PC142 Lab Manual Wilfrid Laurier University

PC142 Lab ManualWilfrid Laurier University

c© Terry SturtevantHasan Shodiev1 2

Winter 2011

1Physics Lab Instructor2This document may be freely copied as long as this page is included.

ii

Winter 2011

Contents

1 Goals for PC142 Labs 1

2 Instructions for PC142 Labs 32.1 Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Workload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Administration . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Plagiarism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Exercise on Introduction to Spreadsheets 93.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3.1 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . 103.3.3 Copying Formulas and Functions: Absolute and Rela-

tive References . . . . . . . . . . . . . . . . . . . . . . 113.3.4 Pasting Options . . . . . . . . . . . . . . . . . . . . . . 133.3.5 Formatting . . . . . . . . . . . . . . . . . . . . . . . . 133.3.6 Print Preview . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . 143.4.2 Investigation . . . . . . . . . . . . . . . . . . . . . . . 143.4.3 Follow-up . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.5 Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Exercise on Linearizing Equations 194.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

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iv CONTENTS

4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3.1 Techniques for Linearization . . . . . . . . . . . . . . . 234.3.2 Procedure for Linearization . . . . . . . . . . . . . . . 234.3.3 Choosing a Particular Linearization . . . . . . . . . . . 264.3.4 Uncertainties in Results . . . . . . . . . . . . . . . . . 28

4.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.4.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . 284.4.2 Investigation . . . . . . . . . . . . . . . . . . . . . . . 294.4.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.5 Bonus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.6 Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.8 Template . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 Exercise on Graphing and Least Squares Fitting in Excel 395.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3.1 Graphing . . . . . . . . . . . . . . . . . . . . . . . . . 405.3.2 Displaying Lines . . . . . . . . . . . . . . . . . . . . . 45

5.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.4.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . 465.4.2 Investigation . . . . . . . . . . . . . . . . . . . . . . . 47

5.5 Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Graphs and Graphical Analysis 516.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2 Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.2.1 Data Tables . . . . . . . . . . . . . . . . . . . . . . . . 526.2.2 Parts of a Graph . . . . . . . . . . . . . . . . . . . . . 53

6.3 Graphical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 586.4 Linearizing Equations . . . . . . . . . . . . . . . . . . . . . . . 596.5 Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.6 Least Squares Fitting . . . . . . . . . . . . . . . . . . . . . . . 62

6.6.1 Correlation coefficient . . . . . . . . . . . . . . . . . . 636.7 Uncertainties in Graphical Quantities . . . . . . . . . . . . . . 63

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CONTENTS v

6.7.1 Small Scatter of Data . . . . . . . . . . . . . . . . . . . 64

6.7.2 Large Scatter of Data . . . . . . . . . . . . . . . . . . . 66

6.7.3 Sample Least Squares Calculations . . . . . . . . . . . 68

6.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7 Standing Waves in an Air Column 71

7.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.3.1 Longitudinal Waves . . . . . . . . . . . . . . . . . . . . 72

7.3.2 Motion of Air Near an Oscillator . . . . . . . . . . . . 75

7.3.3 Speed of Sound in Air . . . . . . . . . . . . . . . . . . 75

7.3.4 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7.4.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . 78

7.4.2 Experimentation . . . . . . . . . . . . . . . . . . . . . 79

7.4.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7.5 Bonus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.6 Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.7 Template . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8 Determining the Heat of Fusion of Ice 85

8.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

8.4.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . 88

8.4.2 Preparation . . . . . . . . . . . . . . . . . . . . . . . . 88

8.4.3 Experimentation . . . . . . . . . . . . . . . . . . . . . 89

8.4.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8.5 Bonus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

8.5.1 Choice of Initial Temperature . . . . . . . . . . . . . . 93

8.5.2 Heat Transfer Method . . . . . . . . . . . . . . . . . . 93

8.5.3 Thermometer Assumptions . . . . . . . . . . . . . . . . 93

8.5.4 Uniformity of Water Temperature . . . . . . . . . . . . 93

8.6 Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

8.7 Template . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

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vi CONTENTS

9 Young’s Modulus and Stretch Measurement by Optical Lever 999.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

9.3.1 The Optical Lever . . . . . . . . . . . . . . . . . . . . 1019.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

9.4.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 1049.4.2 Preparation . . . . . . . . . . . . . . . . . . . . . . . . 1059.4.3 Experimentation . . . . . . . . . . . . . . . . . . . . . 1069.4.4 Determining Young’s Modulus . . . . . . . . . . . . . . 1079.4.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 108

9.5 Bonus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089.6 Template . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

A Review of Uncertainty Calculations 113A.1 Review of uncertainty rules . . . . . . . . . . . . . . . . . . . 113

A.1.1 Repeated measurements . . . . . . . . . . . . . . . . . 113A.1.2 Rules for combining uncertainties . . . . . . . . . . . . 116

A.2 Discussion of Uncertainties . . . . . . . . . . . . . . . . . . . . 118A.2.1 Types of Errors . . . . . . . . . . . . . . . . . . . . . . 118A.2.2 Reducing Errors . . . . . . . . . . . . . . . . . . . . . . 118A.2.3 Ridiculous Errors . . . . . . . . . . . . . . . . . . . . . 118

Index 119

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List of Figures

3.1 Effects of absolute and relative references when copying . . . . 12

4.1 Non-linear equation . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 One linearization . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3 Another linearization . . . . . . . . . . . . . . . . . . . . . . . 27

5.1 Spreadsheet layout with series for x, y, and error bars . . . . . 42

5.2 Maximum and Minimum Slope Coordinates from a Point . . . 44

6.1 Wrong: Data, (not empty space), should fill most of the graph 54

6.2 Point with error bars . . . . . . . . . . . . . . . . . . . . . . . 56

6.3 Graph with unequal error bars in positive and negative directions 57

6.4 A line through the origin is not always the best fit . . . . . . . 58

6.5 Wrong: Graphs should not look like dot-to-dot drawings . . . 60

6.6 Wrong: Graphs should not have meaningless curves just to fitthe data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.7 Small Scatter of Data Points . . . . . . . . . . . . . . . . . . . 64

6.8 Maximum and Minimum Slope Coordinates from a Point . . . 65

6.9 Large Scatter of Data . . . . . . . . . . . . . . . . . . . . . . . 67

7.1 Motion of Air Molecule Near Vibrating Reed . . . . . . . . . . 72

7.2 Variation in Density of Air Molecules Due to Sound . . . . . . 73

7.3 Air Density Near the End of a Tuning Fork . . . . . . . . . . . 76

7.4 Node Pattern in a Tube With an Open End . . . . . . . . . . 77

8.1 A Simple Calorimeter . . . . . . . . . . . . . . . . . . . . . . . 86

8.2 A typical graph of temperature vs. time for the system . . . . 91

9.1 Cross-section of wire . . . . . . . . . . . . . . . . . . . . . . . 101

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viii LIST OF FIGURES

9.2 The Optical Lever . . . . . . . . . . . . . . . . . . . . . . . . . 1029.3 Mirror Stand . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039.4 Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039.5 Reading the Scale . . . . . . . . . . . . . . . . . . . . . . . . . 105

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List of Tables

4.1 λ Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 λ2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 ln (λ) Linearization . . . . . . . . . . . . . . . . . . . . . . . . 37

6.1 Position versus Time for Cart . . . . . . . . . . . . . . . . . . 536.2 Sample Least Squares Fit Data . . . . . . . . . . . . . . . . . 68

7.1 List of quantities . . . . . . . . . . . . . . . . . . . . . . . . . 827.2 Single value quantities . . . . . . . . . . . . . . . . . . . . . . 837.3 Calculated quantities . . . . . . . . . . . . . . . . . . . . . . . 837.4 Experimental factors responsible for effective uncertainties . . 847.5 Resonance data . . . . . . . . . . . . . . . . . . . . . . . . . . 84

8.1 List of quantities . . . . . . . . . . . . . . . . . . . . . . . . . 948.2 Single value quantities . . . . . . . . . . . . . . . . . . . . . . 958.3 Calculated quantities . . . . . . . . . . . . . . . . . . . . . . . 958.4 Experimental factors responsible for effective uncertainties . . 968.5 Timing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

9.1 Experimental Quantities . . . . . . . . . . . . . . . . . . . . . 1069.2 List of quantities . . . . . . . . . . . . . . . . . . . . . . . . . 1099.3 Single value quantities . . . . . . . . . . . . . . . . . . . . . . 1109.4 Calculated quantities . . . . . . . . . . . . . . . . . . . . . . . 1119.5 Experimental factors responsible for effective uncertainties . . 1119.6 Elongation data . . . . . . . . . . . . . . . . . . . . . . . . . . 112

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x LIST OF TABLES

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Chapter 1

Goals for PC142 Labs

The labs in PC142 will build on the skills developed in the labs for PC141.Emphasis will be on learning how to do experimental science, and on

how to communicate well which privide physics knowledge through practicalmeasurements, rather than on illustrating particular physical laws.

For these reasons, the goals of the labs in this course will fall into thesegeneral areas:

• introduce graphing and graphical analysis, which will include

– linearizing equations

– error bars on graphs

• expand abilities in presentation of results in the format of formal labreports ; in particular, the following sections:

– Introduction

– Methods

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2 Goals for PC142 Labs

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Chapter 2

Instructions for PC142 Labs

Students will be divided into sections, each of which will be supervisedby a lab Instructor and a Instructor Assistant. This lab Instructorshould be informed of any reason for absence, such as illness, as soon aspossible. (If the student knows of a potential absence in advance, thenthe lab supervisor should be informed in advance.) A student shouldprovide a doctor’s certificate for absence due to illness. Missed labswill normally have to be made up, and usually this will be scheduledas soon as possible after the lab which was missed while the equipmentis still set up for the experiment in question.

It is up to the student to read over any theory for each experiment andunderstand the procedures and do any required preparation before thelaboratory session begins. This may at times require more time outsidethe lab than the time spent in the lab.

You will be informed by the lab instructor of the location for submis-sion of your reports during your first laboratory period. This reportwill usually be graded and returned to you by the next session. TheInstructor Assistant who marked a particular lab will be identified, andany questions about marking should first be directed to that InstructorAssistant. Such questions must be directed to the Instructor Assistantwithin one week of the lab being returned to the student if any additionalmarks are requested.

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4 Instructions for PC142 Labs

2.1 Expectations

As a student in university, there are certain things expected of you.Some of them are as follows:

– You are expected to come to the lab prepared. This means first ofall that you will ensure that you have all of the information youneed to do the labs. After you have been told what lab you will bedoing, you should read it ahead and be clear on what it requires.You should bring the lab manual, lecture notes, etc. with youto every lab. (Of course you will be on time so you do not missimportant information and instructions.)

– You are expected to be organized This includes recording raw datawith sufficient information so that you can understand it, keep-ing proper backups of data, reports, etc., hanging on to previousreports, and so on. It also means starting work early so there isenough time to clarify points, write up your report and hand it inon time.

– You are expected to be adaptable and use common sense. In labs itis often necessary to change certain details (eg. component valuesor procedures) at lab time from what is written in the manual.You should be alert to changes, and think rationally about thosechanges and react accordingly.

– You are expected to value the time of instructors and lab demon-strators. This means that you make use of the lab time when itis scheduled, and try to make it as productive as possible. Thismeans NOT arriving late or leaving early and then seeking helpat other times for what you missed.

– You are expected to act on feedback from instructors, markers,etc. If you get something wrong, find out how to do it right anddo so.

– You are expected to use all of the resources at your disposal. Thisincludes everything in the lab manual, textbooks for other relatedcourses, the library, etc.

– You are expected to collect your own data. This means that youperform experiments with your partner and no one else. If, due

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2.2 Workload 5

to an emergency, you are forced to use someone else’s data. Oth-erwise, you are committing plagiarism.

– You are expected to do your own work. This means that you pre-pare your reports with no one else. If you ask someone else foradvice about something in the lab, make sure that anything youwrite down is based on your own understanding. If you are basi-cally regurgitating someone else’s ideas, even in your own words,you are committing plagiarism. (See the next point.)

– You are expected to understand your own report. If you discussideas with other people, even your partner, do not use those ideasin your report unless you have adopted them yourself. You areresponsible for all of the information in your report.

– You are expected to be professional about your work. This meansmeeting deadlines, understanding and meeting requirements forlabs, reports, etc. This means doing what should be done, ratherthan what you think you can get away with. This means proof-reading reports for spelling, grammar, etc. before handing themin.

– You are expected to actively participate in your own education.This means that in the lab, you do not leave tasks to your partnerbecause you do not understand them. This means that you try andlearn how and why to do something, rather than merely findingout the result of doing something.

2.2 Workload

Even though the labs are each only worth part of your course mark, theamount of work involved is probably disproportionately higher thanfor assignments, etc. Since most of the “hands-on” portion of youreducation will occur in the labs, this should not be surprising. (Note:skipping lectures or labs to study for tests is a very bad idea. Goodtime management is a much better idea.)

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2.3 Administration

1. Students are advised to have a binder to contain all lab manualsections (if using the printed manual) and all lab reports whichhave been returned. (A 3 hole punch will be in the lab.)

2. Templates will be used in each experiment as follows:

(a) The data in the template must be checked by the Lab Instruc-tor or Instructor Assistant before students leave the lab.

(b) No more than 3 people can use one set of data. If equipmentis tight groups will have to split up. (ie. Only as many peopleas fit the designated places for names on a template may usethe same data.)

(c) The template must be included with lab handed in. Due datefor lab reports is the beginning of the next lab.

(d) If a student misses a lab with valid excuse, a day will bescheduled for make up the lab.

3. Answers to pre-lab questions are to be brought to the lab. Theyare to be handed in at the beginning of the lab.

4. Answers to the in-lab questions are to be handed in at the end ofthe lab.

5. Students are to make notes about pre- and in-lab question answersand keep them in their binders so that the points raised can bediscussed in their reports. Marks for answers to questions will beseparate from marks for the lab. For people who have missed thelab without a doctor’s note and have not made up the lab, thesemarks will be forfeit. The points raised in the answers will still beexpected to be addressed in the lab report.

6. Post-lab questions are to be answered after calculations and em-bedded into the lab report.

7. In-lab tasks are to be performed during the lab, and must getchecked off before you leave the lab.

After the due date, the lab will not be accepted and graded.

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2.4 Plagiarism 7

2.4 Plagiarism

8. Plagiarism includes the following:

– Identical or nearly identical wording in any block of text.

– Identical formatting of lists, calculations, derivations, etc.which suggests a file was copied.

9. You will get 0 zero on the lab report when plagiarism is suspectedfor the first time. After this any suspected plagiarism will be for-warded directly to the course instructor and university plagirismcommittee.

10. If there is a suspected case of plagiarism involving a lab reportof yours, it does not matter whether yours is the original or thecopy. The sanctions are the same.

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Chapter 3

Exercise on Introduction toSpreadsheets

3.1 Purpose

The purpose of this exercise is to introduce you to the use of a spread-sheet as a tool for data analysis.

3.2 Introduction

A spreadsheet is basically a large programmable calculator with somespecial features thrown in. Because of this it is very useful for dataanalysis. It is important that you try to learn how to use the spread-sheet, rather than simply try to “get the work done”. A spreadsheet is ahandy tool, and these labs give you an opportunity to become familiarwith it, if you are not already. It will be well worth the effort to usea spreadsheet for calculations for the time saved and mistakes avoidedas you become more experienced with it. (Ask your lab TA!).

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10 Exercise on Introduction to Spreadsheets

3.3 Theory

Some of the information here will be specific to Microsoft Excel, butmost should be similar for other spreadsheets as well.

3.3.1 Formulas

The value in using cell references instead of numerical values is that ifthe numbers change, the calculations are performed automatically. Ina lab this allows you to set up the spreadsheet and then simply type innew data to see new results. If you do things properly, it is even easyto change the number of data points after the fact.

Never use a number where you can use a cell reference.

3.3.2 Functions

Functions perform commonly used tasks on a cell or block of cells andreturn the result in a cell.1 Some of the most common ones follow.

Common Functions

For blocks of cells

– average()

– count() counts the non-empty cells

Always use the count function instead of typing in the number of data pointsso you can change the number later if needed without editing formulas.

– stdeva() determines the sample standard deviation

– max() finds the maximum

– min() finds the minimum

1There are functions which return a block of cells, which don’t fit with the mathematicalnotion of a “function”, but we won’t get into those here.

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3.3 Theory 11

For individual values

– sqrt() square root

– abs() absolute value

– if() allows a cell to have a value depending on a condition. Forexample, the formula =IF(A1¿¿10,”Over 10”,”10 or less”) returns”Over 10” if A1 is greater than 10, and ”10 or less” if A1 is lessthan or equal to 10

Using the Built-in Automation

In you aren’t sure what parameters are needed by a function, or in whatorder, you can simply type in the function until after the left bracketand then you will be prompted for what needs to be filled in.

3.3.3 Copying Formulas and Functions: Absoluteand Relative References

Never copy values from one part of the spreadsheet to another where a cellreference could be used instead.

Relative References

The value of using formulas in a spreadsheet is that calculations can beeasily repeated. One of the ways this happens is by copying formulas tobe applied to different data. When a formula is copied from one locationto another, references are usually changed relative to the move. In otherwords, if a formula is copied from one cell to the next one on the right(ie. where the column address increases by one), then cell referenceswithin the formula will have their column addresses increased by one aswell. If a formula is copied from one cell to the one above (ie. where therow address decreases by one), then cell references within the formulawill have their row addresses decreased by one as well.2

2Note that when you move a cell, none of the references inside it are changed.

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Figure 3.1 gives an example of both absolute and relative referencesand how they affect copying. If the formula given was in the colouredcell, and then copied to the cell shown, the references in the formulawould become the ones indicated.

A2

=average(A2:A5)HHHj

D4

=average(D4:D7)B10

=average($B10:B$13) PPPq

E11

=average($B11:E$13)

Figure 3.1: Effects of absolute and relative references when copying

Absolute References

Sometimes you don’t want cell references in a formula to change whenit is copied. In this case what you do is to put a $ before the part of acell reference which you do not want to change. in other words, if youhave a formula with a reference to B13 and you do not want the Bto change when you copy, then change the reference to $B13. If youdon’t want the 13 to change when you copy, then change the referenceto B$13. If you don’t want either B or 13 to change, then change thereference to $B$13. (In other words, you make a number a constantby referring to it like $B$13).

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3.3 Theory 13

3.3.4 Pasting Options

Sometimes when you copy a formula, you don’t want to copy everythingabout it. For instance, sometimes you just want to get the numericalresult. In a spreadsheet, you can’t choose what to copy, but you canchoose what to paste. When you go to paste something, you can PasteSpecial and choose to paste whatever you want. Thus you can pastestrings or numbers and not formulas, and you can include formatting(boxes, colours, fonts, etc.) or not.

Always copy formulas from one part of the spreadsheet to another instead ofretyping to avoid making mistakes.

3.3.5 Formatting

You can do many things to a cell or block of cells to change its appear-ance, such as putting boxes around it, changing fonts or how numbersare displayed, changing the background colour, etc.

Format after you’re done copying formulas, etc., to avoid having to alwaysPaste Special to preserve style information.

3.3.6 Print Preview

When printing a spreadsheet, there are lots of options to make theresult more readable. Doing a print preview allows this. You can

– turn the grid off.

– include page headers and footers (or not)

– scale to fit the page

Doing a page preview also helps you figure out which page number(s)you need to print, since it may not be obvious.

Preview the printing to avoid wasting reams of paper printing out stuff youdon’t want.

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3.4 Procedure

3.4.1 Preparation

You are welcome to go ahead and do as much of the exercise on yourown as you wish; you can just bring your spreadsheet with you to thelab and demonstrate the points indicated. If you get it all done inadvance, that’s great.

If you do it on your own, then print a copy of the spreadsheet showingformulas along with the ones indicated in the post-lab questions. (Youonly need to show the formulas for rows mentioned in the instructions.)

Pre-lab Tasks

PT1: Verify that you can log in to a computer in one of the publiclabs on campus with your Novell login before you come to the lab.

3.4.2 Investigation

This exercise can be done outside the lab. Alternative instructions, whereneeded, are boxed like this. You may do that if you wish and bring theanswers to the in-lab questions and the completed in-lab tasks to the lab.

In-lab Tasks

In this exercise, the in-lab tasks are included with each part.

IT1:

Creating the First Formulas

1. Cell B14 should containg the average for the steel ball. In thatcell, begin by typing a “=” so Excel knows you’re typing a for-mula. The function you want to use is average(). Once you

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3.4 Procedure 15

type the left bracket, the function will prompt you for parame-ters.Select the range from B7 to B11, and then type the rightbracket to finish the formula.

2. Cell B15 should contain the standard deviation. Follow the sameprocedure as above, using the function stdeva().

3. Cell B16 going to contain the standard deviation of the mean,so you’ll need to use a formula which takes the standard devia-tion from B15 and divides it by the square root of the numberof measurements. The two functions you’ll need are sqrt() andcount(). Do not type in a value for the number of data points,and do not cut and paste the value of the standard deviation. Youwant to use cell references so if the values change the results willstill be correct.

4. Cell B17 is going to have the uncertainty in the average, which isthe bigger of the standard deviation of the mean and the precisionmeasure. Use the max() function for this.

IT2:

Copying the Formulas

1. Modify the formulas so that they can be copied for the ping pongball. One reference from the formulas in B14 to B17 must bemade absolute in order for them to copy correctly. Change thatone reference.

2. Copy the formulas from B14 to B17, and then use Paste Specialto paste them in C14 to C17. (Copy formulas only). See thatthe results are correct. If not, check to see if any references shouldhave been copied differently. If so, make the changes to the steelball formulas and repeat this step.

3. Notice the block for the brass series has one less value. If you haveset up the formulas correctly, you should be able to copy them forthis ball and the results will be correct. Try this and make changesto the original if necessary so that the copied formulas are correct.

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IT3:

Adapting the Formulas

1. The block for the wood series has one more value. If you copythe formulas, you will notice they do not include the last value.If you want to avoid having to correct the formulas, here’s whatyou can do: Right click on the last value for the steel ball (inB11). Choose Insert so that you push cells down. Now you’ll seethat the formulas have moved down one cell, but they point tothe range that ends in B12, as they should. Now you can copythe formulas over to the column containing the wood values andthe formulas will be correct. Move them up to the correct rowafter that. You can then go back and delete the cell you insertedin B11. This should make the result correct. Try this and see.

IT4:

Comparing Uncertainties

1. The times for the steel and ping pong balls can be compared using

|a− b| ≤ ∆a+ ∆b

(Hint: You can use the if() function for this.)

Save the spreadsheet with the fuctions in separate lines at theend and send it to your TA

3.4.3 Follow-up

Post-lab Tasks

T1: Print two versions of the spreadsheet; one without gridlines or rowand column labels, and one with both gridlines and row and columnlabels.

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3.5 Recap 17

3.5 Recap

By the end of this exercise, you should know how to use spreadsheetfunctions to calculate the :

– mean

– standard deviation

– standard deviation of the mean

3.6 Summary

Item Number Received weight (%)Pre-lab Questions 0 0In-lab Questions 0 0Post-lab Questions 0 0

Pre-lab Tasks 1 10In-lab Tasks 4 90Post-lab Tasks 1 0

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Chapter 4

Exercise on LinearizingEquations

4.1 Purpose

The purpose of the exercise is to develop skills in producing lineargraphs from various types of data and extracting results.

4.2 Introduction

This exercise will develop skills in linearizing data, so that a variety ofrelationships can be graphed as straight lines.

4.3 Theory

Often, the point of a scientific experiment is to try and find empiricalvalues for one or more physical quantities, given measurements of someother quantities and some mathematical relationship between them.For instance, given a marble has a mass of 5 g, and a radius of 0.7 cm,the density of the marble can be calculated given that v = 4/3πr3 andρ = m/v. (For the sake of simplicity, uncertainties will be ignored fornow, although the calculation of those should be familiar by now.)

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20 Exercise on Linearizing Equations

m

r

Figure 4.1: Non-linear equation

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4.3 Theory 21

Many times, however, rather than having one measurement of a quan-tity, or set of quantities, we may have several measurements whichshould all follow the same relationships, (such as if we had several mar-bles made of the same material in the example above), and we wish tocombine the results. The usual way of combining results is to createa graph, and extract information (such as the density) from the slopeand y–intercept of the graph.

One may be tempted to ask why a graph should be better than merelyaveraging all of the data points. The answer is that an average iscompletely unbiased. The variation of any one point from the normis no more or less important than the variation of any other point. Agraph, however, will show any point which differs significantly fromthe general trend. Analysis of the graphical data (such as with a leastsquares fit) will allow such “outliers” to be given either more or lessweight than the rest of the data as the researcher deems appropriate.Depending on the situation, the researcher may wish to verify anyodd point(s), or perhaps the trend will indicate that a linear model isinsufficient. In any case, it is this added interpretive value that a graphhas which makes it preferable.

A plot is better than an average since it may indicate systematic errors inthe data.

The value in fitting the data to an equation is that once the fit has beendone, rather than continuing to work with a large amount of data, wecan simply work with the parameters of our fit and their uncertainties.In the case of a straight line, all of our data can be replaced by fourquantities; m,∆m, b and ∆b.

A fit equation replaces a bunch of data with a few parameters.

The reason a linear graph is so useful is that it’s easier to identifywhether a line is straight than it is to identify whether it looks morelike y = x2 or y = x3, for instance.

A straight line is easy to spot with the unaided eye.

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If the data fits an equation of the form y = mx + b, then it is easy toplot a straight line graph and interpret the slope and y–intercept, but itis rarely that simple. In most cases, the equation must be modified orlinearized so that the variables plotted are different than the variablesmeasured but produce a straight line.

Linearizing equations is this process of modifying an equation toproduce new variables which can be plotted to produce a straight linegraph. In many of your labs, this has been done already.

Look again at y = mx + b. Note that y and x are variables, (as eachcan take on a range of values), while m and b are constants, (as thereis only one value for each for all of the data points). We can linearizean equation if we can get it in the form

variable1 = constant1 × variable2 + constant2

There are a few things to note:

1. Several constants combined together produces another single con-stant.

2. Powers or functions of constants are also constants.

3. Constants may have “special” values of 0 or 1 so they appear“invisible”. For example

y = mx

is still the equation of a straight line, where b = 0. As well,

y = b

is the equation of a line where m = 0.

4. Variables may be combined together to form new variables.

5. Powers or functions of variables are also variables.

Note that linearizing an equation will produce expressions for the slope andy–intercept which depend only on the constants in the original equation, noton the original x and y variables. This means that the constants can berelated to the slope and y–intercept rather than the original variables.

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4.3 Theory 23

4.3.1 Techniques for Linearization

If a relationship involves only multiplication and division, (includingpowers), then logarithms can be used to linearize. Sometimes takingroots or powers of both sides of an equation will help.

4.3.2 Procedure for Linearization

The steps are as follows:

1. Rearrange the equation to get one variable (or a function of it) onthe left side of the equation; this becomes your y variable.

2. Regroup the right side of the equation to create a term containingthe other variable (or some function of it).

3. Use the left-side variable (or the function of it) as your x vari-able, and then your slope should be whatever multiplies it; youry intercept is whatever additive term is left over.

Note: It is important to realize that you don’t need to understandan equation to linearize it; all you have to know is which parametersare variables (ie. things you have data for), and which parameters areconstants (ie. things you want to calculate). Of course different exper-iments involving the same relationship may make different parametersvariable, and so how an equation is linearized will depend on the dataused. To again consider the above example: The original equationswere

v = (4/3)πr3 (4.1)

andρ = m/v (4.2)

where the quantities m and r are measured. (ie. We have severalmarbles of the same material, so we can get several measurements ofm and r, but we expect ρ to be the same for all of them.) Thus for thissituation, m and r are variables, and ρ is a constant. We can combinethe two equations to get

ρ =m

(4/3)πr3(4.3)

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or

ρ =3m

4πr3(4.4)

This equation has a constant on one side, and a mixture of variables andconstants on the other. First we should rearrange it to get a variableon the left hand side. Suppose we rearrange the equation, giving

m = (4/3)πρr3 (4.5)

This leaves a variable on the left. From this point on, there are twomain possibilities for how to proceed: 1

Method I

Now we can create a new variable, Y such that

Y = m

By the rule about powers of variables being variables, then we cancreate a new variable X given by

X = r3

Then equation 4.5 above becomes

Y = (4/3)πρX (4.6)

since π is a constant, and ρ should be, and using the rule that combina-tions of constants produce constants, then we can define M , a constant,(not the same as m), as

M = (4/3)πρ

so equation 4.6 becomes

Y = MX + 0

which is the equation of a straight line. (In the case, B, the y–interceptis zero.)2 So if we plot our “modified” variables, we should get a straight

1Usually the process is not as explicit as this. ie. one doesn’t usually create an X anda Y , but doing this illustrates the procedure.

2Occasionally we can get a situation where the slope is similarly “invisible”, if it is 1or 0.

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4.3 Theory 25

m

r3

Figure 4.2: One linearization

line, passing through the origin with a slope M . How can we get ρ fromthe graph? Well, from above

M = (4/3)πρ

so

ρ =3M

4π

where M is the slope of the graph.

Method II

We can take logarithms of both sides, so that Y such that equation 4.5above becomes

lnm = ln ((4/3)πρ) + ln r3 (4.7)

grouping the terms so one only contains constants (and so the combi-nation should be constant) and one only contains the variable r. We

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can bring down the exponent so equation 4.7 becomes

lnm = ln ((4/3)πρ) + 3 ln r

Now we can create new variables, Y such that

Y = lnm

andX = ln r

which is the equation of a straight line. So if we plot our “modified”variables, we should get a straight line. How can we get ρ from thegraph? Well, from above

B = ln ((4/3)πρ)

so

ρ =3

4πeB

where B is the y-intercept of the graph. (In this case, the value you getfrom the graph for the slope should suggest whether the fit is a goodone.)

Remember that after linearization, our results depend on our graphical quan-tities of the slope and the y-intercept, rather than on the original measuredquantities.

4.3.3 Choosing a Particular Linearization

Often there may be more than one linear form for the equation so theremay be more than one “right answer”. In this case, there are a fewthings which may help you choose.

Simple variables

A preferable linearization is one which most simplifies understandingthe graph or interpreting the results. For instance, in the above ex-ample, it would have been possible to use (4/3)πr3 instead of r3 asour x variable, but that would make confusing axis scales and/or units(although it would have made the slope be ρ with no calculation).

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4.3 Theory 27

ln (m)

ln (r)

Figure 4.3: Another linearization

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Spread of data

The spread of data will be different for each linearization. A graphwith points which are more equally spaced is generally preferable toone where the points are concentrated in one area.

Size of error bars

Like the spread of data, the size of the error bars will be differentfor each linearization. A graph with more equally sized error bars isgenerally preferable to one where the error bars vary greatly in size fordifferent points.

Usually it is preferable to separate variables and constants as much as possiblein your linearization so that graph variables are easily related to experimentalones.

4.3.4 Uncertainties in Results

After determining how equation parameters relate to graphical quanti-ties as above, uncertainties can be determined as usual. In the aboveexample Method I gives

∆ρ =3∆M

4π

while for Method II

∆ρ =3

4πeB∆B

or∆ρ = ρ∆B

4.4 Procedure

4.4.1 Preparation

You are welcome to go ahead and do as much of the exercise on yourown as you wish; you can just bring a sheet with your question answers

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4.4 Procedure 29

to the lab. If you get it all done in advance, that’s great.

Pre-lab Questions

PQ1: Given that F = mg and µ = M`

, write out the equations for ∆Fand ∆µ, given that m, M , and ` have associated uncertainties ∆m,∆M , and ∆` and g is a constant with no given uncertainty.

4.4.2 Investigation

The fundamental wavelength of vibration of a string is given by

λ =1

ν

√F

µ

λ and F , are measured variables and µ is a constant, with an uncer-tainty ∆µ. Determine ν and ∆ν.

There are 3 different ways of doing this. In each case, show whatshould be plotted, how the error bars for your x (independent) and y(dependent) variables are determined, what the slope and y-interceptwill be, and how ν and ∆ν come from the slope and y-intercept andtheir uncertainties.

For instance, from the example above, using Method I we would say:

– Plot m vs. r3. (In other words, the independent (x) variable is r3

and dependent (y) variable is m.)

– The uncertainty in the dependent variable is ∆m.

– The uncertainty in the independent variable is 3r2∆r.

– The slope of the graph will be M = (4/3)πρ.

– The y-intercept should be zero3.

3If the y-intercept turns out to be something other than zero, then there is somesystematic error in our experiment.

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– The density will be determined from the slope by the equationρ = 3M

4π.

– The uncertainty in the density will be determined from the slopeby the equation ∆ρ = 3∆M

4π.

In-lab Questions

For each of the linearizations below, be sure to address all 7 points asin the example above.

IQ0: What is the linearization if our dependent variable is λ? (See theexample above.)

Answer: First we have to rearrange the equation

ν =1

λ

√F

µ

to get the dependent variable on the left, as follows:

λ =1

ν

√F

µ

Then we need to regroup the right side to separate the independentvariable4.

λ =1

ν√µ

√F

So the slope is the part that is multiplied by the independent variable,

slope =1

ν√µ

and the uncertainties in each of the quantities are calculated as usual.(Assume µ and ν have uncertainties ∆µ and ∆ν, and after the leastsquares fit, the slope and y-intercept will simply be quantities m and bwith associated uncertainties ∆m and ∆b.)

4There is another possibility here; see if you can figure out what it is.

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4.4 Procedure 31

IQ1: What is the linearization if our dependent variable is λ2? (Seethe example above.)

IQ2: What is the linearization if our dependent variable is ln (λ)? (Seethe example above.)

IQ3: Without considering the data, which of the above choices wouldyou prefer and why? (In other words, is there a choice that seemsbetter to work with, regardless of how the data fit?)

In-lab Tasks

IT1: Fill in the headings for the columns in Tables 4.1,4.2, and 4.3.

IT2: Fill in the template with the equations, uncertainties, and unitsfor the slope and y-intercept for each linearization.

4.4.3 Analysis

Post-lab Tasks

T1: Open the spreadsheet for “Exercise on Graphing and Least SquaresFitting in Excel” from the web page. On the last three tabbed pages,insert rows as needed and type in your linearized data for each of the 3linearizations. Save the spreadsheet on a floppy, a USB memory stick,or in your Novell account to bring to the lab for that exercise.

Post-lab Discussion Questions

(These are to be done after completing the “Exercise on Graphing andLeast Squares Fitting in Excel” .

Q1: Considering the data, (ie. looking at the plots for each of thelinearizations), which of the above choices did you choose and why?

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4.5 Bonus

You can do one of the questions below. In each of the following ques-tions, state the modified variables to be plotted, and state how theunknown(s) may be determined from the graph.

1. The position of a body starting from rest and subject to uniformacceleration is described by

s =1

2at2

s and t are measured variables. Determine a and ∆a.

2. The fundamental frequency of vibration of a string is given by

ν =1

2l

√T

m

ν, l, and T are measured variables. Determine m and ∆m. (Hint:in this case one of your “modified variables” may incorporate twoof the measured variables.)

3. The period of oscillation of a simple pendulum is

T = 2π

√L

g

where L and T are measured variables. Determine g and ∆g.

4. The Doppler shift of frequency for a moving source is given by

f = f0v

v − v0

f and v0 are measured variables, f0 is fixed and known. Determinev and ∆v. (Note that even though the notation might suggestotherwise, in this case v0 is a variable, not a constant.)

5. The impedance of a series RC circuit is

Z =

√R2 +

1

ω2C2

Z and ω are measured variables. Determine R , ∆R, C, and ∆C.

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4.6 Recap 33

6. The conductivity of an intrinsic semiconductor is given by

σ = Ce−Eg2kT

σ and T are measured variables and k is a known constant. De-termine Eg, ∆Eg, C, and ∆C.

7. The relativistic variation of mass with velocity is

m =m0√1− v2

c2

m and v are measured variables. Determine m0, ∆m0, c, and ∆c.

8. The refraction equation is

µ1 sin θ1 = µ2 sin θ2

θ1 and θ2 are measured variables; µ1 is fixed and known. Deter-mine µ2 and ∆µ2.

4.6 Recap

By the end of this exercise, you should understand the following terms:

– linear graph

– linearized equation

4.7 Summary



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4.8 Template

The distance is measured with:The units of distance are:The smallest division is:The precision measure is:

The distance L is :

independent dependent√F λ

1

2

3

4

5

Table 4.1: λ Linearization

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4.8 Template 35

For this linearization,The equation for the uncertainty in the independent variable is:∆x = ∆λThe equation for the uncertainty in the dependent variable is:

∆y = ∆√F ≈ 1

2√F

∆F (using algebra)

The equation for the slope is: slope = 1ν√µ

The equation for the y-intercept is: y − intercept = 0

The units for the slope are: mN−1/2

The units for the y-intercept are: m

The equation for the frequency, ν, is: ν = 1slope

√µ

The equation for the uncertainty in the frequency, ∆ν, is:

∆ν = ∆(

1slope

√µ

)≈ 1

(slope−∆(slope))(√µ−∆µ)

− 1slope

√µ

(using inspection, given that the slope and µ should be positive)

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independent dependentλ2

1

2

3

4

5

Table 4.2: λ2 Linearization

For this linearization,The equation for the uncertainty in the independent variable is:The equation for the uncertainty in the dependent variable is:

The equation for the slope is:The equation for the y-intercept is:

The units for the slope are:The units for the y-intercept are:

The equation for the frequency, ν, is:The equation for the uncertainty in the frequency, ∆ν, is:

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4.8 Template 37

independent dependentln (λ)

1

2

3

4

5

Table 4.3: ln (λ) Linearization

For this linearization,The equation for the uncertainty in the independent variable is:The equation for the uncertainty in the dependent variable is:

The equation for the slope is:The equation for the y-intercept is:

The units for the slope are:The units for the y-intercept are:

The equation for the frequency, ν, is:The equation for the uncertainty in the frequency, ∆ν, is:

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Chapter 5

Exercise on Graphing andLeast Squares Fitting inExcel

5.1 Purpose

The purpose of this experiment is to become familiar with using Excelto produce graphs and analyze graphical data.

5.2 Introduction

While Excel has a lot of useful features for data analysis, etc. it wasdesigned primarily for use by people in business rather than for scien-tists, and so there are some things which scientists wish to do whichrequire a little effort in Excel. Graphical analysis is one of the areaswhere this is true, as you shall see. (Even with the extra work, it’s alot more convenient than doing it all by hand.)

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40 Exercise on Graphing and Least Squares Fitting in Excel

5.3 Theory

5.3.1 Graphing

Graph Type

The graph type mainly used by scientists is an x-y graph1. Do notchoose a line graph!

Colour

The default grey background in many spreadsheets just looks bad ingraphs; it obscures the data and serves no purpose. Turn the back-ground colour off!

Gridlines

Grid lines should be either removed or in both dimensions. Gridlinesin one direction only look odd on an “xy” graph. Turn the gridlinesoff!

Text

The main text of a graph consists of x− and y− titles, a main title andperhaps a sub–title. All of these may be set in Excel.

Series

Excel allows you to plot several different “series” of (x, y) data. Eachseries can be customized, with choices for many things, including thefollowing:

1 As long as there is some mathematical relationship between the variables, then anx-y graph illustrates the relationship. However, if the independent variable does not havea numerical value, then this doesn’t apply. For instance, if you were graphing reactiontime for men and women, then a bar graph would be the logical choice, since there’s nonumerical relationship between “men” and “women”.

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5.3 Theory 41

– Patterns

Each series can be plotted with lines, symbols, or both.

Do not connect the points like a dot-to-dot drawing!

Do not use an arbitrary function just because it goes through all the datapoints!

– Markers

There are many possible symbols which can be used for each series.

– Lines

There are several line types available for each series.One importantfact about how lines are used to connect points in a series; allpoints in a series are joined by lines, unless the line for that seriesis turned off.

In science, it is almost always wrong to have a dot-to-dot drawing. It isalso wrong to have a curve which has no mathematical significance. For thisreason, data points should not be connected by either line segments or acurve like a polynomial which is made to pass through each data point. Theonly line or curve which should be shown is the result of a fit which is basedon some theoretical mathematical relationship.

Matching up x and y Values

When you create an xy graph is created in Excel, you don’t input datavalues as (x, y) pairs. Instead you select series for each of x and y. Theway the individual x and y values are associated is by where they occurin their respective series.

In Figure 5.1 you can see that the 5th point in each series is highlighted.Even though the series all start in different rows and columns, sincethe number of cells in each is the same, corresponding values can beconsidered to be related. (If a cell is blank, then the correspondingpoint or error bar will not be plotted.)

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A2

x1

D4

∆x1

B10

y1 E11

∆y1

Figure 5.1: Spreadsheet layout with series for x, y, and error bars

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5.3 Theory 43

Error Bars

In Excel, when you choose custom error bars, you can choose seriesfor both x and y, and even potentially different series for the + and −directions.

You don’t necessarily have to put markers on the ends of the error bars; thevalue in doing so is to make it clear that you’re not just using “+” symbolsfor plotting the data points. Also, if you are including a grid on your graph,error bars without markers on the end may be hard to distinguish. Howeverif the error bars will be clearly identifiable without markers, you don’t needto use them.

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For negative slope

For positive slope

( , )

( , )

( , )( , )

Figure 5.2: Maximum and Minimum Slope Coordinates from a Point

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5.3 Theory 45

Determining uncertainties in the slope and y-intercept

Case I: Maximum and minimum slopes If the error bars arelarge enough, then the line of best fit should go through all of theerror bars. In this case, there will be two data points which determinecoordinates for a line of maximum slope which crosses all of the errorbars. Consider the case for positive slope:

If we label two points x1 and x2, where x1 < x2, then we can see fromFigure 5.2 that the steepest line which touches the error bars for both x1

and x2 is the line between (x1 +∆x1,y1−∆y1) and (x2−∆x2,y2 +∆y2).The slope of this line will then be

mmax =(y2 + ∆y2)− (y1 −∆y1)

(x2 −∆x2)− (x1 + ∆x1)

and then the y-intercept is given by

bmin = (y1 −∆y1)−mmax(x1 + ∆x1) = (y2 + ∆y2)−mmax(x2 −∆x2)

Similarly the line with the least slope which touches the error bars forboth x1 and x2 is the line between (x1 − ∆x1,y1 + ∆y1) and (x2 +∆x2,y2 −∆y2). The slope of this line will then be

mmin =(y2 −∆y2)− (y1 + ∆y1)

(x2 + ∆x2)− (x1 −∆x1)


bmax = (y1 + ∆y1)−mmin(x1 −∆x1) = (y2 −∆y2)−mmin(x2 + ∆x2)

The case for a negative slope is shown in Figure 5.2; the analysis is leftto the student.

The points for the maximum and minimum slope will not always be the end-points on the graph.

5.3.2 Displaying Lines

Unless you are going to give the equation of a line or curve, do not show iton a graph!

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Plotting arbitrary lines

To display a line on the graph, such as a best fit line, one can use aseries which has not yet been used. When one knows the equation of aline, all one needs is two endpoints so that a line can be drawn betweenthem. To allow this, include 2 values at the end of your x series, xminand xmax which are the minimum and maximum values from the x data,respectively. Placing the y values calculated from the line equation inthe corresponding cells of another series will allow a line to be plottedbetween those points. (Set the format for that series to lines only.)

Using “trendline”

There is a built-in feature called trendline which allows you to displayvarious fits to data. A linear trendline is, in fact, a least squares fit.Unfortunately, this feature does not automatically display the param-eters for the fit, so it’s not as much use as it could be.

5.4 Procedure

5.4.1 Preparation

You are welcome to go ahead and do as much of the exercise on yourown as you wish; you can just bring your spreadsheet with you to thelab and demonstrate the points indicated. If you get it all done inadvance, that’s great.

If you do it on your own, then print a copy of the spreadsheet showingformulas along with the ones indicated in the post-lab questions. (Youonly need to show the formulas for rows mentioned in the instructions.)Print the graphs as well.

Pre-lab Questions

PQ1: Rewrite the Equations 6.7, 6.8, 6.9, 6.10 for mmax, bmin, mmin,bmax for a line with negative slope.

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5.4 Procedure 47

Pre-lab Tasks

PT1: Fill in the co-ordinates from PQ1 in Figure 5.2.

5.4.2 Investigation

In-lab Tasks

In this exercise the in-lab tasks appear throughout the section.

Part 1: Plotting a Graph

IT1:

Setting up the spreadsheet Use the data which is already in thespreadsheet. Only change to your own data after the formulas have allbeen set up correctly.

1. Load the graph from the lab page.

2. Insert a chart in the box on the first tabbed page.

– Make sure it is an scatter with only markers.

– Select data option for C6 to C9.

– Repeat for x series, using E6 to E9.

3. Click on the “Layout” tab. You should now see the options forerror bars.

– Select more error bar options.

– Select vertical bars, and pick Custom.

– Select series D6 to D9 for both + and -.

– Repeat for y error bars, using F6 to F9.

At this point there should not be a line connecting the data points. If thereis, turn it off.

IT2:

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1. Go to the second tabbed page (FitByFormulas), and replace num-ber cells for everything other than data values with formulas.(Make sure to use a function for N .)

2. Use the values of xmin and xmax in C31 and C32 and the equationof the line of best fit to calculate corresponding y values in F31and F32 Add this series to your graph to show the line of best fit.Right click on one of the points, and format the data series. Youwant to get a line with no endpoints.

IT3:

Part 2: Performing a Least Squares Fit Using Built-in Fea-tures

Here you’re going to identify the values produced by Excel’s LINEST()function by comparing them to the values you got from the formulas.

1. Go to the third tabbed page, and use the LINEST() function todo a least squares fit.

– Highlight C18 to D22

– Put the function in C18, with constant=1 and stats=1.

– Press 〈F2〉 followed by 〈CTRL〉〈SHIFT〉〈ENTER〉.– Fill in B18 to B22 and E18 to E22 with names from the

previous page. (You don’t have to fill in the others.)2

2One of the quantities is the number of degrees of freedom mentioned earlier. Itshould be easy to identify. The two “extra” quantities produced are the Regression Sumof Squares given by

SSR =(∑

(yi − y)2)− S

which can be shown to be given by

SSR =(∑

yi2)−Ny2 − S

and F given by

F =SSRνR

Sν

which you may find out about in a statistics class when discussing Analysis of Variance.

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5.5 Recap 49

2. Select data series as before, and choose linear trendline. See thatit fits on top of best fit line from before, proving it is a least squaresfit.

IT4:

Part 3: Finding Maximum and Minimum Slopes

1. Go to fourth tabbed page (LargeScatter), and put in referenceto third page (ie. page using LINEST() ) to create meaningfultables.

Save the spreadsheet with the fuctions in separate linesat the end and send it to your TA

5.5 Recap

By the end of this exercise, you should understand the followingterms:

– linear graph

– error bars

– least squares fit

– correlation coefficient

– large scatter of data points

– small scatter of data points

In addition, you should, using Excel, be able to:

– plot a linear graph

– add error bars

– perform a least squares fit

– show the least squares fit line on the graph with the data

You should also be able to

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– determine whether the points on a graph classify as either“small” or “large” scatter, and calculate graphical uncertain-ties appropriately in either case;

– compare different linearizations of the same function and toexplain why one may be preferred over others.

5.6 Summary



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Chapter 6

Graphs and GraphicalAnalysis

6.1 Introduction

One of the purposes of a scientific report is to present numeri-cal information, ie. data and calculated results, in concise andmeaningful ways. As with other parts of the report, the goal is tomake the report as self-explanatory as possible. Ideally a personunfamiliar with the experiment should be able to understand thereport without having to read the lab manual. (In your case, thereader can be assumed to be familiar with the general procedureof the experiment, but should not be expected to be intimatelyfamiliar with experiment-specific symbols. For instance, if youmust measure the diameter of an object in the lab, and use thesymbol d for it, be sure to state what d represents the first timeit is used.)

A physical law is a mathematical relationship between measur-able quantities, as has been stated earlier. A graph is a visualrepresentation of such a relationship. In other words, a graph isalways a representation of a particular mathematical relationshipbetween the variables on the two axes; usually these relationshipsare made to be functions.

As a representation of how data are related, a graph will usuallycontain both data points and a fitted curve showing the function

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52 Graphs and Graphical Analysis

which the data should follow. (The term “curve” may include astraight line. In fact, it is often easiest to interpret results whenan equation has been linearized so that the graph should be astraight line. Linearization will be discussed in Chapter 4, “Exer-cise on Linearizing Equations” .)

With single values which are measured or calculated, when thereis an “expected” value, then uncertainties are used to determinehow well the experimental value matches the expected value. Fora set of data which should fit an equation, it is necessary to seehow all points match the function. This is done using error bars,which will be discussed later. In essence, error bars allow one toobserve how well each data point fits the curve or line on thegraph. When parameters of an equation, such as the slope and y-intercept of a straight line, are determined from the data, (as willbe discussed later), then those parameters will have uncertaintieswhich represent the range of values needed to make all of the datapoints fit the curve.

6.2 Graphing

6.2.1 Data Tables

Often the data which is collected in an experiment is in a differentform than that which must be plotted on a graph. (For instance,masses are measured but a graph requires weights.) In this case,the data which is to be plotted should be in a data table of itsown. This is to make it easy for a reader to compare each pointin the data table with its corresponding point on the graph. Thedata table should include the size of error bars for each point, ineach dimension. Units in the table should be the same as on thegraph.

Any graph must be plotted from data, which should be presentedin tables. Tables should

– have ruled lines outside and separating columns, etc. to makeit neat and easy to read

– have meaningful title and column headings

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6.2 Graphing 53

i xi ∆xi ti ∆ti(cm) (cm) (s) (s)

1 0.40 0.03 0.0 0.12 0.77 0.04 2.0 0.13 1.35 0.04 2.7 0.1

Table 6.1: Position versus Time for Cart

– not be split up by page breaks (ie. unless a table is biggerthan a single page, it should all fit on one page.)

– have a number associated with it (such as “Table 1”) for ref-erence elsewhere in the report, and a name, (such as “SteelBall Rolling down Incline”) which makes it self–explanatory

– include the information required for any numerical data, ie.units, uncertainties, etc.

A sample is shown in Table 6.1.

6.2.2 Parts of a Graph

(a) TitleThe title of a graph should make the graph somewhat self–explanatory aside from the lab. Something like “y vs. x” maybe correct but redundant and useless if the person viewing thegraph can read. “Object in Free Fall” would be more helpfulas the reader may be able to figure out the significance of thegraph herself.

(b) Axis LabelsAs above, “m” and “l” are not as useful as “added mass (m)in grams”, and “length of spring (l) in cm”. In this casethe words are meaningful, while the symbols are still shownto make it easy to find them in equations. Units must beincluded with axis labels.

(c) Axis ScalesThe following 3 points are pertinent if you are plotting graphs“by hand”. If you use a spreadsheet, these things are usuallytaken care of automatically.

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Figure 6.1: Wrong: Data, (not empty space), should fill most of the graph

i. Always choose the scales of the axes so that the datapoints will be spread out over as much of the plottingarea as possible.

ii. Choose the scales in a convenient manner. Scales thatare easy to work with are to be preferred over scales suchas ones where every small division corresponds to 0.3967volts, for example. A better choice in such a case wouldbe either 0.25, 0.50, or perhaps even 0.4 volts per division,the decision of which would be determined by the previousconstraint. If you have discrete, i.e. integer, values onone axis, do not use scientific notation to represent thosevalues.

iii. (0,0) does not have to be on graph – data should covermore than 1/4 of the graph area; if you need to extrapo-late, do it numerically.

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6.2 Graphing 55

(d) Plotting PointsOften, results obtained from graphs are slightly suspicious dueto the simple fact that the experimenter has incorrectly plot-ted data points. If plotting by hand, be careful about this.Data points must be fitted with error bars to show uncertain-ties present in the data values. If the uncertainties in eitheror both dimensions are too small to show up on a particulargraph, a note to that effect should be made on the graph sothat the reader is aware of that fact.

Do not connect the points like a dot-to-dot drawing!

(e) Points for SlopeIn calculating parameters from a graph, such as the slope,points on a line should be chosen which are not data points,even if data points appear to fall directly on the line; failureto follow this rule makes the actual line drawn irrelevant andmisleading.When plotting points for the slope, a different symbol shouldbe used from that used for data points to avoid confusion. Theco–ordinates of these data points should be shown near thepoint as well for the reader’s information. If one uses graphpaper with a small enough grid, it may be possible to choosepoints for the slope which fall on the intersection of grid lineswhich simplifies the process of determining their co–ordinates.Of course, points for the slope should always be chosen as farapart as possible to minimize errors in calculation.

(f) Error BarsData points must be fitted with error bars to show uncertain-ties present in the data values. If the uncertainties in eitheror both dimensions are too small to show up on a particulargraph, a note to that effect should be made on the graph sothat the reader is aware of that fact. Uncertainties in quan-tities plotted on a graph are shown by error bars. Figure 6.2shows a point with its error bars. The range of possible val-ues for the data point in question actually includes any pointbounded by the rectangle whose edges fall on the error bars.

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The size of the error bars is given by the uncertainties in bothcoordinates. (Actually, the point’s true value is most likelyto fall within the ellipse whose extents fall on the error bars.This is because it is unlikely that the x and y measurementsare both in error by the maximum amount at the same time.)In fact, error bars may be in one or both directions, and they

x−∆x x x+ ∆x

y −∆y

y

y + ∆y

Figure 6.2: Point with error bars

may even be different in the positive and negative directions.

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6.2 Graphing 57

x−∆x− x x+ ∆x+

y −∆y−

y

y + ∆y+

Figure 6.3: Graph with unequal error bars in positive and negative directions

Is the origin a data point?

Sometimes an experiment produces a graph which is expected togo through (0, 0). In this case, whether to include the origin as adata point or not arises. There is a basic rule: Include (0, 0) as adata point only if you have measured it (like any other data point).Often a graph which is expected to go through the origin will notdo so due to some experimental factor which was not consideredin the derivation of the equation. It is important that the graphshow what really happens so that these unconsidered factors willin fact be noticed and adjusted for. This brings up a second rule:If the origin is a data point, it is no more “sacred” than any otherdata point. In other words, don’t force the graph through (0, 0)any more than you would through any other point. Doing a leastsquares fit will protect you from this temptation.

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Figure 6.4: A line through the origin is not always the best fit

6.3 Graphical Analysis

Usually the point of graphing data is to determine parameters ofthe mathematical relationship between the two quantities. Forinstance, when plotting a straight line graph, the slope and y–intercept are the parameters which describe that relationship.

Note that the slope and y–intercept and their uncertainties should have units.The units of the y–intercept should be the same as the y variable, and theslope should have units of

[slope] =[y]

[x]

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6.4 Linearizing Equations 59

6.4 Linearizing Equations

In many cases, the mathematical model you are testing will sug-gest how the data should be plotted. A great deal of simplifica-tion is achieved if you can linearize your graph, i.e., choose theinformation to be plotted in such a way as to produce a straightline. (This is discussed in Chapter 4, “Exercise on LinearizingEquations” .) For example, suppose a model suggests that therelationship between two parameters is

z = Ke−λt

where K and λ are constants. If a graph of the natural logarithmof z is plotted as a function of t, a straight line given by

ln z = lnK − λt

will be obtained. The parameters K and λ will be much easier todetermine graphically in such a case.

In particular, if we substitute y = ln z and x = t in the aboveequation, and if the slope and y–intercept are measured to be,respectively, m and b, then it should be clear that

m = −λ

andb = lnK

6.5 Curve Fitting

Always draw smooth curves through your data points, unless youhave reason to believe that a discontinuity in slope at some pointis genuine.

Your graphs should not look like a dot–to–dot drawing.

If you are plotting the points using the computer, draw the curveby hand if necessary to avoid this problem. However, do not fit

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Figure 6.5: Wrong: Graphs should not look like dot-to-dot drawings

data to a curve with no physical significance simply so that all ofthe points fit.

Do not use an arbitrary function just because it goes through all the datapoints!

Note that unless a set of data exactly fits a curve, choosing acurve of “best fit” is somewhat arbitrary. (For example, consider4 data points at (-1,1), (1,1), (1,-1) and (-1,-1). What line fitsthese points best?)

Usually, going “by eye” is as good as anything; the advantage to amethod such as the least squares fit is that it is easily automated,and is generally reliable.

If plotting by eye, one should observe that the line of best fit willusually have an equal number of points above and below it. Aswell, as a rule, there should not be several points at either end ofthe graph on the same side of the curve. (If this is the case, the

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6.5 Curve Fitting 61

Figure 6.6: Wrong: Graphs should not have meaningless curves just to fitthe data

curve can be adjusted to avoid this.)

Determining the y-intercept is easy if it is shown on the graph.However if it isn’t, you can determine it from the points you usedfor the slope. If

m =y2 − y1

x2 − x1

andy = mx+ b

for any points on the line, including (x1, y1) and (x2, y2) then

y2 = mx2 + b

sob = y2 −mx2

and finally

b = y2 −(y2 − y1

x2 − x1

)x2

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6.6 Least Squares Fitting

Least Squares Fitting is a procedure for numerically determiningthe equation of a curve which “best approximates” the data beingplotted. If we wish to fit a straight line to data in the form

y = mx+ b

then the least squares fit gives values for b, the y-intercept, andm, the slope, as follows:1

b =(∑yi) (

∑x2i )− (

∑xi) (

∑xiyi)

N (∑x2i )− (

∑xi)

2 (6.3)

and

m =N (∑xiyi)− (

∑xi) (

∑yi)

N (∑x2i )− (

∑xi)

2 (6.4)

(Note: You do not need to calculate uncertainties for m and bduring least squares fit calculations like this. Uncertainties in mand b will be dealt with later.)

If you are interested, an appendix contains a derivation of theleast squares fit. In any case you may use the result above.

One important concept which will come up later is that of degreesof freedom, which is simply the number which is the differencebetween the number of data points, (N above), and the numberof parameters being determined by the fit, (2 for a straight line).Thus, for a linear fit, the number of degrees of freedom, ν, is givenby:

ν = N − 2 (6.5)

1You may notice that a particular quantity comes up a lot. It is

N(∑

x2i

)−(∑

xi

)2

(6.1)

It only takes a couple of lines of algebra to show that this equals

N (N − 1)σx2 (6.2)

where σx2 is the sample standard deviation of the x values.

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6.7 Uncertainties in Graphical Quantities 63

6.6.1 Correlation coefficient

Equation 6.6 gives the square of the Pearson product-momentcorrelation coefficient, which we will refer to simply as thecorrelation coefficient.2

R2 =(N∑xiyi − (

∑xi)(

∑yi))

2

(N∑xi2 − (

∑xi)2)

(N∑yi2 − (

∑yi)

2) (6.6)

The correlation coefficient, R, is a number which has a value be-tween -1 and +1, where a value of -1 indicates a perfect negativecorrelation, +1 indicates a perfect positive correlation, and a valueof zero indicates no correlation. Thus R2 is a value between zeroand 1 indicating just the strength of a correlation. The closer R2

is to one, the stronger the correlation between two variables. Toput it another way, the closer it is to one the better one variablecan be used as a predictor of the other.

6.7 Uncertainties in Graphical Quan-

tities

After the slope and intercept have been calculated, their associ-ated errors are calculated in one of two ways depending on thedata. (This is analogous to the idea that the uncertainty in theaverage is the bigger of the standard deviation of the mean andthe uncertainty in the individual values.) The two possible casesare outlined below.

Unless the points fit exactly on a straight line, any graph with big enougherror bars will fit the first case, and any graph with small enough error barswill fit the second. It is the relative size of the error bars which determineswhich case it is.

2As long as we’re dealing with a linear fit, this is the quantity that would commonlybe used.

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6.7.1 Small Scatter of Data

If the scatter in the data points is small, a straight line whichpasses through every error bar on the graph can be found, asshown in Figure 6.7. This indicates that the uncertainties in yourresults are primarily due to the uncertainties of the measuringinstruments used.

The slope and intercept can be found graphically, by eye or usingthe least squares fit method.

Line of minimum slope

Line of maximum slope

Figure 6.7: Small Scatter of Data Points

To obtain error estimates in these quantities, one draws two lines:a line with the maximum slope passing through all the error bars;and the line with the minimum slope passing through all the er-ror bars. These extremes will determine the required uncertaintiesin the slope and intercept. (In this graph and the one following,boxes have been drawn around each point and its error bars to indi-cate the “uncertainty region” around each point. These would notusually be on a graph, but they are shown here for illustration.)

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The points for the maximum and minimum slope will not always be the end-points on the graph. Also, the data points providing the endpoints for the twolines will not usually be the same for both. If the maximum and minimum

slope are not symmetric about the average, you can calculate

∆m ≈ mmax −mmin

2

and

∆b ≈ bmax − bmin2

For negative slope

For positive slope

( x1 −∆x1, y1 + ∆y1)( x2 + ∆x2, y2 −∆y2)

( x1 + ∆x1, y1 −∆y1)

( x2 −∆x2, y2 + ∆y2)

( x1 −∆x1, y1 −∆y1)

( x2 −∆x2, y2 −∆y2)

( x2 + ∆x2, y2 + ∆y2)( x1 + ∆x1, y1 + ∆y1)

Figure 6.8: Maximum and Minimum Slope Coordinates from a Point

If we label two points x1 and x2, where x1 < x2, then we can seefrom Figure 6.8 that the steepest line which touches the error barsfor both x1 and x2 is the line between (x1 + ∆x1,y1 − ∆y1) and

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(x2 −∆x2,y2 + ∆y2). The slope of this line will then be

mmax =(y2 + ∆y2)− (y1 −∆y1)

(x2 −∆x2)− (x1 + ∆x1)(6.7)


bmin = (y1−∆y1)−mmax(x1+∆x1) = (y2+∆y2)−mmax(x2−∆x2)(6.8)

Similarly the line with the least slope which touches the error barsfor both x1 and x2 is the line between (x1 − ∆x1,y1 + ∆y1) and(x2 + ∆x2,y2 −∆y2). The slope of this line will then be

mmin =(y2 −∆y2)− (y1 + ∆y1)

(x2 + ∆x2)− (x1 −∆x1)(6.9)


bmax = (y1+∆y1)−mmin(x1−∆x1) = (y2−∆y2)−mmin(x2+∆x2)(6.10)

The case for a negative slope is shown in Figure 6.8; the analysisis left to the student.

The points for the maximum and minimum slope will not always be the end-points on the graph.

6.7.2 Large Scatter of Data

Often, you will not be able to find a line which crosses every errorbar, as with the data in Figure 6.9, and you will have to resortto the numerical method below. In this case, the uncertainties inyour graphical results are primarily due to the random variationsin the data.

Once these values for the slope and intercept are determined, thesum of squares error, S is computed. For the linear case, S canbe shown to have a value of

S =∑

y2i −m

∑xiyi − b

∑yi (6.11)

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Figure 6.9: Large Scatter of Data

In order to estimate the uncertainty in each parameter, the stan-dard deviation σ is computed from

σ =

√S

N − 2(6.12)

where N − 2 is the number of degrees of freedom mentioned ear-lier. (Often the symbol ν is used for degrees of freedom.) Thestandard error (i.e. uncertainty) in the intercept is

σb = σ

√ ∑x2i

N (∑x2i )− (

∑xi)

2 (6.13)

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i xi x2i xiyi yi y2

i

1 0.1 0.01 0.3 3 92 0.2 0.04 0.8 4 163 0.3 0.09 1.2 4 164 0.4 0.16 2.0 5 25

N∑xi

∑x2i

∑xiyi

∑yi

∑y2i

4 1.0 0.3 4.3 16 66

Table 6.2: Sample Least Squares Fit Data

and the standard error (uncertainty) in the slope is

σm = σ

√N

N (∑x2i )− (

∑xi)

2 (6.14)

As long as our data fit this second case, then we can use σb and σm as theuncertainties in the y–intercept and the slope respectively, and use the sym-bols ∆b and ∆m instead. Keep in mind, however, that if our data fit the firstcase, then these terms are not interchangeable. Note that the uncertaintiesin the slope and y–intercept should have the same units as the slope andy–intercept.

(Note: You do not need to calculate uncertainties for ∆m and ∆bsince these are uncertainties themselves!)

6.7.3 Sample Least Squares Calculations

Following is a calculation of the least squares fit and the standarderror of the slope and intercept for some test data.

N(∑

x2i

)−(∑

xi

)2

= (4)(0.3)− (1)2 = 0.2

b =(∑yi) (

∑x2i )− (

∑xi) (

∑xiyi)

N (∑x2i )− (

∑xi)

2 =(16)(0.3)− (1)(4.3)

0.2= 2.5

m =N (∑xiyi)− (

∑xi) (

∑yi)

N (∑x2i )− (

∑xi)

2 =(4)(4.3)− (1)(16)

0.2= 6.0

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6.8 References 69

S =∑

y2i −m

∑xiyi−b

∑yi = (66)−(6)(4.3)−(2.5)(16) = 0.2

σ =

√S

N − 2=

√0.2

4− 2= 0.316228

σb = σ

√ ∑x2i

N (∑x2i )− (

∑xi)

2 = (0.316228)

√0.3

0.2= (0.3878298)

σm = σ

√N

N (∑x2i )− (

∑xi)

2 = (0.316228)

√4

0.2= (1.414214)

Thus, if our data are such that σb and σm are the uncertainties inthe y–intercept and the slope, and thus ∆b and ∆m, then

b = 2.5± 0.4

andm = 6± 1

6.8 References

– The Analysis of Physical Measurements, Emerson M. Pughand George H. Winslow, Addison-Wesley Series in Physics,1966, QC39.P8

– Errors of Observation and Their Treatment, J. Topping, Chap-man and Hall Science Paperbacks, 1972(4th Ed.)

– Statistics, Murray R. Speigel, Schaum’s Outline Series in Math-ematics, McGraw-Hill, 1961

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Chapter 7

Standing Waves in an AirColumn

7.1 Purpose

In this experiment sound waves are studied and a measurementof the velocity of sound in air at room temperature is performed.Providing an understanding of the meaning of the term longitu-dinal or compressional waves is also a goal.

7.2 Introduction

This experiment will provide experience in graphing and leastsquares fitting.

NOTE: A tuning fork is a delicate scientific instrument. Pleasedo not rap the fork on hard surfaces or objects such as table–topedges; to make the fork “sound” one of the best methods is to tapthe fork on your knee or the heel of your shoe.

7.3 Theory

Waves in air may be defined as periodic disturbances propagatedthrough air by virtue of the oscillations of the air particles. Unlike

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72 Standing Waves in an Air Column

(a)

(b)

(c)

Figure 7.1: Motion of Air Molecule Near Vibrating Reed

the transverse waves generated on a tight string such as ona guitar, sound waves in air are an example of a longitudinalwave. In such a wave, the motion of an individual air molecule isalong the direction of travel of the wave; the mean position of anyparticle is fixed, however it oscillates back and forth about thisposition due to passage of the wave.

7.3.1 Longitudinal Waves

Consider a vibrating reed of some kind, fixed at its base and stand-ing vertically. If pulled to one side and released it will oscillateback and forth periodically. Let us consider what happens to asingle air particle when the reed is set in motion. Firstly, natureabhors a vacuum and air will rush into any space where the air

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7.3 Theory 73

λ

λ

R C R C

Direction of travel

Figure 7.2: Variation in Density of Air Molecules Due to Sound

density is less than that of its surroundings. In Figure 7.1(a) ourmolecule is considered to be at rest. In Figure 7.1(b), the reed hasmoved to the left creating a partial vacuum at the site where itwas a moment before. Our molecule charges into the rarefied spacethus caused. Now, in Figure 7.1(c), the reed swings back to theright and pushes our particle or molecule to the right. This cyclerepeats periodically buffeting our molecule back and forth. Ourmolecule has the same effect on its neighboring particles as doesthe reed on itself; buffeting them back and forth as well. A soundwave is thus being generated, traveling off to the right. There isone off to the left also, but let us neglect it for the present.

The disturbance propagated from the reed to our molecule isslightly ahead in time of that executed by the molecule itself. Sim-ilarly, the disturbance propagated by our molecule to its neighboron the right is also slightly ahead in time of that actually executedby its neighbor This occurs because of the finite velocity of thedisturbances in air.

The overall result of the (slightly delayed) oscillations of the par-ticles relative to their neighbors, all oscillating along the directionof propagation of the disturbance (longitudinally) is a sound wavepropagating at the speed of sound. If we were able to take a“snapshot” of this traveling wave, and actually see the individualair molecules, we would see something like that in the Figure 7.2.

– C represents a compression; a region where the density of air

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molecules is higher than that the air would have if there wereno sound wave passing.

– R represents a rarefaction; a region in which the density ofair molecules is less than that which the air would have in theabsence of the wave.

– λ is the wavelength of the sound wave; the distance betweenconsecutive compressions (or rarefactions).

Another snapshot taken very quickly after the first would showexactly the same picture, except that the compressions and rar-efactions would all have moved a slight distance to the right. Asuccession of such snaps would reveal the same thing; the wholewave is moving, without change, to the right, at speed v.

Note that the wave moves to the right. The individual particlesdo not; they simply oscillate back and forth along the direction oftravel of the wave, with a fixed mean position.

The frequency ν of this wave is the same as that of the reed whichgenerates it, or the frequency with which an ear drum would beset into oscillation as the wave hits it. Frequency, wavelength, andvelocity of this wave are related as in the previous experiment by

v = νλ (7.1)

If we know ν, the oscillation frequency of the fork, and λ can bemeasured, we thus can measure v, the speed of sound in air atroom temperature.

If our wave is generated at the top of a column of air, the wavewill be propagated down the column from the fork, and will bereflected by the fixed end. The motion of the individual airmolecules will thus be determined by the passage of both inci-dent and reflected waves. At the resonance condition in whicha standing wave is generated in the column, there are certainequally spaced points along the column at which the effects ofincident and reflected waves exactly cancel each other, and theair molecules don’t move at all due to the waves passing. Suchpoints we call nodes. Midway between each neighboring pairof nodes the interaction of the two waves passing is such as tocreate a maximum motion of the air particles; the action of one

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7.3 Theory 75

wave being reinforced by that of the other wave. These regionsare called anti-nodes. The amplitude of molecular oscillationincreases gradually from zero at a node to a maximum at an anti-node.

This succession of nodes and anti-nodes is called a standingwave. The distance between nodes is equal to one half–wavelengthof the waves passing in either direction.

7.3.2 Motion of Air Near an Oscillator

Sound the tuning fork and hold it (handle pointing down) closeto your ear, slowly rotating it. You will notice that maxima andminima occur in the emerging sound. This occurs because thewave generated between the fork prongs is of different phase thanthat generated outside the prongs.

With one simple diagram, (Figure 7.3), this should be understood.The fork is viewed from above, during an instant of its oscillationcycle. Rest positions of the prongs are dashed, while the actualposition is drawn with solid lines (note: the fork prongs are com-ing together). Rarefactions R are produced outside the prongs,and a compression C is produced between them. As these wavespropagate outward from the fork they will cancel each other alongthe directions indicated by the arrows in the diagram. You shouldbe able to verify this by the simple experiment suggested above.

7.3.3 Speed of Sound in Air

The velocity of sound in air is related to the physical propertiesof air by

v =

√kp

d(7.2)

where

– p is the pressure (atmospheric)

– d is air density (at given pressure and room temperature)

– k is the adiabatic constant = 1.403, related to its specific heatsat constant pressure and volume.

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CR R

Figure 7.3: Air Density Near the End of a Tuning Fork

For pressure in N/m2 and density in kg/m3, the velocity will bein m/s

Because air density decreases as air temperature increases, thespeed of sound increases as temperature rises. We can write

vT = v0(1 + αT ) (7.3)

where vT is the velocity of sound in air at temperature T (◦C).

– v0 is velocity of sound in air at 0◦C,

– α is the coefficient of expansion of the gas and α = 12(273.15)

1

– T is the gas temperature (◦C).

7.3.4 Resonance

Resonance, as the term is used in the theory of sound, is theintensification of the sound wave from a source by means of thesympathetic vibration of another body, called resonator, “tuned”to the same frequency. When an organ pipe is used as a resonancechamber, standing waves are set up in the air column of the pipe.

1 Note that absolute zero is at −273.15◦C. That’s where this number comes from.

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7.3 Theory 77

N

A

N

A

N

A

N

A

N

A

N

A

λ/4

λ/2

λ/2

3λ/4

5λ/4

Figure 7.4: Node Pattern in a Tube With an Open End

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The simplest resonant chamber is a tube open at one end andclosed at the other, the sounding body being placed near the openend. If the tube is of the proper length, standing waves are setup which reinforce the sound emitted by the sounding source. Insuch a resonator there is a node at the closed end (since the airmolecules can’t move there) and an anti-node at the open end.i.e. the shortest closed pipe which will resonate with a source ofgiven frequency (thus of given wavelength in air) is one whoselength is one quarter wavelength. A pipe 3 or 5 times as longwill also act as a resonator for this source as seen in Figure 7.4.A little thought will show that any closed pipe whose length isan odd number of multiples of a quarter wavelength will produceresonance. Owing to the fact that the maximum air moleculeoscillations do not occur exactly at the open end of the pipe, thedistance AN in the shortest pipe of Figure 7.4 is not exactly λ/4.In fact, the length of pipe which will have a resonance is given by

Ln = (2n− 1)λ

4− ccr (7.4)

where r is the inner radius of the pipe, n = 1, 2, 3, . . ., and cc is atheoretical correction factor, with a value of about 0.61.

7.4 Procedure

7.4.1 Preparation

Pre-lab Tasks

PT1: Find values for p and d in Equation 7.2 and put them inTable 7.2.

Pre-lab Questions

PQ1: What linearization of the graph is suggested by Equa-tion 7.4? In other words, what should the independent (i.e. x)variable be? What should the dependent (i.e. y) variable be?

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7.4 Procedure 79

What is the slope? What is the y-intercept? What are λ, ∆λ, cc,and ∆cc in terms of the slope and the y-intercept?

PQ2: What is the uncertainty in n and thus what is the uncer-tainty in (2n− 1)? Explain.

7.4.2 Experimentation

Apparatus

– resonant air column setup

– tuning forks

– wave (sound) generator and speaker

– thermometer

Method

A sound of known frequency is produced at the mouth of a verti-cal column of air in a glass tube partially filled with water. Thelength of the air column from “open” end to “solid” bottom, (i.e.the water surface), is adjustable by variation of the water reservoirlevel. Several resonance positions are to be determined for a sin-gle frequency, and from the average distance between successivepositions of this resonance the wavelength of sound in air is deter-mined and thus the velocity of sound in air at room temperatureis determined as well.

(a) The apparatus should be self-explanatory. Use an erasablemarker on the column to mark the water level. Fill the res-onance tube with water as high as possible by raising thereservoir.

(b) Turn on the sound generator over the open end of the column.Slowly lower the reservoir and listen for an intensification ofthe sound as the resonance chamber gets slowly longer. Markthe position on the tube where the resonance is observed,and then record the reading at that position. Repeat this 5times and use the data to determine an average value and itsuncertainty.

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(c) Lower the reservoir further to locate the second resonant lengthof the column and measure it as above. Continue to lower thereservoir in this manner until you have obtained as many res-onance lengths as the air column will permit, taking 5 mea-surements of each as above.

(d) Record the room temperature, T .

In-lab Tasks

IT1: Do an order of magnitude calculation of the velocity usingEquations 7.4, 7.1, and one data point to show that it is in theright range.

In-lab Questions

IQ1: Can you tell if the effective uncertainty in the resonanceposition is due to difficulty in holding the reservoir steady or indetecting the loudest sound? How big is it?

IQ2: Do the resonance positions get harder to find lower down inthe tube?

IQ3: Does the effective uncertainty depend on the frequency?

7.4.3 Analysis

(a) Plot a graph of Ln vs. (2n−1) and from the graph determinevalues for both λ and cc.

(b) Compute v for the λ calculated above, and check it againstthe known velocity which is obtained from Equation 7.3.


Q1: Compare the values for λ, cc, and vT with the expectedvalues. Do they agree?

Q2: Find references for Equations 7.2 and 7.3.

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7.5 Bonus 81

Q3: Would more than five measurements have reduced the uncer-tainty in Ln? If so, how many measurements would have reducedthe uncertainty as much as possible?

7.5 Bonus

7.6 Recap

By the time you have finished this lab report, you should knowhow to :

– Perform graphical analysis of data where there are error barsin one direction only.

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7.7 Template

My name:My partner’s name:My other partner’s name:My lab section:My lab demonstrator:Today’s date:

quantity symbol single/given/ repeated/mine constant

air pressure

air density

constant k

pipe innerradiustemperature

Not in equations

Table 7.1: List of quantities

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7.7 Template 83

symbol value units instrument effectivereference precision zero uncertainty

( e.g. A.1) measure error

k

r

T

ν

Not in equations

Table 7.2: Single value quantities

quantity symbol equation uncertainty

Table 7.3: Calculated quantities

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symbol factor bound units s/r

Table 7.4: Experimental factors responsible for effective uncertainties

Instrumentreference(or name)

units

precisionmeasure

zeroerror

resonance position ( Ln )number trial #

( n ) 1 2 3 4 5

1

2

3

4

5

Table 7.5: Resonance data

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Chapter 8

Determining the Heat ofFusion of Ice

8.1 Purpose

The objectives of this experiment are to understand the idea ofheat of transformation when ice undergoes a phase change and tolearn how to measure heat of fusion for ice

8.2 Introduction

This experiment will develop your skill in graphical analysis andleast squares fitting. In particular, you will have a single data setwhich must be divided into three regions, where each region willneed to have a separate least squares fit performed on it.

8.3 Theory

When a sample of ice of mass m completely undergoes a phasechange from solid into liquid, i.e. melts to water, the total energyQ which the ice absorbs from its environment is proportional tothe heat of fusion Lf , the heat transfer for a unit mass, and the

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86 Determining the Heat of Fusion of Ice

ice and water

insulator ring

cork thermometer

plastic lid

stirrer

reservoir

aluminum container

Figure 8.1: A Simple Calorimeter

mass m:

Q = Lfm (8.1)

In the other direction, when the phase change is from liquid tosolid, the sample must release the same amount of energy. Forwater at its normal freezing or melting temperature,

Lf = 79.5 cal/g = 333 kJ/kg (8.2)

This lab employs a double-wall calorimeter as shown in Figure 8.1to measure the heat of fusion for ice. The calorimeter consists ofan aluminum container, a reservoir, a plastic lid and an insulatorring. The reservoir holds a maximum of 150 ml water. The clearplastic lid has 3 access holes. It includes a cork with a hole forholding a thermometer, and a hole for a stirrer.

Water is poured into the reservoir, and the initial equilibrium tem-perature T1 is reached after heat transfer among all the devices is

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8.3 Theory 87

completed. Ice of mass m is then put into the water, and then ab-sorbs heat from water. In the meanwhile, the water releases heat,and as a result, the temperature of the whole system decreasesuntil the system starts to absorb heat from air surrounding thesystem. At this turning point, the temperature is T2. The tem-perature then increases again.

The total heat transfer can be broken into two parts: heat givenoff and heat absorbed (if we assume the system is closed). Theheat absorbed goes into

– the ice, to turn it into ice water

– the resulting ice water, to raise it to T2.

The heat given off comes from

– the reservoir

– the water in the reservoir

– the thermometer

all of which start out at T1. Assuming the system is closed, all ofthe heat given off must be absorbed. The heat equation for thesystem can then be expressed as:

Lfm+ cm (T2 − 0) = cmw (T1 − T2) + cAmA (T1 − T2) + q (8.3)

where

– m is the mass of ice, as previously stated

– mw is the mass of water

– mA is the mass of the reservoir and stirrer (assuming they areboth made of the same material)

– c is the specific heat of water

– cA is the specific heat of the (aluminum) reservoir

– q is the heat released by the thermometer which is equal to:

q = 1.93V (T1 − T2) (8.4)

where V is the immersed volume of the thermometer in mlwhich can be determined by Archimedes’ Principle.1

1In case you’re wondering where the magical 1.93 came from, it’s the product of cg and

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8.4 Procedure

8.4.1 Preparation

Pre-lab Questions

PQ1: What is the equation for the uncertainty in Lf from Equa-tion 8.3?

PQ2: Why is it only the immersed volume of the thermometerthat matters?

PQ3: What assumptions have been made in calculating the heatgiven off by the thermometer? Will this introduce a random or asystematic error in Lf? If systematic, will it make the calculatedvalue of Lf higher or lower than it should be?

PQ4: What assumptions have been made about the temperatureof the stirrer? Will this introduce a random or a systematic errorin Lf? If systematic, will it make the calculated value of Lf higheror lower than it should be?

8.4.2 Preparation

Pre-lab Tasks

PT1: Look up the specific heats for aluminum and water and putthem in your template.

ρg since ρ = mV , so that

q = cgmg (T1 − T2)

where

∗ cg is the specific heat of glass ≈ 837J/kg

∗ ρg is the density of glass ≈ 2.3g/cm3

ie. it is the same form as the terms for water and the metal can.

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8.4 Procedure 89


Apparatus

– Calorimeter, which consists of several parts, shown in Fig-ure 8.1.

– Stop watch

– Beaker

– Scale

– Graduated cylinder

Materials

– Ice

– Warm water

– Room temperature or cooler water

Hints

To get the best results, you need to keep in mind that:

(a) It is important to prevent any unnecessary heat transfer be-tween the environment and the system. When you stir thewater with the ice don’t do it too fast because you don’t wantto give your energy to the system!

(b) It is also important to choose an appropriate initial tempera-ture T1. If T1 is too low, or as low as room temperature, thesystem will absorb energy from the environment. The bestinitial temperature T1 is 10◦C→ 15◦C above room tempera-ture.

Method

Read over the following instructions carefully before you start. Be sure youunderstand what has to be done before you begin, because once you add theice, things will happen very quickly!

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(a) Identify all components from the diagram. Do not confusethe container with the reservoir.

(b) Measure the masses of the stirrer and the reservoir.

(c) Heat the water up to about 45◦C.

(d) Pour the water into the reservoir. Make sure the water fillsat least half of the reservoir of the calorimeter.

(e) Weigh the reservoir again to get the mass of the water.

(f) Start the stopwatch. Once you start the stopwatch, do notstop it until you are done the whole experiment.

(g) Cover the lid of the reservoir, and begin stirring. Continue tostir until the experiment is done.

(h) Measure the temperature of the system every 20 seconds.

(i) Continue for a couple of minutes until the rate of temperaturechange is constant.

(j) Put the ice into the reservoir, and cover the lid promptly.

(k) Measure the temperature of the system every 10 seconds, untilthe temperature has stopped decreasing for one minute, andremains unchanged or starts to rise.

(l) When you have stopped taking temperature measurements,weigh the reservoir, and subtract the previous mass of reser-voir and water to get the mass of the ice added.

(m) Measure the immersed volume of the thermometer by use ofthe graduated cylinder. Don’t try to remove the thermometerfrom the stopper because you might break the thermometer!

In-lab Tasks

IT1: Do an order of magnitude calculation of the heat of fusionusing Equation 8.3, estimating T1 and T2 from the data, to showthat it is in the right range. Calculate each of the terms separatelybecause that will simplify answering in-lab questions 1 and 2.

In-lab Questions

IQ1: Given the assumptions made in calculating the heat givenoff by the thermometer, what bounds can be placed on error in

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8.4 Procedure 91

Temperature (◦ C)

Time (s)

50

40

30

20

10

T1

T2

100 200 300 400

Figure 8.2: A typical graph of temperature vs. time for the system

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the total heat due to the thermometer? (What percentage of thetotal heat lost is due to the thermometer?)

IQ2: Given the assumptions made about the temperature of thestirrer, what bounds can be placed on the error in the total heatdue to the stirrer? (What percentage of the total heat lost is dueto the stirrer?)

IQ3: How long does it take to take a measurement? Does thisaffect the realistic uncertainty in the time?

IQ4: How does the amount of ice used affect the results? Whatwould happen if a large amount of ice were used with a smallamount of water and vice versa?

8.4.4 Analysis

(a) Plot a graph of temperature vs. time. A typical graph oftemperature vs. time for the system is shown in Figure 8.2.

(b) Extrapolate from the graph as shown to get T1, T2, and theiruncertainties. If the two end regions on the graph are nothorizontal, proceed as follows:

i. Get the times of the two “corner points” on the graph.

ii. At the time which is the average of the corner point times,draw a vertical line.

iii. Extrapolate the two end lines from the graph until theyintersect the vertical line.

iv. Draw horizontal lines from the intersection points to thevertical axis to get T1 and T2 and their uncertainties.

(c) Use these values in Equation 8.3 to determine Lf , and com-pare it to the expected value.


Q1: Based on your data, how realistic was the assumption thatthe reservoir was thermally isolated?

Q2: Find references for the specific heats of glass and aluminum.

Q3: Did the calculated value of Lf agree with the expected value?

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8.5 Bonus 93

8.5 Bonus

You can do one of the questions below.

8.5.1 Choice of Initial Temperature

In the second hint above, consider the effect of different valuesfor T1 on the shape of the graph. Repeat the experiment using adifferent value for T1 and comment on the results.

8.5.2 Heat Transfer Method

Can you determine whether the heat transfer between the reservoirand the environment is primarily by convection, conduction, orradiation? Modify the experiment and test your hypothesis.

8.5.3 Thermometer Assumptions

The term for heat given off by the thermometer depends on thetype of glass used in the thermometer. Look up values for c and ρfor different types of glass, and discuss whether the type of glassused has a significant effect on your results.

8.5.4 Uniformity of Water Temperature

How important is the stirring action to keep the water temperatureuniform? Modify the experiment to test this and discuss what youobserve.

8.6 Recap

By the time you have finished this lab report, you should knowhow to :

– select regions of data for least squares fitting

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8.7 Template

My name:My partner’s name:My other partner’s name:My lab section:My lab demonstrator:Today’s date:


Not in equations


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8.7 Template 95



mass ofstirrer

mass ofreservoirmass ofreservoir

and watermass ofreservoir

and waterand ice

immersedthermometer

volume

Not in equations




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symbol factor effective units s/runcertainty

bound

Single measurement quantities

Variable quantities


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8.7 Template 97

Instrument

reference(or name)

units

precisionmeasure

zeroerror

Time Temperature(hh:mm:ss) (◦ C)

Table 8.5: Timing data

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Chapter 9

Young’s Modulus andStretch Measurement byOptical Lever

9.1 Purpose

The purpose of this experiment is to determine Young’s modulusof elasticity of a wire by stretching.

9.2 Introduction

Note: this experiment refers to both r and R and they are NOT the same.Similarly, there are both d and D. Be sure you keep them straight in yourmind and in your calculations.

This experiment will develop skills of graphing and doing calcula-tions with repeated dependent measurements.

9.3 Theory

It has been found by experiment that the deformation of a bodydue to applied forces is proportional to the magnitude of these

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100 Young’s Modulus and Stretch Measurement by Optical Lever

forces over a wide range of values for these forces. This pro-portionality between deformation and applied force is known asHooke’s Law and is the basis of the theory of elasticity. Thevalue of the stress at which Hooke’s Law ceases to hold is calledthe elastic limit of the substance. ( Stress is defined as the ratioof applied force to cross-sectional area.) The elastic limit may alsobe defined as the magnitude of the applied stress which producesthe maximum amount of recoverable deformation. The fractionalchange in the dimension of a body produced by a system of forcesin equilibrium is called a strain.

Even within the limits of perfect elasticity, different bodies showdistinct differences in their behaviour. Some recover their formimmediately after the removal of the force, while others requirea considerable amount of time to recover completely. This delayin recovering the original condition of the substance is called theelastic after-effect, or elastic lag.

When a wire is stretched beyond its elastic limit and its cross-section is reduced, as in drawing through a die, its structure isbroken down and the surface appears to be an amorphous layerof flowed material. This flowed layer becomes proportionatelythicker with repeated drawings and the density, hardness, andelasticity of the material are profoundly changed. For instance,when a wire of Swedish iron has its diameter reduced from 0.75 mmto 0.10 mm by repeated drawings, the breaking strength measuredin force per unit area is doubled. Keeping in mind these changes,the student will not expect an exact check of results with acceptedvalues given in tables for the modulus of elasticity.

A wire which is stretched experiences not only a change in itslength but also a much smaller change in its diameter. Young’smodulus of elasticity Y takes into account only the change inlength – the longitudinal strain which occurs. This strain is thechange in length per unit length. The longitudinal stress producedby the applied forces is measured in terms of force per unit area.Young’s modulus is defined as the ratio of the longitudinal stressto the longitudinal strain.

If l represents the initial length of the wire, (Figure 9.1), r, itsaverage radius, and e, the stretch produced by the weight of a

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9.3 Theory 101

e

l

A = πr2

Figure 9.1: Cross-section of wire

mass M , then

Young’s modulus =stress

strain=

(force/area)

(elongation/length)(9.1)

or

Y =(Mg/πr2)

(e/l)=Mgl

πr2e(9.2)

9.3.1 The Optical Lever

The optical lever furnishes an interesting application of the law ofreflection of light. If a beam of light TO (Fig. 9.2) originating ata point T on the scale S meets the mirror perpendicularly, it isreflected back on its path; but if the mirror it turned through anangle (NOT ), an incident beam PO originating at a point P onthe scale S will on reflection take the direction OT where the angleof incidence (NOP ) is equal to the angle of reflection, ON beingthe normal to the mirror in the position (OM ′). By the motion ofthe mirror through the angle θ, the reflected beam has thereforebeen turned through an angle 2θ. If a laser is mounted adjacent

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Figure 9.2: The Optical Lever

to the scale at T and if the mirror is properly adjusted, the laserbeam will be reflected on the scale and the angle 2θ may easily bedetermined from the distance D (which is the difference betweenthe scale reading with a load applied and the reading obtained withno load applied) and the scale distance R.

In use the optical lever stands upon three hardened steel points,A,B, and C (Fig. 9.2). Two of these, A and B, resting in agroove, form the fulcrum of the lever, while the third, C, rests onthe chuck H which grips the lower end of the wire W under test.

The lever (Fig. 9.4) carries a mirror which is adjustable about ahorizontal axis.

From Fig. 9.3 it is seen that the elongation e is given by therelation

e = d sin(θ) (9.3)

where d is the perpendicular distance from the line AB to thepoint C.

The double angle 2θ may be determined from the distance R fromthe mirror to the scale and the average distance D between read-

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9.3 Theory 103

Figure 9.3: Mirror Stand

Figure 9.4: Mirror

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ings produced by adding masses M of 1000 gm each to the wireby means of the relation

tan(2θ) =D

R(9.4)

Since the angle θ is small, 12

tan(2θ) ≈ tan(θ) ≈ sin(θ). In thiscase,

e ≈ dD

2R(9.5)

9.4 Procedure

9.4.1 Apparatus

The apparatus consists of a heavy tripod base with leveling screws(Fig. 9.5). On the base are mounted two nickel-plated steel sup-port rods 140 cm long with a yoke clamped to their upper ends.This yoke carries a chuck for supporting the wire to be tested. Aweight hanger supports the load by means of which the tension isapplied to the wire. Near its lower end the wire passes througha hole in the adjustable bridge or platform which is clamped tothe support rods. The measuring apparatus consists of an opticallever (Fig. 9.4) which rests in the groove on the bridge and on theface of the chuck which is tightly clamped to the wire. The chuckrides easily in the hole on the bridge. As weights are applied toload the wire, the extension tilts the optical lever so that the laserreflects on different positions of the scale.

Ten slotted 1 kg weights, a laser and scale, micrometer caliper,and a metre stick are also needed.

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9.4 Procedure 105

Figure 9.5: Reading the Scale

9.4.2 Preparation

Pre-lab Tasks

PT1: In the following table, fill in the symbol for each quantitybeside its definition, and indicate whether it is:

(a) given or assumed before the lab begins, or looked up from anexternal source

(b) selected (or adjusted) by you when you perform the experi-ment, (even if you measure its value subsequently); an inde-pendent variable

(c) measured using an instrument in the lab, as a result of theexperiment; a dependent variable

(d) calculated from quantities measured, selected, and/or given

The list of symbols for this experiment is: M , Y ,R,r,D,d,g,e,A,l

PT2: Look up Young’s modulus for different materials (aluminum,steel, etc.) and write their values in Table 9.3.

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definition symbol g/s/m/c

hanging masswire length

perpendicular distancefrom mirror to scale

perpendicular distancefrom mirror to chuck

gravitational accelerationwire radiusincrease in

scale distancedue to loadincrease inwire length

cross-sectionalarea of wire

Young’s modulus

Table 9.1: Experimental Quantities

Pre-lab Questions

PQ1: Why can we write

tan θ ≈ sin θ ≈ θ

for a small angle?


If the system is already aligned, skip the next part. If not, read onfor the alignment procedure.

Alignment

(a) Place the optical lever on the apparatus with the mirror ver-tical. Set the laser and scale at least a metre away from themirror and at about the same height.

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9.4 Procedure 107

(b) Adjust the laser and mirror so that the reflected beam fallson the scale.

(c) Tilt the mirror until the reflected beam hits the middle of thescale.

9.4.4 Determining Young’s Modulus

(a) Carefully place 8 kg on the weight hanger, keeping one handunder the weight hanger as each weight is removed lightly andcarefully so that the laser beam isn’t shifted by unneccesaryimpact or vibration.

(b) Read the scale position of the reflected laser beam. Wait afull minute and see whether this reading slowly changes. Ifit does, wait another minute and see if the reading changesagain. In this way you can determine the elastic lag of thewire.

Having determined the elastic lag, be sure to wait this amountof time after adjusting the weights before taking readings.

(c) Take a series of readings, decreasing the load in steps of 1 kguntil the hanger is empty. Take a reading with the hangeralone. Record the data in Table 9.6.

(d) Repeat the readings 2 more times as in step 1c above and addthese to Table 9.6 so there are 3 trials for each weight.

(e) With the metre stick, measure the length of the wire from theunder side of the upper chuck to the upper side of the lowerchuck.

(f) With the micrometer calipers, make four determinations ofthe diameter of the wire, two measurements at right anglesto each other at two different points along the wire. Recordthese measurements in centimetres.

(g) Measure the distance R from the mirror to the scale.

(h) Using a sheet of lined paper, place points A and B of theoptical lever on a line and press all three points into the paper.Measure in centimetres the distance d from line AB to pointC.

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In-lab Questions

IQ1: Is there a noticeable elastic lag?

IQ2: What is a reasonable uncertainty for D?

9.4.5 Analysis

(a) Average the scale readings for each load and plot a graph ofelongation e vs. load.

(b) Draw the straight line which best represents the plotted points,disregarding if necessary the first one or two points. Fromthe plotted results determine e/M . Substitute this value intoEq. 9.2 and thus calculate Young’s modulus for the wire.


Q1: Why should the distance R be large?

Q2: Why should you wait so long to observe the elongation?

Q3: How can you reduce your uncertainty in the lab?

9.5 Bonus

Determine whether or not the elastic lag depends on load. Be sureto explain how you did it. Were your results as expected?

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9.6 Template 109

9.6 Template

My name:My student number:My partner’s name:My other partner’s name:My lab section:My lab demonstrator:Today’s date:


Not in equations


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Not in equations


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9.6 Template 111



symbol factor bound units s/r


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Instrumentreference(or name)

units

precisionmeasure

zeroerror

hanging scalemass reading

Trial number1 2 3

Table 9.6: Elongation data

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Appendix A

Review of UncertaintyCalculations

A.1 Review of uncertainty rules

These are from the PC131 lab manual.

A.1.1 Repeated measurements

Arithmetic Mean (Average)

Note: In the following sections, each measurement xi can be as-sumed to have an uncertainty pm, (i.e. the precision measure ofthe instrument used), due to measurement uncertainty.

The arithmetic mean (or average) represents the best value ob-tainable from a series of observations from “normally” distributeddata.

Arithmetic mean = x =∑n

i=1 xi

n

= x1+x2+···+xn

n

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114 Review of Uncertainty Calculations

Standard Deviation

The standard deviation of a number of measurements is a mea-surement of the uncertainty in an experiment due to reproducibil-ity. The standard deviation is given by

Standard Deviation = σ =

√∑ni=1(xi − x)2

n− 1

=1√n− 1

√√√√ n∑i=1

x2i −

(∑n

i=1 xi)2

n

With random variations in the measurements, about 2/3 of themeasurements should fall within the region given by x ± σ, andabout 95% of the measurements should fall within the region givenby x± 2σ. (If this is not the case, then either uncertainties werenot random or not enough measurements were taken to make thisstatistically valid.)

This occurs because the value calculated for x, called the sam-ple mean, may not be very close to the “actual” populationmean, µ, which one would get by taking an infinite number ofmeasurements.

Rule of thumb: For normally distributed data, an order of magnitude ap-proximation for the standard deviation is 1/4 the range of the data. (In otherwords, take the difference between the maximum and minimum values anddivide by 4 to get an approximate value for the standard deviation.)

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Standard Deviation of the Mean

(In some texts this quantity is called the “standard error of themean”.) It is an interval around the calculated mean, x, in whichthe population mean, µ, can be reasonably assumed to be found.This region is given by the standard deviation of the mean,

Standard deviation of the mean = α =σ√n

and one can give the value of the measured quantity as x±α. (Inother words, µ should fall within the range of x± α.)

Uncertainty in the average

The uncertainty in the average is the greater of the uncertainty ofthe individual measurements, (i.e. pm, the precision measure ofthe instrument used), and α; i.e.

∆x = max (pm, α)

If possible, when doing an experiment, enough measurements of aquantity should be taken so that the uncertainty in the measure-ment due to instrumental precision is greater than or equal to α.This is so that the random variations in data values at some pointbecome less significant than the instrument precision. (In prac-tice this may require a number of data values to be taken whichis simply not reasonable, but sometimes this condition will not betoo difficult to achieve.)

In any case, the uncertainty used in subsequent calculations should be thegreater of the uncertainty of the individual measurements and α.

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A.1.2 Rules for combining uncertainties

Basic arithmetic rules

The uncertainty in results can usually be calculated as in thefollowing examples (if the percentage uncertainties in the data aresmall):

(a) ∆(A+B) = (∆A+ ∆B)

(b) ∆(A−B) = (∆A+ ∆B)

(c) ∆(A×B) ≈ |AB|(∣∣∣∣∆AA

∣∣∣∣+

∣∣∣∣∆BB∣∣∣∣)

(d) ∆(A

B) ≈

∣∣∣∣AB∣∣∣∣ (∣∣∣∣∆AA

∣∣∣∣+

∣∣∣∣∆BB∣∣∣∣)

Note that the first two rules above always hold true.

Uncertainties in functions, by algebra

∆f(x) ≈ |f ′(x)|∆x (A.1)

Uncertainties in functions, by inspection

∆f(x) ≈ fmax − f (A.2)

or∆f(x) ≈ f − fmin (A.3)

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Sensitivity of Total Uncertainty to Individual Uncertain-ties

If f = f(x, y), then to find the proportion of ∆f due to each ofthe individual uncertainties, ∆x and ∆y, proceed as follows:

– To find ∆fx, let ∆y = 0 and calculate ∆f .

– To find ∆f y, let ∆x = 0 and calculate ∆f .

Uncertainties and Final Results

Mathematically, if two quantities a and b, with uncertainties ∆a and ∆b arecompared, they can be considered to agree within their uncertainties if

|a− b| ≤ ∆a+ ∆b (A.4)

A value with no uncertainty given can be assumed to have an uncertainty ofzero.

If two numbers do not agree within experimental error, then the percentagedifference between the experimental and theoretical values must be calcu-lated as follows:

Percent Difference =

∣∣∣∣theoretical − experimentaltheoretical

∣∣∣∣× 100% (A.5)

Remember: Only calculate the percent difference if your results do not agreewithin experimental error.

Significant Figures in Final Results

Always quote final answers with one significant digit of uncer-tainty, and round the answers so that the least significant digitquoted is the uncertain one.

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A.2 Discussion of Uncertainties

– Spend most time discussing the factors which contribute mostto uncertainties in your results.

– Always give a measured value or a numerical bound on anuncertainty.

– State whether any particular factor leads to a systematic un-certainty or a random one. If it’s systematic, indicate whetherit would tend to increase or decrease your result.

A.2.1 Types of Errors

– Measurable uncertainties

– Bounded uncertainties

– Blatant filler

Don’t use “human error”; it’s far too vague.

A.2.2 Reducing Errors

(a) Avoid mistakes.

(b) Repeat for consistency, if possible.

(c) Change technique

(d) Observe other factors as well; including ones which you mayhave assumed were not changing or shouldn’t matter.

(e) Repeat and do statistical analysis.

(f) Change equipment; the last resort.

A.2.3 Ridiculous Errors

Anything which amounts to a mistake is not a valid source oferror. A serious scientist will attempt to ensure no mistakes weremade before considering reporting on results.

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Index

barserror, 52, 55

compression, of air, 73curve

fitted, 51curve fitting, 59

datascatter of, 63

degrees of freedom, 62Discussion of Errors, 113Discussion of Uncertainties, 113displaying lines in Excel, 45

equationslinearizing, 19, 58

errorstandard

in y-intercept, 66in slope, 66

error bars, 52, 55in Excel, 41

errorsdiscussion of, 113

Exceldisplaying lines, 45error bars, 41graphs, 39

fitted curve, 51fitting

curve, 59least squares, 61

freedomdegrees of, 62

fusionheat of, 85

graphlinearized, 51origin, 56

graphsExcel, 39

heat of fusion, 85

least squares fitting, 61linearized equations

uncertainties, 28linearized graph, 51linearizing equations, 19, 58

maximum slope, 63minimum slope, 63

origin of graph as a data point, 56

rarefaction, of air, 73

scatter of data, 63slope

maximum, 63minimum, 63

spreadsheet, 9

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120 INDEX

standard error in y-intercept, 66standard error in slope, 66standing waves, 74

uncertaintiesdiscussion of, 113

uncertainties in linearized equations, 28

wavelength, 73waves

compression, 73longitudinal, 71rarefaction, 73standing, 74transverse, 71

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PC142 Lab Manual Wilfrid Laurier University

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