Uncertainty or Absolute Error is an
estimate (or educated guess). The observer assigns a value for the error based
on a number of factors such as the fineness of the graduation of the measuring
scale, the degree of estimation involved in determining the final digit in the
value, how the device is used to measure, etc. A small absolute error indicates a
precise determination. If the measurement is repeated enough times, a statistical
determination, such as the standard deviation of the mean may be used instead
of the estimation. This uncertainty can also be expressed as a percent. We
define the Percent Error as:
P ercent Error = (Absolute Error)/(Quantity) ∗ 100%
(1.2)
1.5. PROPAGATION OF ERROR
1.5.2
5
Rules Used To Calculate Uncertainty
For an indirect measurement, the determination is more complicated. Since this
type of determination is not direct, it must be calculated. There are two rules
by which this can be accomplished.
Rule I When two or more quantities are added and/or subtracted, the
Absolute Error in the calculated quantity is found by ADDING the
Absolute Errors of the values used in the calculation.
Rule II When two or more quantities are multiplied, divided and/or raised
to a power, the Percent Error in the calculated quantity is found by
ADDING the Percent Errors of the values used in the calculation,
and if a value is raised to a power, that particular value’s percent error is
first multiplied by that power before the individual percent errors are
added.
In calculations that involve combinations of addition, subtraction, multiplication, divison and raising to a power, the standard algebraic rules used to perform
the calculation dictate when the above rules are applied.
1.5.3
Propagation of Error Examples
So, how do we use these rules? Let’s look at a simple case: a triangular tabletop.
The tabletop is shaped as a right triangle with sides measuring 3.0m ± 0.2m,
4.0m ± 0.2m and 5.0m ± 0.2m. What are the perimeter and area of the tabletop
with their respective uncertainties (ie. errors)?
Perimeter is defined as the addition of the lengths of the sides, so
P = a + b + c = 3.0m + 4.0m + 5.0m = 12.0m
The absolute error or absolute uncertainty for the perimeter is found by
using the rule for addition which is Rule I,
δP = δa + δb + δc = 0.2m + 0.2m + 0.2m = 0.6m
The percent error or percent uncertainty is,
(δP/P ) · 100% = (0.6m/12.0m) · 100% = 5.0%
The area is,
A = 1/2ab = (1/2)(3.0m)(4.0m) = 6.0m2
and the percent uncertainty for the area is given by Rule II,
(δA/A) · 100% = (0.2/3.0) · 100% + (0.2/4.0) · 100% = 11.6%
and the absolute uncertainty is,
(11.6%/100%) · 6.0m2 = 0.7m2
6
EXPERIMENT 1. P1-1: UNCERTAINTY IN MEASUREMENTS
So, we would quote the values for the perimeter as,
12.0m ± 0.6m or 12.0m ± 5.0%
and for the area as,
6.0m2 ± 0.7m2 or 6.0m2 ± 11.6%
Notice, we have adjusted for the proper number of significant figures.
Now, let us consider a more complicated situation. Suppose we want to
find the magnitude of a two dimensional vector given the vector’s x and y
components. The magnitude in the x direction is Fx = 5.0N ± 0.5N and in the
y direction Fy = 12.0N ± 0.5N . The magnitude of the two components is given
by,
q
p
|F | = Fx2 + Fy2 = (5.0)2 + (12.0)2 = 13.0N
Now the uncertainty calculation is more complicated, because of the addition
of two squared quantities and then a square root. To apply both Rule I and
Rule II, we follow the rules of algebra. Algebraic rules dictate that each square
be calculated first, then added, and finally the square root is found. So the
uncertainty is found as follows; percent error for Fx and Fy (using the definition
of percent error),
(δFx /Fx ) · 100% = (0.5N/5.0N ) = 10.0%
(δFy /Fy ) · 100% = (0.5N/12.0N ) = 4.2%
The percent error in Fx2 (using Rule II),
2 · 10% = 20%
The percent error in Fy2 (using Rule II),
2 · 4.2% = 8.4%
The absolute error for Fx2 and Fy2 (using the definition of percent error
backwards),
2
(20.0%/100%) · (5.0N ) = 5.0N 2
2
(8.4%/100%) · (12.0N ) = 12.0N 2
The absolute error in the quantity, Fx2 + Fy2 (using Rule I),
5.0N 2 + 12.0N 2 = 17.0N 2
The percent error in the quantity, Fx2 + Fy2 (using the definition of percent
error),
17.0N 2 /169.0N 2 · 100% = 10.0%
1.6. EXPERIMENTAL PROCEDURE
The percent error in the quantity,
q
7
Fx2 + Fy2 (using Rule II)
1/2 · 10.0% = 5.0%
And, finally, the absolute error in |F | (using the definition of percent error
backwards),
(5.0%/100%) · 13.0N = 0.7N
1.6
1.6.1
Experimental Procedure
Volume and Density of a Cylinder
The volume of a uniform cylinder is given by
V = π D2 /4 L
(1.3)
where V is the volume, D is the diameter and L is the length. The density is
given by
m
(1.4)
ρ=
V
where ρ is the density and V is the volume. Measuring D, L and m directly,
we will assign the uncertainty for each (see the above section ”Uncertainty or
Error”). Then we can calculate V , ρ, δV and δρ; the latter two being indirect
determinations.
1.6.2
Equipment Usage
Devices needed for measuring the lengths and diameters are: a ruler, a vernier
caliper and a micrometer caliper. For measuring the mass, a triple beam balance
will be used. Each of these instruments used to measure length has a different
degree of precision; the ruler having the least and the micrometer having the
most.
Using a ruler is rather easy, line up one edge of the object in question with
a tick mark on the ruler and then find the corresponding tick mark on the ruler
for the remaining edge. The difference between these two tick marks gives the
length. Please note that this involves some degree of estimation and this will
help dictate the uncertainty assigned. The only other precaution one need take
is to avoid using the edges of the ruler, since they may be worn down and hence
introduce a systematic error in the measurement.
The vernier caliper and micrometer caliper are a bit more complicated. A
vernier caliper is made up of a fixed jaw and a movable jaw (see figure 1.1).
The fixed jaw has a metric scale, called the main scale, which starts at 0mm
and ends at 120mm. The movable jaw has a metric scale, called the vernier
scale which starts at 0 and ends at 10. The leftmost mark on the vernier scale
has the value 0 (zero). When the jaws are closed, the zero mark (left most tick
mark) on the vernier scale should line up with the zero mark on the main scale.
8
EXPERIMENT 1. P1-1: UNCERTAINTY IN MEASUREMENTS
Figure 1.1: Vernier Caliper
If this is not the case, the vernier caliper is out of calibration. Do not use it and
inform your instructor of this condition. To measure the length of an object,
open the jaws of the caliper wide enough for the object to fit in between and
then gently close the jaws so that fit snugly against it. DO NOT TIGHTEN
DOWN ON THE CALIPER!!! It is not a ’C’ clamp! This will break
it or put it out of calibration.
To determine the distance, as indicated by the spread of the jaws, look for
the zero mark on the vernier scale. In the accompanying figure ( figure 1.1) it
lies between 13mm and 14mm. This means that the caliper reading is 13 plus
”something” mm. The ”something” part is found by finding the tick mark on
the vernier scale that lines up with a tick mark on the main scale. This will be
unique for any given measurement. In the figure, the 5 mark lines up the best
with a mark on the main scale, hence the caliper reading is 13 + 0.5 = 13.5mm.
Note that determining which mark on the vernier scale lines up best requires a
degree of estimation and implies what the uncertainty might be.
The micrometer caliper also uses movable jaws and a vernier type scale. The
movable part is called a thimble. The scale on the fixed jaw or main scale has
tick marks every 1mm above the horizontal line, and below the ticks marks half
way between. The thimble is graduated with 50 tick marks (on some types 100)
and each rotation of the thimble moves it 0.5mm (or 1.0mm), so each mark on
the thimble is 0.5mm/50 = 0.01mm. To determine the distance between the
jaws, rotate the thimble until the jaws are spread far enough apart to allow the
object to be inserted and then gently rotate the thimble back until the jaws fit
snugly against it. DO NOT TIGHTEN DOWN ON THE CALIPER!!!
It is not a ’C’ clamp! This will break it or put it out of calibration.
To read the indicated value, locate the edge of the thimble against the scale
above the horizontal line on the main scale. In the accompanying figure (figure
1.2) the edge is between 3 and 4, therefore the reading is 3 plus ”something”
1.6. EXPERIMENTAL PROCEDURE
9
Figure 1.2: Micrometer Caliper
mm. Further, the edge is also to the right of one of the tick marks below the
horizontal line, so add 0.5mm to the 3mm to obtain 3.5 plus ”something” mm.
Lastly, the horizontal line intersects the thimble at 37.5 (between thimble tick
mark 37 and 38), so the ”something” is 37.5 times 0.01mm = 0.375mm. This
makes the measurement 3.5 + 0.375 = 3.875mm. Note that the last digit is an
estimate and gives a hint at the uncertainty.
1.6.3
Experimental Procedure
1. Record the alphanumeric identification stamped on your cylinder.
2. Using the following data sheet, measure and record the mass, diameter
and length of the cylinder using the ruler, vernier and micrometer. Also, record
your estimates of the uncertainties (ie. absolute errors).
3. Make sure you record the measurements with the units common to the
particular device. Also, convert these to SI units and enter them on your data
sheet.
1.6.4
Calculations
On a separate sheet of paper, calculate the Volume, V , the density δρ and the
absolute and percent errors for each. Present each calculation neatly in three
steps: write down the equation, then show how the values are substituted and
then show the final numerical answer.
10
EXPERIMENT 1. P1-1: UNCERTAINTY IN MEASUREMENTS
1.6.5
Questions
1. Compare your most precise value for the density of the cylinder to the values
in the following table.
Metal
Coloring
Density (kg/m3 )
A
Iron
grayish-silver
7.87 × 103
B
Copper
reddish
8.94 × 103
C
Aluminum
silver-white
2.85 × 103
D
Brass
yellow-red
8.51 × 103
Cylinder Letter
To make the comparison, note that the density value can be written as
density ± (absolute error in density)
This implies a range of possible values for the actual density of the cylinder
which is as small as (density) - (absolute error in density) and as large as (density) + (absolute error in density). Does the accepted value fall in this range
for the density? State if your results are consistent, and, if they are not, discuss
possible reasons why your results are not consistent with what a careful experimenter would expect. Are your remaining values for the density consistent with
the accepted value? Discuss along the same lines.
2. In the calculation section, you calculated the volume, V using
V = π D2 /4 L
(1.5)
An alternate formula that could have been used includes the radius, R,
instead of the diameter:
V = π R2 L
(1.6)
If this had been used instead, would your uncertainties have been different?
Why? (Show all work in support of your answer.)
1.6. EXPERIMENTAL PROCEDURE
1.6.6
11
P111/112 Data Sheet - Experiment 1: Error Analysis
Student’s Name:
Partner’s Name:
Lab Period:
Date:
Instructor’s Signature or Initials:
Quantity
Mass
Length
Diameter
Measuring Device
Value
Absolute Error
Balance
g
kg
Ruler
cm
m
Vernier
cm
m
Micrometer
mm
m
Ruler
cm
m
Vernier
cm
m
Micrometer
mm
m
Cylinder Identification:
Cylinder Metal:
12
EXPERIMENT 1. P1-1: UNCERTAINTY IN MEASUREMENTS
Experiment 2
Statistical Analysis
2.1
Objectives
What is a Gaussian Distribution? What is the Standard Deviation? What
is the Standard Deviation of the Mean? How is the Standard Deviation and
Standard Deviation of the Mean used? How is a measured answer judged to be
acceptable?
Pre-lab preparation: read chapters 4, 5, 6, 11 and 12 in Taylor.
2.2
Statistical Distributions
If the error is truly random and the data are plotted on a Frequency Distribution
Graph we get one of two distributions: the Gaussian (a.k.a. Normal or Bell
Curve) or the Poisson.(see figure 2.1 and figure 2.2).
We note that the Gaussian Distribution is symmetrical and the that Poisson
Distribution is asymmetrical. The Poisson Distribution is asymmetrical due to
the mean of the distribution being bounded on one side.
2.2.1
Some Important Parameters of the Gaussian Distribution
1) Average or Mean
x=
N
X
(x1 + x2 + x3 + ... + xi )/N
(2.1)
i=1
where N is the number of data points.
2) Deviation
Devi = (xi − x)
13
(2.2)
14
EXPERIMENT 2. STATISTICAL ANALYSIS
Figure 2.1: Gaussian Distribution (Bell Curve)
Figure 2.2: Poisson Distribution
2.2. STATISTICAL DISTRIBUTIONS
15
3) Standard Deviation (SD)
v
v
uN
uN
uX
uX
t
2
SD =
(Devi ) /(N − 1) = t (xi − x)2 /(N − 1)
i=1
(2.3)
i=1
where N is the number of data points.
Physically, the standard deviation for a Gaussian Distribution means the
following: If one additional measurement of the quantity being measured is
taken, the probability is 2/3 that the value of the measurement will lie in the
range: mean value ± one SD.
4) Standard Deviation of Mean
√
(2.4)
SDM = SD/ N
5) The Gauss or Normal Distribution
GX,σ (x) =
2
2
1
√ e(x−X) /2σ
σ 2π
(2.5)
X = x̄ after many samplings.
Physically, the standard deviation of the mean for a Gaussian Distribution
means the following: If an additional set of measurements were taken, the probability is 2/3 that the mean value for this second set of measurements would lie
in the range: previous mean ± one SDM .
2.2.2
An Example of Standard Deviation of the Mean
The accompanying table shows the results obtained when a student measured
the length of an object seven times. The student then used a spreadsheet
and generated the standard deviation. Verify for yourself that the mean, the
deviations of each datum, the squares of each deviation, the sum of the squared
deviations and the standard deviation are as shown.
Data(cm)
12.32
12.35
12.34
12.38
12.32
12.36
12.38
N=7
M ean
Dev
Dev 2
ΣDev 2
StdDev
12.35
-0.03
0
-0.01
0.03
-0.03
0.01
0.03
0.0009
0
0.0001
0.0009
0.0009
0.0001
0.0009
0.0038
0.025
16
EXPERIMENT 2. STATISTICAL ANALYSIS
The standard deviation of the mean follows from equation (2.4) as:
√
√
SDM = SD/ N = (0.025)/ 7 = 0.0094491 = 0.009
2.2.3
(2.6)
Some Important Parameters of the Poisson Distribution
1) Average Number of Counts Expected
ν̄ = µ
(2.7)
The expected average number of events in time T is µ = rate × time = RT .
2) Standard Deviation
√
σν = µ
(2.8)
3) The Poisson Distribution
Pµ (ν) = e−µ
µν
ν!
(2.9)
where µ is positive1 .
2.3
t-Test
The vast majority of experiments carried out by Physicists involve comparing
two quantities. For example, a theory may predict that a certain nuclear particle
has a certain mass or charge and an experiment is performed that will measure
it. Or, we may have the case where two independent experiments designed
to measure the same quantity are performed. The question that is always of
interest is: Do these two results represent the same thing? That is, do these
two quantities agree and how close in agreement are they? One simple method
of choice would be to compare the one quantity to the experimental result plus
or minus the uncertainty. In other words, does the theoretical or experimental
result fall in the range specified by the other experiment’s uncertainty? If it
does, then it is likely that the two values are in agreement. If not, then perhaps
they do not agree or perhaps something in the experiment was overlooked and
needs to be refined.
A more sophisticated method compares the two values using a statistical
test called a ”t-Test”. A t-Test compares the means of two populations or two
samples using an appropriate hypothesis2 . The hypothesis that we make is
called the Null Hypothesis, or H0 . We call it null because we hypothesize
that there is no difference between the quantities in question. To illustrate this,
suppose two groups set out to measure the same quantity and let us call it ξ.
Our hypothesis is that these two values of ξ are the same. We then make one
1 In
a counting experiment one cannot have less than zero counts!
chapter 5 and 6 in Taylor and the handout by Freund (page 2).
2 see
2.4. CHI-SQUARED TEST
17
of three possible Alternate Hypotheses, H1 . These are:
(1) Group B’s ξ has a value greater than that of Group A (a one-tail test);
(2) Group B’s ξ has a value smaller than that of Group A (a one-tail test);
(3) Group B’s ξ is different from that of Group A (ie. either larger or smaller)
(a two-tail test). Alternate Hypotheses 1 and 2 call for one-tail tests, since
we are only interested in deciding whether Group B’s ξ is larger or smaller.
Alternate Hypothesis 3 calls for a two-tail test, since we are saying that the
Group B value for ξ is simply different. An infinite sample size is required to
prove that the Group B ξ is exactly identical to the Group A ξ, so we ignore
this and concentrate on whether there is any basis to reject the Null Hypothesis,
H0 . That is, if it can be shown that one of the Alternate Hypotheses are true,
then we can reject the Null Hypothesis. Otherwise we accept it. Of course, we
also must assume that the two groups data were independently obtained and
that each group’s data belong to normal populations or are of sufficiently large
sample sizes so that it may be assumed that they are approximately normal.
The next step is to pick the level of significance (α-risk). This is the boundary
between acceptability and unacceptability. This is a judgment call; many regard
the 5% level (α-risk = 0.05) as being the boundary and it is not an unreasonable
choice3 . Once this has been decided, the t value is calculated. In this case, we
have:
ξB − ξA
(2.10)
t=
sp
where sp is the pooled standard deviation of the mean
s
(nA − 1)(sA )2 + (nB − 1)(sB )2
,
(2.11)
sp =
nA + nB − 2
nA − 1 and nB − 1 are the degrees of freedom for each sample. The degree of
freedom is defined as n−1 with n being the number of data points in the sample.
We use this because we do not know the population standard deviation of the
mean for each group’s sample, only the sample standard deviation of the means.
Furthermore, it also accounts for the variation in both samples especially if the
number of data points is not equal between the groups. To judge whether the
Null Hypothesis is accepted we now compare the sample t to tα,(np −1) with
α = 0.05 and np = nA + nB − 2.4
2.4
Chi-Squared Test
A Chi-Squared (χ2 ) test is used to determine if a set of data is normally distributed (ie. random). The Chi-Squared test is similar in idea to that of the
t-Test. The χ2 value is calculated for the sample data using
χ2 =
n
X
(Ok − Ek )2
k=1
3 See
4 See
Ek
.
Freund, page 4.
section 1.9 in Freund for the appropriate table and it’s use.
(2.12)
18
EXPERIMENT 2. STATISTICAL ANALYSIS
where Ok is the observed number and Ek is the expected number (Ek =
N (P robk ), with N being the number of data points). If χ2 is equal to zero,
perfect agreement is indicated; that is Ok − Ek = 0 for all bins k. If χ2 is
of order n (the number of bins) or less, there is no reason to believe that the
distribution is not what was expected (ie. χ2 . n). If the chi-squared value is
significantly greater than the number of bins (ie. χ2 n) then we have reason
to believe that the distribution is not what was expected. The trick in all this is
picking the number of bins. Once the number of bins is selected, the probability
(from under the Gaussian) can be determined and Ek calculated5 .
2.5
Procedure
1) Position the Geiger counter approximately 5 cm above the 90
38 Sr columnator. Make sure you have secured the columnator to the table-top with masking
tape.
2) Start the Logger Pro 3.3 software.
3) Under File click OPEN and choose Nuclear Radiation with Computers
and Statistics.
4) Set-up the data collection as follows. Under Experiment select Data Collection and Sampling Rate set the Experiment Length to 3 minutes and
the ”minutes/sample” to 0.0825 (12.12121 samples/minute). This will store 36
samples based on 36 total points collected. Then click ”Done” or ”OK”.
5) You are now ready to collect data. Do Not DISTURB the Geiger
counter and source during data collection. If you do, this will invalidate your sample by altering the cross-section sampled.
6) To start data collection, click the Collect icon.
7) When the data collection is complete, Do Not DISTURB the Geiger
counter and source.
8) Select the data Table window by clicking the top of its window. Click File,
Print Preview and Print if the table of data is shown; if it is not, you do not
have the data table selected. Fill in the Print Options sub-menu, and call this
data set Trial 1.
9) For Trial 2, repeat steps 4 through 8 without disturbing the Geiger counter
and source.
10) Move the Geiger counter up 5cm making the total distance between the
5 See
Chapter 12 section 1 of Taylor
2.6. ANALYSIS
19
source and counter 10 cm.
11) Collect two more samples just as you did previously and label them Trial 3
and Trial 4 respectively.
12) Move the Geiger counter up another 5 cm so that the total distance between the source and counter is 15 cm and collect two more sets of data (Trials
5 and 6).
2.6
Analysis
1) Plot histograms (ie. Frequency Distribution Graphs) for all six trials. Which
look like Gauss distributions and which look like Poisson distributions?
2) Perform a t-Test to compare trials 1 and 2. Since the apparatus was not
disturbed, the measured cross-section did not change; what should you expect?
3) Perform a t-Test to compare either trial 1 (or 2) to trial 3 (or 4). Now
the measured cross-section has changed; what should you expect?
4) Perform a χ2 test on one of your first four trials. What do you conclude?
5) Perform a χ2 test on either trial 5 or 6. What do you conclude?
20
EXPERIMENT 2. STATISTICAL ANALYSIS
ν|α
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
60
120
∞
ν|2α
PERCENTAGE POINTS, STUDENTS t DISTRIBUTION
(Upper-tail probabilities)
0.40
0.25
0.10
0.05
0.025
0.01
0.005
0.0005
0.325 1.000 3.078 6.314 12.706 31.821 63.657 636.619
0.289 0.816 1.886 2.920
4.303
6.965
9.925
31.598
0.277 0.765 1.638 2.353
3.182
4.541
5.841
12.941
0.271 0.741 1.533 2.132
2.776
3.747
4.604
8.610
0.267 0.727 1.476 2.015
2.571
3.365
4.032
6.859
0.265 0.718 1.440 1.943
2.447
3.143
3.707
5.959
0.263 0.711 1.415 1.895
2.365
2.998
3.499
5.405
0.262 0.706 1.397 1.860
2.306
2.896
3.355
5.041
0.261 0.703 1.383 1.833
2.262
2.821
3.250
4.781
0.260 0.700 1.372 1.812
2.228
2.764
3.169
4.587
0.260 0.697 1.363 1.796
2.201
2.718
3.106
4.437
0.259 0.695 1.356 1.782
2.179
2.681
3.055
4.318
0.259 0.694 1.350 1.771
2.160
2.650
3.012
4.221
0.258 0.692 1.345 1.761
2.145
2.624
2.977
4.140
0.258 0.691 1.341 1.753
2.131
2.602
2.947
4.073
0.258 0.690 1.337 1.746
2.120
2.583
2.921
4.015
0.257 0.689 1.333 1.740
2.110
2.567
2.898
3.965
0.257 0.688 1.330 1.734
2.101
2.552
2.878
3.922
0.257 0.688 1.328 1.729
2.093
2.539
2.861
3.883
0.257 0.687 1.325 1.725
2.086
2.528
2.845
3.850
0.257 0.686 1.323 1.721
2.080
2.518
2.831
3.819
0.256 0.686 1.321 1.717
2.074
2.508
2.819
3.792
0.256 0.685 1.319 1.714
2.069
2.500
2.807
3.767
0.256 0.685 1.318 1.711
2.064
2.492
2.797
3.745
0.256 0.684 1.316 1.708
2.060
2.485
2.787
3.725
0.256 0.684 1.315 1.706
2.056
2.479
2.779
3.707
0.256 0.684 1.314 1.703
2.052
2.473
2.771
3.690
0.256 0.683 1.313 1.701
2.048
2.467
2.763
3.674
0.256 0.683 1.311 1.699
2.045
2.462
2.756
3.659
0.256 0.683 1.310 1.691
2.042
2.457
2.750
3.646
0.255 0.681 1.303 1.684
2.021
2.423
2.704
3.551
0.254 0.679 1.296 1.671
2.000
2.390
2.660
3.460
0.254 0.677 1.289 1.658
1.980
2.358
2.617
3.373
0.253 0.674 1.282 1.645
1.960
2.326
2.576
3.291
0.80
0.50
0.20
0.10
0.05
0.02
0.01
0.001
(Two-tail probabilities)
Experiment 3
Simple Harmonic Motion
3.1
Objectives
To study Hooke’s Law and Simple Harmonic Motion of translation and, in
addition determine the period of said motion. Additionally, the Hooke’s Law
spring constant and the mass factor will also be determined.
3.2
3.2.1
Introduction
Simple Harmonic Motion
Periodic motion (also called harmonic motion) is any motion that repeats itself
at regular intervals. The time of one full cycle of this motion is called the
period of motion. Simple Harmonic Motion is periodic motion in which the
acceleration is proportional to the displacement but opposite in direction. This
is expressed as
a = −Cx
(3.1)
where a is the acceleration, x it the displacement and C is an arbitrary constant
of proportionality.
3.2.2
Hooke’s Law
When a properly made spring is stretched by an applied force, it is know that the
elongation is directly proportional to the applied force, as long as the stretching
does not exceed the elastic limit. This is known as Hooke’s Law and is expressed
as
F = −ky
(3.2)
where k is a constant of proportionality and is called the spring constant. Note
that this expression is identical in form to that of simple harmonic motion.
21
22
EXPERIMENT 3. SIMPLE HARMONIC MOTION
Figure 3.1: Mass on a spring.
3.2.3
Period Of An Oscillating Mass On A Spring
Suppose the mass attached to a Hooke’s Law spring is pulled down such that
when it is released the mass oscillates up and down. It will then move in
accordance with Simple Harmonic Motion. Then, by Newton’s Second Law, the
net force may be written as
F = ma = −ky
(3.3)
ma + ky = 0
(3.4)
which implies
2
but the acceleration is also the second derivative with respect to time ddt2y . Rewriting the above, we have
d2 y
m 2 + ky = 0
(3.5)
dt
which is a second order differential equation linear in y and it can be solved by
assuming a solution of
y = Acos(ωt)
(3.6)
where A is the maximum amplitude of the oscillation, t is the time and ω is the
angular frequency of the oscillation. The above differential equation is called
the equation of motion of the simple harmonic oscillator. Our assumed solution
implies
d2 y
= −ω 2 Acos(ωt)
(3.7)
dt2
and substituting in for y and
d2 y
dt2
we arrive at
m(−ω 2 )Acos(ωt) + kAcos(ωt) = 0
(3.8)
and when like terms are canceled, we are left with
ω2 =
k
.
m
(3.9)
3.2. INTRODUCTION
23
To find an expression that relates the angular frequency to the period of
oscillation let us consider our assumed solution. We note that Acos(ωt) repeats
itself (i.e. one cycle occurs) after a time 2π/ω and is therefore the period of
motion T. Since
k
ω2 =
(3.10)
m
the period is simply
r
2π
m
T=
= 2π
.
(3.11)
ω
k
It would appear that we have finished the derivation of the period of motion
except for one important point - we have neglected the mass of the spring ms . As
the spring stretches from its rest position some of the spring mass contributes
to the overall mass that pulls or pushes the spring. Let us assume that this
additional mass is some fraction of the spring mass so our corrected mass is
m + f ms
hence a more reasonable expression for the period is
r
m + f ms
.
T = 2π
k
3.2.4
(3.12)
(3.13)
An Alternate Derivation Of The Period
In the previous section, we used Newton’s Second Law to write the force equation. Now, we will consider the method of Lagrange which uses scalar quantities
not vectors (ie. energy instead of forces)1 . The Lagrangian function L is composed of the kinetic and potential energies of the system under consideration
and is written as
L=T −V
(3.14)
where T represents the Kinetic energy and V is Potential energy. If we consider
the total mass as previously treated, then the Lagrangian is
1
L = (m + f ms )
2
dy
dt
2
1
1
1
+ ky 2 = (m + f ms )(ẏ)2 + ky 2 .
2
2
2
(3.15)
The equation of motion is found by using the Lagrangian equation for the general
force
d ∂L ∂L
Qy =
−
=0
(3.16)
dt ∂ ẏ
∂y
where ẏ is defined as
dy
dt .
Hence, the equation of motion is then
(m + f ms )ÿ + ky = 0
(3.17)
1 A full treatment of this method is encountered in more advanced courses but is very
accessible to any beginning student who can calculate derivatives.
24
EXPERIMENT 3. SIMPLE HARMONIC MOTION
or
d2 y
+ ky = 0.
(3.18)
dt2
From this point on, the derivation of the period T is identical to the steps
previously taken.
(m + f ms )
3.3
Experimental Concept
The spring is suspended from a rod clamped to a bench-stand and as the mass
is increased the spring stretches down. When set in motion, the mass oscillates
up and down and the period of motion can be found using a timer or stopwatch.
By measuring the time for 30 oscillations, the uncertainty in the period due to
human reaction time can be mitigated. The expression for the period in the
previous derivation suggests that both the spring constant k and the spring
mass factor f may be determined experimentally by measuring the mass m and
the period (T) and graphing the period squared (T2 ) versus m. The slope of
such a graph is
4π 2
(3.19)
k
and the y-intercept is
4π 2 f ms
= (slope)f ms .
(3.20)
k
The slope, y-intercept and their associated uncertainties are found by performing a linear regression on the best fit line.
To determine the period, we can use a timer.
Some studies2 have asserted that the mass factor f has a value of about 1/3
for values where ms /m is small and about 1/2 for values where ms /m is large.
An appropriate range of masses will be employed to examine these two limiting
conditions.
3.4
3.4.1
Experimental Procedure
Materials
Bench clamp, 1 - 1 meter long 3/4 in. dia. rod, 1 - right angle clamp, 1 - timer,
1 - 1/2 in. dia. rod, 1 - 2meterstick, 1 - spiral spring, 1 - mass hanger and 1 slotted mass set.
3.4.2
Period Measurement
1) Measure the mass ms of the spring.
2 See - H.L. Armstrong, ”The Oscillating Spring and Weight - An Experiment Often Misinterpreted”, Am. J. Phys. 37(4), 447-449 (Apr. 1969).
3.5. ANALYSIS AND CALCULATIONS
25
2) Set-up the apparatus according to the description in the previous section.
3) Choose an appropriately large range of masses keeping the ratio
ms
m
in mind.
4) Determine the period for each added mass by measuring the time t it takes
t
).
for 30 oscillations ( remember T = 30
5) Repeat steps 1) through 4) for a total of three or more trials. Then reverse ends and repeat for the same number of trials. (The more trials, the
better.)
3.5
Analysis and Calculations
1) Graph the period squared (T2 ) versus the hanging mass (m). Keep in mind
that you have three or more trials - what is the best way to represent them? Remember - you will need to determine the Hooke’s Law spring constant k and the
mass factor f with their associated uncertainties from the slope and y-intercept
of the line(s). A weighted average might be useful.
2) Compare your value of f to that suggested by Armstrong.
3.6
Report Requirements
See Appendix A and C on report writing.
26
EXPERIMENT 3. SIMPLE HARMONIC MOTION
Experiment 4
Conservation of Angular
Momentum
4.1
Objectives
To determine the moment of inertia for a disk and ring system rotating about
an axis of symmetry and to verify conservation of angular momentum for this
system.
4.2
4.2.1
Introduction
Relating Torque to Moment of Inertia
When a particle moves in a circular path it experiences a centripetal acceleration. In addition to this centripetal acceleration it also experiences a tangential
acceleration the magnitude of which is
αr
(4.1)
where α is the angular acceleration. Applying Newton’s Second Law, the magnitude of the torque experienced becomes:
τ = r⊥ F = r(mαr) = mr2 α = Iα
(4.2)
where mr2 = I is the corresponding moment of inertia. This relation, τ = Iα,
holds true for systems of particles as well.
4.2.2
Angular Momentum
The angular momentum1 for a particle of constant mass m is defined as:
→
−
−
−
−
−
L =→
r ×→
p =→
r × m→
v
(4.3)
1 Units
of kg · m2 /s.
27
28
EXPERIMENT 4. CONSERVATION OF ANGULAR MOMENTUM
−
−
−
where →
v is the particle’s velocity, →
p = m→
v the particle’s linear momentum
→
−
and r is the position vector relative to the origin O of an inertial reference
→
−
frame. The value of L is dependent on the choice of the origin O.
→
−
Since L is the result of a cross product, it is perpendicular to the plane of
−
→
−
2 →
motion , L will be perpendicular to it.. Hence the magnitude of L will be
L = mvr · sin(φ).
(4.4)
To find the angular momentum of a rigid body, we can exploit the above result.
Consider a rotating body lying in the xy-plane. Consider further, a thin slice
Figure 4.1: Slice of rigid body rotating about z-axis
of the body lying in the xy-plane (see figure 4.1). Each ith particle in the slice
−
moves around the axis of rotation in a circle. The velocity →
v i at each instant
−
is perpendicular to its position vector →
r , thus the angle φ is 90◦ . A particle of
mass mi at a distance ri from the axis of rotation has a speed vi equal to ri ω.
According to equation (4.4), then the angular momentum for the ith particle
must be
Li = mi vi ri = mi ri2 ω.
(4.5)
Summing over all the particles, we have
L=
2 ie.
X
Li =
if the particle moves in the xy-plane
X
(mi ri2 )ω = Iω
(4.6)
4.2. INTRODUCTION
29
Figure 4.2: Edge view of off-axis particles rotating about z-axis.
where I is the moment of inertia about the z-axis for that slice. The same
calculation can be made for points off the xy-plane, but a complication arises
−
due to the non-zero components in the x,y and z directions for the →
r vectors
(see figure 4.2).
However, if the z-axis is an axis of symmetry, the perpendicular components
for particles on opposite sides of this axis sum to zero. So when a rigid body
→
−
rotates about an axis of symmetry, L lies along this axis and the magnitude of
→
−
L is L = Iω.
4.2.3
Conservation of Angular Momentum
When the net external torque acting on a system is zero, the total angular
momentum of the system is constant. This is the principle of conservation of
angular momentum. It is a universal conservation law, just like those of energy
and linear momentum.
To see why this is so, let us take the angular momentum of a particle and
apply the chain rule for the derivative of a vector product. Starting with
→
−
−
−
−
−
L =→
r ×→
p =→
r × m→
v
(4.7)
30
EXPERIMENT 4. CONSERVATION OF ANGULAR MOMENTUM
c 1994 PASCO Scientific)
Figure 4.3: Apparatus with rotating disk. (
c 1994 PASCO Scientific)
Figure 4.4: Apparatus with rotating disk. (
4.2. INTRODUCTION
31
the time derivative is
→
→
−
−
dL
d−
r
d→
v
−
−
−
−
−
−
r ×m
= (→
v × m→
v ) + (→
r × m→
a ).
=
× m→
v + →
dt
dt
dt
(4.8)
−
−
The vector product of any vector with itself is zero, so the (→
v × m→
v ) terms
drops out and we are left with
→
−
dL
→
−
−
−
−
=→
r × m→
a =→
r ×F =τ
dt
(4.9)
which is the definition of torque. Thus, if the net sum of torques acting on a
→
−
rotating body is zero, this implies L is a constant.
4.2.4
Conservation of Angular Momentum - Alternate Derivation
Consider the motion a particle in a plane when the potential energy is a function
of the radial distance V = V (r). The Lagrangian is
L=T −V =
1 2 1
mṙ + m(rθ̇)2 − V (r)
2
2
(4.10)
then the general force equation for the radial component is
Qr = 0 =
d ∂L ∂L
∂V
d
∂V
−
= mr̈ − mrθ̇2 +
= (mṙ) − mrθ̇2 +
dt ∂ ṙ
∂r
∂r
dt
∂r
(4.11)
and for the angular component
Qθ = 0 =
d ∂L ∂L
d
= mr2 θ̈ = (mr2 θ̇).
−
dt ∂ θ̇
∂θ
dt
(4.12)
Note that mr2 θ̈ is torque. Integrating with respect to time implies that mr2 θ̇
is a constant hence angular momentum is conserved3 .
4.2.5
Experimental Concept
The apparatus consists of a stand with an axle mounted vertically and a disk
mounted on top (see figure 4.3). A string is wrapped around the axle and run
across a ”smart pulley”. A mass m is attached to the end of the string and falls
vertically when released. The acceleration produced by the falling mass is then
measured with the ”smart pulley”.
Let the radius of the axle plus half the thickness of the string be r, the
tension in the string T , and frictional torque exerted by the bearings be f r
(where f is the frictional force). Applying Newton’s Second Law to the free
body diagram of the hanging mass, we get
mg − T = ma
3 Just
as was done in equation (4.9).
(4.13)
32
EXPERIMENT 4. CONSERVATION OF ANGULAR MOMENTUM
and for rotational motion, the torque equation is
T r − f r = Iα = I(a/r).
(4.14)
Eliminating T between the two previous equations, we obtain
mg − f = ma + I(a/r2 )
(4.15)
(g − a)(m + I/r2 ) = gI/r2 + f
(4.16)
which can be factored as
and if we assume that the frictional torque is constant, the right hand side of
equation (4.16) is also constant. Substituting C for the right hand side we have
(g − a)(m + I/r2 ) = C
(4.17)
and if we keep the acceleration a very small compared to g, we can write equation
(4.17) as
C
C(1 + a/g)
C
=
=
.
(4.18)
m + I/r2 =
g−a
g(1 − a/g)
g
If we plot m versus (1 + a/g), we get a straight line whose slope is C/g and
y-intercept is −I/r2 .4 We can now determine I experimentally for any object
or combination of objects.
Once the moments of inertia are determined, the angular momentum of the
disk and disk-ring combination can now be determined. To do this, the disk is
rotated by hand to impart a speed (no need for hanging masses) and the ring,
having no rotation, is dropped onto the disk. When the ring is dropped, there
is no net torque on the system since the torque on the ring is equal to that of
the disk. There is no net change in angular momentum, hence we have
L = Ii wi = If wf
(4.19)
where the subscripts i and f indicate the initial and final states. The angular
speeds are determined from the graph of ω versus t obtained using a photogate
in place of the smart pulley.
4.3
4.3.1
Experimental Procedure
Materials
You will need thread, a blue mass and hanger set, rotational inertia apparatus,
smart pulley, disk and a ring. Additionally, you will need a meter stick, vernier
caliper, large size triple beam balance and a computer to interface with the
smart pulley.
4 See - F.W. Van Name, ”Graphical Analysis of the Moment of Inertia Experiment”, Am.
J. Phys. 26(9), 605-606 (Dec. 1958).
4.3. EXPERIMENTAL PROCEDURE
33
c 1994 PASCO Scientific)
Figure 4.5: Rotating disk with ring. (
c 1994
Figure 4.6: Smart pulley with pulley removed and in new position. (
PASCO Scientific)
34
EXPERIMENT 4. CONSERVATION OF ANGULAR MOMENTUM
c 1994 PASCO ScienFigure 4.7: Location of modified smart pulley mount. (
tific)
4.3.2
Data Collection - Part A
This data will be used to calculate the experimental moments of inertia.
1. Mount the disk on top of the axle.
2. To find the acceleration start the ”Logger Pro 3.3” program.
3. Put a total of 50 grams on the end of the thread and run the thread over the
smart pulley. Note, the mass hanger is 5 grams.
4. Wind the thread up on to the axle making sure that the thread does not
wind on top of itself; winding the thread on top of itself changes the radius r
and hence the torque.
5. Make sure that the red LED on top of the smart pulley is not lit; if it
is, turn it gently until it goes out.
6. Click the green ’Collect’ icon. The timer will start timing when the photodiode is first blocked by the moving pulley. Let the mass fall and stop the
timing by clicking the red ’Stop’ icon just before the thread completely runs out.
7. Referring to the velocity versus time graph: To get the acceleration, click
’Analyze’, ’Linear Fit’ and a box with the linear regression will be displayed on
the graph. To obtain the proper statistical uncertainties for the fit, right click
the regression output box and select ’Linear Fit’ options and click the check
4.3. EXPERIMENTAL PROCEDURE
35
boxes for the slope and y-intercept standard deviations.
8. The velocity versus time graph should now be displayed with the statistics at the top of the screen. The slope of the graph is the acceleration. Record
the slope and all relevant statistical values. Print the graph with the regression
output.
9. Repeat steps 1 through 8 increasing the hanging mass by 5 grams each
time up to 100 grams.
10. Place the ring on top of the disk so that it sits in the groove cut into
the top of the disk as illustrated in Figure 4.4.
11. Determine the accelerations for the same range of hanging masses as you
just did for the disk alone.
12. Unwind the thread and measure the diameter of the axle with a vernier
caliper. Note - there is more than one place that the thread can be wound,
make sure you measure the diameter where the thread was wrapped.
4.3.3
Data Collection - Part B
This data will be used to calculate the change in angular momentum.
1. Detach the thread from the axle. Remove the pulley from the smart pulley
and mount the photogate portion so that the beam of the photogate is interrupted by the 10 spoke pulley on the vertical axle (see Figure 4.5 and Figure 4.6).
2. Open the Logger Pro 3.3 software. Click ”FILE”, ”OPEN” and the folder
called ”Physics with Vernier”. Scroll all the way to the end of the listings and
select the file called ”angularspeed”. Your display should look like Figure 4.8
3. Check the alignment of the photogate with respect to the 10-spoke wheel,
located on the axle just below the disk. Make sure that the photogate sees all
10 openings and does not interfere with the rotation of the apparatus.
4. While holding the ring just above the center of the disk, give the disk a spin
using your free hand. To start the data collection, click the green ”START”
icon. Drop the ring onto the spinning disk from a point very close to the top of
the disk after approximately 25 data points have been taken. If the ring does
not land in the groove cut into the disk, start this step again.
5. Continue collecting data after the ring is dropped for about the same number
of data points and stop the collection process by clicking the red ”STOP” icon
or hitting the space-bar. Note that there is a period of transition between the
initial and final speeds. Keep this to a minimum.
36
EXPERIMENT 4. CONSERVATION OF ANGULAR MOMENTUM
Figure 4.8: Display of Logger Pro table and graph.
4.4. ANALYSIS
37
6. Determine the angular speed just before the ring makes contact with the
disk (wi ) and just after (wf ).
7. Print the table and graph
SEPARATELY.
8. Repeat this for a total of ten or more good trials.
4.4
Analysis
1. Graph m versus (1 + a/g) for each set of data in Part A. Perform a linear
regression for each set of graphed data and extract a moment of inertia for the
disk (Idisk ) and for the disk+ring (Idisk+ring ) combination from the y-intercepts.
2. Calculate the absolute uncertainties for Idisk and Idisk+ring .
3. Calculate the angular momentum before and after the collision and the
uncertainty for each.
4. Calculate the rotational kinetic energy lost in the collision using
1
1
2
2
2 Ii ωi − 2 If ωf
1
2
2 Ii ωi
(4.20)
5. Graph L1 versus Lf . What special line should you obtain? Do your data fit
this expectation? (How and why?)
6. Perform a t-test on ∆L. Your expected value is zero.
4.5
Questions
1. In section (2.2.4) we defined r as the radius of the axle plus 1/2 the thickness
of the string. In the calculations for I we neglected the thickness of the thread.
Suppose the diameter of the thread was not small as compared to the diameter
of the axle, how would this effect your results?
2. What, if any, impact does the loss in Kinetic Energy have on your result? How
could this experiment be redesigned to reduce or eliminate this effect? NOTE:
you may be asked to redo this experiment while accounting for your suggested
improvements for extra credit or maybe even a poster session at Quest.
38
EXPERIMENT 4. CONSERVATION OF ANGULAR MOMENTUM
Experiment 5
Coupled Oscillations and
Normal Modes
5.1
Objectives
To identify and measure the normal modes of vibration and the corresponding
angular frequencies of the vibration for a system of coupled oscillators.
5.2
5.2.1
Introduction
Normal Mode Vibrations
Many systems when displaced from a stable equilibrium position will oscillate.
One such system, both simple and familiar, is that of a mass on a spring. Such
a system may be expanded by adding masses and springs to create a system
of coupled oscillators. A single oscillator has a single natural frequency and
will oscillate forever in the absence of outside forces. Coupled oscillators have
several natural or normal frequencies at which they vibrate and the sinusoidal
motion associated with each frequency is called a normal mode. If all springs
obey Hooke’s law, then all the equations of motion are linear. Although this is
a special case, it has wide applications in Physics, Chemistry and Engineering.
Consider the two masses as shown. Mass m1 is suspended from a spring
with spring constant k1 which is attached to a rigid support. The second mass
m2 is suspended below m1 and the two masses are coupled by a spring with
constant k2 . In the static equilibrium case, the masses just hang in space.1
If m2 is displaced downward by a distance y2 , then m1 will also experience a
displacement y1 due to a force in the negative y-direction and a force in the
1 In fact, we need not consider the gravitational forces acting on the masses because they
are independent of the displacements and do not contribute to the restoring forces that cause
the oscillations. Their only impact is that of a shift in the equilibrium positions of the masses.
39
40 EXPERIMENT 5. COUPLED OSCILLATIONS AND NORMAL MODES
Figure 5.1: Coupled Masses
positive y-direction. Hence, the net force acting on m1 is
m1 a1 = m1
d2 y1
= −k1 y1 + k2 (y2 − y1 )
dt2
(5.1)
The force of spring 2 pulls m2 in the negative y-direction hence
m2 a2 = m2
d 2 y2
= −k2 (y2 − y1 )
dt2
(5.2)
These two equations are referred to as a second order homogeneous linear2
system of two equations and two unknown functions with constant coefficients.3
The solution of this system is straight forward and relatively simple.
To simplify things, let k1 = k2 = k and m1 = m2 = m. Let us assume
that y1 = A1 cos(ωt) and y2 = A2 cos(ωt) where A1 and A2 are the maximum
2 This is called linear because it contains no higher powers of y or its derivatives than the
first power; homogeneous because every term is a first power (ie. there is no term independent
of y and its derivatives).
3 It is also known as an eigenvalue problem.
5.2. INTRODUCTION
41
amplitudes of the displacement and ω is the angular frequency of the vibration.
Then
dy1
dy2
= −ωA1 sin(ωt) and
= −ωA2 sin(ωt)
(5.3)
dt
dt
and
d2 y1
d2 y2
= −ω 2 A1 cos(ωt) and
= −ω 2 A2 cos(ωt)
(5.4)
dt
dt
which, after a bit of algebra, we obtain
[(2k − mω 2 )A1 − kA2 ]cos(wt) = 0
(5.5)
[kA1 + (mω 2 − k)A2 ]cos(wt) = 0
(5.6)
and
The amplitudes A1 and A2 must be non-zero4 ; if they were zero this would be
trivial and uninteresting. There is a theorem from differential equations that
says for
A system of n homogeneous linear algebraic equations in n unknowns has
a nontrivial solution if and only if the determinant of coefficients of the
system is equal to zero.
So equations (5.5) and (5.6) leads to
(2k − mω 2 )
−k
det(A) = k
(mω 2 − k)
=0
(5.7)
which when evaluated yields
m2 ω 4 − 3kmω 2 + k 2 = 0.
(5.8)
The roots of equation (5.8) are
√
5 k
3k
±
ω =
2m
2 m
2
(5.9)
and therefore, the angular frequencies will be
ω=
! 21
√
3+ 5 k
and
2
m
! 12
√
3− 5 k
2
m
and the ratio of the normal mode frequencies is
√
( 5 + 1)
√
( 5 − 1)
(5.10)
(5.11)
4 These two constants are arbitrary constants because the general solution of a second order
D.E. contains precisely two arbitrary constants.
42 EXPERIMENT 5. COUPLED OSCILLATIONS AND NORMAL MODES
5.2.2
Fast Fourier Transform
A Fast Fourier Transform is a mathematical method which decomposes a sequence of values into components of different frequencies. In our case, we transform the force versus time data into amplitude versus frequency. The frequency
with the maximum amplitude corresponds to the normal mode frequency. Since
we have two frequencies, we will have two maximum amplitudes. The details of
how the Fourier Transform is calculated is left for higher level math and physics
courses.
5.3
5.3.1
Experimental Procedure
Materials
1 - Dual Range Force Sensor, 1 - LabPro interface, 1 - Mechanical Wave Driver,
1 - Digital Function Generator, 1 - bench clamp, 2 - right angle clamps, 1 - 3/4
in. diameter bench clamp rod, 1 - 1/2 in. diameter rod, 1 - 10 gram hooked
mass, 2 - 20 gram hooked masses, 2 - 50 gram hooked mass, 2 - 5N/m springs,
2 - 8N/m springs, 1- spring set containing: 2 - 10N/m, 2 - 20N/m, 2 - 40N/m
1 - metric ruler.
5.3.2
Part A - Determining Spring Constant k.
1) The spring constant k may be determined from a plot of force versus stretch
for a Hooke’s Law spring. This implies that k is the slope of the best fit line.
2) Mount the spring vertically and measure the static elongation x for approximately ten different masses. Be careful not to exceed the elastic limit of the
spring! Repeat this for all the springs used.
3) The corrected mass is equal to the hanging mass plus 1/3 the mass of the
spring.
5.3.3
Part B - Determining the Angular Frequency ω Fast Fourier Transform (FFT).
1) Mount the Force Sensor on a 1/2 inch diameter rod and suspend it with a
right angle clamp from a vertically mounted 3/4 inch diameter rod in a bench
clamp with the hook pointed down.
2) Plug the Force Sensor into CH1 on the LabPro interface. The interface
should already be connect to the computer for you.
3) Suspend two springs with k = 5N/m and two 20 gram hooked masses from
the hook on the Force Sensor as shown in figure 5.1.
5.3. EXPERIMENTAL PROCEDURE
43
4) Start the LoggerPro software. Under File choose Open and click Physics
with Vernier and then click Normal-Modes. This file will have two graphs,
one showing force versus time and the other will have amplitude versus frequency. The latter graph is generated by the Fast Fourier Transform routine.
5) Gently displace the bottom mass downward and release it. The masses should
oscillate up and down without a lot of pendulum like motion (ie. without left
to right motion)5 .
6) Click the green ”Start” icon to start data collection. The data collection
will stop automatically after 30 seconds.
7) You should now see two sets of histogram-like peaks on the FFT graph.
These represent the two normal mode frequencies. To obtain the frequency values select ”Analyze” then ”Examine”. This will open a dialog box on the graph.
Move the cursor over to the desired peak and read the frequency directly from
the dialog box. The maximum peak for each set is probably the best choice.
You may have to ”split the difference” if there are two maximums in a set.
Record both frequencies.
8) Repeat steps 1) to 7) for the following combinations:
m1 (kg)
0.020
0.050
0.050
0.050
0.050
0.020
0.020
5.3.4
k1 (N/m)
5
5
5
8
8
5
5
m2 (kg)
0.050
0.020
0.050
0.050
0.020
0.050
0.030
k2 (N/m)
5
5
5
8
5
8
5
Part C - Determining the Angular Frequency ω Fast Fourier Transform (FFT) Continued.
In this part you will be continuing Part B with larger masses and spring constants using the following combinations:
m1 (kg)
0.200
0.200
0.200
0.200
0.200
0.200
0.300
5 There
k1 (N/m)
10
10
20
20
40
40
40
m2 (kg)
0.200
0.300
0.200
0.300
0.300
0.400
0.400
k2 (N/m)
10
10
20
20
40
40
40
may be some torsion (twisting) present too. This may be unavoidable.
44 EXPERIMENT 5. COUPLED OSCILLATIONS AND NORMAL MODES
5.3.5
Part D - Determining the Angular Frequency ω - By
Excitation.
In this part you will be verifying the results in Part B. Do not use the springs
and masses from Part C as the vibrator will be damaged!
Figure 5.2: Coupled Masses Set-Up
1) Mount the Mechanical Wave Drive on a 1/2 inch diameter rod and suspend it using a right angle clamp from the 3/4 inch diameter rod which has
been clamped to the bench and point the drive arm down. Make sure the banana plug with hook accessory is inserted into the drive arm. Caution: Never
touch the drive arm without the drive arm locking tab in the lock
position; the driver may be damaged by excessive force.
2) Suspend two springs with k = 5N/m and two 20 gram hooked masses from
the hook on the Mechanical Driver Arm as shown in figure 5.2.
3) Connect the Digital Function Generator to the Mechanical Wave Driver with
banana plug leads. Use the LO and GND outputs. (The Mechanical Wave
Driver is essentially an 8Ω speaker, a low impedance device.)
4) Before turning on the Digital Function Generator, make sure the amplitude
is turned to its lowest setting.
5) Turn on the Digital Function Generator and set the frequency below the lowest expected normal mode frequency. These were found using the FFT graph
in Part B.
6) Move the drive arm locking tab to the unlocked position and increase the
amplitude a small amount until you get a visible displacement of the masses.
5.4. ANALYSIS
45
As you approach the normal mode frequency, you will have to decrease the
amplitude or things will fly apart. In fact, when you are close or right on the
normal frequency you will only need a very small amplitude to get things excited.
7) Adjust the frequency up and down slowly until you are sure you have the
normal mode. Note: this search for the frequency should give you an idea of
how to assign uncertainty for this measurement. The first normal mode occurs
when the two masses move up and down in step. This is sometimes called the
symmetric mode.
8) Increase the frequency to find the second normal mode. This is sometimes
called the anti-symmetric mode because the masses move in opposition (ie.
out of step).
9) Repeat steps 2) - 8) for the same combinations in Part B.
5.4
Analysis
1) Plot the data (Force vs. Displacement) for Part A and calculate the spring
constants with uncertainties. How does your value for k compare to what the
manufacturer stated?
2) Calculate the normal mode frequencies for all the combinations using your
values for k and those supplied by the manufacturer. (Do not forget to convert
to f from ω!) Note that in the general case, the square of the angular frequency
is given by
1 p
(k1 + k2 )m2 + m1 k2
±
[−m2 (k1 + k2 ) − m1 k2 ]2 − 4m1 m2 k1 k2 .
2m1 m2
2m1 m2
(5.12)
The uncertainty associated with this particular calculation is found using the
general formula for uncertainty given in equation (3.47) on page 75 of Taylor,
2nd edition. Before attempting to calculate this uncertainty, consider whether
or not you need to do it.
ω2 =
3) In the derivation for the normal mode frequencies, we assumed that the top
spring is attached to a fixed surface. Obviously, with the springs and masses
hanging from the Mechanical Vibrator, this is not the case. Does your data
indicate a significant departure from the theory to indicate that this fact may
come into play? Are there other effects observed that may also contribute to a
departure from theory?
4) Bonus - derive the expression for ω 2 that includes the Mechanical Vibrator effect.
46 EXPERIMENT 5. COUPLED OSCILLATIONS AND NORMAL MODES
Experiment 6
Simple Pendulum
6.1
Objectives
To determine the motion of a simple pendulum and examine several approximations for the period. Additionally, the local acceleration due to gravity will
also be determined.
6.2
6.2.1
Introduction
Period for the Small Angle Approximation
The simple pendulum consists of a point mass suspended by a light inextensible
string (ie. the mass of the string is negligible and the string does not stretch).
When it is pulled to one side from its rest position and let go, it swings back and
forth in a vertical plane under gravity’s influence. Resolving the forces acting
on the pendulum bob, we see that the tangential component FT is
FT = mg · sin(θ)
(6.1)
FR = mg · cos(θ)
(6.2)
and the radial
The radial component supplies the necessary force to keep the bob moving in
a circular arc (ie. centripetal force). The tangential component supplies the
restoring force; it tends to bring the bob to equilibrium, hence
Frestore = −mg · sin(θ)
(6.3)
Since the restoring force for the pendulum goes as sin(θ) (and not θ) the resulting oscillatory motion is not simple harmonic motion. Approximating sin(θ) by
θ (for small angles), we have
−mgθ = −mg
47
x
l
(6.4)
48
EXPERIMENT 6. SIMPLE PENDULUM
Figure 6.1: Simple Pendulum.
which resembles −kx, the restoring force of a mass on a spring. By letting
mg
l = k and substituting this into the expression for the period T of a mass on
a spring, we arrive at
s
l
T0 = 2π
(6.5)
g
for the small angle approximation for the period of a simple pendulum. From
the slope of a graph of T02 versus l, the local acceleration due to gravity may be
determined.
6.2.2
The Average Acceleration Approximation
Consider a pendulum of length l and initial amplitude θ. When the bob is
released, it will swing towards the equilibrium point. In a time equal to one
quarter of the period (ie. t = T4 ), the bob will swing along an arc of distance
s = lθ. As the bob travels from angle θ to angle 0, the distance traveled over
the arc is also
s=
1 2
āt
2
(6.6)
6.2. INTRODUCTION
49
where ā is the average acceleration1 . Therefore, the period will be
r
2s
T = 4t = 4
.
ā
(6.7)
Since the acceleration of the bob goes as g · sinθ, it will decrease as θ decreases.
At some angle smaller than the initial θ, the instantaneous acceleration a will
equal the average acceleration ā. Let us denote this angle as Cθ, where C is a
proper fraction. We can now express ā as
ā = g · sin(Cθ)
The period for the average acceleration approximation is therefore
s
2lθ
T =4
g · sin(Cθ)
(6.8)
(6.9)
and when sin(Cθ) approaches Cθ, the small angle period T0 can be written as
s
2l
T0 = 4
(6.10)
gC
The ratio of (6.9) to (6.10) is given by
s
21
T
Cθ
Cθ
R=
=
=
T0
sin(Cθ)
sin(Cθ)
(6.11)
Since this approximation relies on the assumption that C is a constant, it would
appear that this is an exact solution; it is not. Because C is actually a function
of θ, the ratio R should be
s
C0 θ
(6.12)
sin(Cθ)
where C0 is the limiting value of C as θ approaches zero. However, if C is a
slowly varying function then the approximation should be a good one.
To complete this approximation, it is necessary to evaluate C; a series expansion will aid us in this task. Rewriting (6.11) as
R=
sin(Cθ)
Cθ
− 21
(6.13)
we can then substitute the Maclaurin series (ie. Taylor series about a = 0) for
sin(θ), hence
R=
sin(Cθ)
Cθ
− 21
− 12
(Cθ)2
(Cθ)4
(Cθ)6
= 1−
+
−
+ ...
.
3!
5!
7!
(6.14)
1 See - W.P. Ganley, ”Simple pendulum approximation”, Am. J. Phys. 53(1), 73-76 (Jan.
1985).
50
EXPERIMENT 6. SIMPLE PENDULUM
Now, to evaluate the square root, we re-write R with a binomial expansion by
2 2
4 4
letting x = −C3! θ + C5!θ and p = −1/2 we arrive at
C 2 θ2
C 4 θ4
R= 1+
+
+ ... .
(6.15)
12
160
All that is left now is to evaluate the proper fraction C. This can be determined
experimentally or by comparison as we shall soon see.
6.2.3
Period When Angles Are Not Small
Returning to equation (6.1) and canceling m we have
Let
dθ
dt
d2 θ
g
= − sin(θ)
2
dt
l
(6.16)
d2 θ
du
du dθ
du
=
=
=u
2
dt
dt
dθ dt
dθ
(6.17)
= u then
and we have
du
g
= − sin(θ).
dθ
l
(6.18)
u2
g
= cos(θ) + c
2
l
(6.19)
u
Integrating by θ we obtain
and when θ = θ0 , u = 0 so that c = − gl cos(θ0 ). Hence (6.19) can be written
s 2g
2g
dθ
2
u =
=±
(cosθ − cosθ0 ) or
(cosθ − cosθ0 )
(6.20)
l
dt
l
Over a time equal to one-quarter of the period (ie. θ = θ0 to θ = 0) then (6.20)
becomes
s 2g
dθ
=−
(cosθ − cosθ0 )
(6.21)
dt
l
Separating the variables and noting that t = 0 at θ = θ0 and t =
obtain (upon integration)
s Z
θ0
l
dθ
p
T =4
2g 0
(cosθ − cosθ0 )
T
4
at θ = 0 we
and exploiting the trigonometric identity cosθ = 2sin2 ( θ2 ) − 1 we get
s Z
l θ0
dθ
q
T =2
g 0
θ
0
(sin2 ( ) − sin2 ( θ ))
2
2
(6.22)
(6.23)
6.3. EVALUATION OF THE CONSTANT C
51
Let sin( θ2 ) = sin( θ20 )sin(φ) and take the derivative of both sides then 12 cos( θ2 )dθ =
sin( θ20 )cos(φ)dφ and calling k = sin( θ20 ) we arrive at
2sin( θ0 )cos(φ)dφ
.
dθ = p 2
1 − k 2 sin2 (φ)
(6.24)
and (from the above trigonometric identity) when θ = 0, φ = 0 and when θ = θ0 ,
φ = π2 ; therefore (6.23) becomes
s Z π
2
l
dφ
p
T =4
g 0
1 − k 2 sin2 (φ)
(6.25)
which is an elliptic integral and cannot be evaluated exactly in terms of elementary functions; it can be solved for θ in terms of elliptic functions which
are generalized trigonometric functions2 . Note that if θ is small then k is very
nearly zero and (6.25) reduces to
s Z π
s
2
l
l
T =4
dφ = 2π
(6.26)
g 0
g
which is the small angle approximation solution. Equation (6.25) can be written
as
s "
#
2
2
2
l
1
1·3
1·3·5
2
4
6
T = 2π
1+
k +
k +
k + ...
(6.27)
g
2
2·4
2·4·6
and the square brackett terms are
1
θ0
9
θ0
1 + sin2
+ sin4
+ ...
4
2
64
2
6.3
(6.28)
Evaluation of the Constant C
A ballpark estimate for C can be found by setting the periods for the small angle
and average acceleration approximations equal to each other and solving for C
yielding C = π82 ≈ 0.811. Alternatively, one can generate tables (or graphs) of
R versus θ for various values of C and then the experimental results may be
used to interpolate the value of C. Yet another method 3 exploits the Maclaurin
expansion of equation (6.28). Re-writing (6.28) in terms of powers of θ we get
√
θ2
+ ... and setting equal to (6.15) and solving for C gives C = 3/2
R = 1 + 16
or C ≈ 0.866.
2 see
a good set of standard math tables
assumes knowledge of the exact solution for (6.1)
3 which
52
EXPERIMENT 6. SIMPLE PENDULUM
R as a Function of θ for Various Values of C
θ (Deg.)
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
(Con’t.)
θ (Deg.)
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
θ (Rad.) — C =
0.0873
0.1745
0.2618
0.3491
0.4363
0.5236
0.6109
0.6981
0.7854
0.8727
0.9599
1.0472
1.1345
1.2217
1.3090
1.3963
1.4835
1.5708
0.85
1.0005
1.0018
1.0041
1.0074
1.0116
1.0168
1.0229
1.0301
1.0384
1.0477
1.0583
1.0699
1.0829
1.0971
1.1127
1.1298
1.1483
1.1684
0.855
1.0005
1.0019
1.0042
1.0075
1.0117
1.0170
1.0232
1.0305
1.0388
1.0483
1.0590
1.0708
1.0839
1.0984
1.1142
1.1315
1.1503
1.1706
0.86
1.0005
1.0019
1.0042
1.0076
1.0119
1.0172
1.0235
1.0309
1.0393
1.0489
1.0597
1.0717
1.0850
1.0996
1.1156
1.1332
1.1522
1.1729
0.865
1.0005
1.0019
1.0043
1.0076
1.0120
1.0174
1.0238
1.0312
1.0398
1.0495
1.0604
1.0726
1.0860
1.1009
1.1171
1.1349
1.1542
1.1751
0.866
1.0005
1.0019
1.0043
1.0077
1.0120
1.0174
1.0238
1.0313
1.0399
1.0496
1.0606
1.0728
1.0863
1.1011
1.1174
1.1352
1.1546
1.1756
θ (Rad.) — C =
0.0873
0.1745
0.2618
0.3491
0.4363
0.5236
0.6109
0.6981
0.7854
0.8727
0.9599
1.0472
1.1345
1.2217
1.3090
1.3963
1.4835
1.5708
0.868
1.0005
1.0019
1.0043
1.0077
1.0121
1.0175
1.0239
1.0314
1.0401
1.0499
1.0609
1.0731
1.0867
1.1016
1.1180
1.1359
1.1554
1.1765
0.869
1.0005
1.0019
1.0043
1.0077
1.0121
1.0175
1.0240
1.0315
1.0402
1.0500
1.0610
1.0733
1.0869
1.1019
1.1183
1.1362
1.1558
1.1770
0.87
1.0005
1.0019
1.0043
1.0077
1.0121
1.0176
1.0240
1.0316
1.0403
1.0501
1.0612
1.0735
1.0871
1.1021
1.1186
1.1366
1.1562
1.1774
0.89
1.0005
1.0020
1.0045
1.0081
1.0127
1.0184
1.0252
1.0331
1.0422
1.0525
1.0642
1.0771
1.0914
1.1073
1.1246
1.1436
1.1643
1.1867
0.90
1.0005
1.0021
1.0046
1.0083
1.0130
1.0188
1.0258
1.0339
1.0432
1.0538
1.0657
1.0790
1.0937
1.1099
1.1277
1.1472
1.1684
1.1915
0.867
1.0005
1.0019
1.0043
1.0077
1.0121
1.0174
1.0239
1.0314
1.0400
1.0498
1.0607
1.0729
1.0865
1.1014
1.1177
1.1355
1.1550
1.1761
6.4. EXPERIMENTAL PROCEDURE
6.4
6.4.1
53
Experimental Procedure
Materials
1-Logger Pro interface, 1-photogate, 1-large angle protractor, 1-bob, 1-two meter long length of light string, 1-two meterstick, 1-vernier caliper.
6.4.2
Data Collection
Part A of the experiment is capable of determining g to an accuracy of 0.5%
or so if you are careful with the measurements. This is remarkable indeed.
We challenge you to do your best experimentation so that you can meet this
standard of accuracy. Although we only have gold sticky stars to pass around
for such accuracy, the real reward is the self-satisfaction of a challenge well met.
Part B is capable of differences between the measured and theoretical periods
of 1% or less. Given the difficulty of this part of the experiment, this too is a
challenge. Can you meet it?
Procedure A
1. You can determine the local acceleration due to gravity in Part A by measuring the period of the pendulum for various values of the pendulum length.
Because it is difficult to set the bob swinging in a plane using a single string
attached to the bob, we actually use two strings harnessed to the bob. However,
the bob still swings as though there were only one string connected between the
center of the bob and the support frame directly above the bob. This length is
the length l.
2. Select either the one-meter stick or the two-meter stick as appropriate. Use
the two movable fingers on the stick to measure the distance from the bottom of
the support arm to the top of the bob. These fingers act like the jaws of a giant
caliper, allowing you to measure distances to the nearest half millimeter. Be
sure not to stretch the strings as you make the measurement. Use the vernier
caliper to determine the diameter of the bob. The length l is the distance to
the top of the bob plus the radius of the bob. Use 9 values, starting at about
30 cm and ending at about 195 cm, for the pendulum length. Space the values
about 18 cm apart.
3. Determine the periods corresponding to the various pendulum lengths. For
each pendulum length, position the photogate timer so that only the lower part
of the bob cuts the light beam. Make sure that the hole in the bob does not
pass through the light beam.
4. Load the photogate software by clicking on the icon for Logger Pro (make
sure it is version 3 or higher). Under File choose Open and click Physics
with Vernier and then click 14 Pendulum Periods
54
EXPERIMENT 6. SIMPLE PENDULUM
5. For your smallest pendulum length, start the pendulum swinging with an
amplitude of 5 or less. Make sure that there is no twisting motion as the pendulum swings and that the pendulum is swinging in a plane. The computer will
start timing when the Collect button is clicked. Allow the computer to time
10 periods; then click the Stop button after the tenth data entry.
6. Repeat steps 2, 3, and 5 for the remaining pendulum lengths. Keep the
amplitude less that 5 degrees for each measurement. Do not repeat step 4.
Procedure B
7. When the angle is not small, we will measure the period of the pendulum for
one value of the pendulum length l, but seven values of the amplitude θ Adjust
the length of the pendulum
to roughly 30 cm, but do not measure this length bep
cause the quantity 2π l/g in equation (6.27) is just the period of the pendulum
when the amplitude is small. Determine this period in the same way as in Part
A. Be sure that the amplitude is 5 degrees or less. Call the value of the period T0 .
8. With the pendulum at rest, rotate the protractor until the two strings line
up with the 0 degree mark. This aligns the 0 degree mark on the protractor
with the vertical. Now determine the period for the large angles of swing.
9. Because frictional forces damp out the amplitude, you will have to measure
the period on the fly, so cooperation between you and your partner is essential.
We suggest that you start the pendulum swinging from an amplitude of about
85 degrees. IMPORTANT: Be sure that the pendulum swings in a plane and
that the bob does not hit the photogate. Practice starting the pendulum without the photogate present until you can reliably make it swing in a plane. One
way of starting the pendulum is to hold the bob in the palm of your hand with
the strings taut and the pendulum at 85. Quickly drop your hand to release the
pendulum. You will note that the strings go slack for these large amplitudes,
so no useful data is possible for angles above about 82 degrees. When the pendulum decays to an amplitude of 80 degrees it should be swinging properly.
10. It is important to position your eye properly in order to determine the
amplitude of the pendulum. Your eye is positioned properly when the two
strings and the scratch mark on the protractor are aligned. Of course, you will
have to reposition your eye for each angle because the amplitude of the motion
is decaying.
11. As the pendulum’s amplitude decays, you call out the angle and your
partner records the period of the pendulum from the computer screen. Alternatively, your partner can record the row number of the correct period and later
retrieve the corresponding period after the run is over. Select seven angles 5
degrees apart starting at 80 degrees and finishing at 50 degrees . Practice this
6.5. CALCULATIONS
55
procedure until you and your partner can obtain a good set of data. Don’t worry
if you miss some angles because you can repeat the procedure for the missing
data points. Note that you have to use your judgment when you call out the
angle because the pendulum decays at a rate of about 0.5 degree every swing.
12. Repeat step 11 two more times. Average the periods for each angle.
6.5
Calculations
From procedure A, calculate g the local acceleration due to gravity and specify
the uncertainty for your experimental result. A graph and linear regression will
be required to find g. In your report, you should also use the experimental T0
for both the small angle and average acceleration approximations to estimate C.
From procedure B, determine R and C. Also from procedure B, calculate the
periods (TT ) for each angle.
56
EXPERIMENT 6. SIMPLE PENDULUM
Experiment 7
Physical Pendulum
7.1
Objectives
To determine the motion of a physical pendulum. Additionally, the local acceleration due to gravity will also be determined.
7.2
Introduction
Recall that a simple pendulum consists of a point mass suspended by a light
inextensible string. This means that the mass of the string is negligible and the
string does not stretch. If the mass of the string was not negligible, one would
have what is termed a physical pendulum and this condition must be taken into
account. Let us start with the derivation of the simple pendulum and see where
it leads. Refer to Figure 7.1. Resolving the forces acting on the pendulum bob,
we have for the radial
mg · cos(θ)
(7.1)
and the tangential
mg · sin(θ).
(7.2)
The tangential component acts like the restoring force of a spring and it produces
a torque about the pivot point. Recall that the torque can be written as
τ = r⊥ F
(7.3)
where r⊥ is the moment arm of the force that produces the torque, hence
τ = −Lmg · sin(θ)
(7.4)
with L being the moment arm of the tangential component. But, the torque is
also related to the moment of inertia I by
τ = Iα
57
(7.5)
58
EXPERIMENT 7. PHYSICAL PENDULUM
Figure 7.1: Simple Pendulum.
with α being the angular acceleration about the pivot point. Setting (7.4) equal
to (7.5) we have
mgL
α=−
sin(θ).
(7.6)
I
Applying the small angle approximation sin(θ) = θ we arrive at
α=−
mgL
θ.
I
(7.7)
We note that (7.7) is an equation of the same form as that for the restoring
force of a mass on a spring, ie. we have simple harmonic motion present. The
period for a simple harmonic oscillator is
r
m
T = 2π
(7.8)
k
so we can by inspection write the period as
s
T = 2π
I
mgL
(7.9)
As a check, let us substitute for I. Since the pendulum bob is a uniform mass,
we can treat it as a particle and the moment of inertia is
I = mr2 .
(7.10)
7.3. EXPERIMENTAL PROCEDURE
59
Substituting for I 1 we get the period of the simple pendulum as
s
L
T = 2π
g
(7.11)
which is just what we expect. Now, if we replace the pendulum bob and string
with a uniform rod of length L, the moment of inertia is2
2
1
L
(7.12)
I= m
3
2
which yields the period for this physical pendulum as
s
2L
T = 2π
3g
(7.13)
and knowing T and L the local acceleration due to gravity g can be calculated.
7.3
Experimental Procedure
7.3.1
Materials
1-Logger Pro interface, 1-photogate, 1-one meterstick, 1-Rotary Motion Sensor,
2-right angle clamps, 1-bench clamp, 1-6 foot long 3/4 in. dia. rod, 1- 1/2 in.
rod (for mounting Rotary Sensor) and 1-one meter long rod with hole drilled
in one end and mounted on Rotary Motion Sensor (to be used as the physical
pendulum).
Procedure - Part A
1.) If not already done, set-up the equipment listed above.
2.) You will collect the period of the physical pendulum for one length for
several trials.
3.) You are to determine the best possible method for determining the length
and period with the given tools at hand. Draw on your experience from previous experiments. Keep in mind you want to minimize the uncertainties present.
You will have to estimate the uncertainty in the length but the uncertainty in
the period can be determined using statistical methods.
4.) In addition to determining the local acceleration of gravity g from the
length and period measurements, you are to conduct a Repeatability Study on
the period. Collect at least three sets of data with a minimum sample size of
1 The
2 It
is
center of mass is at a distance L from the pivot point.
L
because it is a uniform mass and the center of mass is located at
2
L
2
60
EXPERIMENT 7. PHYSICAL PENDULUM
N = 10.
5a.) Now disconnect the photogate from the LabPro interface and connect
the Rotary Motion Sensor.
5b.) Insert a Fast Fourier Transform (FFT) graph. Choose ”Insert” and ”Additional Graphs” from the menu at the top of the screen.
5c.) Also adjust the experiment time to 30 seconds. You will find the setting under ”Experiment” and ”Data Collection”.
5d.) The FFT will give you the frequency. You must convert this to period
for later comparison.
Procedure - Part B
1) Repeat for the rod from the Pasco Physical Pendulum kit.
7.4
Calculations and Analysis
1.) Calculate g and δg for all trials.
2.) In your analysis, compare the values of g obtained in this experiment to
that obtained in the Simple Pendulum (Small Angle Approximation) experiment. Remember, both of these values have uncertainty. (Would a weighted
average be appropriate for your g obtained in this experiment?) Also, in your
analysis, discuss the repeatability of the period. (Maybe you should use a ttest?)
3.) Compare the average period from the photogate data to the period determined from the FFT. You should also calculate g from the FFT period and
compare to the values of g determined previously.
Experiment 8
Coupled Pendulums
8.1
Objectives
To measure the angular frequencies of the normal modes for a system consisting
of two coupled pendulums.
8.2
Introduction
For a simple pendulum, if the angle through which it swings is kept small, the
motion can be considered as being simple harmonic motion. The displacement
experienced by the pendulum bob, given as a function of time is:
x = Acos(ωt)
(8.1)
where A is the maximum displacement and ω is the angular frequency. The
second derivative with respect to time gives the acceleration:
a = −ω 2 Acos(ωt) = −ω 2 x
(8.2)
Combining this with Hooke’s law, the force will be
F = mω 2 x
(8.3)
with m being the mass of the bob.
Consider two simple pendulums of equal mass coupled by a spring. The
spring’s relaxed length is equal to the spacing between the pendulums. When
the pendulums are set in motion, at some arbitrary time mass A will be displaced
from the equilibrium position by xA and mass B will be displaced by xB . The
spring is stretched by an amount xA − xB and pulls on mass A and mass B with
a force of magnitude k(xA − xB ), hence the motion of mass A and B is governed
by:
d2 xA
+ mω 2 xA + k(xA − xB ) = 0
(8.4)
m
dt
61
62
EXPERIMENT 8. COUPLED PENDULUMS
Figure 8.1: Two Coupled Pendulums
and
m
8.3
Experimental Procedure
8.3.1
Materials
8.3.2
Procedure
8.4
d2 xB
+ mω 2 xB − k(xA − xB ) = 0
dt
Calculations
(8.5)
Experiment 9
The Oscilloscope
9.1
Objective
To learn the basic function of the oscilloscope and its uses.
9.2
Introduction
The oscilloscope is one of the most important and versatile test instruments in
the laboratory. This experiment will introduce you to its care and feeding. At
its most basic level, the oscilloscope consists of a cathode ray tube (CRT) and
external signal processors which control the movement of a beam of electrons
within the CRT. When the electron beam hits the phosphor coating on the inside
front face of the CRT, it produces a spot of light. This gives the operator of the
oscilloscope a visual display of the various signals applied to the oscilloscope’s
control circuits.
9.2.1
Horizontal Amplifier
One of the signal processors is the horizontal amplifier. It controls the horizontal
motion of the electron beam. This amplifier has a built-in sweep circuit which
automatically moves the electron beam across the face of the CRT. Alternatively,
the operator can apply external signals to the horizontal amplifier to control
the electron beam. When the operator selects the sweep circuit, the horizontal
amplifier applies a sawtooth voltage (or ramp voltage) to the CRT which causes
the electron beam to start at the left side of the CRT and move to the right
across the face of the CRT. When the beam reaches the right side of the CRT, the
horizontal amplifier turns off (blanks) the beam and returns the sweep voltage
to zero. The horizontal amplifier then repeats the process. Depending on the
setting of the external knob which controls the sweep circuit, the beam can cross
the CRT slowly (say in 5 seconds) or quickly (in 0.5µs = 0.5×10−6 s). Thus the
sweep circuit serves as a time base so that the operator can see the shape of an
63
64
EXPERIMENT 9. THE OSCILLOSCOPE
external signal applied to the vertical amplifier as a function of time. Normally
the operator uses the built-in sweep circuit; however, for certain measurements,
the operator must use external signals to control the horizontal motion of the
beam. We show the utility of this mode of operation later in the experiment.
9.2.2
Vertical Amplifier
The second important signal processor is the vertical amplifier. Normally the
vertical amplifier has two channels (inputs) so that it can separately process
two input signals, but display them simultaneously. Each channel has an external knob to control the gain of the vertical amplifier, which ranges from
5mV /division to 20V /division. A division is the distance between adjacent lines
on the front face of the scope (normally 1 cm). Thus the vertical amplifier can
accommodate input signals with a wide range of amplitudes. Depending on the
scope, there is a front panel control which selects what the scope displays: (1)
chan 1, (2) chan 2, (3) chan 2 inverted, (4) chan 1 added to chan 2, (5) chan 1
alternated with chan 2, or (6) chan 1 and chan 2 in chopped mode. This lat-
Figure 9.1: Chopped Mode.
ter display mode is quite useful since it preserves the phase (time) relationship
between the two signals on chan 1 and chan 2. In the chopped mode the scope
displays a small (time) slice of chan 1, then the next (time) slice of chan 2,
then the next slice of chan 1 again, etc. The display would then look like that
in figure 9.1, except for the following. The CRT does not display the vertical
lines and the chopping is quite rapid, so the operator sees the two waveforms as
though they were continuous lines rather than two lines comprised of individual,
closely spaced segments.
9.2.3
Trigger Circuit
The third important signal processor is the trigger circuit. It controls the conditions under which the electron beam starts its horizontal motion. This circuit
prevents the sweep circuit in the horizontal amplifier from operating until the
9.3. PROCEDURE
65
input signal applied to the vertical amplifier meets the trigger circuit parameters. The most important of these parameters are the slope of the input signal
( + or -) and its size or level. Thus the electron beam does not start across the
CRT until the input signal has the proper slope and the input signal voltage
level is larger than the setting of the trigger circuit’s level control. The trigger
circuit can also accept an external signal and the above then applies to this
external signal.
This is a brief introduction to the scope. The best way to fully understand
and appreciate it is to play with all the pretty knobs and see what happens.
One caution, however: you can burn out the phosphor on the CRT face by
leaving the electron beam focused on one spot with full intensity. Otherwise
the scope is fairly rugged and will withstand the onslaught of novice electron
jockeys such as yourself.
9.3
Procedure
1. Turn on the oscilloscope. Set the TRIGGER section controls as follows:
SLOPE
+ SLOPE (step symbol on scope shows a rise)
LEVEL
midway between + and MODE
PP AUTO
HOLDOFF MINIMUM
SOURCE
CH 1/EXT
COUPLING AC
2. Set the VERTICAL or amplitude section controls as follows:
MODE BOTH/NORM/ALT
CH 1
GND
CH 2
GND
3. Set the HORIZONTAL or time section controls as follows:
MODE X1
SEC/DIV 50 µs
4. Make sure all gain adjustment controls are in their CAL or calibrated mode
(fully clockwise).
5. See what happens when you play with the following knobs: (1) FOCUS
(2) INTENSITY (3) horizontal POSITION (4) vertical POSITION (each chan)
6. Set the DC/GND/AC switches on CH 1 and CH 2 to DC.
7. Adjust the vertical POSITION controls for CH 1 and CH 2 to put both
traces on the screen.
66
EXPERIMENT 9. THE OSCILLOSCOPE
8. Set the TRIGGER MODE to NORM.
9. Set the VOLTS/DIV switch on CH 1 to 2 V/DIV.
10. Set the SLOPE to +.
11. Connect the sine wave output of the oscillator to CH 1. Set the oscillator to about 10 kHz. Be sure you can see the beginning of the pattern. Slowly
vary the TRIG LEVEL to get a stable pattern on the screen. Adjust the height
of the sine wave with the oscillator amplitude knob until the amplitude is 4 V.
The trace will span a vertical distance of 4 cm on the screen.
12. Move the trace with the horizontal POSITION control until you can see
the beginning of the trace. Vary the TRIG LEVEL. Note that the function of
the TRIG LEVEL is to establish a voltage level or threshold. This means that
the trace will only start across the screen when the CH 1 input voltage reaches
this threshold voltage. For example, if the TRIG LEVEL were set for 3 V, the
trace would start across the screen only when the input voltage reaches 3 V.
As a result, the trace would show only the top portion of the first sine wave
maximum and then the rest of the sine wave. On the other hand, if the TRIG
LEVEL were set for -3 V, the trace would start just after the bottom of the first
sine wave minimum. In both instances, however, the trace would begin with
a positive slope. Establish with the TRIG LEVEL both positive and negative
threshold voltages and convince yourself that the beginning of the trace behaves
as it should. Call your instructor if you are not sure what you should be seeing
on the scope.
13. Obtain a stable trace. Use the TIME/DIV switch to expand or compress the time/division on the horizontal axis. Note that if you compress the
time/division, the trace shows more sine wave bumps, while if you expand the
time/division, the trace shows fewer bumps.
14. Change the CH 1 vertical amplifier gain with the VOLTS/DIV switch.
This function allows you to expand or compress the trace vertically. Note that
you do not have to change the trigger LEVEL to keep the trace.
15. Set the VOLTS/DIV switch back to 2 V/div. Vary the amplitude of the
input signal. Note that if the amplitude falls below the TRIG LEVEL, the trace
disappears. Check this carefully.
16. Expand the pattern horizontally as much as possible, but still have one
complete wave on the screen. Use the setting on the TIME/DIV switch to determine the frequency of the applied signal. Recall that the frequency of a wave
is the reciprocal of its period and its period is the time for one complete wave.
Note that the period for the wave is also the time interval between adjacent
9.3. PROCEDURE
67
maxima (or adjacent minima); consequently, vary the TIME/DIV to put two
maxima on the screen, and then move the trace vertically to put the maxima
on the fine tick marks at the middle of the screen. Remember, you paid for all
the knobs on the scope, so use them!
Figure 9.2: Series RC Circuit.
17. Set up the circuit in figure 9.2. The wiggly line in the circle represents
the oscillator. The resistor/ capacitor combination is housed in a plastic box
with terminals for easy attachment of leads. Use one of the three cables provided to connect the oscillator to the RC circuit. Connect the red lead on the
cable to the free end of the capacitor and the black lead to the free end of the
resistor. The black lead is ground. Use the other two cables to connect the
scope to the circuit. Connect the red leads of the two cables to the two points
marked chan 1 and chan 2. Connect the black leads to the common ground point.
18. Set the oscillator to about 700 Hz and its output to 2 V amplitude. Set
the TIME/DIV switch to 0.2 ms/div and the DC/GND/AC switch on CH
2 to DC. Set the source to VERT MODE/EXT. Set the vertical MODE to
BOTH/NORM/ALT. Set the horizontal MODE to X1.
19. Display CH 1 alone by moving the MODE switch above the CH 1 VOLTS/DIV
switch from BOTH to CH 1. Next, display CH 2 alone. Now display BOTH
again. Note that the two waves have different amplitudes, but are in the same
location on the screen. Thus we can’t see if there is a phase shift between them
using these scope settings.
20. The RC circuit in figure 9.2 introduces a phase shift (time shift) between
the wave on CH 1 and the wave on CH 2. To display this phase shift, switch the
vertical MODE to CHOP. Note that the CH 2 waveform reaches a peak before
the CH 1 waveform; however, both are still sine waves. Draw the trace to scale
as accurately as possible on your data page.
21. Determine the phase shift in degrees between the two waves from the trace
on the scope. To do so you must calibrate the horizontal axis in degrees with
68
EXPERIMENT 9. THE OSCILLOSCOPE
the TIME/DIV switch and its CAL knob. The CAL feature allows you to expand or compress the trace in order to fit one complete wave into a convenient
number of scale divisions, such as 8. Recall that one complete cycle of a sine
wave is 360 degrees, and 360 divided by 8 is a whole number. Now move the
trace vertically so that you can use the fine tick marks to improve the precision
of the measurement. Return the CAL knob to its calibrated position after you
finish the measurement.
Figure 9.3: Lissajous Pattern.
22. You can obtain the phase shift in another way. Set the TIME/DIV switch
to X-Y (rotate it fully counter-clockwise). Now the signal on CH 1 controls the
movement of the electron beam in the horizontal direction, and CH 2 controls
the movement of the electron beam in the vertical, direction. The scope should
show a pattern similar to figure 9.3. This pattern is called a Lissajous figure, and
you can use it to determine the phase shift . Center the pattern on the middle
of the screen. Use the CH 2 POSITION control to move the pattern vertically
and use the horizontal POSITION control to move the pattern horizontally.
Adjust the CH 1 and CH 2 VOLTS/DIV switch and/or the oscillator amplitude
to make the pattern as large as possible, but still keep it on the screen. Measure
A and B (see 9.3). The phase shift φ is given by φ = sin−1 (A/B) Determine φ
and record your value. Your value should agree with that found from using the
trace method.
23. Use the time remaining to better understand the scope. For example, if
you switch the oscillator output from a sine wave to a square wave, CH 2 will
not show a square wave as well. Furthermore, the degree of distortion will
change with a change in frequency of the applied signal.
9.4
Report Requirements
1. Submit your data and results.
2. Submit your answers to the questions.
9.5. QUESTIONS
9.5
69
Questions
Use a separate sheet of paper for your answers to these questions.
1. The beginning of a sine wave pattern on the scope depends on the settings
of the trigger circuit. Suppose you display a sine wave whose amplitude is 10
V. Sketch the three scope patterns corresponding to the following three settings
of the trigger circuit: the trigger is set for positive slope and the trigger level is
first +4 V, then 0 V, and then -8 V.
2. Repeat Question 1 for the trigger set for negative slope.
3. What would you observe if in Question 1 the trigger level were set for +12
V? State a reason for your answer.
4. Suppose that you have two sine waves which represent two voltages v1 and
v2 . Let v1 be ahead of v2 by 90 degrees; using the standard terminology, we
say that there is a phase shift of 90 degrees between them with v1 leading v2 .
Sketch the scope display in chopped mode and clearly identify v1 and v2
5. Sketch the scope display in chopped mode for the same two sine waves if
v2 now leads v1 by 180 degrees Clearly identify the two voltage waveforms.
70
EXPERIMENT 9. THE OSCILLOSCOPE
Experiment 10
AC and the Series Circuit
10.1
Objective
The magnitude of the alternating current (AC) in a series circuit containing a
resistor, inductor, and capacitor passes through a maximum as the frequency
of the current changes. We call this behavior resonance. This experiment uses
resonance to determine the inductance and resistance of an inductor and to
determine the ”Q” of an LCR series circuit.
10.2
Introduction
10.2.1
Impedance and the LCR Series Circuit
For a resistor, the voltage across it and the current through it are ”in phase”.
That is, when the voltage reaches a maximum, so does the current; when the
voltage passes through zero volts, the current passes through zero amperes, etc.
The same is not true for a capacitor or inductor. For these circuit elements, the
current waveform either leads (for the capacitor) or lags (for the inductor) the
voltage waveform by 90◦ . As a consequence, the analysis of an AC circuit is
somewhat different than that of a DC circuit.
Fortunately, we can use our standard techniques for circuit analysis, such as
Ohm’s Law, if we use ”impedance” instead of ”resistance” in our analysis. Thus
Ohm’s Law becomes
V = IZ
(10.1)
where V is the voltage (potential difference) across the circuit element in question, I is the current through the element, and Z is the impedance of the element.
By properly using the impedance we can account for the phase difference between the current and voltage for inductors and capacitors. As it turns out, we
can do so with our standard techniques for vector manipulation. Before showing
how this is accomplished, we need to specify the magnitudes of the impedance
vectors for inductors and capacitors. These quantities are called reactances.
71
72
EXPERIMENT 10. AC AND THE SERIES CIRCUIT
Figure 10.1: Series LCR circuit.
Figure 10.2: Circuit components vector (phasor) relationships.
The inductive reactance XL for an inductor is given by
XL = 2πf L
(10.2)
where f is the frequency in hertz (Hz), and L is the inductance in henrys (H).
XL is in ohms (Ω). For a capacitor, the relationship is
XC =
1
2πf C
(10.3)
where C is in farads (F). As with the inductive reactance, XC is in ohms Ω.
The quantities XL and XC play a role in AC circuit analysis similar to the role
played by R in DC circuit analysis.
The simplest arrangement of a resistor, inductor, and capacitor is the series
circuit in figure 10.1. The sine wave oscillator is connected to an inductor L,
capacitor C, and resistor R. Because the inductor is made from many turns of
wire, its resistance r1 must be included in our analysis. The total impedance Z
1 Note:
this is an AC resistance, not a DC resistance
10.2. INTRODUCTION
73
Figure 10.3: Circuit component vector addition.
of the circuit depends on the impedances of the circuit components. However, we
can’t simply add the impedances algebraically because the voltage and current
waveforms are not in phase for the inductor or capacitor. A detailed analysis
shows that we must plot the impedance of the circuit components as vectors,
and then use the usual techniques for vector addition to calculate Z. Figure 10.2
shows the vector2 relationship between the circuit components and figure 10.3
their vector addition.
The vector addition yields the following relation for Z
p
Z = (R + r)2 + (XL − XC )2
(10.4)
Using equation (10.1)), the current in the series circuit is
I=
V
Z
(10.5)
Here V is the oscillator output voltage. We see from equation (10.4), however,
that Z is not a fixed value, but depends on the frequency of the oscillator.
Thus I depends on the frequency as well. For one special frequency, called the
resonant frequency f0 we have
XL = XC
(at the resonant f requency only)
or
1
2πf0 C
(10.7)
1
1
·
(2π)2 LC
(10.8)
2πf0 L =
so
f02 =
(10.6)
If we rewrite equation (10.8), we obtain
f02 =
1
1
·
(2π)2 L C
(10.9)
2 or phasor as it is sometimes called because it accounts for the phase angle of the quantity
in question.
74
EXPERIMENT 10. AC AND THE SERIES CIRCUIT
and if we plot f02 on the y-axis vs. C1 on the x-axis, we get a straight line whose
slope is related to the inductance L of the circuit. We will use this technique to
determine the inductance of an unknown inductor.
We can also determine the resistance r of the unknown inductor. At resonance, the circuit impedance is
Z0 = R + r
(10.10)
V
R+r
(10.11)
VR = I0 R
(10.12)
From equation (10.5) we have
I0 =
But
where VR is the voltage across the resistor R. Combining equations (10.11) and
(10.12) we have
V
VR
=
(10.13)
R
R+r
or
VR
R+r =
(10.14)
VR
Finally
V
r=R
−1
(at resonance only)
(10.15)
VR
Thus we can calculate r from a single measurement of V and VR , and a knowledge of R.
10.2.2
More on Resonance
It is useful to plot a response curve or resonance curve. A resonance curve is
a graph which shows how the voltage, current, power, or impedance is affected
by varying the frequency in the region near resonance.The shape of the resonance curve is determined by component values of the circuit and is a convenient
method of determining how sharply a circuit will tune. There are several features worth noting (see figure 10.4). The resonance is denoted as f0 and is the
frequency at which the curve has its maximum amplitude or maximum value.
The half-power points are where the amplitude has been reduced to one half of
its maximum value. The distance between the half-power points is called the
bandwidth.
As has been previously stated, when a series LCR circuit is at resonance,
the reactances are equal, ie.
XL = XC
(10.16)
Dividing through by R, we arrive at a quantity called the Quality Factor or Q
Q=
XC
XL
=
.
R
R
(10.17)
10.2. INTRODUCTION
Figure 10.4: Response curve - Lorentzian.
75
76
EXPERIMENT 10. AC AND THE SERIES CIRCUIT
It follows from equations (10.2), (10.3) and (10.7) that
r
ω0 L
1
1 L
Q=
=
=
R
ω0 CR
R C
(10.18)
where ω0 = 2πf0 3 . The Q may also be expressed as a function of the bandwidth.
At each of the half-power points, the magnitude of the reactive and resistive
parts of the impedance are equal. For the series LCR circuit this implies
ω+ L −
1
=R
ω+ C
and
1
− ω− L = R
ω− C
(10.19)
where ω+ = 2πf+ and ω− = 2πf− . Solving for ω+ and ω− , we find that
p
CR ± (CR)2 + 4LC
(10.20)
ω+ =
2LC
and
p
(CR)2 + 4LC
(10.21)
2LC
and we disregard the negative values of ω+ and ω− because they are non-physical
solutions (ie. you cannot have a negative frequency). The bandwidth is the
difference between the half-power points so
ω− =
−CR ±
R
L
(10.22)
ω0
.
ω+ − ω−
(10.23)
ω+ − ω− =
but Q is also
ω0 L
R
hence
Q=
Dividing both top and bottom by 2π, we obtain
Q=
f0
.
f+ − f−
10.3
Procedure
10.3.1
Part A - Determination of L and r
(10.24)
1. Connect the circuit elements as shown in figure 10.5. Note that the capacitor
C is a variable decade capacitance box.
2. Dial in 0.01 µF on the decade capacitance box. Note that the prefix symbol on the box for ”micro” is M. This is the terminology used 50+ years ago
to denote ”micro”, so you see your grandparents’ tax dollars are still working.
3 The far right-hand expression of equation (10.18) comes from solving equation (10.7) for
f0 , dividing by 2π to obtain ω0 and substituting into the left-hand expression of equation
(10.18)
10.3. PROCEDURE
77
Figure 10.5: (a) LCR series circuit (b) same circuit with DSA and DSG/A
attached.
(Most likely, your tax dollars will still be working at Oswego 40 years from now!)
3. Turn on the Digital Function Generator/Amplifier (DFG/A).
4. Turn on the Oscilloscope (’SCOPE).
5. Connect CH1 (channel 1) across the input to the circuit and CH2 (channel 2) across the resistor R.
6. Set the output voltage of the DFG to 2 volts peak-to-peak (or as close
as possible). Note, you may need to adjust the Volts/Div for CH1 in order to
view your signal4 . You will also want to adjust the vertical position of the CH1
signal; use the CH1 position knob to move is up and down.
7. Adjust the output frequency of the DFG until the circuit is in resonance.
This occurs when the waveforms on the display are in phase; the voltage across
the resistor R (displayed on CH2) will also be at a maximum. You may need
to adjust the horizontal display for easier viewing.
8. Record the frequency from the DFG display and the peak-to-peak voltage.
9. Repeat Procedure steps 7 and 8 for the following values of C: 0.013, 0.015,
0.017, 0.02, 0.025, 0.03, 0.035, 0.04, 0.045, 0.05, 0.06, 0.07, 0.08, 0.09, and 0.1
µF. These data will be used to obtain L.
4 This
does not change the signal, only how it is displayed.
78
10.3.2
EXPERIMENT 10. AC AND THE SERIES CIRCUIT
Part B - Resonance Curves and Q
1. Re-obtain the resonance frequency for your circuit with C = 0.01 µF and
measure the voltage across the resistor (VR ) and the input signal voltage (Vin ).
2. For five to ten points on both sides of f0 (in 1 Hz steps) measure the VR and
Vin . This will help define the maximum of your resonance curve.
3. Continue taking data (on both sides of f0 ) in 10 Hz steps until VR /Vin
is less than 5% of the maximum or until fluctuations in your voltage measurements become to large for reliable measurements.
4. Repeat steps 1 through 3 for C = 0.10µF .
5. Replace the resistor with a 9.5 ohm resistor and repeat the resonance curve
measurements for C = 0.01µF and C = 0.10µF .
6. Bring your inductor to your instructor who will use a meter to measure the
inductance L and the resistance r of the inductor. Also measure both resistors
used. Record these values.
10.4
Calculations
1. Plot f02 on the y-axis and 1/C on the x-axis. The frequency should be in
units of Hz and the capacitance should be in units of F. Data that do not seem
to fit the best fit line should be remeasured. (You should also try Chauvenet’s
test on the suspect point.)
2. Perform a regression to obtain the slope of the best line through your data.
Determine the inductance L of your inductor.
3. Use equation (10.15) to calculate the resistance r of the inductor.
4. Calculate the uncertainties in your values of L and r, and compare to the
meter measurements made by the instructor.
5. Calculate the voltage across the capacitor in Procedure step 8. Don’t panic,
its easy! From Ohm’s Law we know that the voltage across the capacitor VC ,
the current through it, and its capacitive reactance XC are related by
VC = IXC
(10.25)
You can readily calculate XC using equation (10.3), so all that is left is I. The
current through the capacitor is the same as the current through the inductor
or the series resistor because each of these components is in series. In Procedure
step 8 you carefully measured the voltage across the series resistor R and you
10.5. QUESTIONS
79
know its resistance, so you can calculate I also. Now calculate VC . Surprised
that the value is much larger than the input voltage?! Don’t be. The result is
another property of resonant circuits. The voltage across inductors or capacitors
in resonant circuits can be much larger than the voltage applied to the circuit.
Just be careful around circuits containing inductors and capacitors, especially
those supplied with 120 V AC from the wall outlet5 .
6. Using Peakfit, Plot the resonance curves (VR /Vin versus f ) for the four
trials in part B of the procedure. Obtain the best fit using the ’Gaussian Deconvolution Method (third derivative)’. Choose ’Spectroscopy’ and ’Lorentz
Amp’ for the peak type with ’Baseline’ set at ’Constant, D2’. The bandwidth
is given by the a2 from the best fit parameters.
7. Calculate Q using equations (10.18) and (10.24).
10.5
Questions
l. Consult equation (10.11) again. What would happen to the current in the
circuit if there were no series resistor R and you used an ideal inductor which
has no resistance? Why doesn’t the current behave this way in practical situations where real inductors must be used?
2. Although your data are not complete, the result of your calculation for
VC indicates that the algebraic sum of the voltages across the series resistor,
inductor, and capacitor does not equal the oscillator voltage V . So the question arises, how would you add the voltages across the series resistor, inductor,
and capacitor to obtain the oscillator voltage? Consult your text, and Figs.
(2) and (3) for a hint. Recall that if we multiply each impedance in Fig. 2
or Fig. 3 by the current I, we obtain the voltage across that impedance. Now
state how these voltages have to be added as part of your answer to the question.
3. Suppose you did not use the series resistor R. How could you then tell
when the oscillator was at the resonant frequency? In responding, be sure to
draw the circuit and the connections to the oscilloscope. Note, however, that
the scope can only measure the potential difference between some point in a
circuit and ground, and there can be only one ground point in a circuit. The
oscillator already has one terminal at ground potential, so the scope leads must
be positioned properly.
5I
don’t want to attend your funeral. No parent should have to bury their child!
80
EXPERIMENT 10. AC AND THE SERIES CIRCUIT
Experiment 11
Double Pendulum
11.1
Objectives
To identify and measure the normal modes of vibration and the corresponding angular frequencies for a double pendulum system with one pendulum suspended below the other in a gravitational field.
11.2
Introduction
In previous experiments, we have seen that when a system is displaced from a
stable equilibrium position it will oscillate. The simple pendulum is one such
system. A natural extension of this system would be to suspend another simple
pendulum directly below (See figure 11.1).
To analyze this system, instead of using Newton’s method, we will use the
method of Lagrange. Lagrange’s method is not a different theory, it is derived
from Newton’s second law and is simply a different formulation. It uses energy
to derive the equations of motion and it has the advantage of dealing with scalars
rather than vectors.
11.2.1
Generalized Coordinates and Lagrange’s Equations
Any set of quantities that describes completely the configuration/state of a
system is called generalized coordinates. The Lagrange function, or simply the
Lagrangian is defined as:
L=T −V
(11.1)
where T and V are the kinetic and potential energies respectively. If the potential energy is derivable from a conservative force, then the generalized force
equation is
d ∂L
∂L
−
= Qi = 0
(11.2)
dt ∂ q̇i
∂qi
where qi represents the generalized coordinates.
81
82
EXPERIMENT 11. DOUBLE PENDULUM
Figure 11.1: A double pendulum, one mass suspended below the other.
Figure 11.2: A double pendulum - heights displaced shown.
11.2. INTRODUCTION
11.2.2
83
The Double Pendulum
Large Angle
The following refers to figure 11.2. When θ1 = θ2 = 0 then the potential energy
V is zero. When m1 and m2 are displaced from rest, m1 will be raised in the
vertical direction a distance of h1 and m2 by h2 . The rectangular coordinates
of m1 and m2 are
x1 = l1 sinθ1
y1 = l1 cosθ1
(11.3)
x2 = l1 sinθ1 + l2 sinθ2
(11.4)
y2 = l1 cosθ1 + l2 cosθ2
(11.5)
and
Since l1 and l2 are constants, we have
ẏ1 = −l1 θ̇1 sinθ1
ẋ1 = l1 θ̇1 cosθ1
(11.6)
ẋ2 = l1 θ̇1 cosθ1 + l2 θ̇2 cosθ2
(11.7)
ẏ2 = (−l1 θ̇1 sinθ1 − l2 θ̇2 sinθ2 ).
(11.8)
and
The kinetic and potential energies are:
h
i
1
1
T = m1 l12 θ̇12 + m2 l12 θ̇12 + l22 θ̇22 + 2l1 l2 θ̇1 θ̇2 cos(θ1 − θ2 )
2
2
V = m1 gl1 (1 − cosθ1 ) + m2 gl1 (1 − cosθ1 ) + m2 gl2 (1 − cosθ2 )
(11.9)
(11.10)
and since gravity is a conservative force then the Lagrangian function is
L=T −V
(11.11)
which leads to the Lagrangian equations
∂L
d ∂L
−
= Qθ1 = 0
dt ∂ θ̇1
∂θ1
∂L
d ∂L
−
= Qθ2 = 0.
dt ∂ θ̇2
∂θ2
(11.12)
Hence Qθ1 = 0 implies
∂L
d ∂L
=
dt ∂ θ̇1
∂θ1
(11.13)
which leads to
(m1 +m2 )l12 θ̈1 +m2 l1 l2 cos(θ1 −θ2 )θ̈2 +m2 l1 l2 θ̇22 sin(θ1 −θ2 ) = −(m1 +m2 )gl1 sinθ1
(11.14)
84
EXPERIMENT 11. DOUBLE PENDULUM
and Qθ2 = 0 implies
d ∂L
∂L
=
dt ∂ θ̇2
∂θ2
(11.15)
then
m2 l22 θ̈2 − m2 l1 l2 θ̇12 sin(θ1 − θ2 ) + m2 l1 l2 cos(θ1 − θ2 )θ̈1 = −m2 gl2 sinθ2 . (11.16)
Solving 11.14 for θ̈1 , we obtain
m2 l2
m2 l2
g
cos(θ1 − θ2 )θ̈2 −
θ̇2 sin(θ1 − θ2 ) − sinθ1
(m1 + m2 )l1
(m1 + m2 )l1 2
l1
(11.17)
and likewise from 11.16
θ̈1 = −
θ̈2 =
l1
g
l1 2
θ̇ sin(θ1 − θ2 ) − cos(θ1 − θ2 )θ̈1 − sinθ2
l2 1
l2
l2
Letting µ = 1 +
θ̈1 = −
and
θ̈2 =
m1
m2
(11.18)
then the expressions for θ̈1 and θ̈2 become
l2
l2 2
g
cos(θ1 − θ2 )θ̈2 −
θ̇2 sin(θ1 − θ2 ) − sinθ1
µl1
µl1
l1
(11.19)
l1 2
l1
g
θ̇1 sin(θ1 − θ2 ) − cos(θ1 − θ2 )θ̈1 − sinθ2 .
l2
l2
l2
(11.20)
The solution of these equations is best handled by numerical methods.
Small Angle Approximation
If θ1 and θ2 are small then θ1 − θ2 is very small and hence we can approximate
cos(θ1 − θ2 ) as 1. Then the kinetic energy becomes
T =
1
1
(m1 + m2 )l12 θ̇12 + m2 l22 θ̇22 + m2 l1 l2 θ̇1 θ̇2 .
2
2
(11.21)
The potential energy can be approximated1 as
V =
1
1
(m1 + m2 )gl1 θ12 + m2 gl2 θ22
2
2
(11.22)
and then the Lagrangian becomes
1
1
1
1
(m1 + m2 )l12 θ̇12 + m2 l22 θ̇22 + m2 l1 l2 θ̇1 θ̇2 − (m1 + m2 )gl1 θ12 − m2 gl2 θ22 .
2
2
2
2
(11.23)
Calculating
∂L
d ∂L
∂L
d ∂L
,
,
and
(11.24)
∂θ1
dt ∂ θ̇1
∂θ2
dt ∂ θ̇2
L=
1 Because
cos(θ) ≈ 1 +
θ2
2!
11.2. INTRODUCTION
85
gives
Qθ1 = 0 = (m1 + m2 )l12 θ̈1 + m2 l1 l2 θ̈2 + (m1 + m2 )gl1 θ1
(11.25)
and
Qθ2 = 0 = m2 l1 l2 θ̈1 + m2 l22 θ̈2 + m2 gl2 θ2
(11.26)
If we assume that the motion is simple harmonic in nature then a solution
is of the form
θn = An sin(ωt)
(11.27)
which implies
θ̈n = −ω 2 An sin(ωt) = −ω 2 θn .
(11.28)
Substituting for θ and θ̈ in equations (11.25) and (11.26) and putting in to
matrix form
(m1 + m2 )l12 m2 l1 l2
θ1
(m1 + m2 )gl1
0
θ1
2
−ω
+
=0
m2 l1 l2
m2 l22
θ2
0
m2 l2 g
θ2
(11.29)
Multiplying through by
#
"
1
0
(m1 +m2 )gl1
(11.30)
1
0
m2 l2 g
yields
"
−ω
2
l1
g
l1
g
m2 l2
(m1 +m2 )g
l2
g
which can be reduced to
"
1−
# ω 2 l1
g
−ω 2 l1
g
The determinant is
θ1
θ2
+
−ω 2 m2 l2
(m1 +m2 )g
2
1 − ω gl2
1 − ω2 l1
−ω2 lg1
g
1
0
# 0
1
θ1
θ2
−ω 2 m2 l2
(m1 +m2 )g
2
1 − ω gl2
θ1
θ2
=0
= 0.
(11.32)
(11.33)
and when set to zero and evaluated is
ω 4 l1 l2
m2
ω 2 (l1 + l2 )
1
−
−
+ 1 = 0.
g2
(m1 + m2 )
g
The roots after some algebra are
h
q
g
(l
+
l
)
±
(l1 − l2 )2 +
1
2
2l1 l2
h
i
ω2 =
2
1 − (m1m+m
2)
(11.31)
4m2 l1 l2
(m1 +m2 )
(11.34)
i
.
(11.35)
86
EXPERIMENT 11. DOUBLE PENDULUM
This represents the natural frequencies for a double pendulum with l1 6= l2 and
m1 6= m2 using the small angle approximation. Now let us examine limiting
cases of (11.35).
Case 1: l1 = l2 and m1 6= m2
Equation (11.35) reduces to
ω2 =
g
l
q
m2
1 ± (m1 +m2 )
h
i
2
1 − (m1m+m
)
2
(11.36)
Case 2: l1 6= l2 and m1 = m2 Equation (11.35) reduces to
ω2 =
i
p
g h
(l1 + l2 ) ± (l1 − l2 )2 + 2l1 l2 .
l1 l2
(11.37)
Note that the mass dependence drops out.
Case 3: l1 = l2 and m1 = m2 Equation (11.35) reduces to
ω2 =
√ i
gh
2± 2 .
l
(11.38)
Note that the mass dependence drops out.
11.3
Experimental Concept
A double pendulum is suspended from a force transducer and set into motion. A
Fast Fourier Transform can be used to analyze the data from the force transducer
and determine the frequencies of vibration. The pendulums do not have to be
vibrating in their normal modes in order for the FFT to work.
11.4
Experimental Procedure
11.4.1
Materials
1 - Force transducer, 1 - Vernier LabPro interface with software and computer,
2 - pendulum bobs, 1 - meterstick, 3 - small binder clips, 2 - pendulum ”knife
edges”, 1 - bench clamp, 1 - right angle clamp,1 - 6 foot long 1/2 inch diameter
rod, 1 - 1/4 inch rod, light string.
11.4.2
Frequency Measurement
1) Measure and record m1 and m2 .
11.5. ANALYSIS
87
2) Set-up the double pendulum according to figure 11.2.
3) Measure and record l1 and l2 .
4) Determine the two fundamental frequencies of vibration and their harmonics.
The two peaks with the largest amplitudes may be the fundamental frequencies.
The two peaks of lesser amplitude may correspond to the first harmonics. The
any remaining peaks are probably unrelated to the fundamental modes.
5) Change l1 to a different length and repeat step 4.
11.5
Analysis
Calculate the two fundamental frequencies and their first harmonics of vibration
and compare to your experimental results. A spreadsheet using the general
formula [ie. equation (11.35)] may make it easier to calculate. Do not forget
your uncertainty calculations!
11.6
Report Requirements
This will be a ”short” report. Along with calculations and data, provide a
discussion of your results and any other observations you may have made.
88
EXPERIMENT 11. DOUBLE PENDULUM
Appendix A - Reports
11.7
Report Writing Tips
You may be asking yourself, ”Why do I have to write a ’Lab Report’ ?” The answer to that question is simple. As a budding Scientist, Engineer or Professional
you need to learn how to communicate the results of your research and technical developments to your colleagues and other interested parties; and learning
how to do this well, takes lots of practice. You will never perfect this skill, but
you will be refining it throughout your professional life. The purpose of this
handout is to provide you with a few tips that will help you get started. The
instructors written comments, on the graded lab reports, draw your attention
to deficiencies present in the write-up. Please review the instructors comments
and seek out the instructor for clarifications. Try not to make the same mistakes
in succeeding reports.
A laboratory report should communicate, as clearly and concisely as possible,
the rational basis for the experiment. It should also communicate what results
were obtained and what significance they hold. While writing your reports, you
must assume that the reader comes from outside your area of expertise, hence
you must assume they are completely ignorant of what your experiment entails,
even if this is known not to be the case. To this end, we have adopted a report
format that is very similar to that of a scientific journal article.
The lab report consists of:
• A Title Page or Cover Sheet
• An Abstract
• An Introduction, consisting of a Theory subsection and Procedure subsection
• A Discussion of Results
• A Conclusion
• A Calculation Section
• and any Appendices that you believe will help clarify your presentation.
89
90
APPENDIX A - REPORTS
Title Page
On your title page, identify the experiment by name and indicate the date
the data were collected. Also, put your name (first, middle initial, and last) and
then that of your partner(s). Make sure to underline your name to indicate
that the report is yours and not that of any other. Also indicate your lab section,
either by day and time it meets or by the designation in the course offerings
paper (ie. L50, L51, etc.).
Abstract
The Abstract provides a short synopsis of the purpose of the experiment and
your principle results. It should also include what parameters were measured
and it should list the percent difference between the theoretical and measured
principle results. Do not present raw data nor give details to how the measurements were obtained; those details will be discussed in the Procedure and
Results sections. This can all be accomplished in three or four sentences.
Introduction
The Introduction is made up of two subsections; Theory and Procedure.
The Theory provides the Physics behind the experiment. An easy way to
organize this section can be found in the following; Start the section by stating
the reason for the experiment. Follow this with the physical reasoning and
essential ideas behind the experiment (this includes stating what features and
numerical values are predicted). Use 2D diagrams to help clarify your discussion.
Include only the equations that are important to the experiment and provide a
derivation of these equations. Make sure you state what the variables in each
equation represent.
The Procedure tells the reader how you made the measurements. Start this
section by describing the apparatus used. Use 2D diagrams to help describe
how the apparatus is assembled and tell how the apparatus was manipulated
to collect the data. Be very detailed and describe the measurements in the
sequence performed. It is best not to refer to the lab manual when writing this
section because this usually results in prose that tells the reader ’what to do’
instead of telling ’how the measurements were made’. In others words, do not
present the procedure as a ’cookbook recipe’; it will result in a significant loss
of points towards the final grade given for the report in question. To get an idea
of how the prose should be crafted, read the sample report by ’Yutz and Yutz’
located in Appendix C.
Discussion and Interpretation of Results
In this section, you will present the data and results. These should be displayed in an organized form. Make use of tables, charts and graphs. Make sure
that the SI units are stated and all numbers are quoted to the proper significant
figures. Do not use the data sheets from your lab manual for this purpose. All
data should be recopied and reformatted. Theoretical predictions and calculations should be compared to the data. If the experiment requires a graph and/or
curve fitting (ie. Linear Regression Analysis) make sure you discuss the shape
of the resultant curve, the slope of the line, the y-intercept and the Regression
Coefficient. Also discuss the uncertainties in the measured quantities and any
statistical quantities that may arise from a Regression Analysis. It is essential to
11.7. REPORT WRITING TIPS
91
note how the uncertainties in the Principle Results compare to the Theoretical
Predictions.
Conclusions
The Conclusions are a summary of what you conclude from the results
and whether they are what you expect. Restate the results with the Theoretical expectations and include the percent error when appropriate. Do not
use terms such as ”fairly close” and ”pretty good” or ”the experiment was a
success;” give explicit quantitative differences from the expected result. State
whether these differences fall within your expected errors and state possible
explanations for unusual differences. If possible, note any sources of uncertainty that were not specifically determined in the measurements. Give plausible evidence of the magnitude of the same and (if you are brave) quoted
the calculable effects this unaccounted uncertainty may have on the interpretation of the results. If you do such calculations, these must be shown in the
Calculations Section. Do not blame poor results on ”human error”. Human
error is a mistake. If your poor results are possibly defective due to a mistake,
then the experiment has to be redone.
If you have any constructive criticism or ideas on how to improve the
experiment, please include them here.
Calculations
Give one sample calculation for each different type of calculation (this includes uncertainty calculations) that you make in the course of your data processing. You need not show how a Linear Regression Analysis is done (that is
why computers were invented), but you must show how any results from such
a calculation were used. A sample calculation consists of three lines; put the
equation used on the first line, the numbers substituted into the equation on the
second, and the numerical result on the third line. Be sure to include the units
for each result and use the proper significant figures. Do not submit scratch
work!
Regression Analysis - The Method of Least Squares Additional
Example
Suppose a student does an experiment to measure the local acceleration due
to gravity, g, using a simple pendulum. The period of the pendulum is given by
the equation,
s
l
T = 2π
(eqn.3)
g
where,
• T : is the period
• l: is the length of the pendulum
• and g: is the local acceleration due to gravity.
92
APPENDIX A - REPORTS
Squaring both sides we get:
T2 =
(2π)2 l
(eqn.4)
g
which is the same as
T2 =
4π 2 l
(eqn.5)
g
which, if we plot T 2 on the vertical axis and l on the horizontal axis, implies
2
that the slope of the resulting line will be 4πg with a y-intercept equal to zero.
Or, symbolically
m=
4π 2
(eqn.6)
g
Now, all that is left is to take the student’s data and obtain the slope using
the Least Squares method. Once the slope has been found, we can calculate g.
Using a spreadsheet, with a built-in utility to perform a Least Squares fit, we
get the following:
Period Squared (sec2 )
1.35909
1.572516
1.755625
1.993744
2.226064
2.4336
2.621161
2.965284
3.305124
3.728761
4.227136
5.076009
5.597956
Pendulum Length (m)
0.335
0.396
0.440
0.505
0.562
0.609
0.658
0.740
0.830
0.940
1.049
1.262
1.394
Constant
Std Err of Y Est
R Squared
No. of Observations
Degrees of Freedom
X Coefficient(s)
Std Err of Coef.
-0.002391
0.019168
0.999812
13
11
4.030133
0.016645
The first table contains the data the student obtained in the simple pendulum
experiment; Period Squared and Length. The second table is the output from
our spreadsheet regression. It contains values for, the y-intercept (ie. Constant),
a value for the statistical error for the y-intercept, the R-Squared value, the
11.7. REPORT WRITING TIPS
93
number of data points (ie. No. of Observations), the slope (ie. X coefficient(s)),
and a statistical error for the slope (ie. Std Err of Coef.). Since g has units
of m/s2 , then the slope must have units of s2 /m [see (eqn.6)]. Now (eqn.6)
implies:
4π 2
(eqn.7)
m
which, when we substitute the values into (eqn.7) gives:
g=
4π 2
= 9.796 m/s2
4.030
Okay, we now have a value for g, but how reliable is it? Well, if we look at
the graph of Period Squared versus Length, we see that it looks like a straight
line and if we check the R-Squared value, 0.999812, we see that it is very close
to 1. This implies that the student’s data fit a straight line which we expect
according to (eqn.4), so at a first glance it appears the student has a reliable
value for g; but this is not quite enough. We need to find an error for g before
we can compare it to a standard value or another student’s experimental value.
g=
Statistical and Estimated Errors for Experimental Data
What do we mean by error? This question is best answered in two parts. The
first part describes what it is not; the term ”error”, as used by those working
in Natural Science, does not mean mistake! Mistakes can be eliminated;
measurement error cannot. If you forget to zero your ohmmeter before you
make a resistance measurement - that is a mistake, not a scientific error. When
we say a value has error, we mean that it has a specific uncertainty associated
with it; it has a fuzziness. This ”fuzziness” means the measurement has a range
of values that it could be. By choosing the appropriate techniques and tools, we
can reduce the size of this error, but we can never eliminate it. This is because
measurement error, provided that it is not systematic, is random or statistical
in nature. Going back to the table with the regression output, we see that the
regression has calculated an error for the slope of the line. This means that the
slope may be as small as 4.013 (if I keep only 3 decimal places) but no larger
than 4.047; it can be any value within this range. We state this as:
slope = m = 4.030 ± 0.0167 s2 /m
This range has a very important place in Experimental Natural Science. It
is called the ’Absolute Error’ or the ’Absolute Uncertainty’. If an experimenter makes a measurement directly, then the Absolute Error is an estimate
of the uncertainty in the measurement. In other words, it is an educated guess.
Another way to express this error is to state it as a percent. The ’Percent Error’
is defined as:
P ercent Error =
Absolute Error Of Quantity
Quantity
· 100 %
94
APPENDIX A - REPORTS
So, the percent error for the slope is:
P ercent Error =
0.0167
4.030
· 100 % = 0.414 %
So we can specify the uncertainty in the slope as:
slope = m = 4.030 ± 0.0167 s2 /m
(11.39)
slope = m = 4.030 s2 /m ± 0.414 %
(11.40)
or as:
This gives us an uncertainty for the slope, but we still need to determine the
uncertainty in g.
When an experimenter calculates a quantity using values that have uncertainty associated with them, there are very specific rules that govern how the
error for the calculated quantity is found. The absolute error for the slope is
a statistical error, but we treat it in just the same manner as we would for an
estimated error, so we can treat the slope as if we had measured it and use the
following rules to find the uncertainty in the calculated value of g.
Rules For The Propagation Of Error
When the computation involves Addition and/or Subtraction, we ADD
the Absolute Errors of the measured quantities involved.
When the computation involves Multiplication, Division and/or Raising To A Power we ADD the Percent Errors of the measured quantities, and if the quantity is Raised To A Power, the individual Percent Errors are first multiplied by their respective powers, BEFORE
adding all the Percent Errors.
Now since g is found by taking the inverse of the slope, m, [see (eqn.7)] we
use the second rule to find its uncertainty, and since the slope, m, is the only
quantity with uncertainty in (eqn.7), this says that the Percent Error in g is
equal to the Percent Error in m. So, the student should report the value of g
found in the Pendulum experiment as:
g = 9.796 m/s2 ± 0.041 m/s2
(11.41)
g = 9.796m/s2 ± 0.414%
(11.42)
or as:
Now we have a figure of merit to judge how well the student’s value of g
compares to another determination (experimental or theoretical). The Absolute
Error just determined implies the value may be as small as 9.755 m/s2 , but not
more than 9.837 m/s2 . Comparing the student’s laboratory standard value for
g of 9.805 m/s2 , we see that this value falls within the range of uncertainty
in the student’s experiment. This says that, within the error of this particular
experiment, these two values, 9.796 m/s2 and 9.805 m/s2 , are the same.
Appendix C - Sample
Report
The Sample Report
Ima Yutz and Urah Yutz
January 10, 1999
Whatsamatta U.
ABSTRACT
In this experiment, the motion of a body falling freely in the near earth gravitational field is studied. The apparatus used in this experiment gives a record
of the position of the body as a function of time. By analysis of this record, we
show that the motion is one of uniform or constant acceleration and we determine its value to be: 9.78 m/s2 . This acceleration has been determined, using
a different, independent technique, to be 9.805 m/s2 , which is a difference of no
more than 0.3%.
INTRODUCTION
Theory
In order to determine the type of motion a falling body undergoes, it is important
to use a method that is independent of any assumptions about that type of
motion. The equations of uniform motion do not fulfill this requirement in this
case and so may not be used. Instead, the position of the falling body with
respect to time and the definitions of average velocity and acceleration will be
used. Consider three consecutive positions of the falling body; positions a, b
95
96
APPENDIX C - SAMPLE REPORT
Figure 11.3: Illustration of marks made in spark recording paper by spark timer.
Figure 11.4: Left: free-fall aparatus with falling body. Right - schematic of
free-fall aparatus, showing electric circuit.
97
and c (see figure 15.8). The average velocity of the body at position b is given
by:
vb = (sc − sa)/(tc − ta)
(11.43)
where, sc − sa is the displacement between points c and a and tc − ta is the
time it takes to go from a to c. If we determine the average velocity for each
position and plot it against the time, the slope of the resulting curve will give
the average acceleration experienced by the body. Further, if the acceleration is
constant, then the line will be straight and the motion observed will be uniform.
Procedure
The apparatus (see figure 15.9) used to record the position and time of the falling
body consists of a metal support column with an electromagnet at the top, which
is used to hold and release the falling body (in this case a metal cylinder). The
column also supports two wires, one just in front and the other just behind the
falling cylinder, and a long strip of wax paper (located between the cylinder
and the wire in back). A spark timer generates a spark at a constant rate (1/60
sec) and these sparks jump from the wire in front of the cylinder to a tapered
metal collar on the cylinder and then through the wax paper to the wire in
back. Hence, the position of the body is known at precise time intervals, by the
marks made by the spark in the wax coated paper. The accurate measurement
of the displacement of the dots is now one of prime importance. A dot several
millimeters below the initial dot was chosen as the starting point. This was done
because the first few dots occur so close together that they cannot be resolved.
With the wax coated paper rolled out over a table, a 2-meter stick was then laid
along the line of dots in order to measure the distance from the chosen starting
point to each succeeding dot. The time is an integer multiple of the spark timer
rate.
RESULTS
Figure 15.10 is a table of the displacement, velocity and time for the falling
object. Figure 15.11 is a graph of Average Velocity versus Time. The graph
shows a straight line, indicating that the falling body underwent a uniform
acceleration. A least squares fit of the data gives the slope (ie. acceleration)
of the line to be 9.78 m/s2 ± 0.08 m/s2 with a R-squared value of 0.999. The
98
APPENDIX C - SAMPLE REPORT
Table of Displacement
Disp. (m)
0.0000
0.0415
0.0860
0.1325
0.1825
0.2345
0.2895
0.3470
0.4075
0.4710
0.5370
Ave. Vel. (m/s)
2.409 *
2.580
2.730
2.890
3.060
3.210
3.370
3.540
3.720
3.880
**
Time (s)
0.0000
0.01667
0.03334
0.5001
0.06668
0.08335
0.10002
0.11669
0.13336
0.15003
0.16667
Figure 11.5: * Determined from least squares fit. ** Cannot be determined
without an additional data point.
Figure 11.6:
99
error for the slope is the standard error of the coefficient calculated by the least
squares fit. The data point deviations from this model can be attributed to a
Gaussian distribution of errors.
Through independent means, the acceleration due to gravity (g) in Oswego is
known to be 9.805 m/s2 which differs from this measured value by only 0.3% (an
absolute difference of 0.03 m/s2 . We note that this is well within the estimated
error from the least squares fit of the acceleration and that of the measurement
error of the displacements (see Appendix A of this report).
CONCLUSIONS
This experiment illustrates a method for determining the type of motion a
falling body experiences in a near earth gravitational field independent of any
assumptions about the type of motion. This motion is uniform (ie. constant)
and the acceleration is determined to be 9.78 m/s2 . The experimental value
for the acceleration differs from the known gravitational acceleration in Oswego
by only 0.3 %. This is well within the error attributable to the displacement
measurements.
APPENDIX A
Measurement error calculations
Assuming the error in the time is small compared to that of the displacement
measurements, then the percent error in the average velocity is given by: δv/v ∗
100 % = δ(sc − sa)/(sc − sa) ∗ 100 % = (δsc + δsa)/(sc − sa) ∗ 100 % where,
the symbol, δ, denotes the absolute error of the quantity in question. The
percent error in the average acceleration is given by:
δa/a ∗ 100% =
δ(vb − va)/(vb − va) ∗ 100% = (δvb + δva)/(vb − va) ∗ 100 % Figure 15.12 is a
table of the errors due to the displacement measurements. It is clear that the
error decreases the farther the mark in the waxed paper is from the reference
point, hence we need only consider the propagated error in the end points.
The slope of the line from v = 3.880 m/s to v = 2.580 m/s is 9.748 m/s2 .
(We are ignoring the initial velocity at the reference point because that value
was calculated from the least squares fit and not directly from the displacement
measurements). Using the rules for propagation of error, with the above assumptions, gives an absolute error for this slope of 0.5 m/s2 , which implies a
percent error of 5.2 %. This method tends to overestimate the error in acceleration; the error calculated from the least squared fit is the more realistic (and
more stringent) value.
100
APPENDIX C - SAMPLE REPORT
Table of Errors
Disp.(m)
0.0000
0.0415
0.0860
0.1325
0.1825
0.2345
0.2895
0.3470
0.4075
0.4710
0.5370
Abs Err Disp. (m)
0.0005
0.0010
0.0010
0.0010
0.0010
0.0010
0.0010
0.0010
0.0010
0.0010
0.0010
Ave. Vel. (m/s)
2.409
2.580
2.730
2.890
3.060
3.210
3.370
3.540
3.720
3.880
**
Ave. Vel. (m/s)
0.010*
0.062
0.032
0.022
0.016
0.014
0.012
0.011
0.009
0.008
**
Figure 11.7: * Determined from the least squares fit. ** Cannot be calculated
(by definition).
Index
Q
of circuit, 74
acceleration
centripetal, 27
tangential, 27
alternate hypothesis, 17
angular momentum, 29
angular momentum
conservation of, 29
Average, 13
bandwidth, 74, 76
Bell Curve, 13
centripetal acceleration, 27
Chi-Squared test, 17
Deviation, 13
Error, 1
Absolute, 1
Percent, 1
Error
Absolute, 4
error
meaning of, 3
Errors of Judgment, 3
estimation
degree of, 4
experimental value
merit of, 4
Gaussian, 13
Gaussian Distribution, 13
graduation
fineness of, 4
half-power points, 74, 76
impedance, 71
impedance
total, 72
inductance
unknown inductor value, 74
Lab Report
Abstract, 90
Calculations, 91
Conclusions, 91
journal style, 89
Results, 90
Title Page, 90
Lab Report
Introduction, 90
Lab Report Writing Tips, 89
Lack of Definition
in quantity measured, 3
Least Squares Method, 91
Mean, 13
micrometer caliper, 7, 8
micrometer caliper
usage, 8
mistake, 3
Fluctuation Conditions, 3
Frequency Distribution, 13
fuzziness, 93
Normal curve, 13
Normal Distribution, 15
null hypothesis, 16
Gauss Distribution, 15
Ohm’s Law
101
102
INDEX
LCR series AC circuit, 71
Percent Difference, 4
Percent Error, 4
Poisson, 13
Poisson Distribution, 13
Precision and Accuracy, 4
Propagation of Error, 4
Propagation of Error
Examples, 5
quality factor, Q, 74
Random error, 3
random error factors, 3
random nature, 3
Randomness
in quantity measured, 4
reactance, 71
reactance
inductive, 72
Regression Analysis, 91
Report Writing Tips, 89
resistance, 71
resonance, 74
resonance curve, 74
resonant frequency, 73
Rule I, 5
Rule II, 5
ruler, 7
ruler
usage, 7
Significant Figures, 1
Significant Figures
in calculations, 2
summary of rules, 2
Use Of, 2
Small Disturbances, 3
Standard Deviation, 15
standard deviation, 15
Standard Deviation
meaning of, 15
Standard Deviation of Mean, 15
Standard Deviation of Mean
meaning of, 15
statistical error, 93
Systematic error, 3
systematic error, 4
systematic error
example, 7
Systematic Errors
’true’ value, 4
t-Test, 16
torque
frictional, 32
Uncertainty, 1
Absolute, 1
Percent, 1
Uncertainty
Absolute, 4
rules to calculate, 5
uncertainty
meaning of, 3
uniform cylinder
density, 7
volume, 7
vernier caliper, 7
vernier caliper
main scale, 7
usage, 7
vernier scale, 7