Embry-Riddle Aeronautical University

Fall 2010

PS 195Q Laboratory Manual

Editor:

Ellie Jesse

Editor:

Dr. Andri Gretarsson

August 30, 2010

Contents

General Laboratory Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiGuidelines for Laboratory Notebooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiFormal Report Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vVector Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiLab 0: Error in Measurements - Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiLab 1: Experimental Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Appendix A: Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Appendix B: Statistics and Frequency Plots . . . . . . . . . . . . . . . . . . . . . . . 11

Lab 2: Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Lab 3: Gravitational Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Lab 4: Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Appendix C: A Detailed Description of Friction . . . . . . . . . . . . . . . . . . . . . 31Lab 5: Projectile Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Appendix D: A Detailed Description of Kinematic Equations . . . . . . . . . . . . . 38Lab 6: Centripetal Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Lab 7: Elasticity of Materials and Young’s Modulus . . . . . . . . . . . . . . . . . . . . . 44Lab 8: Buoyancy and Archimedes Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Appendix E: A Detailed Explanation of Bouyancy . . . . . . . . . . . . . . . . . . . 51Lab 9: Thermal Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Lab 10: Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Lab 11: Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Lab 12: Magnetic Fields and Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . 63

i

General Laboratory Requirements

Required Materials for the Laboratory:

1. PS 103 Laboratory Manual (this booklet).

2. Two good quality laboratory notebooks: 8 1/2” x 11” acid free paper, quadrille ruled witha sown binding. You may continue to use the same notebook from one semester to the nextand acquire a new one when the first is used up. You must have two lab books so that onecan be turned in while the other is used for that weeks experiment and then vice versa.

3. Pen. Pencils may only be used for plotting data. Many students have found it helpful to havemultiple colors for color coding and graphing, although this is not required

4. Scientific calculator.

General Laboratory Procedures:

1. Read the description of the experiment before entering the laboratory. You will learn moreand enjoy the experience better if you come to class prepared. Your laboratory partner andyour instructor will expect you to have done this! Read the experiment over the night beforeto allow it to sink in and to give yourself time to organize your work.

2. Record all of your data and observations in your laboratory notebook, as described in detailbelow.

3. No food allowed in the laboratory at any time. Water bottles are allowed provided they arestored at the front of the lab on the designated table and are not removed from that area.

4. Closed-toe shoes are required in the laboratory.

5. Lab coats may be used but are not required. However, loose-fitting and/or highly flammableitems of clothing should not be worn in the laboratory. Sleeves must be close-fitting. Studentswho prefer to use a lab coat to cover otherwise unacceptable items of clothing, may do so.Lab coats should be of professional quality and fit well. They should not be baggy or looseand reach to at least the mid-thigh.

6. Bags and other loose items must be stowed out of the walkway in the area(s) designated byyour instructor. In a crowded laboratory, bags and other items underfoot can be a significanthazard.

ii

Guidelines for Laboratory Notebooks

The primary requirement for your laboratory notebook is the following: Another physicist, with

a similar level of general physics knowledge as yourself should be able to duplicate the experiment

exactly as you performed it, using ONLY your lab notebook. Indeed, you will be graded by just such

a person using only your lab notebook.

Note: You must use the lab-book style designated for this class! Other notebook styles are notacceptable.

The following materials are required for this lab:

• Lab notebooks - must be hardbound with pages that cannot easily fall out. This is a recordof your work and as such its integrity must be maintained. Books with removable pages orpages missing/added are not accepted in the physics community. In general, the laboratorynotebook is considered to be the primary and most permanent record of any experiment.Research laboratories often keep lab books for many decades after the work is completed.Therefore, it is important to use a proper bound laboratory notebook (which should haveacid-free paper)

• Page numbers - if your lab book does not have them, write them in by hand in pen. Thisallows you to reference other sections of your lab book easily.

• Millimeter ruled graph paper- all handmade plots and graphs should be executed on appropri-ate millimeter-ruled graph paper and must be pasted into the lab book with archival qualityglue.

• Ink pens- all writing should be done in ink that is not water soluble, except for diagramswhich may be done in pencil. Many labs involve water and any drips will erase your work.Note: it is often times useful, but not required, to have several colors of ink so that you candifferentiate different sections, results, or data sets, especially while graphing.

Your laboratory notebook is your primary record of an experiment. In your lab notes, you willrecord everything you do while working on an experiment. The idea is to capture the most accuratepicture possible of how the experiment is carried out. If you feel the need for extra space, it isacceptable to use only the front half the page for your formal work and the backs of the pages forscratch work. The following are guides for how you should layout your notebook:

• When you start writing each day, write down the date and the time as well as who is usingthe notebook. This will allow you to keep track of when you did things, but also keeps arecord for others viewing your notebook. Often times research notebooks will be shared byall people involved in a project, thus keeping track of who did what can be valuable.

• It is important to enter any procedures and results into the lab book as they occur. Don’t tryto store anything in your head with the intention of writing it into the lab book later. Evenif you do manage to remember to make the entry, accuracy is always compromised. There isa reduced level of confidence in any information recorded late.

• All entries should be in chronological order. Never leave blank space to fill in later (someexception can be made for printed out graphs intended to be pasted into the lab notebook ifan appropriate printer is not handy).

iii

• Graphs, hand drawn or computer generated, must be pasted securely into the notebook andshould take up at least 1

2 page. On any graph, make sure that you clearly label the axes inwords, with appropriate and correct units. An example is Friction Force [N]. All data sets andlines must be labeled and all data should have error bars. Do not connect the dots with yourdata unless there is a very good reason to do so, i.e. you are looking for trends. Connectingthe dots implies there is data or a fit line between the shown data points, thus, arbitrary linessuggest false data. Fit lines should have a physical basis (such as a reason why it would bex2 etc. or whether or not it should go through the origin) and should be explained in thetext and labeled appropriately.

iv

Formal Report Instructions

The most important requirement is that your report needs to be written so clearly and completelythat someone who hasn’t done the experiment can read your report and gain a clear understandingof what you did. Needless to say, the entire report must be written in full sentences with correctgrammar, spelling, and punctuation. All pages must be numbered except for the title page whichtraditionally is not numbered. This page and the following page contain a short description of thesections you should include in your report.

1. TITLE PAGE WITH ABSTRACT: An example of a title page is shown on the page followingthese instructions. Please follow the format of this title page. Data taken during the lab shouldbe recorded in ink.

2. INTRODUCTION: The introduction includes the background material needed to put theexperiment you are about to explain into the broader context. The introduction shouldexplain the goals of the experiment and summarize in a few sentences the way in which theexperiment was performed.

3. PROCEDURE: This is probably the most important section of the report. The proceduresection gives the details of how you performed the experiment. It should be written in thepast tense. The description of how you performed the experiment should be sufficientlydetailed that a person of your own skill level (such as a classmate) who has not performedthe experiment, could reproduce what you did with only your report as a guide. Some crucialparts of the procedure section:

• A diagram of the apparatus with all important parts clearly marked.

• A story (a paragraph or two) of how you carried out the experiment. If you found thatit was important to set up some part of the experiment in some particular way thatmight not be clear from the diagram alone, explain that here. Also explain any pitfallsor mistakes that you made and subsequently corrected.

• Explain what you did to minimize the sources of systematic error and uncertainty in theexperiment.

4. RESULTS AND ANALYSIS: Explain the final results of your experiment and present allyour data. Present and carefully explain (in words) any calculations and analysis required toobtain the final result.

Also, you must attach the date-stamped raw data sheets from your lab book.Otherwise, nocredit will be given for this report. This is the only page to be included from your lab book..

5. DISCUSSION: In this section, you should reflect on your results. How do the results com-pare with other known results/handbook values. What procedures could be improved in theexperiment to further reduce sources of error. When writing this section, remember thatequipment is always less good than the experimenter would like. Equipment funds are alwayslimited, even for a well funded research laboratory. Therefore, while a well-funded govern-ment or private research laboratory might have better or more modern equipment than youare using, the scientists at such institutions are still keenly aware of the limitations of their

v

equipment. Their job is to transcend those limitations. In fact, one might argue that exper-imental physics is on the whole an endeavor to transcend equipment limitations by creativeexperiment design and care. Therefore, statements like ”this experiment could be improvedif only the school would buy non-crappy equipment” are not useful and will not be perceivedwell by your instructor. Instead, explain how you could make the experiment better with theresources available to you.

6. QUESTIONS: This section includes answers to ALL questions.

vi

Vector Review

Introduction

In physics we frequently use two numeric data types: scalar quantities and vector quantities. Ascalar describes a simple characteristic of some quantity like temperature or volume. It will usuallyhave a numerical magnitude and units associated with it. A vector is similar to a scalar exceptthat it includes a direction in addition to magnitude.

Example 1: The distance between Prescott and Phoenix d=100 miles. This is a scalar value,it does not give a direction. If you drive 100 miles away from Phoenix, you will probably go thewrong way and may end up in Tucson instead of Prescott. However, = (100 miles North) is avector quantity because it include a direction. If you get in a car and drive 100 miles north fromPhoenix, you will end up in Prescott. You are given both the magnitude (100 miles) and thedirection (North) of the vector.

Calculating Vectors

A vector has a magnitude (length) and a direction (e.g. an angle or coordinate direction like”North-West”). Variables describing vectors are often written with little arrows above them, to tellus that the quantity is a vector rather than a scalar. In the diagram below, a vector ~V is placed atthe origin of a set of Cartesian coordinates.

Figure 1: Vector V with its components labeled.

We can now define the x and y components of the vector by the lengths of the projections(”shadows”) of the vector onto the respective axes. With the x and y components thus defined, wecan write the vector in component form as:

~V = Vxi+ Vy j

where the Vx and Vy are the x and y components, and the i and j are so-called unit vectors in the xand y directions respectively ( and may also be denoted by ex and ey). Unit vectors have a length

vii

of 1. If the length of vector is V then we can determine the magnitude of the x and y componentsas follows:

Vx = V cos θ

Vy = V sin θ

Note that this means the length is related to the x and y components by the Pythagorean Theorem,the magnitude of the vector is:

V =√

V 2x + V 2

y

Example 2: If the magnitude of the vector in the figure above is 10.00, and the angle it makeswith the x-axis is 60◦, what is the component form of the vector?

The components of the vector are:

Vx = V cos θ = 10 cos 60◦ = 5.00

Vy = V sin θ = 10 sin 60◦ = 8.66

So, our vector expressed in component form is:

~V = 5.00i+ 8.66j

10.002 = 8.662 + 5.002

We can check whether this is consistent with the Pythagorean Theorem. Since we have a righttriangle, with hypotenuse V and legs Vx and Vy, the Pythagorean Theorem says:

V 2 = V 2x + V 2

y

10.002 = 8.662 + 5.002

Try calculating this to verify that it is correct to within rounding error.

Adding Multiple Vectors

Supposing there are multiple vectors which need to be added. We add vectors by placing themall ”head-to-tail”. For example, adding two vectors, ~Va and ~Vb would look something like Figure 2.

The left-hand set of axes shows the two vectors separately, based at the origin of a Cartesiancoordinate system. The right-hand set of axes shows the addition of these two vectors with theresulting vector drawn with dashes. To add the two vectors we place them tail to head and thenconnect the tail of the first vector with the head of the last vector to find the resulting vector (alsoknown as the resultant). Note that it does not matter which order we connect the vectors; we getthe same resultant as long as we do not change the angle or length of either of the constituentvectors during the operation.

We can also add vectors analytically as opposed to graphically by breaking the vectors intocomponents and adding them as demonstrated in the next example.

Example 3: If vector a is the same vector as in the first example with magnitude 10, andvector b is added to it, which has magnitude 5 at 135 degrees, what is the resulting vector? Note:

viii

Figure 2: Adding two vectors head to tail.

the angle is always measured counterclockwise from the + x-axis. What is the resulting vectorslength? What is its direction?

Vector a was already calculated to be:

~Va = 88.6i+ 5.00j

Repeating the same steps for vector b:

Vb,x = V cos θ = 5 cos 135◦ = −3.54

Vb,y = V sin θ = 5 sin 135◦ = −3.54Note that the x-component is negative. This is correct because vector be points in the negative

x direction. So our vector is:~Vb = 3.54i− 3.54j

Next, add the components:V(a+b),x = Va,x + Vb,x

V(a+b),x = 8.66 + 3.54 = 12.20

V(a+b),y = Va,y + Vb,y

V(a+b),y = 5.00− 3.54 = 1.46

The resulting vector is:~Va+b = 12.20i+ 1.46j

The magnitude is (the || around a vector denote its magnitude):

| Va+b |=√

V 2a+b,x + V 2

a+b,y

| Va+b |=√

12.202 + 1.462 = 12.29

The direction is:

θ = tan−1Vx

Vy

ix

θ = tan−112.20

1.46= 83.18

Use of Coordinate Systems

Sometimes problems can be simplified by defining coordinate systems in creative ways, as shownin Figure 3.

Figure 3: Choosing your coordinate system can greatly simplify some problems.

The coordinate system S′ is rotated by with respect to coordinate system S. If both vectorshave a length of 1, or are unit length, then in coordinate system S

V1 = 0.707i+ 0.707j

V2 = −0.707i+ 0.707j

while in coordinate system S′

Vx = 1i+ 0j

Vy = 0i+ 1j

The latter set of vectors is much easier to deal with! You should verify that | ~V1 + ~V2 | givesthe same result in both coordinate systems!

x

Lab 0: Error in Measurements - Theory

Description

This chapter describes how to deal with the fact that all measurements are imperfect. To makeour measurements useful, despite their imperfection, we need to quantify how incorrect any mea-surement we make is likely to be. This paper describes how to do this and how to report the resultsof measurements.

Introduction

The understanding of physical phenomena is invariably accomplished with the help of quantita-tive observations called measurements. Measurements are used in conjunction with a careful storyknown as a theory, to explain physical phenomena. In this way, over the centuries, our understand-ing of the physical world has grown, and provided a foundation for our world-view and for mostmodern technologies.

Measurements constitute the principal activity of experiments conducted in the laboratory.Therefore, it is appropriate and helpful to discuss the methods of measurement, the propagationof uncertainties and the procedure for error analysis at this time.

First of all, lab technicians or experimenters should never be content to make a measurementto the ”nearest” division on the instrument, but instead should include an estimate of the nearestfraction of the smallest division. That number would then become the last significant figure inthe measurement. For example, if a meter stick (graduated in millimeters) is used to measure alength, the measurement should include the (estimated) nearest tenth of a millimeter. For exampleL = 622.3mm, where the decimal ”0.3” represents three tenths of a millimeter. (Note that onemight also chose to write this as 62.23 cm.) Similarly, for a thermometer graduated in one degreedivisions, the temperature measurement should always include the nearest (estimated) tenth ofa degree. While the estimate might not be accurate to the nearest tenth of a division, it wouldprobably be good to the nearest two tenths of a division. In the example above, the length can bewritten, along with its uncertainty, as L±∆L = (62.23± 0.02)cm. Quoting a measurement and itsuncertainty is important for two reasons. First, it communicates the number of significant digitsthat can be justifiably used in subsequent calculations. Secondly, the quantity ∆L

L(sometimes

called the relative uncertainty of L) describes the precision of the measurement. In this case,∆LL

is 0.00048, or 0.048% percentage uncertainty. A scientist would quote the precision of thismeasurement as ”0.05%”, a very precise measurement.

For example, when measuring the volume of a rectangular solid ( V = L ×W × H) the ex-perimenter must measure the length L, width W, and height H of the object. In order to quotethe final result as V ±∆V , the experimenter must first record the measurements along with theiruncertainties (e.g., L±∆L,W ±∆W , and H±∆H). The method for propagating the uncertainties(∆L, ∆W , and ∆H) to determine the final uncertainty ∆V will be described later in this chapter.

Vernier scales and magnifying glasses are also helpful when making measurements; your LabAssistant will be explaining the use of Vernier and micrometer scales in the lab.

xi

Three Kinds of Error

There are three kinds of error that can affect the final result:

• Uncertainties (also known as Random Errors)

• Systematic Errors

• Mistakes!

Uncertainty refers to the randomness inherent in measurements. Uncertainties naturally occurdue to the finite precision of all measurements. Sometimes, but not always, it is the measurementdevice itself that causes the uncertainty. For example, if a meter stick with 1 mm gradations) is usedto measure the length of a solid cylinder (say ∼ 5cm long) the length can be quoted with a statisticaluncertainty of ±0.2mm. If, however, a Vernier caliper is used, the statistical uncertainty is reducedto maybe 0.05mm. Meanwhile, the precision of the measurement could be further improved by usinga micrometer which might provide a statistical uncertainty of ±0.01mm. Uncertainties are inherentto any measurement process. When quoting a measurement, an uncertainty in the measurementshould always be stated.

WARNING: Quite often, the uncertainty is not due to the measurement device’s lack of preci-sion, but to some other factor affecting the measurement. The whole measurement process mustbe considered! For example, suppose the cylinder was ”rough cut” such that the end faces werenot smooth. Multiple measurements of the length may result in a range of answers. Measure-ments taken with a micrometer might include 50.11mm, 50.15mm, 50.17mm, 50.12mm, etc. Thesefluctuations in length are well outside the precision of the micrometer (0.01mm). The statisticaluncertainties inherent in the micrometer are not the limiting factor when quoting the uncertaintyin the length of the cylinder (∆L), rather the uncertainty is determined by the roughness of thecylinder’s end faces.

Systematic Errors are errors in measurements that do not vary randomly between measure-ments. As before, these errors can be due to the measuring device. An example of systematicerrors is parallax. A person using a meter stick to measure the length of the table might not read itdirectly above the meter stick. If, for example, the observer takes the reading at the same angle (say45◦ from the overhead position), then they would record a length that’s systematically too largeor too small. Another example would be if someone dropped the micrometer on the floor (heavenforbid!!) and bent the jaws on the device. While the micrometer would still be very precise (as-suming it is still functioning) with uncertainties ∼ 0.01mm, it would not be very accurate becausethe measurements made with misaligned micrometer would be consistently too large. Systematicerrors generally lead to results that are consistently too large or too small. When systematic errorsare present, the results are not randomly distributed about the true value.

The third sort of error is the plain old Mistake! For example: You had the meter stick thewrong way round, You read the inch scale instead of the centimeter scale. You filled in the wrongcolumn in your lab book. You filled in the right column but wrote down the wrong unit. Youunknowingly forgot to turn on a crucial piece of. You had voltmeter leads plugged into the wrongsockets. Your analysis code had a bug, etc. etc. etc. There are many ways to make a mistake andsome of them are very hard to detect. The only way to get rid of mistakes is to be extra careful,to double-check everything and only do laboratory work when you are feeling rested and are not

xii

distracted by conversation, hunger or other external influences.

Fractional Difference

When making measurements, it is quite possible that the quantity under investigation has pre-viously been measured by other experimenters. Sometimes, the results of those other experimentersare of high enough quality to have been included in reference books listing the ”accepted” or bestknown values for a given quantity. When you need to compare your results to that of others, thefractional difference between your value and the value obtained previously can be a useful quantityto know.

fractional difference (%) =previously measured value− your value

previously measured value× 100%

Confusingly, this quantity is sometimes also called the ”percentage error” even though it is notnecessarily at all related to the random or systematic errors in your measurement! Therefore, wewill avoid that term and always call this quantity the fractional difference. The fractional differencecan be of use a s a quick way to compare a new result to an older one. However, remember thatthe quality of any measurement is determined by the errors in the measured values including sys-tematic errors. Thus, if the uncertainties are too small to account for the difference between yourmeasurement and a previous measurement, then the difference must be due to either a mistake inone (or both) of the measurements or to unaccounted for systematic errors in one (or both) of themeasurements.

The Statistical Approach to Making Measurements and Propagat-ing Uncertainties

After a set of measurements has been collected and estimates of the uncertainties recorded, thequestion is, ”How do measured uncertainties affect the uncertainty in the physical quantity beingstudied?” Whether one measurement is made, or many measurements, N , are made, the techniquefor propagating uncertainties is basically the same. Note that the following discussion is limitedto propagating uncertainties which are random in nature. Systematic errors are assumed to benegligible.

Error equations can be derived in a more rigorous setting. However, in this class it will sufficesimply to state the final error equation and the calculation of partial derivatives is not required.

Function Error Equation

f = a+ b ∆f =√

(∆a)2 + (∆b)2

g = a− b ∆g =√

(∆a)2 + (∆− b)2 =√

(∆a)2 + (∆b)2

h = a · b ∆h =√

(b ·∆a)2 + (a ·∆b)2

l = an ∆l =√

(n · an−1 ·∆a)2

q = ab

∆q =√

(∆ab)2 + (a·∆b

b2)2

s = a2·cb

∆s =√

( c·2a·∆ab

)2 + (a2·∆cb

)2 + (a2·∆bb2

)2

t = a sin b ∆t =√

(∆a · sin b)2 + (a ·∆b cos b)2

xiii

Example

Well, enough theory. How about an example? Suppose we want to measure the volume of acylinder as well the uncertainty in the volume. First of all, this requires two measurements, thediameter D and the height, h. The volume can be written as

V = (πD2

4)h (1)

Looking at the table above, we see that s looks similar, however there is no error in b, so wejust set this equal to 0. We then obtain

∆V =

√

(πh2∆D

4)2 + (

πD2∆h

h)2 (2)

We observe a general rule taking form here as we write the relative uncertainty. The relativeuncertainty in the volume is simply the relative uncertainty of the diameter ∆D

Dand the relative

uncertainty of the height ∆hh

”added in quadrature”, that is, the square-root of the ”sum of thesquares”. How did the factor of 2 appear? This is due to the diameter D entering as a squaredvalue in the equation describing the volume of a cylinder.

Exercise: What is the relative uncertainty in the volume of a sphere where and the diameter isthe only measured quantity?

Answer:∆V

V=

3∆D

D

Going back to the volume of a cylinder, how do we quote our final answer for the volume?Whenever a measurement is made of a physical quantity, let’s say the volume, it should be writtenin the following format, V ±∆V , that is, the volume and the uncertainty in the volume. The uncer-tainty in the volume ∆V , or better yet, the relative uncertainty of the volume ∆V

Vcommunicates

something about the quality of the measurement. If one experiment measures ∆VV

= 0.085 (an8.5% measurement), and a second experiment measures ∆V

V= 0.041 (a 4.1% measurement), then

we can easily conclude that the second measurement is more precise than the first measurement.How could you improve your measurement? There is valuable information contained in each of

the terms being added in quadrature. From Equation 1 we can see the relative uncertainty relatedto the diameter D is 1.54% while the relative uncertainty related to the height h is 0.38%. Note:that these uncertainties are different even though the same measuring device was used to make bothmeasurements. If extra effort is going to be expended to improve the overall 1.58% measurementshown in Equation 1, our efforts would be better spent on improving the measurement of thediameter D (i.e., reducing ∆D) than trying to improve the measurement of the height h. It isinformative to write out each of the uncertainty terms as shown in Equation 1 so the experimentercan readily determine how each uncertainty contributes to the final uncertainty. If the goal is toimprove the overall precision of the experiment, effort should be focused on reducing the relativeuncertainties that make the largest contribution to the overall uncertainty ∆V .

Exercise: An approximate value of the acceleration due to gravity g is easily calculated usinga free-fall experiment. A ball is released from rest and allowed to fall a distance s while the timeinterval t is recorded. Using the equation for constant acceleration s = 1

2gt2, the gravitational

xiv

constant g can be shown to be g = 2st2. What is ∆g

gin terms of (s,∆s, t,∆t)?

Addition or Subtraction of Quantities of the Same Dimension

When the measured quantities are simply added or subtracted, propagating the uncertaintiesis particularly easy. As an example, suppose there are two rods with lengths L1 and L2 anduncertainties ∆L1 and ∆L2 respectively. What is the total length (and uncertainty in the totallength) if the rods are placed end-to-end? In other words, what is L ± ∆L? The total length L

is simply the sum of L1 and L2. However, calculating the final uncertainty ∆L is not necessarilyimmediately obvious. The function f = L = L1 + L2 gives us a simple rule for calculating theuncertainty in L

∆L =√

(∆L1)2 + (∆L2)2

Example: Suppose two rods with highly polished end faces are measured using a vernier caliperand found to have the following lengths:

L1 ±∆L1 = 3.051± 0.005cm

L2 ±∆L2 = 5.442± 0.005cm

If the rods are placed end-to-end, what is their final length L and uncertainty ∆L?

Answer: L = 8.4930.007cm

Notice that the uncertainty in the final length (∼ 70 microns) is slightly larger than the indi-vidual uncertainties (50 microns). In fact it is larger by a factor of

√2.

Uncertainties (Random Errors) and the Gaussian Distribution

When multiple measurements are taken of the same quantity, and the error in those measure-ments is dominated by uncertainties, the measured values are distributed according to a well knownfunction called the Gaussian distribution. This distribution occurs repeatedly when makingmeasurements and forms the starting point for discussing the random nature of uncertainties.

When measuring the length of an object with a high precision device (e.g., a vernier caliper),many different lengths could be measured. Suppose 50 measurements xi are collected using a ruler.Suppose also that all the measurements have the same uncertainty ∆x. We would like to agree onsome simple parameters that describe the results of our measurements. One of the easiest quantitiesto calculate is called the mean value. This is sometimes called the average or average value. Ageneral equation describing the mean value (x) can be written as follows:

x =1

N

N∑

i=1

xi (3)

We would also like to quantify how values of xi are spread over the axis. For example, arethe data bunched up close to each other, or are they spread over a large range of x? This can beillustrated by the following equation that describes the standard deviation of the values xi.

σ =

√

√

√

√

1

N − 1

N∑

i=1

(x− xi)2 (4)

xv

The standard deviation σ is small when the data are tightly bunched and large when the dataare spread over a large value of x. If the uncertainties ∆x in the measurement were chosen correctly,then we should find that σ ≈ ∆x. In fact, as N →∞, σ → ∆x. In fact, if one does not know whatthe uncertainty in a quantity is, one very good way of estimating it is to take the standard deviationof many repeated measurements! In Figure 1 we see how 50 measurements are distributed alongthe x axis using a graphing tool called a histogram.

Figure 1: Fifty measurements of the same physical quantity are displayed in a histogram to showtheir distribution along the x-axis. Mean is x = 19.97. The standard deviation is σ = 0.11.

Overlaid on the histogram (i.e., the vertical bars) is a Gaussian curve. The entire area under thecurve accounts for 100% of the measurements. The area in between x−σ to x+σ is approximately0.68, or 68%. Extending the region of integration from x− 2σ to x+2σ gives an answer of 0.95, or95%, and so on. How can we apply these properties to our interpretation of the 50 measurementsshown in Fig. 1? A priori, we do not know the true value of the quantity we are measuring. Allwe have are the 50 measurements. However, we can ask the following question: ”What is theprobability that the ”true” value is contained in the region from x − σ to x + σ.” The answer is68%.

Likewise, we can ask, ”What is the probability that the ”true” value is contained in the regionfrom x − 2σ to x + 2σ?” The answer is 95%. By applying the properties of a Gaussian functionto the distribution of collected data, we can interpret our results for the mean x and the standarddeviation σ.

Uncertainty in the Mean

Once the mean and standard deviation have been determined, there is a third quantity thatcan be calculated: the uncertainty in the mean ∆x. This quantity describes the region over which

xvi

the mean value x can fluctuate and is calculated using the following equation.

∆x =∆x√N

where N is the number of measurements taken. As the number of measurements increase, theuncertainty in the mean is reduced. Since the best estimate of a measured quantity x is generallythe mean value x of all the measurements xi, the more measurements we make, the lower is ouruncertainty ∆x in our best estimate. For this reason, it is usually a good idea to take as manymeasurements of a quantity as possible (or practical).

The uncertainty in the mean can be calculated for any value where many measurements aretaken. For example, if Figure 1 represents 50 measurements of the length of a cylinder, what canwe conclude? First of all, the cylinder has a mean length of 19.977 cm. (assuming the units inFig.1 are in centimeters). Secondly, the standard deviation σ for these measurements is 0.11 cm.Finally, the uncertainty in the mean length is 0.11√

50= 0.016 cm.

How far can we go with this process? Could we take N=1,000,000 measurements and reducethe uncertainty in the mean by a factor of 1,000 (i.e.,

√N)? Using the above example, we would

conclude that if 1,000,000 measurements were taken of the length of the cylinder, then the uncer-tainty in the mean would reduce to 0.00011 cm, or 1.1 microns. This is incredibly small. If thedata are truly randomly distributed about the mean as we assume, there is no limit to how smallthe uncertainty in the mean can be. However, it is usually not useful to take more measurementsthan needed to reduce the uncertainty in the mean to the level of expected or suspected systematicerrors since systematic errors start to dominate at that point.

xvii

Precision versus Accuracy

Sometimes, we use the words precision and accuracy to indicate two different measures ofthe quality of a result. A result that is highly precise is one with a very low uncertainty. A resultthat is accurate, has low systematic error. This is illustrated below, using a dart-board analogy.In this case, the location bull’s-eye corresponds to the actual value of the physical quantity beingmeasured and the locations of the darts correspond to the results of individual measurements ofthat quantity.

Figure 2: The figure to the left illustrates a result that is both precise and accurate.As you can see, the darts are clustered together (precise) and they are on target(accurate).

Figure 3: This figure illustrates precision without accuracy. The darts areclustered together (precise), but not on target (inaccurate). The darts areclustered about a value that is not the target value.

Figure 4: This figure illustrates accuracy without precision. The darts arespread across the board (imprecise), but they are on target (accurate). Eventhough the darts are scattered across the board, they are still scattered aboutthe target.

Figure 5: This figure illustrates a lack of accuracy and precision. The darts arescattered about the board (imprecise) with no discernible target (inaccurate).

xviii