102M Lab Manual

University of Texas at Austin

Last revised: January 6, 2017

Copyright 2017 byThe University of Texas at Austin,

Department of Physics

All rights reserved.

Printed in the United States of America

Laboratory Manual

PHY102M

Laboratory for PHY302K

Spring 2017 Edition

http://www.ph.utexas.edu/

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phy102m

Department of Physics

University of Texas at Austin

Contents

Acknowledgments v

Preface vii

About the Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiUnderstanding Basic Concepts of Physics . . . . . . . . . . . . . . . . . . . . . . . . . vii

Survival Guide viii

Icon Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiEquation Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAnswer Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiPerformance Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

0 Introduction to Experimental Physics 1

0.1 Experimental Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Lab Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Lab Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60.4 Lab Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Prelab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1 Kinematics of Free Fall 17

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2 Background Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3 The Free Fall Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Newton’s Laws and Vector Addition 29

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Vectors in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Newton’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4 The Force Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Newton’s Second law for Translation 39

3.1 Introduction to Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Carts and Pulleys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 The Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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ii CONTENTS

4 Conservation of Energy 474.1 Introduction to Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 The Atwood Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Corrections for the Atwood Machine . . . . . . . . . . . . . . . . . . . . . . . . . 514.4 The Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 Momentum Conservation in Collisions 575.1 Introduction to Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.3 Special Case: One Object Initially Stationary . . . . . . . . . . . . . . . . . . . . 595.4 Carts on a Track . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6 Rigid Body Equilibrium 676.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.3 Conditions for Rigid Body Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 686.4 The Torque Contraption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7 Newton’s Second Law for Rotation 757.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.3 Moment of Inertia for a Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.4 Conservation of Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 777.5 The Rotating Disk: Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.6 Angular Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . 79Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

8 Archimedes’ Principle 858.1 Introduction to Archimedes’ Principle . . . . . . . . . . . . . . . . . . . . . . . . 858.2 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 858.3 Buoyant force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 858.4 Suspended mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 868.5 Geometric Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878.6 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

9 Simple Harmonic Motion 939.1 Introduction to Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 939.2 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 959.3 Pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969.4 Full Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

CONTENTS iii

10 Standing Waves on a String 10310.1 Introduction to Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10310.2 Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10310.3 Waves on a String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10410.4 Waves on a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10610.5 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

11 Heat Energy 11111.1 Introduction to Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11111.2 Heat is Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11111.3 The Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11211.4 Preparation and Warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Appendix A Spreadsheet Basics 121A.1 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121A.2 Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122A.3 Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123A.4 Printing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124A.5 Further Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Appendix B Units and Quantities 125B.1 Common Quantities with Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125B.2 Unitless Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125B.3 Measured Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126B.4 Common Moments of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126B.5 Greek Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Appendix C Capstone Lab Software 128C.1 Capstone Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Appendix D Useful Equations 129

Index 133

Bibliography 134

iv CONTENTS

Acknowledgments

The first edition of this manual was written by P. R. Antoniewicz in 1984 and extensively revised by DavidY. Chao in 1990. During the 1993-94 academic year a new, computer-based version of the laboratorywas developed by J. David Gavenda and Michael D. Foegelle.

Helpful suggestions for improving the experiments and clarifying the manual were made duringthe 1995-96 academic year by Teaching Assistants James Daniel, Daniel Goldman, Nathan Harshman,Robert Luter, and Paul Patuleanu.

The computer-based version of this laboratory course, which was introduced during the 1993-1994academic year, was largely based on an IBM package called Personal Science Laboratory (PSL), con-sisting of probes and software. The original IBM software was revised and rewritten by Foegelle andGavenda specifically for 102M.

In 1997, Team Labs. Inc. introduced new Excel based software called Excelerator to run the originalPSL probes. Beginning in Spring 1998, this new software and some new probes were introduced intothe 102M laboratory course. The manual was extensively revised by Linda Reichl, Robert Luter, andNathan Harshman to accommodate this new software. Some of the experiments have been revised andthe procedures the students must follow have been streamlined.

In spring 2001 an updated version of the software called Excelerator 2001 was introduced in thelab. The hardware interface was also changed from the IBM PSL to the Thinkstation interface madeby Team Labs. Where necessary, the laboratory procedures were revised in the 2002 edition by AnilShaji to be compatible with the new hardware and software with minimal changes to the experimentsthemselves.

In spring 2004 the experiments were extensively re-designed to use the PASCO SCIENCEWORK-SHOP interface and DATASTUDIO software. The new experiments and laboratory procedures weredesigned by Anil Shaji. This edition of the lab manual is designed to be used with the SCIENCE-WORKSHOP interface, DATASTUDIO software and PASCO probes and sensors. Some illustrationsfrom the previous editions have been used in the 2004 edition.

In the Spring and Summer of 2006 further changes were made to the lab manual mainly by MichaelSnyder to accommodate some hardware changes made to labs 3-5 as well as some grammatical corrections.Lab 10 was changed by Nathan Erickson to accommodate new hardware as well. Appendix B was addedby Zhen Wei to help explain the Vernier Calipers. In the Spring and Summer of 2007 further changeswere made to the lab manual. Most of the changes were corrections for consistence and clarification ofthe lab. Appendix C was also added to give the student a reference for units which are used in this lab.These changes were found and corrected by Melissa Jerkins, Megan Creasey, Guru Khalso, Zhen Weiand Nathan Erickson.

In the 2008 edition, Lab 1 was completely changed. Lab 11 was also changed to eliminate mm andkg. Finally, every mention of “slope (y,x)” has been changed to “slope (y vs. x)” to try to eliminate theconfusion with variables. Thanks again to the TAs of the 2007-2008 School year for their suggestionsand corrections as well as the classes that were the guinea pigs of the new version of Lab 1.

In the 2009 editions, most changes were grammar, clarification, or consistency changes. The usefulequations section in Lab 5 was changed to use “pushed (push)” and “stationary (stay)” notation. Therewere also changes to Lab 11 to work out some of the issues. A place for the lab partner’s names wasadded to all the labs. Thanks again to the TA’s of 2008-2009 who were responsible for finding all thecorrections.

In the 2010 editions, most of the changes were again grammar, clarification and consistency. The onemajor change was with Lab 7, where some of the current experiment was reduced and conservation ofangular momentum was added. Thanks to all the TA’s for helping with the changes and to the studentsin the first section of the week who again were guinea pigs for some lab changes.

In the 2011 editions, most of the changes were again grammar, clarification and consistency. Tablenumber blanks were added at the top of the lab pages to assist the TA’s. The Excel addendum wasupdated by Henry Schreiner to reflect the 2010 version of Excel used in the computer labs. Thanks toall the TA’s and students who helped find errors in the lab manual.

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vi ACKNOWLEDGMENTS

In the 2013 edition, lab 0 was rewritten in preperation of a new version of the manual.In the 2014 edition, the manual was heavily revised and reformatted, and several labs were updated.

New equipment was incorporated into the labs. Lab 1 now uses calipers and rulers. Lab 2 was reworkedto enhance comprehension. Lab 5 was rewritten for all new equipment, and restructured for easier dataentry. Lab 11 was reworked. Food for thought and notation corners were added, and an index wasadded. Places for taking notes were added to all labs. A Preface and a Survival Guide were added. Allnew figures and schematic diagrams were added throughout. New software are hardware required manychanges to procedures. New slope fitting was added to several labs. Clear all data runs was removedfrom the labs and the manual. The Excel addendum was changed to be a general spreadsheet tutorial,with individual tutorials for specific programs moved to the website.

In the 2015 and 2016 edition, lab 0 was updated by Shirin Moza↵ari. A question related to graphingwas added to Prelab1. E↵ect of friction was added to the last two parts of lab 3. Some clarificationswere added to lab 6.

In the 2017 edition, new cameras were installed for lab 1 and the lab 1 on the manual was updatedaccordingly. Thanks to all the TA’s and students who helped find errors in the lab manual.

Preface

About the LabThis course, PHY102M, provides an opportunity to test, in a laboratory setting, some of thebasic laws that govern the physical world, and it will help you develop an intuitive understandingof these basic laws. At the same time, you will learn about standard laboratory reportingprocedures and the role of experimentation in testing the basic laws of physics.

PHY102M is designed to be taken concurrently with PHY302K, which provides the theo-retical basis for the experimental work done in PHY102M. However, the course and the lab areseparately run and taught, and the lab is designed to be self-contained. Topics in the lab maybe covered either before or after they are covered in PHY302K.

Understanding Basic Concepts of PhysicsPhysics deals with everyday phenomena, but they must be expressed in terms of certain basicconcepts before the laws of physics can be applied. For example, Newton’s Second Law ofMotion accurately predicts the motion of an object, but it does so by giving the acceleration ofthe object when forces act on it. You cannot hope to use the Second Law until you understandwhat “acceleration” and “force” mean, and you soon find out that simply memorizing theirdefinitions is not very helpful.

In this course you will learn, for example, to find the acceleration of an object from mea-surements of its position at various times. You will learn how to measure the forces acting onobjects in order to apply the Second Law. In the process of carrying out the experiments youshould begin to make the mental connections between the definitions of the concepts and yourreal-world experiments.

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Survival Guide

Icon GuideWhen completing the labs, look for icons to indicate extra pages required. Theœ icon indicatesthat you need to complete this on a computer and print it (in a computer lab or at home), the“ icon indicates a page printed during the lab, and the b icon indicates a handwritten page.These are the lines used to mark pages needed:

œ Include graph: y vs. x ! Fit equation: ! Thistells you to include a graph printed at home, and the order of the variables tells you which oneis on each axis.

“ Include printout: Name of printout ! This tells you to include a printed page from lab.

b Include page: Type of page ! This is for handwritten pages (usually worked examples).

œ Include postlab ! This reminds you about including a postlab printed at home (every lab).

Equation GuideDisplayed equations come in three forms. Numbered equations are important results and oftenare useful for the final exam. Numbered equations with a star are useful for doing the lab (highprobability that you will use this exactly as it appears). Unnumbered equations are there justto explain how other equations are derived.

Answer GuideThere are several kinds of answer blanks.

1. The normal answer blank, , is usually for a single number and a unit.

2. The uncertanty answer blank, ± , expects an uncertainty also.

3. The name and number answer blank, = , expects you to write downthe name used in the manual (a variable, like x or r), and the number with units.

Performance ProblemsMost common causes for poor performance in the lab are:

Not reading the lab beforehand.

Not watching the prelab videos.

Forgetting to fill in an answer.

Forgetting units.

Not looking in the appendix for help with Excel or with units.

Not checking your calculations if you have a large percent error. (If error persists, makea note saying you checked it.)

All TA’s have o�ce hours to help if you don’t understand a question or a graded mistake. Ifyou have trouble, check the UT 102M website: www.ph.utexas.edu/

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phy102m. This has linksto further help on the labs, such as interactive applets. This also has links to the prelab videos.

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Lab 0 Introduction to Experimental Physics

0.1 Experimental Physics

0.1.1 Scientific ProofMost people know that scientific hypotheses or theories are tested by performing appropriateexperiments. It is less well known, however, that experiments can never prove, in the mathemat-ical sense, that a theory might happen to give the same results for a given set of experimentalconditions. At most, experiments can be used to show that a theory is probably false. Thiswould be the case if the experimental results contradicted the predictions of the theory. As aconsequence, scientists spend much of their time trying to show, by a wide variety of experi-ments, that some theory is false. If no one succeeds after many attempts, the theory comes tobe accepted as a law.

0.1.2 Properties of MatterExperiments are not only used to test scientific hypothesis, they can also be used to measurebasic properties of material objects, such mass density, heat capacity, force constants, etc. Wewill do some of that in the laboratory course.

0.1.3 Experimental ConclusionsThe result of any experiment will be one or more numerical values, expressed with an estimateof their uncertainty and units. The final step is to draw a conclusion about the hypothesis ortheory being tested on the basis of these values: Are the results consistent with the predictionsof the theory to within the uncertainty inherent in the experiment? If not, why not? Is thetheory wrong, or is it possible that systematic errors caused the discrepancy? Any statementsabout the e↵ects of a systematic error must be logically consistent. For example, if you saythat friction in the apparatus caused the measured acceleration to di↵er from that predictedby the theory, you must show that friction would lead to a lower acceleration if your measuredvalue is lower than that predicted, and vice versa.

0.2 Lab Methods

0.2.1 Measurement UncertaintyAll measurements have some degree of uncertainty built in simply from the limitations of themeasuring devices. Therefore, every measurement that we make in the lab must be reportedwith the uncertainty due to the measuring device. Below we give “rules” for specifying theerror intrinsic to various types of measurement.

Ruler RuleWe will use the rule that a measurement has an uncertainty equal to 1/2 the smallestinterval on the measurement device. You must guess a digit in the smallest interval touse this rule. This is used, for example, with rulers, meter sticks, and Vernier calipers if youare good at them. For example, if a length is measured with a meter stick, the smallest interval

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2 LAB 0. INTRODUCTION TO EXPERIMENTAL PHYSICS

marks are 1mm apart. Therefore, the uncertainty of a length measurement for this measuringdevice is 0.5mm or �L = ±0.0005m. You must guess an extra digit when you read the meterstick, so the reading might look something like 7.3 ± 0.5mm. You should never report this as7± 0.5mm, as the decimal place of uncertainty and the reading would not match.

Digital RuleMany devices (including some we will use in this lab) give a digital readout of the data theyare measuring. For such devices, the uncertainty is equal to the smallest interval in thedigital readout.

Fluctuating RuleIf the reading fluctuates (some digital readings), the uncertainty is equal to the size ofthe fluctuations. This may occur in two situations. First, if a device can only be read inincrements of 0.03N, for example, it is common in scientific applications to have the devicereport all available digits—but this has an uncertainty of about 0.03N, not 0.01N, since itcan’t read any numbers in between its minimum increment. (Many commercial applications,like phones, will actually display fewer digits to hide this resolution limit).

The second reason you may see fluctuations is because something is actually fluctuating.You are still unsure of the exact reading because it is changing.

Uncertainty is denoted with the small Greek letter delta (�) prefixed before thequantity. If it is part of a measured value, it is listed with a plus/minus symbol(±), and comes after the number but before the unit. If you have best measure-ment x0, and x0 = 2.20m and �x = 0.01m, the correct way to write it is in the formx = x0 ± �x, with the unit last:

x = 2.20± 0.01m

Notation Corner

0.2.2 Significant FiguresThe number of reliably known digits in a number is called the number of significant figures.When making measurements, or when doing calculations, you should not keep more digits inthe final answer than is justified. As a general rule, when adding or multiplying numbers, thefinal result should have only as many significant figures as the least accurate number used inthe calculation. When performing a calculation, always wait until you are writing your finalanswer and then write the final answer with the correct number of significant figures.

The number of significant figures in a stated number may not always be clear. For exam-ple, if someone tells you the distance between two cities is about 920miles, does this mean920± 10miles or 920± 1miles? However, when using scientific notation (numbers expressed inexponential notation), it can always be made clear—the number of decimal places kept is thenumber of significant figures. For example, the number 920 ± 10 is written 9.2 ⇥ 102, indicat-ing that it has two significant figures. The number 920 ± 1 is written 9.20 ⇥ 102, indicatingthat it has three significant figures. The number of decimal places kept tells us the number ofsignificant figures.

Addition/subtraction matches the largest final decimal place, multiplication/division keepsthe smallest number of significant figures.

0.2. LAB METHODS 3

Consider a number w = x + yz which is composed of two numbers y = 1.0 andz = 1.00 which are multiplied together and then added to a third number x = 9.000.Write the number w with the correct number of significant figures (sig figs).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Note that y = 1.0 has 2 sig figs and z = 1.00 has 3 sig figs. The product has 2 sigfigs, so yz = 1.0.a Now add yz to x to get w = 9.000 + 1.0 = 10.0, so w has three sigfigs, since when adding we match the last decimal place.To demonstrate this, we can replace unknown digits with question marks ?. Un-known digits (question marks) stay unknown (since they can be any number from 0to 9) when they are added or multiplied. So, for example,

1 . 0 ? ?+9 . 0 0 01 0 . 0 ? ?

aDon’t remove digits from your calculator! Keep extra digits until you are done if you can.

Significant Figures – Worked Example

0.2.3 Uncertainty in Calculated QuantitiesEvery calculation we make using measured quantities will have an uncertainty associated withit. Let us assume that two quantities, A and B are measured, with measurement uncertainties of�A and �B, respectively. These measured quantities are then written in the form A = A0 ± �Aand B = B0 ± �B, where A0 and B0 are the best measured values of the quantities A andB. Below we show how to add or subtract, multiply or divide, these numbers and obtain theuncertainty �C of the resulting calculated number C = C0 ± �C.

Addition and subtractionIf we calculate the sum, C = A+B, we can replace these with the full expressions for uncertainty,

C = C0 ± �C = A0 +B0 ± �A± �B

Thus, the calculated quantity has value C0 = A0+B0 and its uncertainty is �C = �A+ �B.Note that one must be careful when subtracting two numbers. Never subtract the errors.Always add them. Subtraction would result in C = A�B = A0�B0± �A± �B, and thereforethe same final formula for error.

�C = �A+ �B (0.2.1)?

Multiplication and divisionIf we multiply numbers, C = AB, we can again expand

C0 ± �C = (A0 ± �A)(B0 ± �B)

If this product is multiplied out, we get

�C = |A0| �B + |B0| �A+ �A�B

4 LAB 0. INTRODUCTION TO EXPERIMENTAL PHYSICS

Here, since uncertainties always add, we selected the positive values using the freedom grantedus by the ±. The final term in this equation, �A�B, is very tiny as long as the uncertaintiesare smaller than the best values, so we can safely remove it.

�C = |A0| �B + |B0| �A (0.2.2)

We can use the above formula if needed for multiplication problems, or we can go a few stepsfurther to get a more general form we can use with division too. We divide both sides by A0B0

to get�C

|A0| |B0|=

�A

|A0|+

�B

|B0|Remembering that A0B0 = C0 finally gives us the formula for error:

�C

|C0|=

�A

|A0|+

�B

|B0|

This is sometimes called relative error, because each term is the fractional error of thequantities. You can work this through for division1 This multiplication and division rule ismost commonly written in a more useful (but possibly harder to remember) form:

�C = |C0|✓

�A

|A0|+

�B

|B0|

◆(0.2.3)?

Raising to a powerMost common operations are now accounted for, but let’s include a quick short-cut for raisingto a power (C = An). We can just take the multiplication rule over and over again, and theresult is

�C = nC0�A

A0(0.2.4)?

Let us consider the number W = X + Y Z which is composed of three measurednumbers X = 9.000 ± 0.001, Y = 1.0 ± 0.1 and Z = 1.00 ± 0.01. Compute Wincluding its uncertainty.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Using the notation of Section (0.2.3), X0 = 9.000, �X = 0.001, Y0 = 1.0, �Y = 0.1,Z0 = 1.00, and �Z = 0.01. First multiply Y and Z to get Y Z = Y0Z0 ± (Y0�Z +Z0�Y ) = 1.0± 0.1. Now add this to X to get W = 10.0± 0.1.

Measurement Uncertainty – Worked Example

0.2.4 Error and Percentage ErrorThere may be error sources, both systematic and random, not accounted for by the measurementuncertainty of our measuring devices or by our theory. For example, in experiments which testthe laws of dynamics, we try to eliminate the e↵ects of friction and of air resistance as much aspossible, but they can never be entirely eliminated. Both of these e↵ects will cause systematicdeviations of our results from our theoretical predictions.

1Hint: Any time you have two � quantities multiplied by each other, the result is very small

0.2. LAB METHODS 5

0.2.5 Absolute Percentage ErrorWhen performing a measurement of some physical quantity, we need to find as many ways aspossible to check to see our results are reasonable and can be trusted. Have sources of errorcrept in that we are unaware of? If the experiment measures some quantity, V , for which thereis a known “accepted” value, Vaccept, coming from other careful experiments, then we mightcalculate the percentage error, E

V

(%), of our result, Vmeas, from the “accepted” value, Vaccept.The percentage error is defined as

EV

(%) =

����Vaccept � Vmeas

Vaccept

����⇥ 100% (0.2.5)?

The percent error gives an estimate of the discrepancy between our measured result andthe accepted result. If the discrepancy is larger than expected considering the known inherenterrors in the experiment, then we must re-examine all aspects of the experimental procedureuntil we understand the source of the large discrepancy.

This also is a good way to test error in a conserved quantity; in this case, you would replaceaccepted with initial, and measured with final in this formula.

0.2.6 Relative Percentage ErrorSometimes we do a measurement of some quantity, V , for which there is no known acceptedvalue. However, if we measure V in two di↵erent ways and get results, V1 and V2, then we cancheck the internal consistency of our measurement process by computing a relative percentageerror

EV

(%) =

����V1 � V2

(V1 + V2) /2

����⇥ 100% (0.2.6)?

We will use both the absolute percentage error and the relative percentage error in thislaboratory course.

0.2.7 Units of Physical QuantitiesAll physical quantities have “units” associated with them, and the size of these units is de-termined by international agreement. In this lab, we generally use the S.I. system of units,which is the system of units most appropriate for scientific purposes. In the S.I. system, lengthis measured in meters (m), time in seconds (s), mass in kilograms (kg), and temperature inKelvin (K). We will encounter a number of physical quantities whose units are combinations ofthe basic units mentioned above. For example, energy is measured in units of Joules (J) and1 J = 1 kg·m2/s2.

Whenever a physical quantity is measured, the reported value must include both the nu-merical value and the unit associated with that physical quantity. For example, if an experimentmeasures the acceleration of gravity, the reported value would be of the form g = 9.8± 0.1m/s2

(if the error in the measurement were of order 0.1m/s2). If your numbers do not includeunits, you will lose points!

Three important facts about units:

1. Units can be multiplied and divided according to the laws of algebra.

2. Units that are not equivalent (mass and length, for example) cannot be added or sub-tracted.

6 LAB 0. INTRODUCTION TO EXPERIMENTAL PHYSICS

3. For every equation that you write down, the units of each term in the equation are thesame.

This provides an important way to check if your equation is correct. For example, thedisplacement x at time t of a particle that experiences constant acceleration a can be writtenin terms of the displacement x0 at time t = 0 s, the velocity v0 at time t = 0 s as

x = x0 + v0t+1

2at2 with units ! m+ (m/s) s + (m/s2) s2 = m

Notice that after cancellations (according to the rules of algebra), every term in this equationhas units of meters.

0.3 Lab Tools

0.3.1 Vernier Caliper

0 5 10 15 20 25 30 mm

0 5 100.1mm

Figure 0.1: A closed Vernier Caliper.

A Vernier caliper is a measuring tool designed to get better readings than a ruler. A caliperis just a device with a sliding jaw that you put on something to measure it. Usually there areouter jaws to measure objects, and inner jaws to measure cavities. Most calipers also have aVernier scale. This is a special scale that uses a simple mathematical trick to enable betterreadings.

A Vernier scale is designed by matching two separate scales at slightly di↵erent spacing.For the pictured caliper, the lower (Vernier) scale has 10 marks in 9mm. This means the firstmark is at 0, followed by 0.9mm, then 1.8mm, etc. If you ignore the whole number part andonly look at the decimal, you’ll see that it decreases by 0.1mm each time. This is the trick thatis used; for it is easier for you to see which set of lines matches than to see lines 0.1mm apart.

To read a caliper, first read the main scale as you would a normal scale. The point youneed to read is the location of the 0 vernier line on the normal scale (look at a closed caliper,the reading should be 0, see Figure 0.1). Then, to get another digit, read the vernier scale bymatching lines. If the 3rd line past 0 matches a line on the main scale, the vernier reading is 3.(See Figure 0.2)

0.4. LAB REPORT 7

0 5 10 15 20 25 30 mm

0 5 100.1mm

6.3 mm

6.3 mm

First, read 6mm from main scale

Then get final digit, .3mmby matching the third line

Figure 0.2: An open Vernier Caliper.

Always make sure your reading is reasonable. If you would have guessed the measurementto be between 6mm and 6.5mm, your vernier reading should come out to be between 6mmand 6.5mm.

Even though your first time or two reading a vernier scale may be challenging, with practice,you will find they are fast and easy to read.

0.4 Lab Report

At the end of each chapter there is a Lab Report section which will help guide you in takingdata during the experiment, and in analyzing that data afterwords. This set of sheets canbe torn out of the lab manual and must be turned in as part of your lab report. Your LabInstructor will give you additional questions and topics to discuss in a post-lab that also partof your lab report.

0.4.1 Graphs

t (s) x (m)0.3 1.11.3 1.53.5 2.05.5 2.9

x = 0.333t+ 1

x vs. t

x(m

)

t (s)

1

2

3

1 2 3 4 5 6

Figure 0.3: An example graph.

A graph provides a visual picture of your data, and can show qualitative features not obviousfrom looking at the numbers alone. Figure 0.3 shows a set of data along with it’s matchinggraph, and a linear fit line.

8 LAB 0. INTRODUCTION TO EXPERIMENTAL PHYSICS

Often your labs will require you to make graphs in a program such as Excel. These shouldbe done at home, or on a flash drive or personal computer in lab. You should make your owngraph, you cannot share anything except raw data with your partners. A graph should alwaysinclude the following parts:

Title The “vs.” always indicates vertical axis versus horizontal axis.

Labels on both axes Include units.

Tick marks The tick marks provide numerical increments for the values of plotted quantities(grid optional).

Points Always use distinct points, never use a connect-the-dots form.

Fit line We shall often be asked to plot a curve that best fits the data obtained in an experi-ment. This can be obtained either by hand or by using a software package like Excel.

Fit equation It is often possible to write an equation that represents the curve that best fitsyour data. This is particularly simple if the curve is straight line. If the curve is not astraight line then a software package like Excel can be used to find the fit. Excel uses yand x to refer to the vertical axis and the horizontal axis, respectively.

Your graphs should be as large as possible, usually a full page. Make sure you know whatfit is needed (linear or polynomial order 2 in this lab). It is possible to have points that looklike a line that are described by a second order polynomial equation. You will often comparethe equation of a graph with an equation from lab to find the values of the coe�cients.

For a graph (with vertical axis y and horizontal axis x) whose data is fit by a straight line,the equation describing that empirical curve can be written in the form y = mx+ b, where mis the slope of the best fit curve and b is the y-intercept. You can obtain the average slope mbetween two points y2 = y(x2) and y1 = y(x1) by writing

m =y2 � y1x2 � x1

=�y

�x,

where �y = y2 � y1 and �x = x2 � x1.For example, given the data and straight-line curve in Figure 0.3, we can write x = v0t+x0,

where the slope of the curve is v0 = 0.333m/s and the intercept is x0 = 1m.

Lab 0 Notes

This is for initial comments, work, etc. that you may have from the reading and the prelabvideos. Put down anything you think might help for the upcoming lab.

9

10 LAB 0. INTRODUCTION TO EXPERIMENTAL PHYSICS

Lab 0 Worksheet

Name: Table: Partner(s):Note: This lab worksheet will not be graded. This is intended to prepare you for working inthis lab.

Significant figures

What is the correct way to report d = 0.13724± 0.00247m?

±

Given the following numbers, write the answer with the correct number of significant figures.Include units.

A = 3.000m, B = 1.11m, C = 0.004m, D = 2.02 s

A⇥B =

A⇥ C =

A/B =

A/D =

A5 =

Error Propagation

Given the following numbers, write the answer with the correct error and significant figures.Include units.

A = 3.000± .005m, B = 1.11± .05m, C = 0.004± 0.001m, D = 2.02± .08 s

A⇥B = ±

A⇥ C = ±

A/B = ±

A/D = ±

11

12 Lab 0. Worksheet

A5 = ±

Assume a = 27± 2, b = 3.0± 0.5, c = 2.0± 0.2, and d = a

b

� c. Then, compute the value of d:

d = ±

Now, compute the maximum value that d can be (hint: a should be a = 12+2 = 14. Experimentto maximize d).

dmax =

Do the same to find the minimum value for d.

dmin =

Now, propagate the uncertainty as seen in section 0.

�d =

Compare dmax with d0 + �d, and dmin with d0 � �d.

Lab 0. Worksheet 13

Graphing

Fill in the missing parts of this graph of d (distance in meters) vs. t (time in seconds).

y = 0.500x+ 1

1

2

3

4

5

1 2 3 4 5 6

14 Lab 0. Worksheet

Vernier Calipers

Read the following calipers (include uncertainty and units, as always):

0 5 10 15 20 25 30 mm

0 5 100.1mm

Reading =

Read the following calipers:

0 5 10 15 20 25 30 mm

0 5 100.1mm

Reading =

Measure the length of the object the cylinder in your box. Record that here:

Length =

œ Include postlabTA sign-o↵:

Lab 1 Lab 1 Prelab

Name:

0 0.5 1 1.5 2 2.5 3

0

1

2

3

4

5

6

7

8

9

t (s)

x(m

)x vs. t

1. A ball is rolling along the x-axis as shown above, under constant acceleration. Solve thefollowing. Show your work.

(a) Find its displacement, �x1, during the time from 0 to 1 seconds.

(b) Find its average velocity, v1, during the time from 0 to 1 seconds.

(c) Find its average velocity, v2, during the time from 1 to 2 seconds.

(d) What is t when the object’s velocity, v1, is equal to its average velocity, v1, duringthe first time interval? Think about the shape of the v-t graph.

15

16 Lab 1. Prelab

2. Below is the plot of v (velocity) vs. t (time) for an object. The equation of the fitted lineis written on the graph. Use that equation to find the slope and intercept of the graph.What physical quantity the slope is equal to?

y = 0.50x+ 11

v vs. t

v(ms)

t (s)

11

12

13

14

15

1 2 3 4 5 6

Slope=

Intercept=

Slope represents=

3. Using the table below, calculate and insert the missing values in the following tables.

j xj

(cm)

0 2.5

1 8.2

2 13.0

3 16.4

4 19.0

tj

is the time at which the jth video frame is captured. The position, xj

, is the positionof the steel ball in the jth frame. (The data is taken 30 times per second.)

j tj

(s) xj

(m)

0 0.00 0.0250

1 0.0333 0.0820

2 0.0667 0.130

3

4

j tj

(s) vj

(m/s)

0.5 0.0166 1.71

1.5 0.0500 1.44

2.5 0.0833 1.02

3.5

j a (m/s2)

1 -8.10

2 -12.6

3

Average Acceleration: atable =

Lab 1 Kinematics of Free Fall

1.1 Introduction

Kinematics is the description of the motion of material objects. The description of a movingobject is a record of the position of the object as a function of time. Once this record of positionversus time is obtained, it can be used to obtain the velocity of the object (the rate of changeof position) and the acceleration of the object (the rate of change of velocity).

In this lab, we study the kinematics of an object that is undergoing two particular kinds ofmotion simultaneously, constant velocity in the horizontal direction and “free fall” in the verticaldirection. An object will have constant velocity in the horizontal direction if no horizontalforce acts on it. An object undergoes free fall motion when the only force on the object is thegravitational attraction of the Earth. One goal of this experiment is to show that the horizontaland vertical components of motion are independent of each other.

While we can come close to ideal unaccelerated motion and free fall, neither of these perfectlyoccur on Earth because of air resistance. When an object moves through the air, it experiencesa “drag” force which opposes its motion. This drag force is proportional to the squared speed ofthe object and the cross sectional area perpendicular to the motion. An additional goal of thisexperiment is to show that the e↵ects of air resistance can be made small if the cross sectionalarea is small and speeds are low. Then the horizontal and vertical motions experience verysmall deceleration and we can measure the acceleration due to gravity during free fall with afair degree of accuracy.

The object whose motion we will study is a steel sphere (its small size and large inertia willhelp reduce the e↵ects of air resistance). A web cam will be used to record the position of thesteel ball, at discrete times, as it moves through the air. A grid placed behind the steel ball willenable us to see each position of the steel ball recorded by the webcam (each frame), and therepetition rate of the webcam will enable us to obtain the time at which each frame occurred.

The major errors in this lab will be due to the resolution of the camera and our ability toaccurately find the position of the steel ball as a function of time.

What type of error is introduced by the resolution of the camera?What about the drag force? (Systematic or Random)

Food for thought

1.2 Background Discussion

1.2.1 Constant VelocityFor an object with a constant velocity v

x

along the x-direction, if the initial position is xi

atthe initial time t

i

, the position at a later time tf

is

xf

= xi

+ vx

�t (1.2.1)

17

18 LAB 1. KINEMATICS OF FREE FALL

where �t = tf

� ti

. The quantity �x = xf

� xi

is the displacement of the object during thetime interval �t.

1.2.2 Constant AccelerationLet us now consider a slightly more complicated set of conditions. Assume a small mass movesfreely along the x-direction under the influence of a constant force and, therefore, a constantacceleration, a. This will cause the velocity to change as time passes. However, let us assumethat we know the initial conditions (the velocity, v0 and position, x0 at the initial time t0 = 0).If we are able to record the position x at a time t, we can relate them to the initial conditionsand the acceleration as follows:

x = x0 + v0t+1

2at2 (1.2.2)?

andv = v0 + at. (1.2.3)?

These two equations are only valid if acceleration is constant. We can see thatequations (1.2.2) and (1.2.3) reduce to equation (1.2.1) if the acceleration is zero. Finally, if theonly force is due to gravity then we normally use the variable g to represent the acceleration dueto gravity. Since we usually consider the upward direction to be positive, and the accelerationof gravity is directed downward, we have the common replacement a = �g.

As an example, let us consider a cart rolling on a frictionless track with a fan ontop. The fan provides a constant acceleration of 1m/s2. Starting from rest, how fastis it moving after 10 seconds? How long does it take to go 50m?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .For the first part, we just need to use equation (1.2.3) to calculate velocity given atime,

v = v0 + at = 0 + (1m/s2)(20 s) = 20m/s.

And, for the second part, we can use equation (1.2.2) and solve for time, using thefact that it starts from rest and has moved from x0 = 0m to x = 100m, and we get

x =1

2at2

t =

r2x

a=

r100m

1ms2= 10 s.

Worked Example

1.2.3 Computing AveragesIn order to complete our experiment, we’ll need to compute the average velocity and averageacceleration, from a series of data points, for the case when the acceleration is not constant. Ifwe know the position x

i

if an object at time ti

and its position xf

at the later time tf

, then itsaverage velocity during this time interval is

v =�x

�t=

xf

� xi

tf

� ti

. (1.2.4)?

1.2. BACKGROUND DISCUSSION 19

At what time during the motion between xi

and xf

is this average velocity actually reached?A simple method is to take the average of the two times, t

f

and ti

, called the midpoint time.This has an added bonus; when the acceleration is constant, the average velocity isthe velocity at the midpoint time. If the acceleration is not constant, it is still a betterapproximation than choosing one of the two endpoint times.

v(m/s)

t (s)

1

2

3

4

5

1 2 3 4 5 6 7 8 9

(t i,v i)

(tf, vf

)

�t

�v

v vs. t

Figure 1.1: Average acceleration example graph.

We can also find average acceleration (shown in Figure 1.1):

a =�v

�t=

vf

� vi

tf

� ti

. (1.2.5)

Even if the line is not as straight as the one in Figure 1.1, this still gives you the averageacceleration between the two chosen points.

1.2.4 The j LabelIn this lab, we will follow the motion of the steel sphere with a webcam that records the positionof the sphere at discrete times t

j

(j = 0, 1, ...). Each record of position is called a “frame”and the rate at which the positions are recorded is called the “frame rate”. The frame-numbersubscript j, will increment starting from j = 0 (which is the first usable recorded point) tothe last time recorded. Midpoint times are labeled by using half values for j. To calculate theactual time t

j

for each value of j, you can simply use the frame rate f (in frames per second):

tj

=j

f(1.2.6)?

With this notation, if the label for the initial point is j � 0.5, and the label for the final pointis j0 + 0.5, so that the average velocity equation (1.2.4) becomes

v =�x

�t=

xj

0+0.5 � xj�0.5

tj

0+0.5 � tj�0.5

(1.2.7)

We can write a similar equation a = �v

�t

for the average acceleration (1.2.5).

20 LAB 1. KINEMATICS OF FREE FALL

All of the above equations were written using the position variable x, but they aregenerally rewritten with position variable y when describing motion in the verticaldirection. Will the acceleration a be di↵erent for x and y?

Food for thought

1.3 The Free Fall Ball

We are going to test our theory by watching an object in free fall. Free fall occurs wheneveran object does not have any forces other than gravity acting on it. Air resistance does a↵ectfree fall, but not significantly at slower speeds for small massive objects. We hope to be ableto measure the acceleration due to gravity, and to show that we can measure and calculatevertical displacement y and horizontal displacement x separately for the steel sphere.

Figure 1.2: Bouncing ball and webcam setup.

We are going to film the motion of the steel sphere with a web cam (shown in Figure 1.2),as it is bounced past a sheet of paper with grid lines. We will use the grid lines to record theposition of the sphere at discrete times t

j

. You have a two-sided sheet of graph paper in yourworkbook to use for the two runs. The sheet is attached to the “wall” behind the bouncingsphere.

In the first run we’ll bounce the ball and only analyze the vertical component of the motion.The bounce allows us to track frames for the sphere going down and up, versus dropping it andonly tracking downward frames. As long as nothing touches the ball between the starting timet0 and ending time t

j

max

, we have free fall.In the second run, we will arc the ball across the paper, and track both vertical and horizontal

distances.

1.4. PREPARATION 21

1.4 Preparation

Figure 1.3: Proper settings for the sliders in Webcam Settings.

To do the experiment, you need to open two programs: Photo Booth and WebcamSettings. These programs are in the Application folder in the dock of the MAC computer.Click on the application folder and then double click on each of the programs to open them. Wefirst adjust the settings of the camera using Webcam Settings, and then record a movie ofthe ball as it falls using Photo Booth. On Photo Booth, you should be able to see the gridsheet in front of the camera. Make sure the entire grid behind the ball is visible and reasonablyflat. On Webcam Settings, under “Basic”, click on “Manual”, and change the “ExposureTime” to 30. You can see the actual number for the exposure time by clicking on the word“Exposure Time”. Adjust the rest of parameters (Gain, Brightness, Contrast,...) so that youcan see the fine-grid on the board. The ball is very fast so a very low exposure time helps tohave a less blurry picture of the ball. A slightly too dark image is usually clearer than a slightlytoo bright image. See Figure 1.3 for an example of proper settings. Under “Performance” setboth the “Read auto settings from webcam” and “Write every settings to webcam” to “Every0.5 seconds”. You are now ready for recording:

To take a movie, press the record button on Photo Booth and drop the ball very close tothe grid, but not touching the grid. After you finished recording, drag the file to the Desktopand open the movie file by double clicking. In order to see the movie frame by frame, pause it(by clicking on the I button) and use the arrow key to advance to the next frame. Once theball bounces o↵ the table copy (by hand) the position of the ball at each frame on the providedgrid sheet on the manual. One movie that has at least eight frames of the ball is su�cient. Ifin any of the consecutive frames of the movie, the ball bounced out of the grid try recordinganother movie while dropping the ball at a slightly lower initial height.

The frame rate for the camera is set to 30 frames/second.

22 LAB 1. KINEMATICS OF FREE FALL

Lab 1 Notes

This is for initial comments, work, etc. that you may have from the reading and the prelabvideos. Put down anything you think might help for the upcoming lab.

23

Lab 1 Guidelines

Raw Data

First datasetFilm the ball in free fall for at least 8 consecutive frames, more than 8 is better. You will needto bounce the ball and then track the part after the bounce in order to get enough frames. Youcannot include a bounce for it to be in free fall! Transfer data from the screen to the graphpaper at the end of this lab. Number the points on the graph paper to keep track of direction.Use a ruler to measure the positions and record them in the worksheet.

Looking at the ball diameter on the screen, try to measure its apparent diameter based on thesquares behind it.

Second datasetRepeat the first experiment, only this time bounce the ball across the paper in an arc; recordboth vertical and horizontal motion.

Analysis

TablesYou’ll need to calculate both v and a for midpoint times, though only on your first dataset.

Graphing the datasetsMake the graphs indicated in spreadsheet software and include them with your lab report. Besure your fit lines match the form of the expected equations (1.2.3) and (1.2.2). Copy the fitequations from the graphs, and use the coe�cients to answer the questions below.

24

Lab 1 Worksheet

Name: Table: Partner(s):

Run 1: Vertical

Frame Rate: fframerate = Note: Always include units!

Ball Diameter: dreal = ± Note: Must measure with calipers!

dapperent = Is it di↵erent from the actual diameter?

j tj

(s) yj

(m)

0

1

2

3

4

5

6

7

8

9

10

11

j tj

(s) vj

(m/s)

0.5

1.5

2.5

3.5

4.5

5.5

6.5

7.5

8.5

9.5

10.5

j aj

(m/s2)

1

2

3

4

5

6

7

8

9

10

Average of above values: atable =

Graphing the first datasetœ Include graph: y vs. t ! Fit equation:

Time at the highest peak on the y graph: Tmax =

Use Tmax and the fit line eqn. to get maximum height: Ymax =

From the y vs. t graph coe�cients: v0 = a =

œ Include graph: v vs. t ! Fit equation:

From the v vs. t graph coe�cients: v0 = a =

25

26 Lab 1. Worksheet

Run 2: Both Directions

j tj

(s) xj

(m) yj

(m)

0

1

2

3

4

5

6

7

8

9

10

11

Graphs of the second datasetœ Include graph: y vs. t ! Fit equation:

Using the values of the best fit line coe�cients on the y vs. t graph:

vy,0 = a

y

=

œ Include graph: x vs. t ! Fit equation:

Using the values of the best fit line coe�cients on the x vs. t graph:

vx,0 = a

x

=

b Include page: Work

Make sure you include the following sheet of graph paper, too, with your numbered points.

œ Include postlabTA sign-o↵:

Lab 1. Worksheet 27

28 Lab 1. Worksheet

Lab 2 Newton’s Laws and Vector Addition

2.1 Introduction

We live in a three-dimensional world and, consequently, many of the quantities we work withhave both a magnitude and a three-dimensional direction—we call these vector quantities. Wealso work with quantities that do not have a direction, but only a magnitude - we call thesescalar quantities. For example, velocity is a vector quantity. To fully specify the velocity ofa moving object, we must specify the magnitude of its velocity (its speed), and the directionof the motion. A volume is a scalar quantity; for example, a liter or a cup does not have adirection.

Sometimes our motion is constrained to one or two dimensions. For example, If we walkalone a straight flat road, we are moving in one dimension. There is no vertical or sidewayscomponent to our velocity, so we can neglect those other two dimensions. If your motion hasonly one dimension, it is su�cient to just use a sign to indicate direction. One direction ispositive, and the other is negative.

In this lab, you will learn how to work with vectors in two dimensions. In Lab 1, youactually studied motion in two dimensions, but kept them separate. In this lab, we will learnhow to express two dimensional motion in vector notation.

We often use one of three ways to indicate a vector.Textbooks often use bold upright font, A

Handwritten equations often use an arrow, ~A

On the blackboard, you will sometimes see double lines, AIf a vector is not marked as one, it indicates the magnitude of the vector (A = |A|).

Notation Corner

2.2 Vectors in 2D

In this lab, you will only work with two dimensional vectors (2D vectors). However, everythingwe will cover is fairly simple to extend to 3D with the right mathematics. It is just harder tomeasure and visualize in 3D.

A vector can be thought of as an arrow, where the length is the magnitude of the vector, andthe direction of the arrow indicates the direction of the vector. Vectors do not have a location!The vector arrow can be moved anywhere on the graph without changing its properties as longas one does not change its length or its direction. This should give you an idea of how to addvectors.

As shown in Fig. 2.1, vectors can be drawn on a graph. Their direction is always measuredin terms of the angle they make with the positive x-axis in a counterclockwise direction.

Graphical AdditionTwo vectors, A and B, are added by joining the tail of the B arrow to the head of the A arrow.The resultant vector C = A+B is an arrow drawn from the tail of A (at origin) to the head

29

30 LAB 2. NEWTON’S LAWS AND VECTOR ADDITION

x

y

135

3 cm@135

Figure 2.1: An example vector.

of B. The length of C can be measured with a ruler and the direction of C can be measuredwith a protractor. Always include a scale.

A

BC=A+B

1 cm = 2 N

Figure 2.2: Vector addition example.

Remember that angles are always measured counterclockwise from the x-axis (0 ). Theresultant vector always points from the original starting point to the last vector tip. Thisshould give you a common-sense notion of what vector addition does, it is the shortest wayfrom the beginning to the end. As the crow flies, so to speak.

Algebraic Addition

Algebraic addition is another way to add vectors. To add, simply split up all the vectors intoperpendicular (orthogonal) components (in our case, x and y), and then add the componentslike normal numbers.

A+B = C =)⇢

Cx

= Ax

+Bx

Cy

= Ay

+By

(2.2.1)

If you are given the components Ax

and Ay

of a vector A, you can easily find the lengthA and direction ✓ of the vector. Remember that A

x

= A cos(✓) and Ay

= A sin(✓). Then

A =qA2

x

+A2y

and tan ✓ = A

y

A

x

. However, note that there can be some ambiguity regarding

the angle ✓. If you use ✓ = tan�1 A

y

A

x

, your calculator may only give answers from �90 to90 (in the first and fourth quadrants), even if the correct answer lies in the second or thirdquadrants (between 90 and 270 ). You can recover the correct angle (90 to 270 ) by adding180 to the answer if A

x

< 0. If you want to write a positive angle, you can always add 360 tothe angle without changing it.

2.3. NEWTON’S LAWS 31

As an example, let us consider the vector shown in Fig. 2.1, which we will call A.The length of A is A = 3 cm and the angle it makes with the positive x-axis is ✓ =135 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Thus,

Ax

= (3 cm) cos (135 ) = �2.12 cm and Ay

= (3 cm) sin (135 ) = 2.12 cm.

If we had more vectors, we could add them using these components.

Worked Example

2.3 Newton’s Laws

Armed with vectors, let’s consider Newton’s laws again. If N di↵erent forces act on an object,then Newton’s equation is F

net

= ma , where the net force

Fnet =NX

j=1

Fj

⌘ F1 + · · ·+ FN

is equal to the vector sum of all forces acting on the object. Thus, when you have multipleforces on a single object, they add according to the rules of vector addition. In this lab, wewill only consider an object at rest (in equilibrium), even though several forces act on it. Thismeans that the net force

Fnet = 0 (2.3.1)?

So, in all our experiments in Lab. 2, the combinations of forces should add up to zero.Let’s take a moment to look at the units of force. We know that Fnet = ma, so force must

have units kgm/s2 using S.I. units. This combination of fundamental units is called the Newton.One Newton of force has the value 1N = 1 kg·m/s2.

2.4 The Force Table

A force table is a table that allows masses to hang o↵ the sides on pulleys. The hanging massesadd tension T to the string. The string then exerts a force, with magnitude T , on each objectthat it is attached to. The direction of the force is tangent to the string at the point of contact.In these experiments, the strings all attach to a ring in the center.

When the forces exerted by the strings add up to zero (vectorially), there is no net force,and the ring in the center will stay in the center. When there is a small net force, the ring willshow a tendency to move/stay in one direction when tapped. The friction in the pulleys makesvery small forces undetectable. If there is a fairly large net force on the ring when it is at thecenter, it will be pulled to one side and to a position where the net force is zero.

Free Body DiagramsA useful way to visualize the forces on an object is to draw a free body diagram. One way toconstruct a free body diagram is to draw all the vector forces on an object so the tails of the

32 LAB 2. NEWTON’S LAWS AND VECTOR ADDITION

Figure 2.3: The force table.

0

30

6090

120

150

180

210

240270

300

330

2N @ 0

3N@135

3N

@270

Figure 2.4: A free body diagram.

vectors lie at the same spot. Note that this is not a vector addition; you cannot measure a sumfrom this plot. Here’s an example of a mass with three forces:

This example does not add up to 0. What is the resultant? What will the objectdo?

Food for thought

There were three concepts introduced here:Algebraic addition: Adding vectors by adding components.

Graphical addition: Adding vectors by placing them tip to tail and measuring.

Free body diagram: A way to visualize vectors. You cannot use this to addvectors.

Note that vectors can only be added using one of the described methods. You can-not add vectors by adding magnitudes and angles directly.

Notation Corner

Lab 2 Notes

This is for initial comments, work, etc. that you may have from the reading and the prelabvideos. Put down anything you think might help for the upcoming lab.

33

Lab 2 Guidelines

Raw Data

Two StringAttach two masses on to the ring (via strings). Hang them o↵ at 0 and at 180 . The 0 degreemass should be 200 g, including mass hanger. How much mass do you have to add (1 g at atime) to one of the masses and to get the ring to move when tapped (no longer in equilibrium)?Write down the actual measurements1 of the masses you used. Subtract to find the maximumextra mass you can add while still keeping the system in equilibrium.

Three Strings

You are given two masses and angles for the three strings. Try to calculate algebraically andgraphically what the remaining mass and angle should be (calculation), then test it in lab(measured). Remember that you want to find the sum, and then put a force equal and oppositeto make the final sum 0. Note: to get an opposite vector, add 180 to the angle.

Test your values on the force table and try to observe equilibrium. Tap the ring to make sureit does not move. Write down measured masses.

On the table for the Three Strings part, Expected vector is the vector that is supposed to cancelthe sum vector.

Draw a force diagram, too.

Four Strings

On this one, you will not precalculate the values. Just experiment, and see if you can getthe ring to look like it is in equilibrium. You’ll test the lab answer later. Ask your TA to fill inthe TA blank for your group before starting.

Find the resultant algebraically and graphically. You will be summing four vectors instead oftwo.

You probably missed true equilibrium by a little. Let’s see how close you got. First, how faro↵ do you think you were (maximum force error)? Remember, there are four pulleys, you canuse the results of the two string experiment.

1Use triple beam balance

34

Lab 2 Worksheet

Name: Table: Partner(s):

Two Strings

Angle Measured m (g)

0

180

Maximum extra mass:

Maximum extra force (carefully convert to N): �F2 =

Maximum extra force per pulley: �F

Three Strings

Three String Masses, CalculationRemember: sum the components only, you cannot sum magnitude or direction!

Name Mass(g) Direction Magnitude (N) x-comp (N) y-comp (N)

First 150 g 0

Second 150 g 90

Total (vector sum):

Expected vector:

Three String Masses, Measured

Angle Expected m (g) Measured m (g)

0 150 g

90 150 g

35

36 Lab 2. Worksheet

Three String GraphicalMake vectors as large as possible! Must fill over half the grid. Add forces from first and secondstrings to get the sum. You will not be drawing the expected vector. Make sure you includearrow heads to indicate direction. Always report measured magnitude and direction for anyvector sum. You can’t copy your algebraic results for the sum!

Scale: Sum: @

Lab 2. Worksheet 37

Force Diagrams (3 string, 4 string)

0

30

6090

120

150

180

210

240270

300

330

0

30

6090

120

150

180

210

240270

300

330

Four Strings

Angle Expected m (g) Measured m (g)

0 100

75 100

TA:

Error

Maximum force error, 4 pullies: �F4

How did you calculate this?

Four String Algebraic Resultant

Name Direction Magnitude (N) x-comp (N) y-comp (N)

First 0

Second 75

Third

Fourth

Resultant (sum):

How did the magnitude of the resultant compare with the maximum error? (Smaller/Larger)

38 Lab 2. Worksheet

Four String Graphical ResultantRemember there are 5 vectors on this graph! Four measured vectors and a resultant (sum)vector.

Scale: Sum: @

œ Include postlabTA sign-o↵: